Line Integrals
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Vector Fields
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The Gradient Theorem
0/4 completed

Vector Analysis
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Exercises
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- Exercise 1 Part a
- Exercise 1 Part b
- Exercise 1 Part c
- Exercise 1 Part d
- Exercise 2 Part a
- Exercise 2 Part b
- Exercise 3
- Exercise 4
- Exercise 5 Part a
- Exercise 5 Part b
- Exercise 6 Part a
- Exercise 6 Part b
- Exercise 7 Part a
- Exercise 7 Part b
- Exercise 7 Part c
- Exercise 7 Part d
- Exercise 8
- Exercise 9
- Exercise 10 Part a
- Exercise 10 Part b
- Exercise 11 Part a
- Exercise 11 Part b
- Exercise 12 Part a
- Exercise 12 Part b
- Exercise 13

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[{"Name":"Line Integrals","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Line Integral of Type 1","Duration":"16m 29s","ChapterTopicVideoID":10180,"CourseChapterTopicPlaylistID":82678,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10180.jpeg","UploadDate":"2020-02-26T06:20:25.5830000","DurationForVideoObject":"PT16M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.415","Text":"Continuing with line integrals,"},{"Start":"00:02.415 ","End":"00:06.405","Text":"we just learned about vector fields,"},{"Start":"00:06.405 ","End":"00:11.580","Text":"and now we\u0027re about ready to give a definition."},{"Start":"00:11.580 ","End":"00:17.130","Text":"But before I do that there\u0027s an omission of mine."},{"Start":"00:17.130 ","End":"00:18.915","Text":"I tried to avoid it."},{"Start":"00:18.915 ","End":"00:21.660","Text":"Something called ds."},{"Start":"00:21.660 ","End":"00:22.830","Text":"You\u0027ll see what I mean."},{"Start":"00:22.830 ","End":"00:26.310","Text":"I\u0027m going to have to introduce this concept now,"},{"Start":"00:26.310 ","End":"00:36.510","Text":"because it\u0027s helpful on because not everyone else but it appears a lot in the literature."},{"Start":"00:36.860 ","End":"00:40.065","Text":"This ds relates to curve length,"},{"Start":"00:40.065 ","End":"00:44.710","Text":"and let me for a moment flash back to a previous lecture."},{"Start":"00:44.710 ","End":"00:49.235","Text":"Here we are back in length of curve."},{"Start":"00:49.235 ","End":"00:53.445","Text":"There were actually 3 cases I covered,"},{"Start":"00:53.445 ","End":"00:55.290","Text":"y as the function of x,"},{"Start":"00:55.290 ","End":"00:56.970","Text":"x as the function of y,"},{"Start":"00:56.970 ","End":"01:03.380","Text":"and x and y both as a function of a parameter t. This was all in 2-dimensions."},{"Start":"01:03.380 ","End":"01:06.880","Text":"There was a fourth case, I don\u0027t want to mention that here."},{"Start":"01:06.880 ","End":"01:09.410","Text":"In each of them there was a formula."},{"Start":"01:09.410 ","End":"01:12.185","Text":"I\u0027m taking the parametric one as an example,"},{"Start":"01:12.185 ","End":"01:17.450","Text":"and we reached a formula that looks like this, and of course,"},{"Start":"01:17.450 ","End":"01:24.390","Text":"I could have rewritten this with the Leibnitz notation for derivative,"},{"Start":"01:24.390 ","End":"01:33.095","Text":"instead of x prime I could have put dx by dt and instead of y prime,"},{"Start":"01:33.095 ","End":"01:37.860","Text":"I could have put dy by dt."},{"Start":"01:38.420 ","End":"01:41.565","Text":"I could have called this L,"},{"Start":"01:41.565 ","End":"01:44.645","Text":"and I could have changed the names for these to."},{"Start":"01:44.645 ","End":"01:50.010","Text":"In short I\u0027m going to jump back now,"},{"Start":"01:50.010 ","End":"01:53.805","Text":"and I\u0027m going to show you a diagram."},{"Start":"01:53.805 ","End":"01:57.990","Text":"Here we are. If you just look at this last one,"},{"Start":"01:57.990 ","End":"02:03.635","Text":"notice this expression here is exactly the same."},{"Start":"02:03.635 ","End":"02:05.480","Text":"Let me flashback again."},{"Start":"02:05.480 ","End":"02:09.500","Text":"Same as this expression here with the Leibnitz notation."},{"Start":"02:09.500 ","End":"02:15.470","Text":"If I call this part ds,"},{"Start":"02:15.470 ","End":"02:25.805","Text":"then we could write the integral as just the integral of ds."},{"Start":"02:25.805 ","End":"02:29.215","Text":"There it was from t_a to t_b."},{"Start":"02:29.215 ","End":"02:32.985","Text":"But we could say it\u0027s from Alpha to Beta."},{"Start":"02:32.985 ","End":"02:37.265","Text":"In this case that\u0027s where the parameter goes from."},{"Start":"02:37.265 ","End":"02:42.080","Text":"If we then visited the other 2 cases which I\u0027m not going to,"},{"Start":"02:42.080 ","End":"02:43.700","Text":"where y is the function of x,"},{"Start":"02:43.700 ","End":"02:45.095","Text":"so x is the function of y."},{"Start":"02:45.095 ","End":"02:47.195","Text":"It turns out that in all of the cases,"},{"Start":"02:47.195 ","End":"02:49.895","Text":"L is the integral of ds,"},{"Start":"02:49.895 ","End":"02:55.175","Text":"except that the bounds are different in the case that y is a function of x,"},{"Start":"02:55.175 ","End":"03:00.605","Text":"then the integral is from a to b."},{"Start":"03:00.605 ","End":"03:03.875","Text":"If x is a function of y,"},{"Start":"03:03.875 ","End":"03:11.985","Text":"then the integral is from c to d. But in each case it\u0027s ds,"},{"Start":"03:11.985 ","End":"03:16.730","Text":"ds, where in each case ds is defined correspondingly."},{"Start":"03:16.730 ","End":"03:18.920","Text":"Turns out that these are all equivalent."},{"Start":"03:18.920 ","End":"03:21.979","Text":"I\u0027m not going to go into all the rationale."},{"Start":"03:21.979 ","End":"03:29.110","Text":"There is a logic here and s is an element of arc length and how it relates."},{"Start":"03:29.110 ","End":"03:33.695","Text":"We\u0027ll just leave these as formal definitions for the moment,"},{"Start":"03:33.695 ","End":"03:39.390","Text":"but there is a deeper meaning behind them."},{"Start":"03:41.170 ","End":"03:47.610","Text":"Having said that, we really only going to use the last one,"},{"Start":"03:47.610 ","End":"03:50.670","Text":"I just gave the first two for completeness."},{"Start":"03:50.670 ","End":"03:54.575","Text":"I\u0027m going to just erase these 2 and keep the last one,"},{"Start":"03:54.575 ","End":"04:01.080","Text":"and we\u0027ll change Alpha and Beta back to a and b more friendly."},{"Start":"04:01.520 ","End":"04:07.790","Text":"That\u0027s the ds part and now we need to briefly"},{"Start":"04:07.790 ","End":"04:13.310","Text":"review some equations of parametric curves,"},{"Start":"04:13.310 ","End":"04:17.090","Text":"or rather parametric equations of curves."},{"Start":"04:17.090 ","End":"04:19.910","Text":"Just to remind you of some famous ones,"},{"Start":"04:19.910 ","End":"04:21.665","Text":"and we\u0027ll start with 2d."},{"Start":"04:21.665 ","End":"04:26.794","Text":"Let\u0027s start with, for example, the ellipse,"},{"Start":"04:26.794 ","End":"04:36.815","Text":"which also includes a circle and in the section on quadratic surfaces we learned"},{"Start":"04:36.815 ","End":"04:41.570","Text":"that one way of writing an ellipse in Cartesian"},{"Start":"04:41.570 ","End":"04:49.565","Text":"coordinates is x squared over a squared plus y squared over b squared equals 1."},{"Start":"04:49.565 ","End":"04:55.130","Text":"Now, what we want to do is to have the parametric equivalent."},{"Start":"04:55.130 ","End":"05:02.130","Text":"One way to parameterize this the usual one is as follows."},{"Start":"05:02.130 ","End":"05:05.295","Text":"We say that x equals,"},{"Start":"05:05.295 ","End":"05:12.090","Text":"y equals, x equals a cosine of t,"},{"Start":"05:12.090 ","End":"05:19.610","Text":"and y equals b sine of t. But if we want to just go around the ellipse once,"},{"Start":"05:19.610 ","End":"05:26.060","Text":"starting from the rightmost point then we let t"},{"Start":"05:26.060 ","End":"05:33.130","Text":"go from 0 to 2 Pi."},{"Start":"05:33.130 ","End":"05:38.390","Text":"But it turns out that as t goes from 0 to 2 Pi,"},{"Start":"05:38.390 ","End":"05:43.760","Text":"we go around the ellipse in a counterclockwise direction."},{"Start":"05:43.760 ","End":"05:49.150","Text":"Now, with line integrals in general,"},{"Start":"05:49.150 ","End":"05:54.335","Text":"when we have a curve not necessarily closed, but it could be,"},{"Start":"05:54.335 ","End":"05:58.170","Text":"and we go from one point to another,"},{"Start":"05:58.170 ","End":"05:59.900","Text":"then the direction matters."},{"Start":"05:59.900 ","End":"06:01.250","Text":"We would put an arrow and say,"},{"Start":"06:01.250 ","End":"06:05.075","Text":"we go from point a to point b in this direction."},{"Start":"06:05.075 ","End":"06:06.740","Text":"In the case of the ellipse,"},{"Start":"06:06.740 ","End":"06:12.460","Text":"it actually matters if we go around counterclockwise or clockwise."},{"Start":"06:12.460 ","End":"06:16.165","Text":"I wrote the word counterclockwise."},{"Start":"06:16.165 ","End":"06:19.040","Text":"Turns out that if I wanted to do it clockwise,"},{"Start":"06:19.040 ","End":"06:21.835","Text":"all I have to do is either minus here,"},{"Start":"06:21.835 ","End":"06:25.755","Text":"and here I just basically copied it but notice"},{"Start":"06:25.755 ","End":"06:29.895","Text":"the minus here for when we\u0027re going around clockwise,"},{"Start":"06:29.895 ","End":"06:35.030","Text":"which is not the usual mathematical direction."},{"Start":"06:35.180 ","End":"06:39.210","Text":"As for a circle, well,"},{"Start":"06:39.210 ","End":"06:42.610","Text":"when a and b are equal and they\u0027re both equal to"},{"Start":"06:42.610 ","End":"06:47.710","Text":"r. If a equals b and I then use the letter r instead,"},{"Start":"06:47.710 ","End":"06:55.440","Text":"then I would get x squared plus y squared equals r-squared,"},{"Start":"06:55.440 ","End":"06:58.345","Text":"and the same parametric equations,"},{"Start":"06:58.345 ","End":"07:06.450","Text":"but with a and b replaced everywhere by r. It would be an r cosine t,"},{"Start":"07:06.450 ","End":"07:13.040","Text":"r sine t. I\u0027m not going to write the whole thing again just for the circle."},{"Start":"07:14.040 ","End":"07:19.600","Text":"As another example, let\u0027s take a general curve,"},{"Start":"07:19.600 ","End":"07:22.150","Text":"where y is defined as a function of x."},{"Start":"07:22.150 ","End":"07:24.835","Text":"Let\u0027s say y equals f of x,"},{"Start":"07:24.835 ","End":"07:30.700","Text":"and that x goes between a and b,"},{"Start":"07:30.700 ","End":"07:34.150","Text":"let\u0027s say, a longer curve."},{"Start":"07:34.150 ","End":"07:35.950","Text":"Now we want to parametric version of that."},{"Start":"07:35.950 ","End":"07:37.735","Text":"We learned how to do that."},{"Start":"07:37.735 ","End":"07:41.035","Text":"All you do is replace x by t,"},{"Start":"07:41.035 ","End":"07:47.230","Text":"so what we get is x equals t. We just need a different letter."},{"Start":"07:47.230 ","End":"07:48.610","Text":"We could have kept x,"},{"Start":"07:48.610 ","End":"07:52.460","Text":"that would be funny to add x equals x."},{"Start":"07:52.620 ","End":"07:55.870","Text":"Then y, which is f of x,"},{"Start":"07:55.870 ","End":"08:04.330","Text":"is now f of t. The same limits apply to t between a and b."},{"Start":"08:04.330 ","End":"08:08.890","Text":"This is a parametric equation of the portion of curve defined by this function"},{"Start":"08:08.890 ","End":"08:14.410","Text":"between these limits on x and other way around."},{"Start":"08:14.410 ","End":"08:19.160","Text":"Suppose we have x as a function of y."},{"Start":"08:19.260 ","End":"08:24.169","Text":"Let\u0027s say y goes between c and d,"},{"Start":"08:24.750 ","End":"08:30.980","Text":"then a parametric version of this would be,"},{"Start":"08:32.760 ","End":"08:35.980","Text":"let me just scroll down a bit,"},{"Start":"08:35.980 ","End":"08:41.900","Text":"this time we would take y to be the parameter."},{"Start":"08:44.850 ","End":"08:48.115","Text":"I want to use a different letter."},{"Start":"08:48.115 ","End":"08:51.175","Text":"Use g otherwise it could be misleading."},{"Start":"08:51.175 ","End":"08:57.865","Text":"Then x would equal g of t and still t would go between the same limits."},{"Start":"08:57.865 ","End":"09:02.620","Text":"The t is y between t"},{"Start":"09:02.620 ","End":"09:09.220","Text":"and c and d. This is the general category of curves."},{"Start":"09:09.220 ","End":"09:14.590","Text":"Now, let\u0027s talk about line segments."},{"Start":"09:14.590 ","End":"09:17.815","Text":"Oh, all these curves have a beginning and an end."},{"Start":"09:17.815 ","End":"09:21.070","Text":"We don\u0027t do them infinitely, well, usually."},{"Start":"09:21.070 ","End":"09:26.800","Text":"We\u0027ll be doing either from 1 point to another or in a case of a closed curve,"},{"Start":"09:26.800 ","End":"09:27.970","Text":"back where we started."},{"Start":"09:27.970 ","End":"09:32.485","Text":"Then we have directions of clockwise and counter-clockwise."},{"Start":"09:32.485 ","End":"09:39.325","Text":"The line segment, I actually wanted to do in 3D and in 2D."},{"Start":"09:39.325 ","End":"09:42.350","Text":"Let me start with the 3D case."},{"Start":"09:42.560 ","End":"09:47.025","Text":"The line segment will be from one point."},{"Start":"09:47.025 ","End":"09:51.100","Text":"Let\u0027s say we have the first point,"},{"Start":"09:51.100 ","End":"09:52.870","Text":"I\u0027ll call it x naught,"},{"Start":"09:52.870 ","End":"09:56.275","Text":"y naught, z naught."},{"Start":"09:56.275 ","End":"10:03.370","Text":"The end point will be x_1, y_1, z_1."},{"Start":"10:03.370 ","End":"10:06.235","Text":"This has a direction, from here to here."},{"Start":"10:06.235 ","End":"10:08.950","Text":"This is a curve. It happens to be a line."},{"Start":"10:08.950 ","End":"10:13.120","Text":"The parametric equation for that we already learned,"},{"Start":"10:13.120 ","End":"10:17.275","Text":"but I\u0027m reminding you in vector form,"},{"Start":"10:17.275 ","End":"10:19.615","Text":"I\u0027ll do the vector form first and the parametric,"},{"Start":"10:19.615 ","End":"10:24.205","Text":"we got r of t equals"},{"Start":"10:24.205 ","End":"10:31.120","Text":"1 minus t times the position vector of the first point,"},{"Start":"10:31.120 ","End":"10:33.205","Text":"x naught, y naught,"},{"Start":"10:33.205 ","End":"10:39.970","Text":"z naught plus t times the position vector of the second point."},{"Start":"10:39.970 ","End":"10:43.330","Text":"It\u0027s easier to remember the vector form, but of course,"},{"Start":"10:43.330 ","End":"10:46.960","Text":"it\u0027s easy to get from here to the parametric form."},{"Start":"10:46.960 ","End":"10:48.430","Text":"I forgot to say of course,"},{"Start":"10:48.430 ","End":"10:53.215","Text":"that t goes from 0 to 1,"},{"Start":"10:53.215 ","End":"10:55.840","Text":"that when t is 0, this is 0,"},{"Start":"10:55.840 ","End":"10:57.550","Text":"and this is 1."},{"Start":"10:57.550 ","End":"10:58.810","Text":"We get the first point."},{"Start":"10:58.810 ","End":"11:03.145","Text":"When t is 1, this becomes 0 and we get the second point."},{"Start":"11:03.145 ","End":"11:07.180","Text":"In parametric, this becomes,"},{"Start":"11:07.180 ","End":"11:11.334","Text":"I have to say what x equals or y equals,"},{"Start":"11:11.334 ","End":"11:13.135","Text":"and what z equals."},{"Start":"11:13.135 ","End":"11:16.405","Text":"It\u0027s just component-wise."},{"Start":"11:16.405 ","End":"11:25.210","Text":"It\u0027s going to be 1 minus tx naught plus tx_1."},{"Start":"11:25.210 ","End":"11:27.115","Text":"Similarly for y and z,"},{"Start":"11:27.115 ","End":"11:35.140","Text":"and still t goes from 0 to1."},{"Start":"11:35.140 ","End":"11:37.705","Text":"This is for 3D."},{"Start":"11:37.705 ","End":"11:43.100","Text":"If I want the 2D case,"},{"Start":"11:43.650 ","End":"11:50.210","Text":"then I just have to eliminate the last component."},{"Start":"11:50.550 ","End":"11:54.040","Text":"What I highlighted is what I\u0027m editing out for 2D."},{"Start":"11:54.040 ","End":"11:57.040","Text":"I\u0027d remove this, I\u0027d remove this."},{"Start":"11:57.040 ","End":"12:01.015","Text":"I would remove this and this."},{"Start":"12:01.015 ","End":"12:06.130","Text":"Here I will remove the whole of the last equation."},{"Start":"12:06.130 ","End":"12:11.080","Text":"If you highlight it,"},{"Start":"12:11.080 ","End":"12:15.560","Text":"then you get the 2D case. Very similar."},{"Start":"12:16.590 ","End":"12:19.420","Text":"We\u0027re almost ready for the definition."},{"Start":"12:19.420 ","End":"12:24.100","Text":"We\u0027ve given some typical lines and curves."},{"Start":"12:24.100 ","End":"12:28.525","Text":"Ellipse circle, y is a function of x, x is a function of y,"},{"Start":"12:28.525 ","End":"12:33.610","Text":"and a line segment from one point to another in 2D and 3D."},{"Start":"12:33.610 ","End":"12:39.435","Text":"We\u0027ve also covered what ds is."},{"Start":"12:39.435 ","End":"12:47.785","Text":"Finally, I\u0027m going to get to the definition of a line integral."},{"Start":"12:47.785 ","End":"12:49.615","Text":"For a line integral,"},{"Start":"12:49.615 ","End":"12:51.565","Text":"I need 2 things."},{"Start":"12:51.565 ","End":"13:01.225","Text":"I need a function of 2 variables and I need"},{"Start":"13:01.225 ","End":"13:07.105","Text":"a smooth curve C. I"},{"Start":"13:07.105 ","End":"13:13.795","Text":"forgot to say that we\u0027re starting off with 2D line integral and later we\u0027ll do the 3D."},{"Start":"13:13.795 ","End":"13:22.450","Text":"F of x, y is a function of 2 variables and C is a smooth curve."},{"Start":"13:22.450 ","End":"13:26.605","Text":"I\u0027ll go over again what is a smooth curve in a minute."},{"Start":"13:26.605 ","End":"13:28.810","Text":"Just leave that for a moment,"},{"Start":"13:28.810 ","End":"13:31.765","Text":"because I want to get to the definition already."},{"Start":"13:31.765 ","End":"13:35.605","Text":"Then the line integral will be,"},{"Start":"13:35.605 ","End":"13:38.380","Text":"I\u0027ll just write it, but it will need explaining,"},{"Start":"13:38.380 ","End":"13:42.235","Text":"the integral along the curve"},{"Start":"13:42.235 ","End":"13:51.325","Text":"of f of x, y, ds."},{"Start":"13:51.325 ","End":"13:56.020","Text":"This is the ds that we learned above."},{"Start":"13:56.020 ","End":"13:57.790","Text":"Now, I need some explanations."},{"Start":"13:57.790 ","End":"13:59.574","Text":"First of all, what does it mean?"},{"Start":"13:59.574 ","End":"14:02.139","Text":"Well, a smooth curve,"},{"Start":"14:02.139 ","End":"14:04.330","Text":"first in vector form,"},{"Start":"14:04.330 ","End":"14:06.115","Text":"I\u0027ll give it, then in parametric,"},{"Start":"14:06.115 ","End":"14:15.050","Text":"would be to say that r of t, which will be."}],"ID":10485},{"Watched":false,"Name":"Worked Examples 1-2","Duration":"16m 59s","ChapterTopicVideoID":10181,"CourseChapterTopicPlaylistID":82678,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10181.jpeg","UploadDate":"2020-02-26T06:22:12.1670000","DurationForVideoObject":"PT16M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"I\u0027m going to start the example on a new page,"},{"Start":"00:03.480 ","End":"00:05.805","Text":"I just kept the formula."},{"Start":"00:05.805 ","End":"00:10.035","Text":"I want to remind you that what h and g are,"},{"Start":"00:10.035 ","End":"00:19.155","Text":"the curve C is given in parametric form by x equals h of"},{"Start":"00:19.155 ","End":"00:29.020","Text":"t and y equals g of t. That\u0027s the h and g here and t goes from a to b."},{"Start":"00:30.320 ","End":"00:34.725","Text":"We\u0027re using the Leibniz notation for derivative."},{"Start":"00:34.725 ","End":"00:40.410","Text":"In the example, we want to compute the"},{"Start":"00:40.410 ","End":"00:45.780","Text":"integral over C and I\u0027ll tell you what C is in a minute,"},{"Start":"00:45.780 ","End":"00:53.355","Text":"of x to the 4th yds,"},{"Start":"00:53.355 ","End":"00:56.325","Text":"the x to the 4th y is the f of xy."},{"Start":"00:56.325 ","End":"00:59.320","Text":"But I have to tell you what C is."},{"Start":"00:59.660 ","End":"01:04.760","Text":"Well, C is described as the top 1/2 of the circle,"},{"Start":"01:04.760 ","End":"01:07.220","Text":"x squared plus y squared equals 16."},{"Start":"01:07.220 ","End":"01:08.450","Text":"We know this is a circle."},{"Start":"01:08.450 ","End":"01:11.780","Text":"I\u0027m going to take it in the counterclockwise direction."},{"Start":"01:11.780 ","End":"01:17.580","Text":"What we need to do is write a parametric form of this top 1/2 of the circle."},{"Start":"01:18.350 ","End":"01:21.755","Text":"Here\u0027s a little picture of that circle,"},{"Start":"01:21.755 ","End":"01:24.350","Text":"or 1/2 a circle, the top 1/2."},{"Start":"01:24.350 ","End":"01:27.305","Text":"Notice that 16 is 4 squared,"},{"Start":"01:27.305 ","End":"01:30.405","Text":"so its radius is 4."},{"Start":"01:30.405 ","End":"01:34.600","Text":"The parametric equation is going to be,"},{"Start":"01:34.600 ","End":"01:37.965","Text":"let\u0027s see, x equals,"},{"Start":"01:37.965 ","End":"01:40.260","Text":"normally r sine, sorry,"},{"Start":"01:40.260 ","End":"01:42.600","Text":"r cosine t, but r is 4."},{"Start":"01:42.600 ","End":"01:45.240","Text":"So 4 cosine t,"},{"Start":"01:45.240 ","End":"01:51.230","Text":"y equals 4 sine t. The parameter t"},{"Start":"01:51.230 ","End":"01:57.215","Text":"is actually the angle and for a full circle it goes from 0 to 2Pi,"},{"Start":"01:57.215 ","End":"01:59.555","Text":"but we only have 1/2 a circle,"},{"Start":"01:59.555 ","End":"02:03.590","Text":"so it only goes up to Pi, like 180 degrees."},{"Start":"02:03.590 ","End":"02:08.150","Text":"Now we want to plug in the formula."},{"Start":"02:08.150 ","End":"02:10.830","Text":"Let me just copy this over here."},{"Start":"02:10.870 ","End":"02:15.005","Text":"Here we are, and the ds I already marked it here."},{"Start":"02:15.005 ","End":"02:17.185","Text":"That\u0027s the bit over here."},{"Start":"02:17.185 ","End":"02:20.465","Text":"Let\u0027s see. What do we get?"},{"Start":"02:20.465 ","End":"02:22.610","Text":"We get the integral."},{"Start":"02:22.610 ","End":"02:26.120","Text":"Now, the a and the b are what the parameter goes from,"},{"Start":"02:26.120 ","End":"02:28.175","Text":"so it\u0027s 0 to Pi."},{"Start":"02:28.175 ","End":"02:33.405","Text":"Then x, I just substitute from what it is here,"},{"Start":"02:33.405 ","End":"02:39.390","Text":"so it\u0027s 4 cosine t to the power of"},{"Start":"02:39.390 ","End":"02:45.715","Text":"4 and y is 4 sine t. The ds."},{"Start":"02:45.715 ","End":"02:48.380","Text":"Let\u0027s compute that at the side."},{"Start":"02:48.380 ","End":"02:51.020","Text":"What I get, well,"},{"Start":"02:51.020 ","End":"02:53.100","Text":"only dx over dt."},{"Start":"02:53.590 ","End":"03:04.250","Text":"That is equal to minus 4 sine t. I"},{"Start":"03:04.250 ","End":"03:11.220","Text":"need dy by dt and that is equal to"},{"Start":"03:11.220 ","End":"03:15.525","Text":"4 cosine t. Then"},{"Start":"03:15.525 ","End":"03:23.685","Text":"ds will be the square root of,"},{"Start":"03:23.685 ","End":"03:26.680","Text":"I\u0027ll put the dt so in case I forget later,"},{"Start":"03:26.680 ","End":"03:29.460","Text":"of this squared plus this squared."},{"Start":"03:29.460 ","End":"03:38.240","Text":"What we get is 16 sine squared t minus squared is plus,"},{"Start":"03:38.240 ","End":"03:45.155","Text":"and this thing squared is 16 cosine squared t. Now,"},{"Start":"03:45.155 ","End":"03:49.990","Text":"notice that because sine squared plus cosine squared is 1,"},{"Start":"03:49.990 ","End":"03:53.135","Text":"we know this, we\u0027ve seen it often,"},{"Start":"03:53.135 ","End":"03:54.740","Text":"then under the square root,"},{"Start":"03:54.740 ","End":"03:59.165","Text":"I have just 16 and the square root of 16 is 4."},{"Start":"03:59.165 ","End":"04:01.775","Text":"This thing is just 4dt."},{"Start":"04:01.775 ","End":"04:08.250","Text":"Now back here, I write 4dt."},{"Start":"04:09.710 ","End":"04:15.255","Text":"Note that we have here 4 to the 4th times the 4 times the 4."},{"Start":"04:15.255 ","End":"04:19.465","Text":"If you compute 4 to the power of 6, that\u0027s what we have."},{"Start":"04:19.465 ","End":"04:23.640","Text":"That comes out to be 4,096."},{"Start":"04:23.640 ","End":"04:27.620","Text":"So this equals, I didn\u0027t take the constant out front,"},{"Start":"04:27.620 ","End":"04:33.230","Text":"4,096 integral from 0 to Pi of"},{"Start":"04:33.230 ","End":"04:41.110","Text":"cosine to the 4th t sine t dt."},{"Start":"04:41.940 ","End":"04:45.610","Text":"Now, we can do this by substitution,"},{"Start":"04:45.610 ","End":"04:48.955","Text":"but I don\u0027t want to go into the whole, all the details."},{"Start":"04:48.955 ","End":"04:53.330","Text":"I\u0027m going to substitute cosine to the 4th t. Or we can just say"},{"Start":"04:53.330 ","End":"04:57.740","Text":"that we have the derivative of cosine to the 4th t here. Not quite."},{"Start":"04:57.740 ","End":"05:00.395","Text":"Derivative of cosine is minus sine."},{"Start":"05:00.395 ","End":"05:05.585","Text":"How about if I put a minus here and I put a minus here,"},{"Start":"05:05.585 ","End":"05:08.510","Text":"that should be okay because minus minus is plus."},{"Start":"05:08.510 ","End":"05:13.985","Text":"Now I can say that this is minus 4,096."},{"Start":"05:13.985 ","End":"05:19.880","Text":"Now I know what the integral of this is because the integral of cosine to the 4th with"},{"Start":"05:19.880 ","End":"05:29.245","Text":"its derivative of cosine is just cosine to the fifth of t over 5."},{"Start":"05:29.245 ","End":"05:36.000","Text":"But I need to evaluate this between 0 and Pi."},{"Start":"05:36.310 ","End":"05:40.720","Text":"Constant can come out so what I get is"},{"Start":"05:40.720 ","End":"05:48.400","Text":"minus 4,096 over 5."},{"Start":"05:48.400 ","End":"05:53.700","Text":"Now just a bit more space here."},{"Start":"05:54.940 ","End":"06:04.830","Text":"At the side, cosine to the 5th of Pi is cosine of Pi to the 5th."},{"Start":"06:04.830 ","End":"06:08.430","Text":"Cosine of Pi is minus 1 to the 5th."},{"Start":"06:08.430 ","End":"06:09.540","Text":"5 is an odd number,"},{"Start":"06:09.540 ","End":"06:14.580","Text":"so it\u0027s minus 1 and cosine to the fifth of 0,"},{"Start":"06:14.580 ","End":"06:16.110","Text":"cosine of 0 is 1,"},{"Start":"06:16.110 ","End":"06:19.620","Text":"1 to the 5th is 1."},{"Start":"06:19.620 ","End":"06:28.260","Text":"What I get here is minus 1 from the Pi,"},{"Start":"06:28.260 ","End":"06:31.980","Text":"take away 1 from the 0."},{"Start":"06:31.980 ","End":"06:37.920","Text":"Altogether what we get is, let\u0027s see."},{"Start":"06:37.920 ","End":"06:41.030","Text":"Well this is minus 2 with the minus,"},{"Start":"06:41.030 ","End":"06:49.735","Text":"we just double that and that\u0027s 8,192 over 5."},{"Start":"06:49.735 ","End":"06:54.475","Text":"I\u0027ll leave it like this as an improper fraction, that\u0027s the answer."},{"Start":"06:54.475 ","End":"06:59.305","Text":"The only thing I forgot to do on the diagram was to write the arrow."},{"Start":"06:59.305 ","End":"07:02.350","Text":"One of the thing I forgot to mention that the curve has to be"},{"Start":"07:02.350 ","End":"07:06.940","Text":"smooth and I\u0027m claiming that this curve is smooth."},{"Start":"07:06.940 ","End":"07:10.930","Text":"Smooth means continuous or differentiable"},{"Start":"07:10.930 ","End":"07:17.480","Text":"anyway and the derivatives can\u0027t both be 0. Essentially."},{"Start":"07:17.720 ","End":"07:25.385","Text":"What we had was that if we do it formally, we have r of t,"},{"Start":"07:25.385 ","End":"07:32.555","Text":"we put it in vector form is 4 cosine t, 4 sine t,"},{"Start":"07:32.555 ","End":"07:37.185","Text":"and r prime of t was,"},{"Start":"07:37.185 ","End":"07:40.785","Text":"from here, minus 4 sine t,"},{"Start":"07:40.785 ","End":"07:44.790","Text":"4 cosine t. This is never"},{"Start":"07:44.790 ","End":"07:49.860","Text":"the 0 vector because you can\u0027t have cosine and sine being 0 at the same angle,"},{"Start":"07:49.860 ","End":"07:51.735","Text":"they\u0027ll never both 0."},{"Start":"07:51.735 ","End":"07:54.285","Text":"So yes, it is smooth."},{"Start":"07:54.285 ","End":"07:57.705","Text":"It\u0027s just something you should mentally check."},{"Start":"07:57.705 ","End":"08:02.145","Text":"Now we\u0027re complete with this example, so let\u0027s move on."},{"Start":"08:02.145 ","End":"08:05.985","Text":"Moving on to the next example where I\u0027m going to introduce"},{"Start":"08:05.985 ","End":"08:10.665","Text":"a new concept of piecewise smooth curves."},{"Start":"08:10.665 ","End":"08:13.050","Text":"But first, I\u0027ll clear the board."},{"Start":"08:13.050 ","End":"08:15.795","Text":"I\u0027ll highlight that term,"},{"Start":"08:15.795 ","End":"08:20.145","Text":"which as I say I\u0027ll explain through an example."},{"Start":"08:20.145 ","End":"08:23.940","Text":"Here\u0027s an example of a piecewise smooth curve."},{"Start":"08:23.940 ","End":"08:25.545","Text":"It just has 2 pieces."},{"Start":"08:25.545 ","End":"08:30.530","Text":"I\u0027ll give the equations in a moment and we\u0027ll solve the line integral over this."},{"Start":"08:30.530 ","End":"08:34.625","Text":"I just wanted to show you how it might look as 1 piece and another piece."},{"Start":"08:34.625 ","End":"08:38.400","Text":"In general, it could have a lot of pieces."},{"Start":"08:38.400 ","End":"08:40.500","Text":"Here\u0027s another example with 4 pieces."},{"Start":"08:40.500 ","End":"08:45.030","Text":"Anyway, I want to focus on this one and actually give the equations."},{"Start":"08:45.030 ","End":"08:54.120","Text":"Let me say that this bit here is a parabola and this is the parabola y equals x squared."},{"Start":"08:54.120 ","End":"08:56.940","Text":"This is a horizontal straight line,"},{"Start":"08:56.940 ","End":"09:02.670","Text":"and this is y equals 4."},{"Start":"09:02.670 ","End":"09:05.040","Text":"No, I didn\u0027t quite write this."},{"Start":"09:05.040 ","End":"09:09.075","Text":"What I meant to write was y equals,"},{"Start":"09:09.075 ","End":"09:11.820","Text":"either x squared or 4."},{"Start":"09:11.820 ","End":"09:15.555","Text":"If you noticed that this point is above the x equals 2,"},{"Start":"09:15.555 ","End":"09:21.135","Text":"so it\u0027s x squared when x is less than 2,"},{"Start":"09:21.135 ","End":"09:24.045","Text":"and it\u0027s 4 when x is bigger than 2."},{"Start":"09:24.045 ","End":"09:25.665","Text":"If you don\u0027t want to leave a hole,"},{"Start":"09:25.665 ","End":"09:28.710","Text":"it doesn\u0027t really matter where we put the less than or equal"},{"Start":"09:28.710 ","End":"09:32.040","Text":"2 because either way when you substitute x equals 2,"},{"Start":"09:32.040 ","End":"09:36.525","Text":"you get 4, so this doesn\u0027t matter."},{"Start":"09:36.525 ","End":"09:43.020","Text":"What I want to do is compute a line integral over a portion of this curve,"},{"Start":"09:43.020 ","End":"09:45.660","Text":"which I\u0027m going to call C up to here,"},{"Start":"09:45.660 ","End":"09:52.200","Text":"and up to the point where x equals 4."},{"Start":"09:52.200 ","End":"09:56.040","Text":"The part I\u0027ve highlighted,"},{"Start":"09:56.040 ","End":"09:58.455","Text":"that\u0027s going to be the C,"},{"Start":"09:58.455 ","End":"10:00.885","Text":"and we\u0027re going to go in this direction."},{"Start":"10:00.885 ","End":"10:03.060","Text":"What I want to compute,"},{"Start":"10:03.060 ","End":"10:10.995","Text":"I need to give you a function and the function is going to be 2x and always ds,"},{"Start":"10:10.995 ","End":"10:16.740","Text":"and it\u0027s the integral over the curve C. This is our exercise."},{"Start":"10:16.740 ","End":"10:19.815","Text":"When we have such a curve,"},{"Start":"10:19.815 ","End":"10:22.605","Text":"then we just add up the separate pieces."},{"Start":"10:22.605 ","End":"10:25.365","Text":"We do the line integral over,"},{"Start":"10:25.365 ","End":"10:32.790","Text":"let\u0027s call this piece C_1 and this piece C_2,"},{"Start":"10:32.790 ","End":"10:36.030","Text":"and what we say in this case is that this is equal to the"},{"Start":"10:36.030 ","End":"10:40.020","Text":"integral over C_1 of the same thing"},{"Start":"10:40.020 ","End":"10:48.390","Text":"plus the integral of same thing over C_2."},{"Start":"10:48.390 ","End":"10:50.460","Text":"We break it up into 2 separate pieces in here,"},{"Start":"10:50.460 ","End":"10:54.060","Text":"that would be 4 pieces and so on."},{"Start":"10:54.060 ","End":"10:56.415","Text":"Let\u0027s do the first bit."},{"Start":"10:56.415 ","End":"11:01.020","Text":"I need a parametric equation of the parabola from here to here."},{"Start":"11:01.020 ","End":"11:03.900","Text":"Now whenever y is a function of x,"},{"Start":"11:03.900 ","End":"11:06.810","Text":"we can always get a parametric representation,"},{"Start":"11:06.810 ","End":"11:14.625","Text":"and what we would say for C_1 is that we just let x equals t,"},{"Start":"11:14.625 ","End":"11:23.020","Text":"and then y is equal to t squared and t goes from 0-2."},{"Start":"11:24.830 ","End":"11:28.185","Text":"C_2 is easier."},{"Start":"11:28.185 ","End":"11:31.860","Text":"C_2, to parameterize it,"},{"Start":"11:31.860 ","End":"11:36.345","Text":"we see that y is constantly equal to 4 here,"},{"Start":"11:36.345 ","End":"11:41.355","Text":"so I know that y equals 4 and x will equal, well,"},{"Start":"11:41.355 ","End":"11:44.055","Text":"just any letter, I\u0027ll use t again,"},{"Start":"11:44.055 ","End":"11:47.320","Text":"and t goes from 2-4."},{"Start":"11:51.310 ","End":"11:56.955","Text":"I think I\u0027ll do each one in a separate color so we won\u0027t confuse the 2 integrals."},{"Start":"11:56.955 ","End":"12:00.930","Text":"What we\u0027re going to do, let\u0027s do the hard one first,"},{"Start":"12:00.930 ","End":"12:04.830","Text":"the C_1, and then we get, in this case,"},{"Start":"12:04.830 ","End":"12:13.379","Text":"that ds is equal to the square root."},{"Start":"12:13.379 ","End":"12:18.220","Text":"Now, dx over dt is 1,"},{"Start":"12:18.380 ","End":"12:26.430","Text":"dy over dt is 2t,"},{"Start":"12:26.430 ","End":"12:28.720","Text":"and that the end, dt."},{"Start":"12:33.230 ","End":"12:42.210","Text":"I\u0027ll do it over here. The first integral is the integral from 0-2,"},{"Start":"12:42.210 ","End":"12:46.380","Text":"that\u0027s the limit of the parameter of the function."},{"Start":"12:46.380 ","End":"12:49.935","Text":"The function is actually why it doesn\u0027t appear here,"},{"Start":"12:49.935 ","End":"12:53.625","Text":"but 2x is the function f of xy,"},{"Start":"12:53.625 ","End":"13:01.710","Text":"so 2x in this case is just 2t because x equals t. This is the 2x."},{"Start":"13:01.710 ","End":"13:06.165","Text":"Then I need the ds, which is the square root of"},{"Start":"13:06.165 ","End":"13:12.910","Text":"1 plus 4t squared dt."},{"Start":"13:13.070 ","End":"13:18.040","Text":"You know what, I\u0027ll write the square root as the power of a half."},{"Start":"13:18.290 ","End":"13:22.590","Text":"Now I\u0027m not going to spend a great deal of time on the integration,"},{"Start":"13:22.590 ","End":"13:31.005","Text":"but notice that the derivative of 1 plus 4t squared is almost 2t."},{"Start":"13:31.005 ","End":"13:40.380","Text":"Well, it\u0027s 8t. If I change this 2 to an 8 and I put a 1/4 in front,"},{"Start":"13:40.380 ","End":"13:42.150","Text":"a quarter of 8 is 2,"},{"Start":"13:42.150 ","End":"13:43.800","Text":"so I haven\u0027t changed anything."},{"Start":"13:43.800 ","End":"13:46.335","Text":"That will make the integral easier."},{"Start":"13:46.335 ","End":"13:49.500","Text":"This will equal 1/4."},{"Start":"13:49.500 ","End":"13:52.890","Text":"Now the integral of something to the power of a half would be"},{"Start":"13:52.890 ","End":"13:57.090","Text":"that same thing to the power of 3 over 2,"},{"Start":"13:57.090 ","End":"14:01.440","Text":"so 1 plus 4t squared to the power of 3 over 2."},{"Start":"14:01.440 ","End":"14:04.455","Text":"But then I have to divide by 3 over 2,"},{"Start":"14:04.455 ","End":"14:08.310","Text":"so I\u0027m just multiplying by 2/3 instead."},{"Start":"14:08.310 ","End":"14:15.765","Text":"We don\u0027t need a constant because we\u0027re going to evaluate all this between 0 and 2,"},{"Start":"14:15.765 ","End":"14:18.405","Text":"2 goes into 4 twice,"},{"Start":"14:18.405 ","End":"14:22.480","Text":"so this whole thing is really 1 over 6."},{"Start":"14:23.180 ","End":"14:26.475","Text":"Now we\u0027ll plug in the 2."},{"Start":"14:26.475 ","End":"14:30.150","Text":"2 squared is 4 times 4 is 16 plus 1 is 17,"},{"Start":"14:30.150 ","End":"14:34.275","Text":"so it\u0027s 17^3 over 2."},{"Start":"14:34.275 ","End":"14:38.380","Text":"Wait, I need some more space here."},{"Start":"14:38.840 ","End":"14:42.480","Text":"The 1/6 stays, then we have"},{"Start":"14:42.480 ","End":"14:50.235","Text":"17^3 over 2 minus what I get when I plugin 0."},{"Start":"14:50.235 ","End":"14:53.550","Text":"So if a plugin 0, that\u0027s just 0,"},{"Start":"14:53.550 ","End":"14:57.280","Text":"1^3 over 2 is 1."},{"Start":"14:58.520 ","End":"15:01.745","Text":"I could simplify this but let\u0027s just leave it like this."},{"Start":"15:01.745 ","End":"15:04.070","Text":"This is just the answer for this bit."},{"Start":"15:04.070 ","End":"15:09.155","Text":"What I\u0027ve just done is computed the C_1 part."},{"Start":"15:09.155 ","End":"15:11.255","Text":"Now the second part,"},{"Start":"15:11.255 ","End":"15:18.300","Text":"what we get is that ds in the second case"},{"Start":"15:18.300 ","End":"15:26.639","Text":"is the square root of dx over dt is just 1,"},{"Start":"15:26.639 ","End":"15:29.670","Text":"and dy over dt, it\u0027s a constant,"},{"Start":"15:29.670 ","End":"15:33.180","Text":"so it\u0027s 0 dt,"},{"Start":"15:33.180 ","End":"15:35.980","Text":"so it\u0027s just equal to dt."},{"Start":"15:38.600 ","End":"15:46.470","Text":"What we get is the integral, let\u0027s see,"},{"Start":"15:46.470 ","End":"15:51.810","Text":"from 2-4 of 2x,"},{"Start":"15:51.810 ","End":"15:56.794","Text":"which is 2t, same as before,"},{"Start":"15:56.794 ","End":"16:01.610","Text":"but the ds is now dt."},{"Start":"16:03.170 ","End":"16:12.455","Text":"This is quite easy because this is just t squared from 2-4."},{"Start":"16:12.455 ","End":"16:13.910","Text":"Now if I plug in 4,"},{"Start":"16:13.910 ","End":"16:18.230","Text":"I get 16, I plug in 2,"},{"Start":"16:18.230 ","End":"16:25.260","Text":"I get 4, so this is just equal to 12."},{"Start":"16:25.690 ","End":"16:33.410","Text":"Finally, I combine the results and I say that my answer,"},{"Start":"16:33.410 ","End":"16:37.520","Text":"which is the integral from over"},{"Start":"16:37.520 ","End":"16:47.850","Text":"the curve of 2x ds is the first bit,"},{"Start":"16:48.020 ","End":"16:53.900","Text":"1/6 of 17^3 over 2 minus"},{"Start":"16:53.900 ","End":"17:00.240","Text":"1 plus 12. That\u0027s it."}],"ID":10486},{"Watched":false,"Name":"Worked Example 3","Duration":"14m 59s","ChapterTopicVideoID":10182,"CourseChapterTopicPlaylistID":82678,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10182.jpeg","UploadDate":"2020-02-26T06:21:48.5070000","DurationForVideoObject":"PT14M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.770","Text":"In the next example,"},{"Start":"00:01.770 ","End":"00:05.550","Text":"I\u0027m going to reuse this picture."},{"Start":"00:05.550 ","End":"00:11.995","Text":"What I want to do is compute the integral of the same function 2x,"},{"Start":"00:11.995 ","End":"00:13.590","Text":"but not over the curve C,"},{"Start":"00:13.590 ","End":"00:18.720","Text":"over a different curve that connects this point to this point."},{"Start":"00:18.720 ","End":"00:20.655","Text":"The same start and endpoints,"},{"Start":"00:20.655 ","End":"00:24.015","Text":"I drew the straight line between these 2 points."},{"Start":"00:24.015 ","End":"00:25.875","Text":"This is the direction."},{"Start":"00:25.875 ","End":"00:28.070","Text":"I want to compute it over this curve."},{"Start":"00:28.070 ","End":"00:31.105","Text":"Let me call it C_3."},{"Start":"00:31.105 ","End":"00:35.760","Text":"Let me just note that this is the point 0, 0, of course."},{"Start":"00:35.760 ","End":"00:37.920","Text":"This is the point that C,"},{"Start":"00:37.920 ","End":"00:41.690","Text":"x is 4, 4."},{"Start":"00:41.690 ","End":"00:45.920","Text":"Well, it\u0027s going to be very easy to find the parametric equation for this."},{"Start":"00:45.920 ","End":"00:50.070","Text":"Let me just write down the integral that I want."},{"Start":"00:51.260 ","End":"00:59.060","Text":"What we want this time is the integral over the C_3 line,"},{"Start":"00:59.060 ","End":"01:09.000","Text":"the curve of the same thing, 2x ds."},{"Start":"01:09.000 ","End":"01:11.355","Text":"What do we get this time?"},{"Start":"01:11.355 ","End":"01:15.365","Text":"Well to parametrize this, it\u0027s easy."},{"Start":"01:15.365 ","End":"01:19.620","Text":"What I can say is I\u0027ll write it over here,"},{"Start":"01:19.620 ","End":"01:22.510","Text":"is that x is y."},{"Start":"01:22.510 ","End":"01:26.005","Text":"You can see that x equals y is this line."},{"Start":"01:26.005 ","End":"01:31.700","Text":"Anyway, x equals t, y equals t,"},{"Start":"01:31.700 ","End":"01:38.725","Text":"and t goes from 0 to 4."},{"Start":"01:38.725 ","End":"01:42.800","Text":"It\u0027s intuitive. I didn\u0027t use the general method"},{"Start":"01:42.800 ","End":"01:47.795","Text":"of line between 2 points because we\u0027ve got an easy case here."},{"Start":"01:47.795 ","End":"01:51.660","Text":"What we need is ds,"},{"Start":"01:51.940 ","End":"01:58.205","Text":"ds is equal to the square root."},{"Start":"01:58.205 ","End":"02:08.920","Text":"Now, dx over dt is 1 squared dy over dt is 1 squared dt."},{"Start":"02:08.920 ","End":"02:16.600","Text":"In other words, ds is just equal to square root of 2dt."},{"Start":"02:16.760 ","End":"02:19.980","Text":"Now I can compute the integral,"},{"Start":"02:19.980 ","End":"02:28.950","Text":"and I can say that this is the integral from 0 to 4 of 2x,"},{"Start":"02:28.950 ","End":"02:34.935","Text":"which is 2t times ds,"},{"Start":"02:34.935 ","End":"02:38.590","Text":"which is the square root of 2dt."},{"Start":"02:40.910 ","End":"02:47.295","Text":"Let me bring the square root upfront."},{"Start":"02:47.295 ","End":"02:49.410","Text":"We\u0027ve got the square root of 2."},{"Start":"02:49.410 ","End":"02:53.235","Text":"Now the integral of 2t is t squared."},{"Start":"02:53.235 ","End":"02:58.025","Text":"What we have is t squared taken from 0 to 4."},{"Start":"02:58.025 ","End":"03:00.635","Text":"When I plug in 4, I get 16,"},{"Start":"03:00.635 ","End":"03:02.060","Text":"0 I get nothing."},{"Start":"03:02.060 ","End":"03:08.940","Text":"The answer is just 16 square root of 2."},{"Start":"03:09.230 ","End":"03:14.615","Text":"I\u0027ll highlight it and I\u0027ll also highlight the answer to the previous part."},{"Start":"03:14.615 ","End":"03:19.205","Text":"There we go. Just notice that these 2 answers are different."},{"Start":"03:19.205 ","End":"03:24.770","Text":"I have a reason for contrasting this because in future with another integral,"},{"Start":"03:24.770 ","End":"03:27.920","Text":"it won\u0027t make any difference which path we go along,"},{"Start":"03:27.920 ","End":"03:28.955","Text":"but that\u0027s not for now,"},{"Start":"03:28.955 ","End":"03:31.130","Text":"so you can forget what I said."},{"Start":"03:31.130 ","End":"03:33.860","Text":"I want to do another example now."},{"Start":"03:33.860 ","End":"03:37.230","Text":"I\u0027ll go back to the previous picture."},{"Start":"03:37.840 ","End":"03:46.980","Text":"What would happen if I took the integral in the opposite direction?"},{"Start":"03:46.980 ","End":"03:49.400","Text":"The 2 pictures are the same, just the arrows,"},{"Start":"03:49.400 ","End":"03:52.475","Text":"but I\u0027ll call the 1 going in this direction C_3,"},{"Start":"03:52.475 ","End":"03:54.500","Text":"this 1, I\u0027ll call it C_4,"},{"Start":"03:54.500 ","End":"03:56.600","Text":"same thing in opposite direction."},{"Start":"03:56.600 ","End":"04:01.450","Text":"Now, I can\u0027t use the same parametrization."},{"Start":"04:02.150 ","End":"04:04.785","Text":"This was for C_3."},{"Start":"04:04.785 ","End":"04:08.450","Text":"Now for C_4, I could take the parametrization,"},{"Start":"04:08.450 ","End":"04:09.560","Text":"I\u0027ll just give it to you."},{"Start":"04:09.560 ","End":"04:13.530","Text":"Suppose instead of t I put 4 minus t,"},{"Start":"04:14.540 ","End":"04:17.459","Text":"and the same thing for y,"},{"Start":"04:17.459 ","End":"04:21.450","Text":"notice that when t is 0,"},{"Start":"04:21.450 ","End":"04:23.835","Text":"I get the point 4, 4,"},{"Start":"04:23.835 ","End":"04:27.015","Text":"and when t is 4, I get the point 0, 0."},{"Start":"04:27.015 ","End":"04:32.280","Text":"It\u0027s also between 0 and 4, but different expression."},{"Start":"04:32.280 ","End":"04:38.475","Text":"Here t, here 4 minus t. The integral over"},{"Start":"04:38.475 ","End":"04:46.765","Text":"C_4 which is in the opposite direction of 2x ds is equal to,"},{"Start":"04:46.765 ","End":"04:48.850","Text":"let\u0027s see what is ds in this case?"},{"Start":"04:48.850 ","End":"04:59.500","Text":"Well, square root of dx over dt is minus 1,"},{"Start":"04:59.660 ","End":"05:03.720","Text":"and dy over dt is also minus 1."},{"Start":"05:03.720 ","End":"05:07.470","Text":"We square both, we add, take the square root, add a dt."},{"Start":"05:08.600 ","End":"05:11.370","Text":"We get the same thing because look,"},{"Start":"05:11.370 ","End":"05:14.050","Text":"it doesn\u0027t matter about the minus, it\u0027s squared."},{"Start":"05:14.050 ","End":"05:17.875","Text":"We also get the square root of 2dt."},{"Start":"05:17.875 ","End":"05:26.585","Text":"Now all I have to do is plug in 2x will be, well, I\u0027ll just write it."},{"Start":"05:26.585 ","End":"05:28.970","Text":"We\u0027ve got the integral from 0 to 4,"},{"Start":"05:28.970 ","End":"05:33.720","Text":"2x is 8 minus 2t,"},{"Start":"05:35.990 ","End":"05:40.210","Text":"times the square root of 2dt."},{"Start":"05:42.350 ","End":"05:46.080","Text":"Let\u0027s see what we get this time."},{"Start":"05:46.080 ","End":"05:49.290","Text":"Square root of t is a constant."},{"Start":"05:49.290 ","End":"05:52.365","Text":"The integral of 8 is 8t."},{"Start":"05:52.365 ","End":"05:55.265","Text":"Integral of 2t is t squared,"},{"Start":"05:55.265 ","End":"06:00.895","Text":"so it\u0027s minus t squared taken between 0 and 4."},{"Start":"06:00.895 ","End":"06:04.980","Text":"Now, let\u0027s see. If we plug in 4,"},{"Start":"06:04.980 ","End":"06:13.600","Text":"we get 8 times 4 is 32 minus 16 is 16 root 2."},{"Start":"06:15.380 ","End":"06:18.769","Text":"That\u0027s it because when I put in the 0,"},{"Start":"06:18.769 ","End":"06:21.650","Text":"this is 0, so I don\u0027t have to subtract anything."},{"Start":"06:21.650 ","End":"06:30.045","Text":"Notice that I got the same answer as I did regardless of the direction."},{"Start":"06:30.045 ","End":"06:34.310","Text":"It turns out that we can generalize this,"},{"Start":"06:34.310 ","End":"06:38.780","Text":"that when you reverse the direction of this particular kind of line integral,"},{"Start":"06:38.780 ","End":"06:39.980","Text":"and there will be others,"},{"Start":"06:39.980 ","End":"06:42.185","Text":"but in this case, you\u0027ll get the same answer."},{"Start":"06:42.185 ","End":"06:44.670","Text":"There\u0027s also a notation."},{"Start":"06:44.780 ","End":"06:48.920","Text":"When you have 2 curves that are the same but in opposite directions,"},{"Start":"06:48.920 ","End":"06:50.840","Text":"you say that 1 is minus the other."},{"Start":"06:50.840 ","End":"06:56.340","Text":"I could say that C_4 is equal to minus C_3."},{"Start":"06:56.340 ","End":"07:02.190","Text":"When we put a minus in front of a curve it means that curve but in opposite direction."},{"Start":"07:02.190 ","End":"07:04.485","Text":"There is a general rule,"},{"Start":"07:04.485 ","End":"07:09.375","Text":"and I\u0027ll just scroll and I\u0027ll give it to you."},{"Start":"07:09.375 ","End":"07:15.545","Text":"Here is this general rule which says that this is not a coincidence,"},{"Start":"07:15.545 ","End":"07:20.150","Text":"that when you take this particular line integral over a curve,"},{"Start":"07:20.150 ","End":"07:23.000","Text":"or if you take it over the curve with the opposite direction,"},{"Start":"07:23.000 ","End":"07:26.130","Text":"you\u0027ll get the same answer."},{"Start":"07:27.870 ","End":"07:30.790","Text":"This is important, but in future,"},{"Start":"07:30.790 ","End":"07:35.200","Text":"there will be other kinds of line integral where this will not be the case."},{"Start":"07:35.200 ","End":"07:38.005","Text":"Finally, for this clip,"},{"Start":"07:38.005 ","End":"07:40.569","Text":"I want to just talk about 3 dimensions,"},{"Start":"07:40.569 ","End":"07:43.930","Text":"cause so far we\u0027ve restricted ourselves to 2 dimensions."},{"Start":"07:43.930 ","End":"07:52.310","Text":"Now, we\u0027re going to expand our definition from 2D to 3D."},{"Start":"07:52.740 ","End":"07:55.179","Text":"Just to remark on notation,"},{"Start":"07:55.179 ","End":"07:58.915","Text":"sometimes rather than having to keep inventing letters f,"},{"Start":"07:58.915 ","End":"08:01.120","Text":"h, g. Instead of h of t,"},{"Start":"08:01.120 ","End":"08:03.280","Text":"I could have just said x of t,"},{"Start":"08:03.280 ","End":"08:06.850","Text":"where it means that x is a function of t and then y of t,"},{"Start":"08:06.850 ","End":"08:10.540","Text":"because it\u0027ll save me keep on using up letters of the alphabet."},{"Start":"08:10.540 ","End":"08:14.995","Text":"Now I\u0027m going to generalize this to 3D,"},{"Start":"08:14.995 ","End":"08:19.690","Text":"and here it is, same thing, just longer."},{"Start":"08:19.690 ","End":"08:22.760","Text":"Just added a z component."},{"Start":"08:24.090 ","End":"08:28.630","Text":"Before this, well this is part is the ds and the curve c is"},{"Start":"08:28.630 ","End":"08:32.350","Text":"given in parametric form by x equals,"},{"Start":"08:32.350 ","End":"08:34.090","Text":"well, we had h of t, g of t, now,"},{"Start":"08:34.090 ","End":"08:37.270","Text":"we\u0027ll just call it generically some function of t,"},{"Start":"08:37.270 ","End":"08:40.330","Text":"y is also some function of t,"},{"Start":"08:40.330 ","End":"08:42.880","Text":"and z is a function of t,"},{"Start":"08:42.880 ","End":"08:50.270","Text":"and t goes between 2 numbers, a and b."},{"Start":"08:50.580 ","End":"08:54.565","Text":"Often we write this in vector form in fact."},{"Start":"08:54.565 ","End":"08:58.000","Text":"Instead of saying x, y, and z,"},{"Start":"08:58.000 ","End":"09:03.760","Text":"we would say r of t a vector function."},{"Start":"09:03.760 ","End":"09:08.770","Text":"We know that vector form and parametric form are very similar."},{"Start":"09:08.770 ","End":"09:12.760","Text":"If you spell it out, it\u0027s this, and of course,"},{"Start":"09:12.760 ","End":"09:15.354","Text":"we still have the same interval,"},{"Start":"09:15.354 ","End":"09:20.035","Text":"a, t is between a and b."},{"Start":"09:20.035 ","End":"09:22.450","Text":"If we do use the vector form,"},{"Start":"09:22.450 ","End":"09:24.550","Text":"there is a slightly shorter way of writing this."},{"Start":"09:24.550 ","End":"09:27.880","Text":"Remember that the derivative of the vector"},{"Start":"09:27.880 ","End":"09:32.545","Text":"is just the derivative of each piece separately, dx over dt,"},{"Start":"09:32.545 ","End":"09:38.260","Text":"dy over dt, dz over dt,"},{"Start":"09:38.260 ","End":"09:42.655","Text":"and I hope you remember what is the magnitude of a vector."},{"Start":"09:42.655 ","End":"09:50.095","Text":"If so, I will show you now the slightly shorter version."},{"Start":"09:50.095 ","End":"09:51.430","Text":"Because if you look at it,"},{"Start":"09:51.430 ","End":"09:57.745","Text":"what\u0027s written under this square root is just the magnitude of this vector."},{"Start":"09:57.745 ","End":"10:01.790","Text":"We get a simpler form now for ds."},{"Start":"10:02.370 ","End":"10:05.305","Text":"It\u0027s time for an example."},{"Start":"10:05.305 ","End":"10:08.410","Text":"I\u0027ll do the example over here. I still have space."},{"Start":"10:08.410 ","End":"10:11.335","Text":"Let\u0027s first define the curve and then the function."},{"Start":"10:11.335 ","End":"10:15.700","Text":"The curve C is now going to be a 3D curve,"},{"Start":"10:15.700 ","End":"10:18.895","Text":"and I\u0027ll give it in vector form,"},{"Start":"10:18.895 ","End":"10:29.709","Text":"so r of t equals cosine t, sine t,"},{"Start":"10:29.709 ","End":"10:34.690","Text":"3t, and we\u0027ll take t"},{"Start":"10:34.690 ","End":"10:41.540","Text":"between 0 and 4-Pi."},{"Start":"10:42.330 ","End":"10:47.350","Text":"Here\u0027s a picture, though we don\u0027t need it, of the curve C,"},{"Start":"10:47.350 ","End":"10:49.870","Text":"and actually because of the 4-Pi,"},{"Start":"10:49.870 ","End":"10:54.340","Text":"it\u0027s a helix but it goes twice around and as it goes around twice,"},{"Start":"10:54.340 ","End":"10:55.959","Text":"it\u0027s also going upwards."},{"Start":"10:55.959 ","End":"11:06.055","Text":"Anyway, what I want is the integral along this curve C in 3 dimensions of the function x,"},{"Start":"11:06.055 ","End":"11:13.370","Text":"y, z, and it\u0027s always ds in this kind of integral."},{"Start":"11:14.130 ","End":"11:18.265","Text":"First thing I need to do is compute ds."},{"Start":"11:18.265 ","End":"11:21.235","Text":"If I use this vector form,"},{"Start":"11:21.235 ","End":"11:29.185","Text":"I need r prime of t. R prime of t vector is,"},{"Start":"11:29.185 ","End":"11:33.565","Text":"differentiate each bit separately."},{"Start":"11:33.565 ","End":"11:40.840","Text":"Derivative of cosine t is minus sine t. Derivative of sine t is"},{"Start":"11:40.840 ","End":"11:48.820","Text":"cosine t. Derivative of 3t is 3 over here."},{"Start":"11:48.820 ","End":"11:54.115","Text":"The ds is equal to,"},{"Start":"11:54.115 ","End":"11:57.620","Text":"the magnitude of this is the square root,"},{"Start":"11:58.320 ","End":"12:02.620","Text":"it\u0027s sine squared, I don\u0027t need the minus of t,"},{"Start":"12:02.620 ","End":"12:09.805","Text":"plus cosine squared of t plus 9,"},{"Start":"12:09.805 ","End":"12:13.180","Text":"sine squared plus cosine squared of any angle is 1,"},{"Start":"12:13.180 ","End":"12:17.180","Text":"1 plus 9 is 10, and it\u0027s dt."},{"Start":"12:17.520 ","End":"12:27.690","Text":"What we get is that ds is just root 10 gt."},{"Start":"12:27.690 ","End":"12:31.875","Text":"Now I can go ahead and substitute everything."},{"Start":"12:31.875 ","End":"12:34.950","Text":"What I get is the integral."},{"Start":"12:34.950 ","End":"12:38.165","Text":"Now, my parameter is still from,"},{"Start":"12:38.165 ","End":"12:42.230","Text":"I still have it here from 0 to 4-Pi."},{"Start":"12:42.750 ","End":"12:46.045","Text":"I\u0027d better scroll up, just to remind you,"},{"Start":"12:46.045 ","End":"12:48.145","Text":"here we can see it just about."},{"Start":"12:48.145 ","End":"12:51.220","Text":"Cosine t, sine t 3t, so we get,"},{"Start":"12:51.220 ","End":"12:53.215","Text":"I\u0027ll put the 3 first,"},{"Start":"12:53.215 ","End":"12:54.610","Text":"I\u0027ll put the 3t first,"},{"Start":"12:54.610 ","End":"12:55.990","Text":"I\u0027ll do the z part,"},{"Start":"12:55.990 ","End":"12:58.960","Text":"and then sine t, cosine t,"},{"Start":"12:58.960 ","End":"13:01.405","Text":"it doesn\u0027t matter the order of the multiplication,"},{"Start":"13:01.405 ","End":"13:05.680","Text":"times sine t times cosine t,"},{"Start":"13:05.680 ","End":"13:08.050","Text":"and then I need ds,"},{"Start":"13:08.050 ","End":"13:09.520","Text":"which is root 10,"},{"Start":"13:09.520 ","End":"13:14.270","Text":"which I can put here, and then dt."},{"Start":"13:15.210 ","End":"13:19.000","Text":"I want to remind you of a trigonometrical formula."},{"Start":"13:19.000 ","End":"13:27.340","Text":"Sine of 2 Alpha is 2 sine Alpha cosine Alpha."},{"Start":"13:27.340 ","End":"13:29.245","Text":"If I use that here,"},{"Start":"13:29.245 ","End":"13:35.770","Text":"then I can replace sine t cosine t by a half sine of 2t."},{"Start":"13:35.770 ","End":"13:39.415","Text":"In short, what I get,"},{"Start":"13:39.415 ","End":"13:48.760","Text":"the half and the 3 and the square root of 10 will give me 3 over 2 square root of 10."},{"Start":"13:48.760 ","End":"13:50.890","Text":"I\u0027ll take that outside the integral,"},{"Start":"13:50.890 ","End":"13:59.650","Text":"and the integral will be the t sine of 2t dt."},{"Start":"13:59.650 ","End":"14:02.455","Text":"Now, this requires some work."},{"Start":"14:02.455 ","End":"14:06.115","Text":"Actually, integration by parts will do it."},{"Start":"14:06.115 ","End":"14:09.685","Text":"I\u0027m just going to quote the answer."},{"Start":"14:09.685 ","End":"14:18.520","Text":"The answer to this integral is 1/4 of"},{"Start":"14:18.520 ","End":"14:28.105","Text":"sine 2t minus 1.5t cosine of 2t."},{"Start":"14:28.105 ","End":"14:33.760","Text":"Of course, all this has to be put in a bracket,"},{"Start":"14:33.760 ","End":"14:37.750","Text":"and then we still need the 3 over 2 root 10,"},{"Start":"14:37.750 ","End":"14:44.890","Text":"and then we need to evaluate this whole thing between 0 and 4 Pi."},{"Start":"14:44.890 ","End":"14:46.750","Text":"Cut to the chase,"},{"Start":"14:46.750 ","End":"14:52.540","Text":"the answer is minus 3 root 10 Pi."},{"Start":"14:52.540 ","End":"14:59.750","Text":"I\u0027ll highlight it, and we are done with this clip."}],"ID":10487},{"Watched":false,"Name":"Line Integral of Type 2 in 2D","Duration":"14m 26s","ChapterTopicVideoID":10183,"CourseChapterTopicPlaylistID":82678,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10183.jpeg","UploadDate":"2020-02-26T06:20:45.8830000","DurationForVideoObject":"PT14M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.000","Text":"Continuing with line integrals,"},{"Start":"00:03.000 ","End":"00:05.460","Text":"we only learned about 1 kind,"},{"Start":"00:05.460 ","End":"00:09.270","Text":"which is ds, meaning with respect to arc length."},{"Start":"00:09.270 ","End":"00:11.264","Text":"But in this section,"},{"Start":"00:11.264 ","End":"00:15.000","Text":"we\u0027re going to learn about others,"},{"Start":"00:15.000 ","End":"00:23.370","Text":"dx and dy in 2 dimensions and after that in 3 dimensions."},{"Start":"00:23.370 ","End":"00:26.505","Text":"We\u0027ll also do a dz."},{"Start":"00:26.505 ","End":"00:30.325","Text":"Let\u0027s start with a definition for dx."},{"Start":"00:30.325 ","End":"00:37.710","Text":"It\u0027s written as the integral of f of x and y,"},{"Start":"00:37.710 ","End":"00:40.620","Text":"we\u0027re talking about 2D for the moment,"},{"Start":"00:40.620 ","End":"00:45.300","Text":"later we\u0027ll do the 3D, dx,"},{"Start":"00:45.300 ","End":"00:50.935","Text":"not ds, over a curve c. But I have to tell you what c is."},{"Start":"00:50.935 ","End":"00:57.365","Text":"c is given as before in parametric form,"},{"Start":"00:57.365 ","End":"01:03.440","Text":"where we say that x equals some function x of t and y equals"},{"Start":"01:03.440 ","End":"01:09.410","Text":"some function y of t. t goes in some interval,"},{"Start":"01:09.410 ","End":"01:12.410","Text":"let\u0027s say between a and b."},{"Start":"01:12.410 ","End":"01:14.255","Text":"Now that I\u0027ve given you what c is,"},{"Start":"01:14.255 ","End":"01:17.330","Text":"now I can tell you the definition of this."},{"Start":"01:17.330 ","End":"01:25.460","Text":"This is going to be the integral from a to b of f of x, y."},{"Start":"01:25.460 ","End":"01:27.470","Text":"But if I write it in full,"},{"Start":"01:27.470 ","End":"01:34.640","Text":"x is a function of t and y is a function of t. Now dx,"},{"Start":"01:34.640 ","End":"01:36.140","Text":"well you could guess it from here."},{"Start":"01:36.140 ","End":"01:38.210","Text":"It\u0027s just the derivative of this."},{"Start":"01:38.210 ","End":"01:44.285","Text":"I\u0027ll use the Newton style notation with the prime x prime of t, dt."},{"Start":"01:44.285 ","End":"01:52.400","Text":"This part is the dx and this part is the f. I\u0027ll give a definition for y also,"},{"Start":"01:52.400 ","End":"01:54.505","Text":"and then we\u0027ll do the example."},{"Start":"01:54.505 ","End":"01:56.730","Text":"I just did a copy-paste now,"},{"Start":"01:56.730 ","End":"02:02.570","Text":"but what I\u0027m going to do is erase this x and I\u0027m going to erase this x and I\u0027ll put a y"},{"Start":"02:02.570 ","End":"02:09.635","Text":"here and a y here and then we have the definition for the dy."},{"Start":"02:09.635 ","End":"02:15.950","Text":"An example. I\u0027ll write the example but it will need some explaining."},{"Start":"02:15.950 ","End":"02:18.170","Text":"I want the integral of"},{"Start":"02:18.170 ","End":"02:24.105","Text":"xy dx along the curve c"},{"Start":"02:24.105 ","End":"02:29.240","Text":"plus x minus y dy."},{"Start":"02:29.240 ","End":"02:36.230","Text":"Now, I forgot to mention that these 2 integrals are often given in combined form."},{"Start":"02:36.230 ","End":"02:38.540","Text":"Something that looks like this,"},{"Start":"02:38.540 ","End":"02:40.715","Text":"where instead of f here and here,"},{"Start":"02:40.715 ","End":"02:44.540","Text":"I use a P function for the dx and"},{"Start":"02:44.540 ","End":"02:50.570","Text":"a Q function for the dy and all it means is that we just do each bit separately."},{"Start":"02:50.570 ","End":"02:54.680","Text":"That\u0027s what this equation is saying and you\u0027ll see this in the example."},{"Start":"02:54.680 ","End":"02:57.350","Text":"Now, I still have to tell you what c is."},{"Start":"02:57.350 ","End":"02:59.090","Text":"Otherwise, we can\u0027t solve this."},{"Start":"02:59.090 ","End":"03:02.810","Text":"Let me just say in words that c is"},{"Start":"03:02.810 ","End":"03:10.590","Text":"the line segment from 2, 0-3, 2."},{"Start":"03:12.730 ","End":"03:18.830","Text":"Now, we need to parametrize this curve."},{"Start":"03:18.830 ","End":"03:23.810","Text":"Easiest way to do this is to find y as a function of x,"},{"Start":"03:23.810 ","End":"03:25.775","Text":"so x is a function of y."},{"Start":"03:25.775 ","End":"03:32.930","Text":"In this case, I\u0027ll just give you the answer of how we get the equation of y in terms of"},{"Start":"03:32.930 ","End":"03:40.970","Text":"x for this line segment and we get it that the equation would be,"},{"Start":"03:40.970 ","End":"03:50.915","Text":"well, the equation is y equals 2x minus 4,"},{"Start":"03:50.915 ","End":"03:54.740","Text":"which I have to write in parametric form to use"},{"Start":"03:54.740 ","End":"03:58.430","Text":"this at least in the classic the way I wrote it."},{"Start":"03:58.430 ","End":"04:00.020","Text":"Let\u0027s just write it."},{"Start":"04:00.020 ","End":"04:01.625","Text":"We\u0027ve learned how to do it."},{"Start":"04:01.625 ","End":"04:06.065","Text":"We just let the independent variable x be some other letter,"},{"Start":"04:06.065 ","End":"04:12.395","Text":"and that\u0027s usually t. Then y equals just 2t minus 4."},{"Start":"04:12.395 ","End":"04:14.420","Text":"Because y is a function of x,"},{"Start":"04:14.420 ","End":"04:17.305","Text":"we can just take x from 2-3."},{"Start":"04:17.305 ","End":"04:20.850","Text":"If x was greater than this x,"},{"Start":"04:20.850 ","End":"04:22.340","Text":"we just change the order."},{"Start":"04:22.340 ","End":"04:24.840","Text":"The order is important."},{"Start":"04:26.680 ","End":"04:33.170","Text":"t goes from 2-3 and t is x."},{"Start":"04:33.170 ","End":"04:40.115","Text":"Now we can do this computation."},{"Start":"04:40.115 ","End":"04:43.220","Text":"Let\u0027s just note that in this case,"},{"Start":"04:43.220 ","End":"04:47.330","Text":"we get that dx is equal to dt from here,"},{"Start":"04:47.330 ","End":"04:50.840","Text":"and that dy is equal to 2,"},{"Start":"04:50.840 ","End":"04:54.015","Text":"the derivative of this, dt."},{"Start":"04:54.015 ","End":"04:56.450","Text":"If I put everything here,"},{"Start":"04:56.450 ","End":"04:58.970","Text":"these x and y inside here,"},{"Start":"04:58.970 ","End":"05:06.780","Text":"I get that this is equal to the integral"},{"Start":"05:15.680 ","End":"05:25.260","Text":"from 2-3, x times y is t times 2t minus 4."},{"Start":"05:25.260 ","End":"05:29.715","Text":"Then I still need the dx, which is dt."},{"Start":"05:29.715 ","End":"05:33.200","Text":"But as here we separated into 2 integrals."},{"Start":"05:33.200 ","End":"05:35.795","Text":"We had another integral for the dy bit,"},{"Start":"05:35.795 ","End":"05:42.620","Text":"which is also from 2-3 of x minus yt minus 2,"},{"Start":"05:42.620 ","End":"05:45.575","Text":"t minus 4 is,"},{"Start":"05:45.575 ","End":"05:49.414","Text":"let\u0027s say it\u0027s minus t plus 4,"},{"Start":"05:49.414 ","End":"05:51.710","Text":"and dy is 2 dt."},{"Start":"05:51.710 ","End":"05:55.320","Text":"I can put the 2 here, dt."},{"Start":"05:55.550 ","End":"05:59.075","Text":"Now we just have 2 straightforward integrals."},{"Start":"05:59.075 ","End":"06:01.820","Text":"Let me just note that if I expand them,"},{"Start":"06:01.820 ","End":"06:11.130","Text":"this is 2t squared minus 4t and if I expand this,"},{"Start":"06:11.130 ","End":"06:17.530","Text":"it\u0027s minus 2t plus 8."},{"Start":"06:18.080 ","End":"06:25.915","Text":"Actually, I could recombine these integrals and get the integral from 2-3."},{"Start":"06:25.915 ","End":"06:32.520","Text":"Let\u0027s see, 2t squared minus 4t minus 2t"},{"Start":"06:32.520 ","End":"06:40.290","Text":"minus 60 plus 8 dt."},{"Start":"06:40.290 ","End":"06:45.570","Text":"Now the integral is,"},{"Start":"06:45.570 ","End":"06:50.025","Text":"this would be 2/3t cubed."},{"Start":"06:50.025 ","End":"06:59.175","Text":"Here we would increase the power to 2 divide by 2 minus 3t squared here plus 8t."},{"Start":"06:59.175 ","End":"07:05.190","Text":"This we want to evaluate between 2 and 3."},{"Start":"07:05.580 ","End":"07:08.530","Text":"Let\u0027s see plug-in 3,"},{"Start":"07:08.530 ","End":"07:14.800","Text":"this comes out to be 18 minus 27 plus 16."},{"Start":"07:14.800 ","End":"07:18.220","Text":"I make it 15 minus,"},{"Start":"07:18.220 ","End":"07:24.460","Text":"and if I plug in 2, then 8 times 2 is 16."},{"Start":"07:24.460 ","End":"07:27.475","Text":"This is 12, 16 minus 12 is 4."},{"Start":"07:27.475 ","End":"07:32.260","Text":"Here I get 2 cubed times 2 is 16 over 3,"},{"Start":"07:32.260 ","End":"07:37.390","Text":"5, and a 1/3 plus the 4 is 9 and a 1/3."},{"Start":"07:37.390 ","End":"07:41.335","Text":"Altogether the answer is 5 and 2/3,"},{"Start":"07:41.335 ","End":"07:42.550","Text":"but that\u0027s less important."},{"Start":"07:42.550 ","End":"07:45.550","Text":"How we do it is what matters."},{"Start":"07:45.550 ","End":"07:49.810","Text":"That\u0027s one kind of line integral and yQ,"},{"Start":"07:49.810 ","End":"07:54.250","Text":"not dx, and should always notice what you\u0027re doing,"},{"Start":"07:54.250 ","End":"07:58.060","Text":"whether it\u0027s ds, dx, dy, a combined."},{"Start":"07:58.060 ","End":"08:02.720","Text":"There are different types of line integrals and there will be more."},{"Start":"08:02.910 ","End":"08:07.240","Text":"Now I\u0027d like to take a variation on this example."},{"Start":"08:07.240 ","End":"08:11.890","Text":"What I\u0027d like to do is same integral but"},{"Start":"08:11.890 ","End":"08:16.780","Text":"change the curve to be the other way from 3,2 to 2,0."},{"Start":"08:16.780 ","End":"08:21.075","Text":"I just copied, and I\u0027m going to take the curve minus C,"},{"Start":"08:21.075 ","End":"08:23.190","Text":"I have to switch these 2."},{"Start":"08:23.190 ","End":"08:27.800","Text":"It\u0027s from 3,2 to 2,0."},{"Start":"08:27.800 ","End":"08:34.360","Text":"Now this parametrization won\u0027t work. There are many things we could do."},{"Start":"08:34.360 ","End":"08:41.140","Text":"We could just use the standard formula to do the line segment from here to here."},{"Start":"08:41.140 ","End":"08:44.050","Text":"Let me just give you one possible parameterization"},{"Start":"08:44.050 ","End":"08:47.330","Text":"because I don\u0027t want to spend time doing that."},{"Start":"08:50.820 ","End":"08:57.700","Text":"We could do it as x equals, and y equals."},{"Start":"08:57.700 ","End":"09:00.370","Text":"Now I\u0027m thinking aloud."},{"Start":"09:00.370 ","End":"09:05.660","Text":"I noticed that the distance from here to here is 1."},{"Start":"09:07.170 ","End":"09:11.184","Text":"To get from 3 to 2,"},{"Start":"09:11.184 ","End":"09:19.190","Text":"I could take 3 minus t and have t go from 0 to 1."},{"Start":"09:20.100 ","End":"09:27.235","Text":"That would do it, but as for y it has to go from 2 to 0."},{"Start":"09:27.235 ","End":"09:28.810","Text":"When t is 0,"},{"Start":"09:28.810 ","End":"09:31.340","Text":"it has to be 2."},{"Start":"09:31.800 ","End":"09:36.505","Text":"If I try just t plus 2, it wouldn\u0027t work."},{"Start":"09:36.505 ","End":"09:40.180","Text":"Anyway it turns out if you want it to work for y also,"},{"Start":"09:40.180 ","End":"09:45.775","Text":"we have to take y equals 2,"},{"Start":"09:45.775 ","End":"09:50.480","Text":"if you start off with 2 and then if we subtract 2t,"},{"Start":"09:50.640 ","End":"09:54.400","Text":"when t is 0 will be at 2."},{"Start":"09:54.400 ","End":"09:57.745","Text":"When t is 1, then it will be 0."},{"Start":"09:57.745 ","End":"09:59.185","Text":"This is okay now,"},{"Start":"09:59.185 ","End":"10:01.300","Text":"and this is the curve minus c,"},{"Start":"10:01.300 ","End":"10:03.295","Text":"which is the opposite curve."},{"Start":"10:03.295 ","End":"10:07.300","Text":"Notice that this time dx and dy also change."},{"Start":"10:07.300 ","End":"10:16.730","Text":"In fact, dx is equal to minus dt and dy is equal to minus 2dt."},{"Start":"10:17.100 ","End":"10:19.720","Text":"What we end up with,"},{"Start":"10:19.720 ","End":"10:21.880","Text":"this is the same integral,"},{"Start":"10:21.880 ","End":"10:28.810","Text":"but c minus c. We get this time the"},{"Start":"10:28.810 ","End":"10:36.700","Text":"integral for minus c is t goes from 0 to 1,"},{"Start":"10:36.700 ","End":"10:39.955","Text":"x times y is"},{"Start":"10:39.955 ","End":"10:48.730","Text":"3 minus t. 2 minus 2t and then dx,"},{"Start":"10:48.730 ","End":"10:51.160","Text":"which is minus dt."},{"Start":"10:51.160 ","End":"10:55.075","Text":"I\u0027ll put a dt here and a minus here."},{"Start":"10:55.075 ","End":"10:57.565","Text":"Then the other one,"},{"Start":"10:57.565 ","End":"11:00.670","Text":"x minus y, dy."},{"Start":"11:00.670 ","End":"11:03.820","Text":"Well, we know we can combine them as we saw before."},{"Start":"11:03.820 ","End":"11:10.225","Text":"In fact, I\u0027ll just take this off and put 1dt at the end."},{"Start":"11:10.225 ","End":"11:14.830","Text":"We get x minus y is going to be this minus,"},{"Start":"11:14.830 ","End":"11:17.510","Text":"this is 1 plus t,"},{"Start":"11:19.620 ","End":"11:22.345","Text":"dy is minus 2 dt."},{"Start":"11:22.345 ","End":"11:27.590","Text":"I\u0027ll put the minus 2 here and the dt here."},{"Start":"11:28.110 ","End":"11:30.460","Text":"I take the dt out."},{"Start":"11:30.460 ","End":"11:33.850","Text":"A lot have got regular integral."},{"Start":"11:33.850 ","End":"11:39.565","Text":"Let me just simplify what\u0027s inside just as a quadratic."},{"Start":"11:39.565 ","End":"11:42.115","Text":"We\u0027ve got the integral from 0 to 1. Let\u0027s see."},{"Start":"11:42.115 ","End":"11:44.740","Text":"We\u0027ll collect them as we need them."},{"Start":"11:44.740 ","End":"11:47.605","Text":"Let\u0027s see the t squared, I\u0027ve got from here,"},{"Start":"11:47.605 ","End":"11:51.820","Text":"minus t times minus 2t is 2t squared,"},{"Start":"11:51.820 ","End":"11:53.110","Text":"but there\u0027s another minus,"},{"Start":"11:53.110 ","End":"11:55.780","Text":"so it\u0027s minus 2t squared."},{"Start":"11:55.780 ","End":"12:02.950","Text":"Next, the t minus 2t minus 6t is minus 8t,"},{"Start":"12:02.950 ","End":"12:04.690","Text":"but it\u0027s a minus."},{"Start":"12:04.690 ","End":"12:10.720","Text":"It\u0027s plus 8t plus 8t minus 2t."},{"Start":"12:10.720 ","End":"12:15.110","Text":"We\u0027re left with plus 6t."},{"Start":"12:15.110 ","End":"12:23.790","Text":"Finally, the numbers 3 times 2 is 6 minus 6 minus 2."},{"Start":"12:23.790 ","End":"12:28.320","Text":"That would make it minus 8dt."},{"Start":"12:30.450 ","End":"12:37.510","Text":"Then the integral is minus 2/3t cubed."},{"Start":"12:37.510 ","End":"12:45.280","Text":"Here we get 6 over 2 is 3t squared and minus 8t."},{"Start":"12:45.280 ","End":"12:48.895","Text":"This from 0 to 1,"},{"Start":"12:48.895 ","End":"12:54.010","Text":"0 we get nothing at all."},{"Start":"12:54.010 ","End":"12:57.160","Text":"So we just need to plug-in 1,"},{"Start":"12:57.160 ","End":"13:02.965","Text":"and what we get is, let\u0027s see,"},{"Start":"13:02.965 ","End":"13:07.210","Text":"minus 2/3 plus 3,"},{"Start":"13:07.210 ","End":"13:13.615","Text":"minus 8, 3 minus 8 is minus 5."},{"Start":"13:13.615 ","End":"13:16.585","Text":"So we get minus 5,"},{"Start":"13:16.585 ","End":"13:19.540","Text":"and 2/3 and look,"},{"Start":"13:19.540 ","End":"13:22.960","Text":"here 5 and 2/3 here,"},{"Start":"13:22.960 ","End":"13:25.780","Text":"minus 5 and 2/3."},{"Start":"13:25.780 ","End":"13:28.915","Text":"In the case of this kind of integral,"},{"Start":"13:28.915 ","End":"13:31.405","Text":"it\u0027s not the same as with ds."},{"Start":"13:31.405 ","End":"13:33.265","Text":"We don\u0027t get the same answer,"},{"Start":"13:33.265 ","End":"13:36.955","Text":"we get the opposite and so the negative."},{"Start":"13:36.955 ","End":"13:41.120","Text":"We could express this as a formula."},{"Start":"13:42.720 ","End":"13:49.630","Text":"Here it is, and notice that there\u0027s a minus on the curve,"},{"Start":"13:49.630 ","End":"13:51.610","Text":"meaning the opposite direction curve."},{"Start":"13:51.610 ","End":"13:53.635","Text":"Then just puts a minus in front,"},{"Start":"13:53.635 ","End":"13:55.720","Text":"which is different than the ds case."},{"Start":"13:55.720 ","End":"14:00.490","Text":"You can also write two separate formulas for the dx and for the dy."},{"Start":"14:00.490 ","End":"14:03.100","Text":"Here we are. Once again,"},{"Start":"14:03.100 ","End":"14:11.545","Text":"the reversal on the curve causes a minus for the dx case and for the dy case."},{"Start":"14:11.545 ","End":"14:14.650","Text":"All these are in contrast to ds,"},{"Start":"14:14.650 ","End":"14:19.415","Text":"because in the ds case it did not change direction."},{"Start":"14:19.415 ","End":"14:26.790","Text":"Okay. Now let\u0027s move on to the third dimension, x, y, and z."}],"ID":10488},{"Watched":false,"Name":"Line Integral of Type 2 in 3D","Duration":"9m 51s","ChapterTopicVideoID":10184,"CourseChapterTopicPlaylistID":82678,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10184.jpeg","UploadDate":"2020-02-26T06:21:00.6800000","DurationForVideoObject":"PT9M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.620 ","End":"00:04.860","Text":"Now we\u0027re going to move into the third dimension, another dz."},{"Start":"00:04.860 ","End":"00:08.670","Text":"Let me just rearrange these 2 things."},{"Start":"00:08.670 ","End":"00:11.070","Text":"I put the curve first."},{"Start":"00:11.070 ","End":"00:13.230","Text":"Now to adapt to 3D,"},{"Start":"00:13.230 ","End":"00:14.910","Text":"I\u0027m going to have to change the formulas."},{"Start":"00:14.910 ","End":"00:19.440","Text":"The curve is going to have to have a third component,"},{"Start":"00:19.440 ","End":"00:22.590","Text":"z equals z of t,"},{"Start":"00:22.590 ","End":"00:25.890","Text":"and I\u0027ll just change the curly braces."},{"Start":"00:25.890 ","End":"00:30.225","Text":"It\u0027s still t from some number to some number."},{"Start":"00:30.225 ","End":"00:32.865","Text":"Now instead of these 2,"},{"Start":"00:32.865 ","End":"00:35.560","Text":"we\u0027ll actually have 3."},{"Start":"00:36.110 ","End":"00:41.510","Text":"Here they are, just completely analogous to what we had in 2D,"},{"Start":"00:41.510 ","End":"00:44.150","Text":"so I\u0027m not going to go over it in any particular detail."},{"Start":"00:44.150 ","End":"00:47.150","Text":"Of course this thing changes also,"},{"Start":"00:47.150 ","End":"00:49.250","Text":"is that we\u0027ll often see a dx,"},{"Start":"00:49.250 ","End":"00:51.455","Text":"dy, and a dz all combined."},{"Start":"00:51.455 ","End":"00:53.410","Text":"I\u0027m going to replace this one,"},{"Start":"00:53.410 ","End":"00:55.590","Text":"and here is the combined form,"},{"Start":"00:55.590 ","End":"00:57.510","Text":"instead of just p and q, we have p, q,"},{"Start":"00:57.510 ","End":"01:00.580","Text":"and r, the dx, the dy, and the dz."},{"Start":"01:00.580 ","End":"01:05.490","Text":"We just write it together with one integral sign."},{"Start":"01:05.590 ","End":"01:08.554","Text":"Let\u0027s just do an example."},{"Start":"01:08.554 ","End":"01:14.480","Text":"As an example, we\u0027ll take the integral over the curve c,"},{"Start":"01:14.480 ","End":"01:16.760","Text":"which I still have to specify,"},{"Start":"01:16.760 ","End":"01:21.445","Text":"of z squared dx"},{"Start":"01:21.445 ","End":"01:31.080","Text":"plus ydy plus 2ydz."},{"Start":"01:31.900 ","End":"01:35.375","Text":"It\u0027s one of these mixed variety."},{"Start":"01:35.375 ","End":"01:40.170","Text":"The curve, well, I\u0027ll explain it with a picture."},{"Start":"01:40.170 ","End":"01:42.885","Text":"Here\u0027s a picture."},{"Start":"01:42.885 ","End":"01:45.629","Text":"But this is not the curve exactly,"},{"Start":"01:45.629 ","End":"01:53.715","Text":"this is a circle of radius 2 hovering 3 units above the xy plane."},{"Start":"01:53.715 ","End":"01:57.334","Text":"We\u0027ll find its parametric equation in a moment."},{"Start":"01:57.334 ","End":"02:02.265","Text":"But it goes through the plane, z equals 3."},{"Start":"02:02.265 ","End":"02:08.060","Text":"I wanted from the point directly above here,"},{"Start":"02:08.060 ","End":"02:12.719","Text":"say that somewhere here,"},{"Start":"02:12.719 ","End":"02:16.190","Text":"that would be the point where x is 0,"},{"Start":"02:16.190 ","End":"02:21.990","Text":"y is 2, and z is 3."},{"Start":"02:21.990 ","End":"02:26.585","Text":"Then I want to go to the point above here,"},{"Start":"02:26.585 ","End":"02:29.200","Text":"see where that hits,"},{"Start":"02:29.200 ","End":"02:33.785","Text":"say here, not quite to scale."},{"Start":"02:33.785 ","End":"02:39.930","Text":"Anyway, that would be the point where x is minus 2,"},{"Start":"02:39.930 ","End":"02:43.050","Text":"y is 0, and z,"},{"Start":"02:43.050 ","End":"02:45.280","Text":"as always, is 3."},{"Start":"02:45.280 ","End":"02:47.240","Text":"It\u0027s actually a quarter circle."},{"Start":"02:47.240 ","End":"02:52.490","Text":"If I do the projection of this thing onto the xy plane,"},{"Start":"02:52.490 ","End":"02:57.380","Text":"I\u0027m going from here to here in a quarter circle."},{"Start":"02:57.380 ","End":"02:59.270","Text":"Because of the perspective,"},{"Start":"02:59.270 ","End":"03:01.350","Text":"it looks like an ellipse."},{"Start":"03:03.770 ","End":"03:08.780","Text":"Let\u0027s do the difficult part first, not difficult,"},{"Start":"03:08.780 ","End":"03:11.150","Text":"but it\u0027s to think,"},{"Start":"03:11.150 ","End":"03:15.325","Text":"what is the parameterization of the circle?"},{"Start":"03:15.325 ","End":"03:20.655","Text":"Well, the easiest thing is the z part."},{"Start":"03:20.655 ","End":"03:28.185","Text":"The z is always 3 because it\u0027s in the z equals 3 plane."},{"Start":"03:28.185 ","End":"03:33.950","Text":"Now if it was a circle in the xy plane with radius 2,"},{"Start":"03:33.950 ","End":"03:42.035","Text":"we could parametrize it as x equals cosine t. That would be if the radius was 1,"},{"Start":"03:42.035 ","End":"03:45.550","Text":"but we have to make it 2 cosine t,"},{"Start":"03:45.550 ","End":"03:49.970","Text":"because the radius is 2 and y equals to"},{"Start":"03:49.970 ","End":"03:57.710","Text":"sine t. Now the way this is set up is that t equals 0."},{"Start":"03:57.710 ","End":"04:06.120","Text":"We begin on the positive x-axis and work our way counterclockwise."},{"Start":"04:06.120 ","End":"04:09.765","Text":"We would be going something like this,"},{"Start":"04:09.765 ","End":"04:13.465","Text":"and that would be from 0 to 2Pi."},{"Start":"04:13.465 ","End":"04:18.275","Text":"But we\u0027re only going from here to here."},{"Start":"04:18.275 ","End":"04:23.480","Text":"In other words, that would be 90 degrees to 180 degrees."},{"Start":"04:23.480 ","End":"04:26.465","Text":"Pi over 2, less than or equal to t,"},{"Start":"04:26.465 ","End":"04:29.064","Text":"less than or equal to Pi."},{"Start":"04:29.064 ","End":"04:33.440","Text":"Just highlight that a bit on the xy plane,"},{"Start":"04:33.440 ","End":"04:36.920","Text":"but really up in the air at the height of 3,"},{"Start":"04:36.920 ","End":"04:41.099","Text":"this is our curve c,"},{"Start":"04:44.540 ","End":"04:48.190","Text":"and it goes in this direction."},{"Start":"04:48.190 ","End":"04:50.515","Text":"Anyway, we haven\u0027t parameterized."},{"Start":"04:50.515 ","End":"04:53.400","Text":"Now we\u0027re going to just use the formula."},{"Start":"04:53.400 ","End":"04:58.220","Text":"I\u0027ll just scroll a bit so we get more space but still see the formulas."},{"Start":"04:58.220 ","End":"05:06.970","Text":"What we get is the integral from t equals Pi over 2 to t equals Pi,"},{"Start":"05:06.970 ","End":"05:12.270","Text":"z squared is 3 squared."},{"Start":"05:12.270 ","End":"05:16.980","Text":"Going back here, we need to do dx, dy, and dz."},{"Start":"05:17.900 ","End":"05:24.120","Text":"From here I could get that dx is equal to"},{"Start":"05:24.120 ","End":"05:30.585","Text":"minus 2 sine t dt."},{"Start":"05:30.585 ","End":"05:32.370","Text":"Curly braces here too."},{"Start":"05:32.370 ","End":"05:37.080","Text":"Dy equals 2 cosine t,"},{"Start":"05:37.080 ","End":"05:43.150","Text":"and dz is equal to 0,"},{"Start":"05:43.760 ","End":"05:47.105","Text":"where dt, if you like,"},{"Start":"05:47.105 ","End":"05:48.770","Text":"but it\u0027s just 0."},{"Start":"05:48.770 ","End":"05:51.960","Text":"I\u0027ll write 0dt , but it\u0027s 0."},{"Start":"05:52.060 ","End":"05:54.950","Text":"When we substitute here,"},{"Start":"05:54.950 ","End":"05:56.840","Text":"well, this part\u0027s going to disappear."},{"Start":"05:56.840 ","End":"06:00.389","Text":"We get z squared is 3 squared,"},{"Start":"06:01.150 ","End":"06:08.130","Text":"and the dx is minus 2 sine t,"},{"Start":"06:10.220 ","End":"06:13.305","Text":"and the dt I\u0027ll write at the end."},{"Start":"06:13.305 ","End":"06:17.480","Text":"I\u0027ll put a square brackets here and know that at the end I\u0027ll have a dt."},{"Start":"06:17.480 ","End":"06:24.585","Text":"Then y is 2 sine t,"},{"Start":"06:24.585 ","End":"06:29.770","Text":"and dy is 2 cosine t dt."},{"Start":"06:33.170 ","End":"06:38.510","Text":"Then just to show that I haven\u0027t forgotten that the dz is 0,"},{"Start":"06:38.510 ","End":"06:40.805","Text":"I\u0027ll just write plus 0,"},{"Start":"06:40.805 ","End":"06:43.170","Text":"and all this dt."},{"Start":"06:44.580 ","End":"06:49.929","Text":"Now we just need to do a bit of simplification of the limits,"},{"Start":"06:49.929 ","End":"06:51.490","Text":"that, and let\u0027s see,"},{"Start":"06:51.490 ","End":"07:02.470","Text":"[inaudible] 2 is minus 18 sine t plus."},{"Start":"07:02.470 ","End":"07:07.550","Text":"Now I\u0027m going to use a trigonometric formula that"},{"Start":"07:07.550 ","End":"07:15.150","Text":"2 sine Alpha cosine Alpha is sine of 2 Alpha,"},{"Start":"07:15.150 ","End":"07:18.330","Text":"and if Alpha is t, that\u0027s also true."},{"Start":"07:18.330 ","End":"07:28.170","Text":"Here plus 2 sine of 2t dt."},{"Start":"07:28.170 ","End":"07:33.960","Text":"Remember that the integral of sine is minus cosine."},{"Start":"07:34.040 ","End":"07:38.420","Text":"This equals, if the integral of sine is minus cosine,"},{"Start":"07:38.420 ","End":"07:39.935","Text":"and we already have a minus,"},{"Start":"07:39.935 ","End":"07:44.020","Text":"so here we get 18 cosine t,"},{"Start":"07:44.020 ","End":"07:47.675","Text":"and the integral of sine is minus cosine,"},{"Start":"07:47.675 ","End":"07:50.880","Text":"so we get minus cosine 2t,"},{"Start":"07:50.880 ","End":"07:53.990","Text":"this 2 gets swallowed up because it\u0027s a 2t."},{"Start":"07:53.990 ","End":"07:56.300","Text":"Well, easiest way is just to differentiate"},{"Start":"07:56.300 ","End":"08:00.290","Text":"this and see that you get the 2 that comes out to give you this."},{"Start":"08:00.290 ","End":"08:02.200","Text":"This is what we have,"},{"Start":"08:02.200 ","End":"08:04.070","Text":"that\u0027s the indefinite integral,"},{"Start":"08:04.070 ","End":"08:10.550","Text":"and now we have to evaluate it between Pi over 2 and Pi. Let\u0027s see."},{"Start":"08:10.550 ","End":"08:14.530","Text":"If we substitute Pi, that\u0027s 180 degrees."},{"Start":"08:14.530 ","End":"08:19.500","Text":"Cosine of Pi is minus 1,"},{"Start":"08:19.500 ","End":"08:22.635","Text":"so we have minus 18."},{"Start":"08:22.635 ","End":"08:28.580","Text":"Cosine of 2Pi is 360 degrees, is like 0."},{"Start":"08:28.580 ","End":"08:33.030","Text":"Cosine 0 is 1 so we have minus 1."},{"Start":"08:33.030 ","End":"08:35.710","Text":"Then we have to do the same thing for Pi over 2."},{"Start":"08:35.710 ","End":"08:39.815","Text":"Cosine Pi over 2 is 0,"},{"Start":"08:39.815 ","End":"08:42.600","Text":"and cosine, let\u0027s see,"},{"Start":"08:42.600 ","End":"08:45.065","Text":"twice Pi over 2 is Pi."},{"Start":"08:45.065 ","End":"08:51.695","Text":"Cosine of Pi is minus 1."},{"Start":"08:51.695 ","End":"08:54.320","Text":"We have minus minus 1,"},{"Start":"08:54.320 ","End":"08:56.425","Text":"so it\u0027s plus 1."},{"Start":"08:56.425 ","End":"08:58.710","Text":"All together we have minus 18,"},{"Start":"08:58.710 ","End":"09:00.450","Text":"minus 1, minus 1,"},{"Start":"09:00.450 ","End":"09:03.375","Text":"so minus 20."},{"Start":"09:03.375 ","End":"09:05.370","Text":"That\u0027s the answer."},{"Start":"09:05.370 ","End":"09:06.810","Text":"They\u0027re almost at the end,"},{"Start":"09:06.810 ","End":"09:14.465","Text":"I just wanted to remind you that if instead of a simple smooth curve c,"},{"Start":"09:14.465 ","End":"09:18.764","Text":"c is made up of say several,"},{"Start":"09:18.764 ","End":"09:20.970","Text":"in this case, 4 separate bits."},{"Start":"09:20.970 ","End":"09:24.470","Text":"Then just like in the case of ds,"},{"Start":"09:24.470 ","End":"09:27.140","Text":"same thing works with dx, dy,"},{"Start":"09:27.140 ","End":"09:30.605","Text":"and dz, in the sense that it\u0027s additive."},{"Start":"09:30.605 ","End":"09:34.060","Text":"If you have the integral of something over c,"},{"Start":"09:34.060 ","End":"09:37.220","Text":"we break it up into separate integrals."},{"Start":"09:37.220 ","End":"09:40.355","Text":"Let\u0027s see, 1, 2, 3, 4, plus, plus,"},{"Start":"09:40.355 ","End":"09:44.420","Text":"plus, over c_1, c_2, c_3, and c_4."},{"Start":"09:44.420 ","End":"09:47.670","Text":"In this case, the same principle applies."},{"Start":"09:47.670 ","End":"09:51.420","Text":"We\u0027re at the end of this clip."}],"ID":10489},{"Watched":false,"Name":"Line Integral of Vector Fields","Duration":"15m 33s","ChapterTopicVideoID":10185,"CourseChapterTopicPlaylistID":82678,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10185.jpeg","UploadDate":"2020-02-26T06:21:23.9730000","DurationForVideoObject":"PT15M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.310","Text":"Continuing with line integrals."},{"Start":"00:02.310 ","End":"00:08.475","Text":"In this clip, we\u0027re going to learn about line integrals of vector fields."},{"Start":"00:08.475 ","End":"00:10.125","Text":"Just to remind you,"},{"Start":"00:10.125 ","End":"00:15.870","Text":"we\u0027ve studied line integrals over a curve C,"},{"Start":"00:15.870 ","End":"00:17.670","Text":"where C is a parametrized curve,"},{"Start":"00:17.670 ","End":"00:21.465","Text":"and we\u0027ve learned how to do it with respect to arc length."},{"Start":"00:21.465 ","End":"00:23.850","Text":"Then we also did dx,"},{"Start":"00:23.850 ","End":"00:27.810","Text":"dy, dz, or a combination."},{"Start":"00:27.810 ","End":"00:31.100","Text":"Then there was also a section earlier on vector fields,"},{"Start":"00:31.100 ","End":"00:32.615","Text":"and now we\u0027re going to tie it in."},{"Start":"00:32.615 ","End":"00:35.569","Text":"I get yet another kind of integral."},{"Start":"00:35.569 ","End":"00:39.350","Text":"But all these integrals before were of a function f,"},{"Start":"00:39.350 ","End":"00:40.880","Text":"which was a scalar function,"},{"Start":"00:40.880 ","End":"00:43.040","Text":"whether of x, y or x, y, z."},{"Start":"00:43.040 ","End":"00:45.544","Text":"This time we\u0027re going to take a vector function,"},{"Start":"00:45.544 ","End":"00:46.760","Text":"or a vector field,"},{"Start":"00:46.760 ","End":"00:49.440","Text":"and take a line integral of that."},{"Start":"00:49.790 ","End":"00:52.930","Text":"We\u0027ll start right away in 3 dimensions."},{"Start":"00:52.930 ","End":"00:55.069","Text":"Suppose I have a vector field,"},{"Start":"00:55.069 ","End":"00:58.009","Text":"F of x, y, and z,"},{"Start":"00:58.009 ","End":"01:03.920","Text":"to vector, and we break it up into components."},{"Start":"01:03.920 ","End":"01:06.980","Text":"Let\u0027s say we use the I, j, k notation."},{"Start":"01:06.980 ","End":"01:15.875","Text":"Then we have some function P times standard basis vector I plus Q of x, y,"},{"Start":"01:15.875 ","End":"01:23.105","Text":"z times j plus R of x, y,"},{"Start":"01:23.105 ","End":"01:26.570","Text":"and z times k. Of course,"},{"Start":"01:26.570 ","End":"01:29.540","Text":"we could also use the other notation,"},{"Start":"01:29.540 ","End":"01:32.790","Text":"and say it\u0027s P of x, y,"},{"Start":"01:32.790 ","End":"01:34.800","Text":"z, Q of x, y,"},{"Start":"01:34.800 ","End":"01:36.810","Text":"z, R of x, y, z."},{"Start":"01:36.810 ","End":"01:38.040","Text":"Sometimes we\u0027ll use the I, j,"},{"Start":"01:38.040 ","End":"01:40.115","Text":"k sometimes angular brackets."},{"Start":"01:40.115 ","End":"01:44.210","Text":"Anyway, that\u0027s the function, the vector field."},{"Start":"01:44.210 ","End":"01:46.340","Text":"Now we need a curve C,"},{"Start":"01:46.340 ","End":"01:49.850","Text":"and we\u0027ll also give it in this form."},{"Start":"01:49.850 ","End":"01:54.255","Text":"The curve will now also be in 3D."},{"Start":"01:54.255 ","End":"01:59.525","Text":"We\u0027ll write it as r of a parameter t is equal to,"},{"Start":"01:59.525 ","End":"02:02.330","Text":"it has a component x of t,"},{"Start":"02:02.330 ","End":"02:08.805","Text":"and it has a component y of t. This time j,"},{"Start":"02:08.805 ","End":"02:12.920","Text":"i, and then we have a z which is a function of t,"},{"Start":"02:12.920 ","End":"02:20.180","Text":"k, and of course we could also write this in the form x of t, y of t,"},{"Start":"02:20.180 ","End":"02:23.450","Text":"z of t. But sticking for now with the i, j,"},{"Start":"02:23.450 ","End":"02:26.810","Text":"k. Now I\u0027m going to give the definition of the line integral of"},{"Start":"02:26.810 ","End":"02:32.180","Text":"the vector field F over the curve C. We\u0027re going to define"},{"Start":"02:32.180 ","End":"02:41.910","Text":"the integral of the vector field F. This time we\u0027re going to have dot dr,"},{"Start":"02:41.910 ","End":"02:43.470","Text":"and r is a vector."},{"Start":"02:43.470 ","End":"02:45.885","Text":"Instead of the ds, dx, dy, dz,"},{"Start":"02:45.885 ","End":"02:48.570","Text":"we have a vector, and vector as a scalar,"},{"Start":"02:48.570 ","End":"02:53.945","Text":"and we also indicate here the curve C. I forgot to say that t,"},{"Start":"02:53.945 ","End":"02:58.650","Text":"the parameter goes between 2 numbers, a, and b."},{"Start":"02:58.820 ","End":"03:04.325","Text":"This is equal to the integral from a to b."},{"Start":"03:04.325 ","End":"03:10.215","Text":"Now we take the function F and apply it."},{"Start":"03:10.215 ","End":"03:19.540","Text":"The x, y, z is going to be simply r of t. When I say F of r of t,"},{"Start":"03:19.540 ","End":"03:21.970","Text":"I mean that if r of t is x, y,"},{"Start":"03:21.970 ","End":"03:24.550","Text":"z, we put x, y, z."},{"Start":"03:24.550 ","End":"03:26.575","Text":"It\u0027s just a way of writing that."},{"Start":"03:26.575 ","End":"03:28.450","Text":"We\u0027ll see that in the example."},{"Start":"03:28.450 ","End":"03:37.020","Text":"Dot r prime of t vector dot with a vector,"},{"Start":"03:37.020 ","End":"03:44.275","Text":"and all this dt, and best really is to explain by means of an example."},{"Start":"03:44.275 ","End":"03:47.860","Text":"But before the example, maybe I\u0027ll just stress this point again because it"},{"Start":"03:47.860 ","End":"03:50.895","Text":"looks strange because F is a function of 3 variables."},{"Start":"03:50.895 ","End":"03:58.065","Text":"Well, this bit here is just a shorthand way of saying F of x of t,"},{"Start":"03:58.065 ","End":"04:01.725","Text":"y of t, z of t,"},{"Start":"04:01.725 ","End":"04:05.660","Text":"where these are the x,y,z from R. But instead of writing with commas,"},{"Start":"04:05.660 ","End":"04:07.280","Text":"we just write it as a vector function."},{"Start":"04:07.280 ","End":"04:09.080","Text":"This is a shorthand for this."},{"Start":"04:09.080 ","End":"04:11.180","Text":"Okay. Now the example,"},{"Start":"04:11.180 ","End":"04:13.490","Text":"for the example I need to give you 2 things."},{"Start":"04:13.490 ","End":"04:15.470","Text":"I need to give you the function,"},{"Start":"04:15.470 ","End":"04:17.410","Text":"rather the vector field F,"},{"Start":"04:17.410 ","End":"04:21.890","Text":"and I need to give you the curve C. Let\u0027s start with the curve."},{"Start":"04:21.890 ","End":"04:30.350","Text":"The curve C is given by r of t is equal to"},{"Start":"04:30.350 ","End":"04:39.680","Text":"t times vector I plus t squared times vector j plus"},{"Start":"04:39.680 ","End":"04:49.520","Text":"t cubed times vector k. They also have to tell you the interval t goes from 0-1."},{"Start":"04:49.520 ","End":"04:52.460","Text":"Now, I need to give you what F,"},{"Start":"04:52.460 ","End":"04:59.600","Text":"the vector field is of x, y, and z."},{"Start":"04:59.600 ","End":"05:04.010","Text":"It is equal to, it\u0027s going to be something i,"},{"Start":"05:04.160 ","End":"05:11.550","Text":"and something j and something k. Let\u0027s see,"},{"Start":"05:11.550 ","End":"05:16.700","Text":"8x squared yz here,"},{"Start":"05:16.700 ","End":"05:23.310","Text":"5z here, and minus 4xy here."},{"Start":"05:23.310 ","End":"05:24.950","Text":"What I want, of course,"},{"Start":"05:24.950 ","End":"05:31.800","Text":"is the line integral over the curve C of F.dr,"},{"Start":"05:34.280 ","End":"05:38.380","Text":"and we\u0027re going to use this formula."},{"Start":"05:39.620 ","End":"05:43.295","Text":"What we get is from here,"},{"Start":"05:43.295 ","End":"05:47.600","Text":"the integral from 0-1."},{"Start":"05:47.600 ","End":"05:49.670","Text":"Now to interpret this,"},{"Start":"05:49.670 ","End":"05:52.565","Text":"I have to take this vector function,"},{"Start":"05:52.565 ","End":"05:56.395","Text":"and instead of x, y, and z,"},{"Start":"05:56.395 ","End":"06:00.465","Text":"I have to put in this, this, and this."},{"Start":"06:00.465 ","End":"06:06.805","Text":"This is my x of t. This is the y of t,"},{"Start":"06:06.805 ","End":"06:12.520","Text":"and this is the z of t. What we get"},{"Start":"06:12.520 ","End":"06:19.695","Text":"is 8t squared because x is t,"},{"Start":"06:19.695 ","End":"06:23.565","Text":"and then y is t squared, that\u0027s t squared."},{"Start":"06:23.565 ","End":"06:26.400","Text":"Then z is t cubed."},{"Start":"06:26.400 ","End":"06:29.745","Text":"That\u0027s t cubed, That\u0027s all this bit,"},{"Start":"06:29.745 ","End":"06:36.300","Text":"times i, and then plus 5z,"},{"Start":"06:36.300 ","End":"06:39.330","Text":"z is t cubed,"},{"Start":"06:39.330 ","End":"06:44.680","Text":"so it\u0027s 5t cubed"},{"Start":"06:45.920 ","End":"06:50.200","Text":"j."},{"Start":"06:50.200 ","End":"06:55.980","Text":"Then minus 4xy, see x is t,"},{"Start":"06:55.980 ","End":"07:06.270","Text":"y is t squared 4t t squared k. This is just the F bit."},{"Start":"07:06.270 ","End":"07:13.905","Text":"Then we have to dot product it with the dr. Dr is r prime of t, dt."},{"Start":"07:13.905 ","End":"07:19.425","Text":"We need r prime of t. I\u0027ll just do it straight in here,"},{"Start":"07:19.425 ","End":"07:21.005","Text":"r prime of t,"},{"Start":"07:21.005 ","End":"07:22.730","Text":"derivative of t is 1."},{"Start":"07:22.730 ","End":"07:24.830","Text":"I\u0027m deliberately writing the 1,"},{"Start":"07:24.830 ","End":"07:29.490","Text":"i derivative of t squared is 2tj,"},{"Start":"07:30.160 ","End":"07:35.225","Text":"and derivative of t cubed is 3t squared,"},{"Start":"07:35.225 ","End":"07:41.630","Text":"that\u0027s k. This dot this, and then dt."},{"Start":"07:41.630 ","End":"07:44.735","Text":"Now, with a dot product,"},{"Start":"07:44.735 ","End":"07:47.930","Text":"It\u0027s actually easy with the angular brackets with the i,"},{"Start":"07:47.930 ","End":"07:49.340","Text":"j, k is okay."},{"Start":"07:49.340 ","End":"07:52.205","Text":"We take the i bit with the i bit,"},{"Start":"07:52.205 ","End":"07:53.390","Text":"plus this with this,"},{"Start":"07:53.390 ","End":"07:54.860","Text":"plus this with this."},{"Start":"07:54.860 ","End":"07:58.415","Text":"Let\u0027s see what we get. We get still the integral from 0-1."},{"Start":"07:58.415 ","End":"08:03.190","Text":"Now, this is 8t to the power of,"},{"Start":"08:03.190 ","End":"08:06.445","Text":"let\u0027s see, 2 plus 2 plus 3 is 7,"},{"Start":"08:06.445 ","End":"08:10.960","Text":"8t to the 7th, but times 1,"},{"Start":"08:10.960 ","End":"08:15.705","Text":"it\u0027s just 8t to the power of 7,"},{"Start":"08:15.705 ","End":"08:17.460","Text":"that\u0027s the i part."},{"Start":"08:17.460 ","End":"08:18.915","Text":"Now the j part,"},{"Start":"08:18.915 ","End":"08:26.290","Text":"5t cubed times 2t is 10t to the 4th."},{"Start":"08:27.380 ","End":"08:35.520","Text":"The k part minus 4t cubed and here 3t squared."},{"Start":"08:35.520 ","End":"08:41.130","Text":"It\u0027s minus 12t^5 and"},{"Start":"08:41.130 ","End":"08:46.770","Text":"all this dt equals."},{"Start":"08:46.770 ","End":"08:50.235","Text":"Then the integral of this would just"},{"Start":"08:50.235 ","End":"08:57.150","Text":"be t^8 because I raise the power by 1 is 8 and divide by 8 here,"},{"Start":"08:57.150 ","End":"08:59.744","Text":"t^5, I divide by 5,"},{"Start":"08:59.744 ","End":"09:06.615","Text":"2t^5 and here, raise it by 1 is 6, 12/6 is 2."},{"Start":"09:06.615 ","End":"09:09.150","Text":"That is, the numbers are coming out nicely,"},{"Start":"09:09.150 ","End":"09:11.055","Text":"it was cooked this way."},{"Start":"09:11.055 ","End":"09:15.690","Text":"Sorry t^6 This is the integral,"},{"Start":"09:15.690 ","End":"09:19.005","Text":"but we have to evaluate it between 0 and 1."},{"Start":"09:19.005 ","End":"09:21.840","Text":"Obviously at 0, everything is 0,"},{"Start":"09:21.840 ","End":"09:23.685","Text":"so I just have to put 1 in,"},{"Start":"09:23.685 ","End":"09:27.195","Text":"so all we get is,"},{"Start":"09:27.195 ","End":"09:34.170","Text":"1 plus 2,"},{"Start":"09:34.170 ","End":"09:39.220","Text":"minus 2 and so it equals 1."},{"Start":"09:39.410 ","End":"09:46.005","Text":"This is our answer and that\u0027s the first example."},{"Start":"09:46.005 ","End":"09:48.210","Text":"Let\u0027s do another example,"},{"Start":"09:48.210 ","End":"09:50.265","Text":"I\u0027ll erase the old one."},{"Start":"09:50.265 ","End":"09:55.455","Text":"This example I\u0027ll use the other notation, the angular brackets."},{"Start":"09:55.455 ","End":"09:57.690","Text":"I\u0027ll let f of x, y and z"},{"Start":"09:57.690 ","End":"10:08.205","Text":"equal xz, 0, minus yz."},{"Start":"10:08.205 ","End":"10:13.050","Text":"That\u0027s the vector field and now the curve, let\u0027s call it c,"},{"Start":"10:13.050 ","End":"10:18.315","Text":"is given by r of t equals,"},{"Start":"10:18.315 ","End":"10:20.700","Text":"hold it, I\u0027m going to make things a bit more difficult,"},{"Start":"10:20.700 ","End":"10:22.470","Text":"I\u0027m not going to give it to you explicitly."},{"Start":"10:22.470 ","End":"10:33.000","Text":"I\u0027m going to say on the side here that c is the line segment from the point minus 1,"},{"Start":"10:33.000 ","End":"10:40.650","Text":"2, 0 to 3, 0, 1."},{"Start":"10:40.650 ","End":"10:43.035","Text":"It\u0027s the line segment,"},{"Start":"10:43.035 ","End":"10:46.875","Text":"line segment, and we\u0027ve learned how to do that."},{"Start":"10:46.875 ","End":"10:50.550","Text":"This will be a side exercise and in fact,"},{"Start":"10:50.550 ","End":"10:57.990","Text":"the formula is that r of t is equal to we"},{"Start":"10:57.990 ","End":"11:06.000","Text":"take 1 minus t times the position vector for the first minus 1,"},{"Start":"11:06.000 ","End":"11:14.890","Text":"2, 0 and t times the position vector of the second 3,0, 1."},{"Start":"11:15.560 ","End":"11:19.710","Text":"Now we can get back here and we do the computation mentally."},{"Start":"11:19.710 ","End":"11:21.600","Text":"Let\u0027s see the x component."},{"Start":"11:21.600 ","End":"11:29.595","Text":"We\u0027ve got 1 minus t times minus 1 will be t minus 1 plus 3t,"},{"Start":"11:29.595 ","End":"11:32.880","Text":"that would be 4t minus 1."},{"Start":"11:32.880 ","End":"11:35.715","Text":"Yeah, 4t minus 1."},{"Start":"11:35.715 ","End":"11:37.950","Text":"Now the y component,"},{"Start":"11:37.950 ","End":"11:43.980","Text":"so we have twice 1 minus t plus 0t,"},{"Start":"11:43.980 ","End":"11:46.570","Text":"that\u0027s 2 minus 2t."},{"Start":"11:48.230 ","End":"11:52.035","Text":"The last one, 1 minus t times 0,"},{"Start":"11:52.035 ","End":"11:59.190","Text":"nothing t times 1 is t. That is our curve and when we do it this way,"},{"Start":"11:59.190 ","End":"12:02.475","Text":"it\u0027s always 0 less than or equal to t,"},{"Start":"12:02.475 ","End":"12:05.265","Text":"less than or equal to 1."},{"Start":"12:05.265 ","End":"12:07.260","Text":"Here\u0027s the vector field,"},{"Start":"12:07.260 ","End":"12:08.445","Text":"here is the curve."},{"Start":"12:08.445 ","End":"12:10.425","Text":"Now we use the formula,"},{"Start":"12:10.425 ","End":"12:14.920","Text":"we get the integral from 0-1."},{"Start":"12:15.050 ","End":"12:17.580","Text":"Now when we substitute x,"},{"Start":"12:17.580 ","End":"12:19.020","Text":"y, z into f,"},{"Start":"12:19.020 ","End":"12:24.270","Text":"this is the x, this is the y, and this is the z."},{"Start":"12:24.270 ","End":"12:28.845","Text":"What we get is here,"},{"Start":"12:28.845 ","End":"12:36.960","Text":"we get the vector xz"},{"Start":"12:36.960 ","End":"12:43.180","Text":"is this times this is t times 4t minus 1."},{"Start":"12:43.310 ","End":"12:47.925","Text":"I need 0 and then minus"},{"Start":"12:47.925 ","End":"12:54.540","Text":"yz instead of the minus I\u0027ll just invert the order and make it 2t minus 2,"},{"Start":"12:54.540 ","End":"13:00.280","Text":"it\u0027s t times 2t minus 2.r."},{"Start":"13:01.700 ","End":"13:03.990","Text":"We don\u0027t have r prime,"},{"Start":"13:03.990 ","End":"13:06.135","Text":"let\u0027s just quickly do that."},{"Start":"13:06.135 ","End":"13:10.980","Text":"R prime is equal to,"},{"Start":"13:10.980 ","End":"13:14.265","Text":"I\u0027m taking it from there,"},{"Start":"13:14.265 ","End":"13:17.730","Text":"is derivative of 4t minus 1 is 4."},{"Start":"13:17.730 ","End":"13:20.055","Text":"Everything\u0027s linear, so it\u0027s easy."},{"Start":"13:20.055 ","End":"13:23.880","Text":"Then we have a minus 2 and then we have 1."},{"Start":"13:23.880 ","End":"13:31.725","Text":"Here I write 4 minus 2,"},{"Start":"13:31.725 ","End":"13:35.605","Text":"1, and then dt."},{"Start":"13:35.605 ","End":"13:38.390","Text":"Okay, what is this equal?"},{"Start":"13:38.390 ","End":"13:40.850","Text":"You\u0027ve got the integral of,"},{"Start":"13:40.850 ","End":"13:45.185","Text":"let\u0027s do the dot product 4 times this."},{"Start":"13:45.185 ","End":"13:53.385","Text":"I\u0027ve got 4t times 4t minus 1,"},{"Start":"13:53.385 ","End":"13:56.205","Text":"then minus 2 times 0,"},{"Start":"13:56.205 ","End":"14:00.720","Text":"nothing and then 1 times this so"},{"Start":"14:00.720 ","End":"14:07.050","Text":"plus t times 2t minus 2."},{"Start":"14:07.050 ","End":"14:14.200","Text":"All this, better put it in brackets, dt from 0-1."},{"Start":"14:14.330 ","End":"14:17.970","Text":"Let\u0027s just simplify everything inside,"},{"Start":"14:17.970 ","End":"14:19.950","Text":"we\u0027ll collect the t squared terms."},{"Start":"14:19.950 ","End":"14:25.035","Text":"T squared, I have 16t squared plus 2t squared,"},{"Start":"14:25.035 ","End":"14:27.495","Text":"so it\u0027s 18t squared."},{"Start":"14:27.495 ","End":"14:31.120","Text":"Let\u0027s see how many t\u0027s do I have?"},{"Start":"14:31.190 ","End":"14:40.260","Text":"Minus 4t, and then minus 2t is minus 6t,"},{"Start":"14:40.260 ","End":"14:42.480","Text":"and that\u0027s about it,"},{"Start":"14:42.480 ","End":"14:50.310","Text":"all this dt from 0-1."},{"Start":"14:50.310 ","End":"14:56.925","Text":"Now the integral, this integral raise the power by 1 is 3 divided by 3."},{"Start":"14:56.925 ","End":"15:00.930","Text":"That gives us 6t cubed from here,"},{"Start":"15:00.930 ","End":"15:02.160","Text":"raise the power by 1."},{"Start":"15:02.160 ","End":"15:05.250","Text":"It\u0027s t squared, divide by 2,"},{"Start":"15:05.250 ","End":"15:10.470","Text":"so we\u0027ve got minus 3t squared and then here"},{"Start":"15:10.470 ","End":"15:16.185","Text":"0 and here 1 equals,"},{"Start":"15:16.185 ","End":"15:18.870","Text":"and then at 0, we get nothing,"},{"Start":"15:18.870 ","End":"15:20.490","Text":"so we just plug in 1,"},{"Start":"15:20.490 ","End":"15:23.265","Text":"so it\u0027s 6 minus 3."},{"Start":"15:23.265 ","End":"15:25.530","Text":"No need for a calculator,"},{"Start":"15:25.530 ","End":"15:28.125","Text":"this is equal to 3."},{"Start":"15:28.125 ","End":"15:32.860","Text":"Highlight that that\u0027s the answer to the second example."}],"ID":10490},{"Watched":false,"Name":"Line Integral of Vector Fields - Continued","Duration":"6m 29s","ChapterTopicVideoID":10186,"CourseChapterTopicPlaylistID":82678,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10186.jpeg","UploadDate":"2017-08-21T14:49:48.3070000","DurationForVideoObject":"PT6M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.390","Text":"Now I want to keep this example,"},{"Start":"00:03.390 ","End":"00:06.644","Text":"but erase the work."},{"Start":"00:06.644 ","End":"00:09.870","Text":"I guess I want you to remember the answer because I\u0027m going to do"},{"Start":"00:09.870 ","End":"00:13.935","Text":"it a different way and I want to make sure we get the same result."},{"Start":"00:13.935 ","End":"00:22.740","Text":"We get that the integral of f.dr is equal to 3, in this example."},{"Start":"00:22.740 ","End":"00:28.680","Text":"I\u0027d like to give you another formula, a theorem,"},{"Start":"00:28.680 ","End":"00:33.785","Text":"that if in general if I have f of x,"},{"Start":"00:33.785 ","End":"00:37.340","Text":"y, z, just like in this example,"},{"Start":"00:37.340 ","End":"00:39.260","Text":"but let\u0027s make it general."},{"Start":"00:39.260 ","End":"00:42.650","Text":"P won\u0027t write of x, y, and z,"},{"Start":"00:42.650 ","End":"00:44.810","Text":"but we know it\u0027s function of x, y, z. P,"},{"Start":"00:44.810 ","End":"00:48.175","Text":"Q, and R, just like we have here."},{"Start":"00:48.175 ","End":"00:53.300","Text":"We have that the curve C is given"},{"Start":"00:53.300 ","End":"01:01.865","Text":"by r of t equals just of short write x of t,"},{"Start":"01:01.865 ","End":"01:10.130","Text":"y of t, z of t. I\u0027m just writing an abbreviated and t is goes between sum a and b."},{"Start":"01:11.650 ","End":"01:20.150","Text":"It turns out that the integral of the vector field over the curve,"},{"Start":"01:20.150 ","End":"01:28.385","Text":"that\u0027s this f.dr is equal to,"},{"Start":"01:28.385 ","End":"01:31.765","Text":"remember the dx, dy, dz integrals."},{"Start":"01:31.765 ","End":"01:38.990","Text":"Well, it turns out this is exactly equal to the integral of Pdx plus"},{"Start":"01:38.990 ","End":"01:47.050","Text":"Qdy plus Rdz over the curve."},{"Start":"01:47.050 ","End":"01:51.335","Text":"What I\u0027d like to do is same example"},{"Start":"01:51.335 ","End":"01:55.825","Text":"using this formula and verify that we get the same answer."},{"Start":"01:55.825 ","End":"01:59.610","Text":"In our case, p of x,"},{"Start":"01:59.610 ","End":"02:02.070","Text":"y, z is x,"},{"Start":"02:02.070 ","End":"02:08.040","Text":"z, q is equal to 0,"},{"Start":"02:08.040 ","End":"02:09.420","Text":"r of x, y,"},{"Start":"02:09.420 ","End":"02:12.945","Text":"z is minus y z."},{"Start":"02:12.945 ","End":"02:16.100","Text":"Then if I write this x, y,"},{"Start":"02:16.100 ","End":"02:21.275","Text":"z, I will write it to like in parametric instead of vector,"},{"Start":"02:21.275 ","End":"02:26.940","Text":"we have that x is equal to 4t minus 1,"},{"Start":"02:26.940 ","End":"02:30.250","Text":"y equals 2 minus 2t,"},{"Start":"02:30.250 ","End":"02:35.180","Text":"z equals t. That\u0027s the curve C,"},{"Start":"02:35.180 ","End":"02:40.330","Text":"where t goes from 0-1."},{"Start":"02:40.670 ","End":"02:45.155","Text":"Now let\u0027s evaluate this and see what we get."},{"Start":"02:45.155 ","End":"02:51.765","Text":"We get the integral t goes from 0-1."},{"Start":"02:51.765 ","End":"02:56.025","Text":"Now P is x z,"},{"Start":"02:56.025 ","End":"02:59.610","Text":"and x is 40 minus 1 times t,"},{"Start":"02:59.610 ","End":"03:04.680","Text":"so we get t times 4t minus 1,"},{"Start":"03:04.680 ","End":"03:11.590","Text":"that\u0027s P. Then dx is just 4dt,"},{"Start":"03:12.230 ","End":"03:15.600","Text":"because dx by dt is 4."},{"Start":"03:15.600 ","End":"03:17.310","Text":"That\u0027s the first term."},{"Start":"03:17.310 ","End":"03:21.580","Text":"Next Q."},{"Start":"03:21.580 ","End":"03:23.919","Text":"But Q is 0,"},{"Start":"03:23.919 ","End":"03:28.240","Text":"so I\u0027ll just write it plus 0 to show I haven\u0027t forgotten it."},{"Start":"03:28.240 ","End":"03:31.300","Text":"Then we need R,"},{"Start":"03:31.300 ","End":"03:40.690","Text":"which is minus yz, instead minus yz."},{"Start":"03:40.690 ","End":"03:50.535","Text":"I\u0027ll change the order here to make it to 2t minus 2."},{"Start":"03:50.535 ","End":"03:58.600","Text":"If I change the order it\u0027s like the minus and then the z is t. What do I need here?"},{"Start":"03:58.600 ","End":"04:02.780","Text":"Dz is just dt."},{"Start":"04:04.040 ","End":"04:10.890","Text":"What we get is the integral from 0-1 of"},{"Start":"04:10.890 ","End":"04:18.509","Text":"4t times 4t minus"},{"Start":"04:18.509 ","End":"04:23.715","Text":"1 plus t,"},{"Start":"04:23.715 ","End":"04:29.350","Text":"2t minus 2, all this dt."},{"Start":"04:30.860 ","End":"04:39.965","Text":"If I remember, this is exactly the same expression that we got previously."},{"Start":"04:39.965 ","End":"04:42.035","Text":"I just did a flashback."},{"Start":"04:42.035 ","End":"04:45.695","Text":"Here it is. Now back to the present."},{"Start":"04:45.695 ","End":"04:51.425","Text":"I guess there\u0027s no reason to continue from here because it\u0027s the same expression,"},{"Start":"04:51.425 ","End":"04:54.215","Text":"so it\u0027s also equal to 3."},{"Start":"04:54.215 ","End":"05:00.230","Text":"This is another way of doing line integrals of vector fields."},{"Start":"05:00.230 ","End":"05:02.900","Text":"But normally you wouldn\u0027t do it this way,"},{"Start":"05:02.900 ","End":"05:08.395","Text":"but it\u0027s useful to know that this form can be reduced to this form."},{"Start":"05:08.395 ","End":"05:13.970","Text":"There is just 1 more theoretical thing I\u0027d like to mention."},{"Start":"05:13.970 ","End":"05:16.870","Text":"I won\u0027t do an example or a proof,"},{"Start":"05:16.870 ","End":"05:19.370","Text":"but you might wonder with such an integral,"},{"Start":"05:19.370 ","End":"05:23.225","Text":"what happens if I do the curve in the opposite direction?"},{"Start":"05:23.225 ","End":"05:29.750","Text":"Remember, we did it for ds and showed that it makes no difference."},{"Start":"05:29.750 ","End":"05:33.890","Text":"Then we did it for dx, dy,"},{"Start":"05:33.890 ","End":"05:38.885","Text":"and dz, and in these cases it reversed."},{"Start":"05:38.885 ","End":"05:40.100","Text":"We had a minus."},{"Start":"05:40.100 ","End":"05:45.275","Text":"Now we\u0027ve got a new kind with the.dr, and the question is,"},{"Start":"05:45.275 ","End":"05:48.605","Text":"if I alter the direction of the curve,"},{"Start":"05:48.605 ","End":"05:52.280","Text":"does it stay the same or does it reverse or something else?"},{"Start":"05:52.280 ","End":"05:55.775","Text":"Well, it turns out that this behaves just like these."},{"Start":"05:55.775 ","End":"05:58.670","Text":"In fact, I\u0027ll just write that down."},{"Start":"05:58.670 ","End":"06:03.799","Text":"This will be the last thing that the integral, instead of c,"},{"Start":"06:03.799 ","End":"06:10.530","Text":"if I take the opposite curve minus c of f.dr,"},{"Start":"06:12.050 ","End":"06:21.975","Text":"and this is the minus of the integral over the regular curve of f.dr."},{"Start":"06:21.975 ","End":"06:24.560","Text":"If you reverse the direction,"},{"Start":"06:24.560 ","End":"06:27.275","Text":"you get the negative of the answer."},{"Start":"06:27.275 ","End":"06:29.970","Text":"That\u0027s it for this clip."}],"ID":10491}],"Thumbnail":null,"ID":82678},{"Name":"Vector Fields","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Vector Fields","Duration":"13m 18s","ChapterTopicVideoID":10171,"CourseChapterTopicPlaylistID":82679,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10171.jpeg","UploadDate":"2020-02-26T06:20:57.3630000","DurationForVideoObject":"PT13M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.595","Text":"In this clip, we\u0027re beginning a new topic,"},{"Start":"00:02.595 ","End":"00:04.395","Text":"that of line integrals."},{"Start":"00:04.395 ","End":"00:09.315","Text":"The first subtopic in line integrals is something called vector fields."},{"Start":"00:09.315 ","End":"00:13.260","Text":"Now, I\u0027m assuming that before we start this,"},{"Start":"00:13.260 ","End":"00:18.060","Text":"that you have a concept of vector functions,"},{"Start":"00:18.060 ","End":"00:21.900","Text":"which should have been covered earlier, and if you don\u0027t,"},{"Start":"00:21.900 ","End":"00:25.500","Text":"I suggest you go back and study vector functions,"},{"Start":"00:25.500 ","End":"00:28.560","Text":"especially in more than 1 variable."},{"Start":"00:28.560 ","End":"00:32.970","Text":"I believe I showed it usually in 1 variable,"},{"Start":"00:32.970 ","End":"00:37.220","Text":"for example, we had a vector function of a variable t,"},{"Start":"00:37.220 ","End":"00:40.365","Text":"and this was either a 2D or a 3D vector,"},{"Start":"00:40.365 ","End":"00:45.630","Text":"but then we generalized it to a vector function of 2 variables,"},{"Start":"00:45.630 ","End":"00:47.115","Text":"say, s and t,"},{"Start":"00:47.115 ","End":"00:51.255","Text":"or more commonly, x and y."},{"Start":"00:51.255 ","End":"00:53.790","Text":"Review vector functions,"},{"Start":"00:53.790 ","End":"00:58.530","Text":"at least of 2 variables or more."},{"Start":"00:58.530 ","End":"01:05.255","Text":"This is a topic that is used mostly in applied math,"},{"Start":"01:05.255 ","End":"01:07.600","Text":"and in particular,"},{"Start":"01:07.600 ","End":"01:12.360","Text":"in the area of physics it\u0027s used a lot,"},{"Start":"01:12.360 ","End":"01:16.250","Text":"there are gravitational fields,"},{"Start":"01:16.250 ","End":"01:20.300","Text":"magnetic fields, electrical fields,"},{"Start":"01:20.300 ","End":"01:21.740","Text":"they are all electrical fields, but in other areas too,"},{"Start":"01:21.740 ","End":"01:23.210","Text":"in meteorology, weather,"},{"Start":"01:23.210 ","End":"01:27.695","Text":"the wind has a magnitude and direction at every point, and so on."},{"Start":"01:27.695 ","End":"01:31.670","Text":"Also, we will be covering only 2 cases,"},{"Start":"01:31.670 ","End":"01:33.780","Text":"2D or 3D, although,"},{"Start":"01:33.780 ","End":"01:36.635","Text":"in principle, it could be in any dimension."},{"Start":"01:36.635 ","End":"01:41.260","Text":"I\u0027m actually going to start with the 2D case."},{"Start":"01:41.260 ","End":"01:46.590","Text":"In 2D, we use this form,"},{"Start":"01:46.590 ","End":"01:49.670","Text":"and x and y are not just any parameters,"},{"Start":"01:49.670 ","End":"01:54.720","Text":"they actually represent Euclidean space in 2-dimensions and later x, y,"},{"Start":"01:54.720 ","End":"01:56.790","Text":"z in 3-dimensions,"},{"Start":"01:56.790 ","End":"02:02.390","Text":"and also the vector will have the same dimension as the space."},{"Start":"02:02.390 ","End":"02:05.465","Text":"If it\u0027s a vector function of x and y,"},{"Start":"02:05.465 ","End":"02:07.670","Text":"we\u0027ll also have an x and a y,"},{"Start":"02:07.670 ","End":"02:10.370","Text":"it\u0027ll be a 2-dimensional vector,"},{"Start":"02:10.370 ","End":"02:16.250","Text":"so we would have something like the vector function,"},{"Start":"02:16.250 ","End":"02:18.050","Text":"only I won\u0027t call it v,"},{"Start":"02:18.050 ","End":"02:21.335","Text":"now we\u0027re going to use letter capital F,"},{"Start":"02:21.335 ","End":"02:25.740","Text":"F from the field."},{"Start":"02:26.080 ","End":"02:30.410","Text":"In 2-dimensions, we\u0027ll have a vector function,"},{"Start":"02:30.410 ","End":"02:33.440","Text":"which is a vector field of x and y,"},{"Start":"02:33.440 ","End":"02:36.950","Text":"2-dimensional space, and it\u0027s going to equal,"},{"Start":"02:36.950 ","End":"02:40.550","Text":"there\u0027s 2 notations when we did vectors,"},{"Start":"02:40.550 ","End":"02:43.400","Text":"I\u0027m going to use the notation with the i,"},{"Start":"02:43.400 ","End":"02:44.630","Text":"j, k,"},{"Start":"02:44.630 ","End":"02:45.920","Text":"or in this case i, j,"},{"Start":"02:45.920 ","End":"02:49.624","Text":"so it\u0027s going to be some function of x and y"},{"Start":"02:49.624 ","End":"02:54.680","Text":"times standard basis vector i plus another function,"},{"Start":"02:54.680 ","End":"02:56.660","Text":"I don\u0027t know, g, the letter doesn\u0027t matter,"},{"Start":"02:56.660 ","End":"03:00.395","Text":"just some other function times j."},{"Start":"03:00.395 ","End":"03:03.245","Text":"We\u0027re going to use the i, j notation, although,"},{"Start":"03:03.245 ","End":"03:10.550","Text":"in principle, we could have used the angular brackets notation and said f of x,"},{"Start":"03:10.550 ","End":"03:13.610","Text":"y, g of x,"},{"Start":"03:13.610 ","End":"03:20.120","Text":"y, but in case of vector fields,"},{"Start":"03:20.120 ","End":"03:22.685","Text":"we usually use the i and j."},{"Start":"03:22.685 ","End":"03:27.920","Text":"Similarly, in 3D,"},{"Start":"03:27.920 ","End":"03:31.040","Text":"we\u0027re going to have a vector field F,"},{"Start":"03:31.040 ","End":"03:33.290","Text":"and it will take 3 variables, x,"},{"Start":"03:33.290 ","End":"03:37.085","Text":"y, and z, which are assumed to be points in space,"},{"Start":"03:37.085 ","End":"03:39.960","Text":"usually, might be exceptions I don\u0027t know,"},{"Start":"03:39.960 ","End":"03:40.990","Text":"but anyway, in physics,"},{"Start":"03:40.990 ","End":"03:42.395","Text":"it\u0027s usually space,"},{"Start":"03:42.395 ","End":"03:45.830","Text":"and that\u0027s going to be 3 functions, let\u0027s call them f, g,"},{"Start":"03:45.830 ","End":"03:48.230","Text":"and h: there will be an f of x,"},{"Start":"03:48.230 ","End":"03:51.195","Text":"y, z times i,"},{"Start":"03:51.195 ","End":"03:55.200","Text":"there will be another function of 3 variables, x, y, and z,"},{"Start":"03:55.200 ","End":"03:58.920","Text":"that\u0027s g, and it will be times j,"},{"Start":"03:58.920 ","End":"04:01.095","Text":"and there will also be a third component,"},{"Start":"04:01.095 ","End":"04:03.365","Text":"h, also of x, y,"},{"Start":"04:03.365 ","End":"04:10.590","Text":"and z times the other standard basis vector k. Typically,"},{"Start":"04:10.590 ","End":"04:15.285","Text":"this or this is our vector fields,"},{"Start":"04:15.285 ","End":"04:24.440","Text":"it\u0027s a special case of vector functions and used in typically physics."},{"Start":"04:24.440 ","End":"04:27.830","Text":"That\u0027s the introduction. Now,"},{"Start":"04:27.830 ","End":"04:29.149","Text":"it\u0027s time for an example."},{"Start":"04:29.149 ","End":"04:31.580","Text":"Let\u0027s take an example in 2D."},{"Start":"04:31.580 ","End":"04:33.850","Text":"Let me take F,"},{"Start":"04:33.850 ","End":"04:36.260","Text":"I guess I shouldn\u0027t always be using the same letter F,"},{"Start":"04:36.260 ","End":"04:38.090","Text":"let me just call it F_2D,"},{"Start":"04:38.090 ","End":"04:40.910","Text":"so we know it\u0027s a 2D example,"},{"Start":"04:40.910 ","End":"04:45.575","Text":"of x and y only because it\u0027s 2D,"},{"Start":"04:45.575 ","End":"04:51.120","Text":"is going to equal sine of"},{"Start":"04:51.120 ","End":"04:55.604","Text":"y times i plus"},{"Start":"04:55.604 ","End":"05:02.115","Text":"sine of x times j."},{"Start":"05:02.115 ","End":"05:03.895","Text":"Now, the question is,"},{"Start":"05:03.895 ","End":"05:06.160","Text":"how would you draw such a thing?"},{"Start":"05:06.160 ","End":"05:08.500","Text":"How do we draw a vector field?"},{"Start":"05:08.500 ","End":"05:10.600","Text":"Best way is with a computer,"},{"Start":"05:10.600 ","End":"05:12.295","Text":"with a good computer program,"},{"Start":"05:12.295 ","End":"05:14.320","Text":"but let me just show you the idea."},{"Start":"05:14.320 ","End":"05:16.270","Text":"We take a bunch of points, the more,"},{"Start":"05:16.270 ","End":"05:18.505","Text":"the better, and substitute."},{"Start":"05:18.505 ","End":"05:21.870","Text":"For example, we could compute F, you know what?"},{"Start":"05:21.870 ","End":"05:24.570","Text":"I\u0027ll just forget about the word 2D,"},{"Start":"05:24.570 ","End":"05:29.085","Text":"and let\u0027s say what happens at 0, 0."},{"Start":"05:29.085 ","End":"05:32.215","Text":"Well, if x and y are 0,"},{"Start":"05:32.215 ","End":"05:34.595","Text":"sine x and sine y are also 0,"},{"Start":"05:34.595 ","End":"05:37.025","Text":"so this just gives us the 0 vector,"},{"Start":"05:37.025 ","End":"05:39.110","Text":"which is not so good for drawing,"},{"Start":"05:39.110 ","End":"05:41.615","Text":"so let\u0027s carry on."},{"Start":"05:41.615 ","End":"05:44.925","Text":"Let\u0027s take F of,"},{"Start":"05:44.925 ","End":"05:50.780","Text":"let say Pi over 2,"},{"Start":"05:50.780 ","End":"05:56.055","Text":"0, that would be,"},{"Start":"05:56.055 ","End":"05:58.455","Text":"now x is Pi over 2,"},{"Start":"05:58.455 ","End":"06:05.670","Text":"so this here is 1 and sine of 0 is 0,"},{"Start":"06:05.670 ","End":"06:09.700","Text":"so we get just j,"},{"Start":"06:09.860 ","End":"06:15.130","Text":"0i plus 1j is just the vector j."},{"Start":"06:15.200 ","End":"06:18.255","Text":"Instead of Pi over 2,"},{"Start":"06:18.255 ","End":"06:24.800","Text":"if I took Pi over 6 because I know the sine of 30 degrees,"},{"Start":"06:24.800 ","End":"06:34.220","Text":"and let\u0027s say 0, then I would get the same thing except that sine y is 0,"},{"Start":"06:34.220 ","End":"06:35.765","Text":"but sine x is 1/2,"},{"Start":"06:35.765 ","End":"06:38.160","Text":"so I get a 1/2j."},{"Start":"06:38.380 ","End":"06:42.290","Text":"If I took the other way around,"},{"Start":"06:42.290 ","End":"06:46.235","Text":"suppose I took F of 0,"},{"Start":"06:46.235 ","End":"06:49.040","Text":"Pi over 6, then clearly,"},{"Start":"06:49.040 ","End":"06:53.450","Text":"it\u0027s just reversed and we get 1/2 of i."},{"Start":"06:53.450 ","End":"06:56.625","Text":"If I take F of,"},{"Start":"06:56.625 ","End":"06:59.100","Text":"let\u0027s say both of them Pi over 6,"},{"Start":"06:59.100 ","End":"07:00.915","Text":"Pi over 6,"},{"Start":"07:00.915 ","End":"07:09.750","Text":"then I would get 1/2i plus 1/2j."},{"Start":"07:09.750 ","End":"07:11.855","Text":"I\u0027ll just do a negative example."},{"Start":"07:11.855 ","End":"07:20.505","Text":"Let\u0027s say f of minus Pi over 2,"},{"Start":"07:20.505 ","End":"07:23.670","Text":"minus Pi over 2."},{"Start":"07:23.670 ","End":"07:27.060","Text":"The sine of minus Pi over 2 is minus 1,"},{"Start":"07:27.060 ","End":"07:31.935","Text":"so we get minus i minus j,"},{"Start":"07:31.935 ","End":"07:35.320","Text":"and so on, and so on."},{"Start":"07:38.060 ","End":"07:41.825","Text":"well, I\u0027ll show you the picture first and I\u0027ll explain it."},{"Start":"07:41.825 ","End":"07:45.810","Text":"I got this example from the Wikipedia."},{"Start":"07:46.460 ","End":"07:49.800","Text":"The picture came without axes,"},{"Start":"07:49.800 ","End":"07:51.320","Text":"so I added some axes."},{"Start":"07:51.320 ","End":"07:54.920","Text":"I\u0027m guessing that this is the origin,"},{"Start":"07:54.920 ","End":"07:58.470","Text":"it\u0027s where it seems to be symmetrical about."},{"Start":"07:59.300 ","End":"08:04.715","Text":"Well, let\u0027s say that this here is minus Pi over 2,"},{"Start":"08:04.715 ","End":"08:10.230","Text":"and this is minus Pi over 2 for y,"},{"Start":"08:10.230 ","End":"08:11.850","Text":"we don\u0027t have enough for Pi over 2,"},{"Start":"08:11.850 ","End":"08:14.970","Text":"never mind, this is Pi over 4, let\u0027s say,"},{"Start":"08:14.970 ","End":"08:16.845","Text":"and this is Pi over 4,"},{"Start":"08:16.845 ","End":"08:19.040","Text":"I\u0027m just giving you the general idea so nothing has to"},{"Start":"08:19.040 ","End":"08:21.590","Text":"be precise here, this is the origin."},{"Start":"08:21.590 ","End":"08:26.640","Text":"Let\u0027s start plotting some vectors."},{"Start":"08:26.640 ","End":"08:30.510","Text":"Well, at 0, 0 we have the 0 vector,"},{"Start":"08:30.510 ","End":"08:36.270","Text":"so actually, that\u0027s a bad example because it\u0027s just a point, there\u0027s no arrow."},{"Start":"08:37.670 ","End":"08:45.495","Text":"The next one, let me change this from Pi over 2 to minus Pi over 2,"},{"Start":"08:45.495 ","End":"08:49.300","Text":"and then I\u0027ll get minus j."},{"Start":"08:49.400 ","End":"08:51.780","Text":"That means that this point,"},{"Start":"08:51.780 ","End":"08:59.630","Text":"we have to draw a vertical arrow through this with length, well,"},{"Start":"08:59.630 ","End":"09:00.710","Text":"it would be 1,"},{"Start":"09:00.710 ","End":"09:02.950","Text":"but these are usually scaled,"},{"Start":"09:02.950 ","End":"09:05.610","Text":"they don\u0027t actually take the real length,"},{"Start":"09:05.610 ","End":"09:06.930","Text":"you scale them down,"},{"Start":"09:06.930 ","End":"09:09.150","Text":"so it\u0027s intelligible in the picture,"},{"Start":"09:09.150 ","End":"09:10.515","Text":"otherwise it would be a mess,"},{"Start":"09:10.515 ","End":"09:14.100","Text":"so that it\u0027s 1 unit as far as the vectors go."},{"Start":"09:14.100 ","End":"09:16.740","Text":"Take the next one, Pi over 6,"},{"Start":"09:16.740 ","End":"09:20.340","Text":"0, this is Pi over 4,"},{"Start":"09:20.340 ","End":"09:23.430","Text":"then Pi over 6 is somewhere here,"},{"Start":"09:23.430 ","End":"09:26.115","Text":"and so Pi over 6, 0,"},{"Start":"09:26.115 ","End":"09:28.805","Text":"and we\u0027d get a vertical arrow,"},{"Start":"09:28.805 ","End":"09:31.195","Text":"half the length of this one,"},{"Start":"09:31.195 ","End":"09:34.140","Text":"and that would be the 1/2j."},{"Start":"09:34.140 ","End":"09:38.370","Text":"Similarly, 0, Pi over 6, somewhere here,"},{"Start":"09:38.370 ","End":"09:48.785","Text":"we\u0027d get an arrow half the length going to the right, that\u0027s 1/2i,"},{"Start":"09:48.785 ","End":"09:51.830","Text":"Pi over 6, Pi over 6,"},{"Start":"09:51.830 ","End":"09:53.815","Text":"that\u0027s probably here,"},{"Start":"09:53.815 ","End":"09:56.540","Text":"and that\u0027s a diagonal,"},{"Start":"09:56.540 ","End":"09:59.550","Text":"half this way and half up,"},{"Start":"09:59.550 ","End":"10:01.010","Text":"and then minus Pi over 2,"},{"Start":"10:01.010 ","End":"10:02.320","Text":"minus Pi over 2,"},{"Start":"10:02.320 ","End":"10:05.020","Text":"that would be just at the edge here,"},{"Start":"10:05.020 ","End":"10:06.395","Text":"it might be this one,"},{"Start":"10:06.395 ","End":"10:10.610","Text":"it might be this one which is minus i minus j."},{"Start":"10:10.610 ","End":"10:13.895","Text":"Anyway, the more points you draw with little arrows,"},{"Start":"10:13.895 ","End":"10:16.430","Text":"it\u0027s a nightmare to do by hand,"},{"Start":"10:16.430 ","End":"10:20.690","Text":"and that\u0027s why there are computer programs to generate these things."},{"Start":"10:20.690 ","End":"10:23.510","Text":"Anyway, this gives you the idea."},{"Start":"10:23.510 ","End":"10:26.270","Text":"Next, I\u0027m going to do a 3D example,"},{"Start":"10:26.270 ","End":"10:28.705","Text":"so I\u0027ll erase what I don\u0027t need."},{"Start":"10:28.705 ","End":"10:33.180","Text":"This time, the example will be F of x,"},{"Start":"10:33.180 ","End":"10:35.460","Text":"y, z, a 3D example,"},{"Start":"10:35.460 ","End":"10:45.560","Text":"is equal to 2x times vector i minus 2y times unit"},{"Start":"10:45.560 ","End":"10:48.335","Text":"vector j and minus"},{"Start":"10:48.335 ","End":"10:57.180","Text":"2x again times vector k. As before,"},{"Start":"10:57.180 ","End":"11:02.680","Text":"we\u0027ll plug in a few examples just to get the idea."},{"Start":"11:02.680 ","End":"11:05.430","Text":"Let\u0027s just try any number, say 1,"},{"Start":"11:05.430 ","End":"11:07.860","Text":"2, minus 3,"},{"Start":"11:07.860 ","End":"11:10.065","Text":"and see what we get."},{"Start":"11:10.065 ","End":"11:15.480","Text":"X is 1, so 2x is 2,"},{"Start":"11:15.480 ","End":"11:17.430","Text":"so we get 2i,"},{"Start":"11:17.430 ","End":"11:22.210","Text":"and then minus 2y is minus 4j,"},{"Start":"11:23.060 ","End":"11:28.260","Text":"and then minus 2k."},{"Start":"11:28.260 ","End":"11:30.920","Text":"Actually, the number here is not important."},{"Start":"11:30.920 ","End":"11:32.285","Text":"Let\u0027s take another example."},{"Start":"11:32.285 ","End":"11:33.770","Text":"F of, I don\u0027t know,"},{"Start":"11:33.770 ","End":"11:40.190","Text":"minus 5, 0,"},{"Start":"11:40.190 ","End":"11:46.890","Text":"9 is equal to 2x is minus 10i,"},{"Start":"11:46.940 ","End":"11:50.160","Text":"2y is nothing,"},{"Start":"11:50.160 ","End":"11:55.815","Text":"and then we have minus 2x is plus 10k."},{"Start":"11:55.815 ","End":"11:58.960","Text":"I left a gap there because there is no j."},{"Start":"11:58.960 ","End":"12:00.570","Text":"Maybe one more example."},{"Start":"12:00.570 ","End":"12:03.950","Text":"F of 0, 1,"},{"Start":"12:03.950 ","End":"12:06.430","Text":"2 is equal to,"},{"Start":"12:06.430 ","End":"12:08.530","Text":"let\u0027s see, x is 0,"},{"Start":"12:08.530 ","End":"12:10.990","Text":"so that\u0027s 0i,"},{"Start":"12:10.990 ","End":"12:13.645","Text":"y is 1,"},{"Start":"12:13.645 ","End":"12:17.930","Text":"so it\u0027s minus 2j,"},{"Start":"12:17.930 ","End":"12:19.920","Text":"and again, x is 0,"},{"Start":"12:19.920 ","End":"12:22.320","Text":"so this is just minus 2j."},{"Start":"12:22.320 ","End":"12:23.790","Text":"At the point 0,"},{"Start":"12:23.790 ","End":"12:24.950","Text":"1, 2 in space,"},{"Start":"12:24.950 ","End":"12:31.820","Text":"I put a vector facing 2 units in the direction of the negative y-axis,"},{"Start":"12:31.820 ","End":"12:33.725","Text":"it\u0027s hard to picture."},{"Start":"12:33.725 ","End":"12:36.275","Text":"I happen to have with me,"},{"Start":"12:36.275 ","End":"12:37.490","Text":"again, from the Internet,"},{"Start":"12:37.490 ","End":"12:39.900","Text":"I find these pictures,"},{"Start":"12:40.160 ","End":"12:44.100","Text":"and here is a computer-plotted image."},{"Start":"12:44.100 ","End":"12:46.830","Text":"This might be a front view."},{"Start":"12:46.830 ","End":"12:49.050","Text":"It doesn\u0027t really matter,"},{"Start":"12:49.050 ","End":"12:51.785","Text":"I just wanted to give you an idea that there are computer programs,"},{"Start":"12:51.785 ","End":"12:55.800","Text":"and I\u0027ll show you another view from above."},{"Start":"12:56.360 ","End":"13:00.439","Text":"Anyway, they\u0027re a real mess even to understand,"},{"Start":"13:00.439 ","End":"13:02.765","Text":"even to see let alone to plot."},{"Start":"13:02.765 ","End":"13:04.580","Text":"But it gives you an idea."},{"Start":"13:04.580 ","End":"13:10.630","Text":"I think you\u0027ve got the idea of what a vector field is by now in 2D and in 3D."},{"Start":"13:10.630 ","End":"13:14.150","Text":"I\u0027m going to continue on a new page."},{"Start":"13:14.150 ","End":"13:18.030","Text":"The next subject will be the gradient."}],"ID":10476},{"Watched":false,"Name":"Gradient Vector Field","Duration":"15m 46s","ChapterTopicVideoID":10172,"CourseChapterTopicPlaylistID":82679,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10172.jpeg","UploadDate":"2020-02-26T06:21:20.9630000","DurationForVideoObject":"PT15M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.415","Text":"Talk about the concept of gradient,"},{"Start":"00:02.415 ","End":"00:04.949","Text":"in the context of vector fields."},{"Start":"00:04.949 ","End":"00:08.835","Text":"Now suppose I have a regular function,"},{"Start":"00:08.835 ","End":"00:11.430","Text":"f of x, y, and z."},{"Start":"00:11.430 ","End":"00:13.725","Text":"When I say regular, I mean not vector,"},{"Start":"00:13.725 ","End":"00:21.210","Text":"and we sometimes use the word scalar to distinguish it to say it\u0027s not a vector,"},{"Start":"00:21.210 ","End":"00:23.205","Text":"it just gives a number."},{"Start":"00:23.205 ","End":"00:28.665","Text":"Then from this scalar function of x, y, z,"},{"Start":"00:28.665 ","End":"00:34.110","Text":"we can get a vector field and it\u0027s written as follows,"},{"Start":"00:34.110 ","End":"00:36.615","Text":"an upside-down triangle,"},{"Start":"00:36.615 ","End":"00:42.110","Text":"and then f, also of x, y, and z."},{"Start":"00:42.110 ","End":"00:46.610","Text":"But this time it\u0027s going to be a vector field and it\u0027s going to"},{"Start":"00:46.610 ","End":"00:54.125","Text":"be the partial derivative of f with respect to x times i,"},{"Start":"00:54.125 ","End":"00:56.840","Text":"I\u0027ll say more about this in a moment,"},{"Start":"00:56.840 ","End":"01:03.810","Text":"plus the partial derivative with respect to y times j, plus,"},{"Start":"01:03.810 ","End":"01:07.655","Text":"you can guess, partial derivative of f with respect to z,"},{"Start":"01:07.655 ","End":"01:13.305","Text":"k. We are writing it this way,"},{"Start":"01:13.305 ","End":"01:15.860","Text":"but if you really like the angular brackets,"},{"Start":"01:15.860 ","End":"01:19.200","Text":"we could write it like this;"},{"Start":"01:19.200 ","End":"01:21.590","Text":"f with respect to x,"},{"Start":"01:21.590 ","End":"01:23.794","Text":"f with respect to y,"},{"Start":"01:23.794 ","End":"01:25.940","Text":"f with respect to z."},{"Start":"01:25.940 ","End":"01:30.300","Text":"Just want to make a note on notation."},{"Start":"01:30.790 ","End":"01:37.985","Text":"F with respect to x is sometimes written as f prime with respect to x."},{"Start":"01:37.985 ","End":"01:40.280","Text":"Sometimes with Leibniz\u0027s notation,"},{"Start":"01:40.280 ","End":"01:48.270","Text":"it\u0027s a funny d partial derivative with respect to x."},{"Start":"01:48.860 ","End":"01:51.860","Text":"Although I omitted it also a function of x,"},{"Start":"01:51.860 ","End":"01:53.120","Text":"y, and z,"},{"Start":"01:53.120 ","End":"01:58.085","Text":"so this is really longhand would be f with respect to x,"},{"Start":"01:58.085 ","End":"02:01.235","Text":"partial derivative of x, y, and z."},{"Start":"02:01.235 ","End":"02:04.850","Text":"But we don\u0027t want to write this out every time."},{"Start":"02:04.850 ","End":"02:08.180","Text":"Anyway, the way we pronounce this,"},{"Start":"02:08.180 ","End":"02:16.005","Text":"this symbol is actually called a nabla,"},{"Start":"02:16.005 ","End":"02:19.940","Text":"sometimes it\u0027s from Hebrew or Arabic,"},{"Start":"02:19.940 ","End":"02:23.270","Text":"and we sometimes call it a del,"},{"Start":"02:23.270 ","End":"02:26.650","Text":"and sometimes we just say grad."},{"Start":"02:26.650 ","End":"02:31.560","Text":"I might say grad f or nabla f or del"},{"Start":"02:31.560 ","End":"02:36.245","Text":"f. I\u0027d like to use the word grad because it\u0027s like gradient,"},{"Start":"02:36.245 ","End":"02:38.730","Text":"but just so you\u0027ll know."},{"Start":"02:39.970 ","End":"02:47.200","Text":"Here\u0027s something I found when I did a search on this term,"},{"Start":"02:47.450 ","End":"02:50.850","Text":"del or nabla, but it\u0027s also grad."},{"Start":"02:50.850 ","End":"02:55.900","Text":"It doesn\u0027t just apply to functions of 3 variables,"},{"Start":"02:55.900 ","End":"02:57.305","Text":"any number of variables,"},{"Start":"02:57.305 ","End":"03:00.140","Text":"even function of 1 variable,"},{"Start":"03:00.140 ","End":"03:03.515","Text":"and then it\u0027s just the regular derivative,"},{"Start":"03:03.515 ","End":"03:05.480","Text":"and that\u0027s what it says here."},{"Start":"03:05.480 ","End":"03:11.100","Text":"Basically, if it\u0027s a function 1-dimension,"},{"Start":"03:11.100 ","End":"03:12.735","Text":"it\u0027s a standard derivative,"},{"Start":"03:12.735 ","End":"03:15.220","Text":"maybe I\u0027ll write it in 2 dimensions."},{"Start":"03:15.220 ","End":"03:18.350","Text":"If I have a scalar function,"},{"Start":"03:18.350 ","End":"03:21.410","Text":"I\u0027ll use the same letter again, but a different color."},{"Start":"03:21.410 ","End":"03:24.890","Text":"In the 2D case, if I have a function of x and y,"},{"Start":"03:24.890 ","End":"03:28.805","Text":"which is a scalar function of 2 variables,"},{"Start":"03:28.805 ","End":"03:34.950","Text":"then grad f of x and y is"},{"Start":"03:34.950 ","End":"03:39.770","Text":"just fx of x and"},{"Start":"03:39.770 ","End":"03:48.680","Text":"y times i plus the partial derivative of f with respect to y times j."},{"Start":"03:48.680 ","End":"03:51.515","Text":"Or if you like the angular brackets,"},{"Start":"03:51.515 ","End":"03:53.855","Text":"then we can write it like this."},{"Start":"03:53.855 ","End":"03:56.045","Text":"Many possibilities."},{"Start":"03:56.045 ","End":"03:58.940","Text":"We\u0027re going to be working in 2D and 3D,"},{"Start":"03:58.940 ","End":"04:01.175","Text":"not in 1D and not in 4D."},{"Start":"04:01.175 ","End":"04:03.245","Text":"Just 2D and 3D."},{"Start":"04:03.245 ","End":"04:05.900","Text":"Now examples."},{"Start":"04:05.900 ","End":"04:08.285","Text":"Let\u0027s start in 2D."},{"Start":"04:08.285 ","End":"04:10.310","Text":"If I have f of x,"},{"Start":"04:10.310 ","End":"04:13.865","Text":"y is equal to, let\u0027s say,"},{"Start":"04:13.865 ","End":"04:21.155","Text":"I don\u0027t know x cubed sine of 2y,"},{"Start":"04:21.155 ","End":"04:26.920","Text":"then I will get that grad f,"},{"Start":"04:26.920 ","End":"04:31.935","Text":"and I\u0027ll use the angular bracket."},{"Start":"04:31.935 ","End":"04:37.220","Text":"The partial derivative of f with respect to x and y is a constant,"},{"Start":"04:37.220 ","End":"04:38.674","Text":"and all this is a constant,"},{"Start":"04:38.674 ","End":"04:40.700","Text":"so it just sticks here."},{"Start":"04:40.700 ","End":"04:47.280","Text":"We get 3x squared sine of 2y,"},{"Start":"04:47.800 ","End":"04:52.140","Text":"or this thing times i."},{"Start":"04:52.480 ","End":"04:56.990","Text":"There we are. Plus the other 1,"},{"Start":"04:56.990 ","End":"04:58.370","Text":"derivative with respect to y,"},{"Start":"04:58.370 ","End":"04:59.410","Text":"then x is a constant,"},{"Start":"04:59.410 ","End":"05:01.230","Text":"so the x cubed is a constant."},{"Start":"05:01.230 ","End":"05:06.285","Text":"It just sticks and the derivative of sine is cosine,"},{"Start":"05:06.285 ","End":"05:08.665","Text":"so it\u0027s cosine 2y."},{"Start":"05:08.665 ","End":"05:10.040","Text":"Well, not quite."},{"Start":"05:10.040 ","End":"05:12.140","Text":"We have the derivative which is 2,"},{"Start":"05:12.140 ","End":"05:15.635","Text":"so let me stick that in front and close the brackets."},{"Start":"05:15.635 ","End":"05:17.610","Text":"That\u0027s 1 example."},{"Start":"05:17.610 ","End":"05:22.100","Text":"I\u0027m just reminding you that this of course is a vector field."},{"Start":"05:22.100 ","End":"05:29.904","Text":"I just remember that some people write an arrow over the symbol."},{"Start":"05:29.904 ","End":"05:34.280","Text":"Here they might put an arrow, I won\u0027t,"},{"Start":"05:34.280 ","End":"05:39.425","Text":"just to remind us that we\u0027re getting a vector field."},{"Start":"05:39.425 ","End":"05:42.440","Text":"It just gets tedious with all the arrows."},{"Start":"05:42.440 ","End":"05:44.800","Text":"Now a 3D example."},{"Start":"05:44.800 ","End":"05:49.650","Text":"I\u0027ll change the letter, let\u0027s call it g of x,"},{"Start":"05:49.650 ","End":"05:52.785","Text":"y, and z scalar function,"},{"Start":"05:52.785 ","End":"05:59.775","Text":"is equal to z e to the power of x y,"},{"Start":"05:59.775 ","End":"06:03.870","Text":"and so we get that grad g,"},{"Start":"06:03.870 ","End":"06:08.460","Text":"del g is equal to,"},{"Start":"06:08.460 ","End":"06:12.920","Text":"I take the 3 partial derivatives first with respect to x,"},{"Start":"06:12.920 ","End":"06:15.010","Text":"and then I will get,"},{"Start":"06:15.010 ","End":"06:17.655","Text":"z is a constant."},{"Start":"06:17.655 ","End":"06:23.330","Text":"Then the derivative of e to the x y would be e to the x y,"},{"Start":"06:23.330 ","End":"06:26.480","Text":"except that it\u0027s not x."},{"Start":"06:26.480 ","End":"06:31.915","Text":"It\u0027s some constant y times x or y comes in front."},{"Start":"06:31.915 ","End":"06:35.250","Text":"Then with respect to y, very similar."},{"Start":"06:35.250 ","End":"06:38.975","Text":"We would just get an x here in place of the y,"},{"Start":"06:38.975 ","End":"06:42.020","Text":"but still e to the power of x y."},{"Start":"06:42.020 ","End":"06:44.060","Text":"Finally, with respect to z,"},{"Start":"06:44.060 ","End":"06:45.500","Text":"this whole thing is a constant,"},{"Start":"06:45.500 ","End":"06:48.600","Text":"so it\u0027s e to the power of x y."},{"Start":"06:48.600 ","End":"06:52.390","Text":"This is a 3-dimensional vector field."},{"Start":"06:52.850 ","End":"06:57.620","Text":"I\u0027m going to move on and clear the board everything I don\u0027t need."},{"Start":"06:57.620 ","End":"07:07.310","Text":"I just kept the definitions in 2D and in 3D of the del, grad operator."},{"Start":"07:07.310 ","End":"07:13.930","Text":"I want to relate this concept to another concept we\u0027ve seen before."},{"Start":"07:13.930 ","End":"07:16.270","Text":"That would be the concept,"},{"Start":"07:16.270 ","End":"07:17.920","Text":"what we gave it 2 names."},{"Start":"07:17.920 ","End":"07:21.925","Text":"We gave it the name contours,"},{"Start":"07:21.925 ","End":"07:25.030","Text":"but we also called them level curves."},{"Start":"07:25.030 ","End":"07:28.945","Text":"Especially in economics, this is what they\u0027re called,"},{"Start":"07:28.945 ","End":"07:31.310","Text":"but the same thing."},{"Start":"07:31.530 ","End":"07:34.315","Text":"I\u0027m going to illustrate the concept,"},{"Start":"07:34.315 ","End":"07:37.975","Text":"the relationship between the 2 by means of an example."},{"Start":"07:37.975 ","End":"07:42.355","Text":"I\u0027m going to take the example in 2 dimensions, f of x,"},{"Start":"07:42.355 ","End":"07:50.690","Text":"y equals x squared plus y squared."},{"Start":"07:50.940 ","End":"07:58.210","Text":"First of all, I\u0027ll do the gradient and then we\u0027ll talk about contours."},{"Start":"07:58.210 ","End":"08:08.500","Text":"The gradient grad f also of x and y is equal to,"},{"Start":"08:08.500 ","End":"08:15.220","Text":"first I differentiate with respect to x and I get 2x times vector i."},{"Start":"08:15.220 ","End":"08:20.830","Text":"Actually I prefer the angular bracket notation that even though in the beginning"},{"Start":"08:20.830 ","End":"08:26.770","Text":"I said abusing the other and the derivative with respect to y,"},{"Start":"08:26.770 ","End":"08:30.560","Text":"then x is a constant and it\u0027s just 2y."},{"Start":"08:31.500 ","End":"08:39.040","Text":"Just notice that this is actually equal to twice x,"},{"Start":"08:39.040 ","End":"08:42.805","Text":"y, which is the position vector of the point x, y."},{"Start":"08:42.805 ","End":"08:46.630","Text":"When we take a vector and multiply it by a scalar,"},{"Start":"08:46.630 ","End":"08:50.000","Text":"is a parallel vector."},{"Start":"08:50.550 ","End":"08:53.590","Text":"In the plane, for example,"},{"Start":"08:53.590 ","End":"08:57.110","Text":"suppose I just take a quick pair of x\u0027s,"},{"Start":"08:59.340 ","End":"09:02.230","Text":"and suppose I took a point,"},{"Start":"09:02.230 ","End":"09:05.935","Text":"let\u0027s say 1, 2."},{"Start":"09:05.935 ","End":"09:10.510","Text":"Let\u0027s say that was here 1,2."},{"Start":"09:10.510 ","End":"09:15.760","Text":"Then the grad would be 2,"},{"Start":"09:15.760 ","End":"09:19.540","Text":"4, which if you notice, is parallel."},{"Start":"09:19.540 ","End":"09:22.735","Text":"If this is the position vector 1,2,"},{"Start":"09:22.735 ","End":"09:24.940","Text":"2,4 is the same thing,"},{"Start":"09:24.940 ","End":"09:27.835","Text":"just twice in length."},{"Start":"09:27.835 ","End":"09:31.315","Text":"When we sketch the vector field,"},{"Start":"09:31.315 ","End":"09:34.075","Text":"we\u0027ll sketch a lot of arrows like this,"},{"Start":"09:34.075 ","End":"09:36.370","Text":"but we also scale them down sometimes."},{"Start":"09:36.370 ","End":"09:40.075","Text":"Well, I\u0027ll show you in a moment the sketch."},{"Start":"09:40.075 ","End":"09:43.495","Text":"But basically, what I wanted to say is in this particular case,"},{"Start":"09:43.495 ","End":"09:47.575","Text":"the grad is parallel to the position vector."},{"Start":"09:47.575 ","End":"09:50.335","Text":"That aside for the moment."},{"Start":"09:50.335 ","End":"09:55.300","Text":"When I go to contours and these are level curves,"},{"Start":"09:55.300 ","End":"09:57.355","Text":"which means that we take f of x,"},{"Start":"09:57.355 ","End":"09:59.980","Text":"y equals a constant."},{"Start":"09:59.980 ","End":"10:07.660","Text":"In other words, I get that x squared plus y squared equals a constant."},{"Start":"10:07.660 ","End":"10:10.105","Text":"We\u0027d have to have a positive constant."},{"Start":"10:10.105 ","End":"10:13.000","Text":"This is the equation of the circle."},{"Start":"10:13.000 ","End":"10:21.230","Text":"The circle has radius actually of the square root of k circle."},{"Start":"10:21.510 ","End":"10:26.140","Text":"Because yeah, x squared plus y squared equals r squared and if r squared is k,"},{"Start":"10:26.140 ","End":"10:30.700","Text":"then r is the square root of k. I get a whole bunch of circles."},{"Start":"10:30.700 ","End":"10:37.975","Text":"I\u0027m going to bring the picture and show you what I meant here and here it is."},{"Start":"10:37.975 ","End":"10:42.820","Text":"The circles are the level curves for"},{"Start":"10:42.820 ","End":"10:47.815","Text":"different values of k. I would imagine that this 1 would be,"},{"Start":"10:47.815 ","End":"10:53.860","Text":"say, k equals 2 because this looks like square root of 2,"},{"Start":"10:53.860 ","End":"10:58.000","Text":"like 1.4 and so on."},{"Start":"10:58.000 ","End":"11:01.090","Text":"Different values of k give different circles."},{"Start":"11:01.090 ","End":"11:04.195","Text":"Also the vector field,"},{"Start":"11:04.195 ","End":"11:09.260","Text":"which is this scale down,"},{"Start":"11:09.840 ","End":"11:12.610","Text":"making the arrows smaller,"},{"Start":"11:12.610 ","End":"11:14.350","Text":"otherwise it\u0027s a mess like I mentioned,"},{"Start":"11:14.350 ","End":"11:16.270","Text":"but they are at each point,"},{"Start":"11:16.270 ","End":"11:22.225","Text":"the direction is continuation of the vector from the origin to that point."},{"Start":"11:22.225 ","End":"11:25.615","Text":"Like here, if I join the origin to this point,"},{"Start":"11:25.615 ","End":"11:28.015","Text":"then this would just be a continuation."},{"Start":"11:28.015 ","End":"11:31.100","Text":"It\u0027s this but scaled down."},{"Start":"11:31.500 ","End":"11:36.250","Text":"The important thing now that you\u0027ve registered"},{"Start":"11:36.250 ","End":"11:41.290","Text":"the picture is that it turns out that they are orthogonal,"},{"Start":"11:41.290 ","End":"11:43.000","Text":"that at every point,"},{"Start":"11:43.000 ","End":"11:46.675","Text":"the tangent to the level curve,"},{"Start":"11:46.675 ","End":"11:49.340","Text":"for example here,"},{"Start":"11:51.060 ","End":"11:53.320","Text":"this looks like a good place,"},{"Start":"11:53.320 ","End":"12:00.670","Text":"but take this point here than the well,"},{"Start":"12:00.670 ","End":"12:04.390","Text":"no, it\u0027s fine 1 with the arrow, yeah, this 1 here."},{"Start":"12:04.390 ","End":"12:14.714","Text":"The tangent to this red curve and if I just amplify this vector,"},{"Start":"12:14.714 ","End":"12:18.970","Text":"these things are 90 degrees to each other."},{"Start":"12:19.290 ","End":"12:25.225","Text":"The relationship between this and this is that at any given point,"},{"Start":"12:25.225 ","End":"12:29.155","Text":"the gradient field is perpendicular or orthogonal."},{"Start":"12:29.155 ","End":"12:34.600","Text":"I\u0027ll just write that. Orthogonal. Didn\u0027t write a full sentence,"},{"Start":"12:34.600 ","End":"12:37.525","Text":"but I think I illustrated what I meant."},{"Start":"12:37.525 ","End":"12:44.210","Text":"I\u0027ll just write in brackets in case you forgot that orthogonal means perpendicular."},{"Start":"12:45.480 ","End":"12:48.340","Text":"Just this picture illustrates it."},{"Start":"12:48.340 ","End":"12:52.210","Text":"I won\u0027t go into any more depth than that."},{"Start":"12:52.210 ","End":"12:54.685","Text":"As a similar thing in 3D,"},{"Start":"12:54.685 ","End":"13:00.940","Text":"only in 3D we have level surfaces."},{"Start":"13:00.940 ","End":"13:03.190","Text":"If I have a function of x,"},{"Start":"13:03.190 ","End":"13:05.560","Text":"y, and z, and let it be a constant,"},{"Start":"13:05.560 ","End":"13:09.700","Text":"then we get level surfaces but the same principle applies that"},{"Start":"13:09.700 ","End":"13:15.760","Text":"the gradient vector field is perpendicular to level surfaces."},{"Start":"13:15.760 ","End":"13:18.490","Text":"Anyways, we\u0027re staying in 2D meanwhile."},{"Start":"13:18.490 ","End":"13:22.930","Text":"Before I finished there\u0027s another couple of terms I have to introduce."},{"Start":"13:22.930 ","End":"13:25.915","Text":"Let\u0027s continue with this example."},{"Start":"13:25.915 ","End":"13:29.769","Text":"When I take the grad of this function,"},{"Start":"13:29.769 ","End":"13:32.350","Text":"this is now a vector field and I could give it a name,"},{"Start":"13:32.350 ","End":"13:33.985","Text":"I could say, \"Okay.\""},{"Start":"13:33.985 ","End":"13:36.670","Text":"In fact, I might have even started from here and said,"},{"Start":"13:36.670 ","End":"13:40.990","Text":"let\u0027s consider the vector field F of x,"},{"Start":"13:40.990 ","End":"13:45.550","Text":"y is equal to this tangent for a change,"},{"Start":"13:45.550 ","End":"13:52.760","Text":"we\u0027ll take the other notation to xi plus 2yj."},{"Start":"13:54.360 ","End":"13:57.220","Text":"Because of what we saw before,"},{"Start":"13:57.220 ","End":"14:03.370","Text":"it so happens that F is equal to grad of"},{"Start":"14:03.370 ","End":"14:09.700","Text":"little f. I\u0027m just writing this in condensed form without the xy."},{"Start":"14:09.700 ","End":"14:13.720","Text":"There\u0027s a special pair of terms when this happens."},{"Start":"14:13.720 ","End":"14:20.965","Text":"If a vector field happens to be the gradient of a scalar function,"},{"Start":"14:20.965 ","End":"14:24.385","Text":"I forgot the arrows,"},{"Start":"14:24.385 ","End":"14:33.160","Text":"then we say that F is a conservative vector field."},{"Start":"14:33.160 ","End":"14:34.340","Text":"When I say conservative,"},{"Start":"14:34.340 ","End":"14:36.020","Text":"I don\u0027t mean in the political sense,"},{"Start":"14:36.020 ","End":"14:43.275","Text":"I mean it conserves but I won\u0027t go into the reason semantics of this."},{"Start":"14:43.275 ","End":"14:46.850","Text":"So there\u0027d be 2 terms. 1 of them is conservative and"},{"Start":"14:46.850 ","End":"14:51.440","Text":"the other term is a potential function."},{"Start":"14:51.440 ","End":"14:54.770","Text":"If F is a conservative field and it means there\u0027s some"},{"Start":"14:54.770 ","End":"14:59.780","Text":"f such that this thing holds and this little f is called"},{"Start":"14:59.780 ","End":"15:03.875","Text":"the potential function for"},{"Start":"15:03.875 ","End":"15:10.190","Text":"F. I say a potential function because for example,"},{"Start":"15:10.190 ","End":"15:15.620","Text":"in this case, if I took instead of x squared plus y squared,"},{"Start":"15:15.620 ","End":"15:20.510","Text":"I took x squared plus y squared plus 3 plus a constant and that would"},{"Start":"15:20.510 ","End":"15:25.160","Text":"also be a potential function for"},{"Start":"15:25.160 ","End":"15:31.010","Text":"F here."},{"Start":"15:31.010 ","End":"15:39.800","Text":"I\u0027ll settle for that. We\u0027re done with this introduction to vector fields."},{"Start":"15:41.730 ","End":"15:46.310","Text":"Coming up next line integrals."}],"ID":10477}],"Thumbnail":null,"ID":82679},{"Name":"The Gradient Theorem","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Gradient Theorem Part a","Duration":"10m 56s","ChapterTopicVideoID":10173,"CourseChapterTopicPlaylistID":82680,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10173.jpeg","UploadDate":"2017-08-21T14:38:12.4170000","DurationForVideoObject":"PT10M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.385","Text":"In this clip, I\u0027m going to talk about the fundamental theorem for line integrals."},{"Start":"00:05.385 ","End":"00:06.930","Text":"It also has another name,"},{"Start":"00:06.930 ","End":"00:09.285","Text":"it\u0027s also called the gradient theorem,"},{"Start":"00:09.285 ","End":"00:11.040","Text":"so that\u0027s both names."},{"Start":"00:11.040 ","End":"00:12.300","Text":"But before I begin,"},{"Start":"00:12.300 ","End":"00:16.950","Text":"I want to remind you of some basic concepts from vector fields."},{"Start":"00:16.950 ","End":"00:20.745","Text":"The 3 concepts that I want to mention are"},{"Start":"00:20.745 ","End":"00:30.315","Text":"gradient, conservative, and potential."},{"Start":"00:30.315 ","End":"00:36.765","Text":"The gradient is what we call the grad or del or nabla."},{"Start":"00:36.765 ","End":"00:42.195","Text":"Anyway, let me just flashback to the previous clip."},{"Start":"00:42.195 ","End":"00:44.420","Text":"Here we are back in the past."},{"Start":"00:44.420 ","End":"00:48.125","Text":"Remember, we defined the gradient,"},{"Start":"00:48.125 ","End":"00:51.545","Text":"which we denoted with an upside down triangle."},{"Start":"00:51.545 ","End":"00:53.390","Text":"For a function in 2 dimensions,"},{"Start":"00:53.390 ","End":"00:57.110","Text":"we take a derivative with respect to x and the derivative with respect to"},{"Start":"00:57.110 ","End":"01:01.965","Text":"y in the i direction and in the j direction."},{"Start":"01:01.965 ","End":"01:04.860","Text":"Similarly, in 3-dimensions,"},{"Start":"01:04.860 ","End":"01:10.915","Text":"and you could also write it with the angular bracket notation."},{"Start":"01:10.915 ","End":"01:12.830","Text":"In our particular example,"},{"Start":"01:12.830 ","End":"01:15.810","Text":"we took f, little f,"},{"Start":"01:15.810 ","End":"01:21.460","Text":"to be this, and it\u0027s gradient in the angular bracket form was 2x, 2y."},{"Start":"01:21.460 ","End":"01:23.890","Text":"Or in the ij form,"},{"Start":"01:23.890 ","End":"01:27.855","Text":"2x of i plus 2yj."},{"Start":"01:27.855 ","End":"01:30.715","Text":"We also said that big F,"},{"Start":"01:30.715 ","End":"01:37.445","Text":"the vector function, is the gradient of little f, the scalar function."},{"Start":"01:37.445 ","End":"01:42.665","Text":"Whenever this happens, when we take the gradient of something,"},{"Start":"01:42.665 ","End":"01:48.620","Text":"then the vector field is called conservative."},{"Start":"01:48.620 ","End":"01:53.795","Text":"If it comes from the gradient of a little f, that\u0027s conservative."},{"Start":"01:53.795 ","End":"01:55.220","Text":"In this case also,"},{"Start":"01:55.220 ","End":"01:59.345","Text":"we say that little f is a potential function"},{"Start":"01:59.345 ","End":"02:05.540","Text":"for the vector field big F. Similarly,"},{"Start":"02:05.540 ","End":"02:08.465","Text":"in 3D or in any number of dimensions."},{"Start":"02:08.465 ","End":"02:11.470","Text":"Okay. Now let\u0027s get back to the present."},{"Start":"02:11.470 ","End":"02:14.955","Text":"Okay. That\u0027s these 3 terms reviewed."},{"Start":"02:14.955 ","End":"02:18.590","Text":"Now, I want to give an analogy before I give this theorem."},{"Start":"02:18.590 ","End":"02:22.880","Text":"In regular calculus, there\u0027s a fundamental theorem"},{"Start":"02:22.880 ","End":"02:29.700","Text":"also that let\u0027s say we have a function f of x,"},{"Start":"02:29.700 ","End":"02:34.545","Text":"and we have its derivative f prime of x."},{"Start":"02:34.545 ","End":"02:42.460","Text":"Then f is also a primitive or indefinite integral of f prime,"},{"Start":"02:42.460 ","End":"02:46.105","Text":"and the fundamental theorem talks about the definite integral,"},{"Start":"02:46.105 ","End":"02:50.520","Text":"and says that if we take the definite integral from a to"},{"Start":"02:50.520 ","End":"02:56.075","Text":"b of some function which is already a derivative of something,"},{"Start":"02:56.075 ","End":"02:58.775","Text":"we take its primitive,"},{"Start":"02:58.775 ","End":"03:04.410","Text":"which is f without the prime of x,"},{"Start":"03:04.410 ","End":"03:07.050","Text":"and evaluate it from a to b,"},{"Start":"03:07.050 ","End":"03:11.560","Text":"which in other words is f of b minus f of a."},{"Start":"03:11.560 ","End":"03:15.995","Text":"Now, this is just an analogy for what I\u0027m about to say."},{"Start":"03:15.995 ","End":"03:19.910","Text":"This is the fundamental theorem of calculus."},{"Start":"03:19.910 ","End":"03:24.550","Text":"Now, we\u0027re going to do it for line integrals, so here goes."},{"Start":"03:24.550 ","End":"03:26.520","Text":"Well, I\u0027ve written it,"},{"Start":"03:26.520 ","End":"03:28.830","Text":"now I need to explain it."},{"Start":"03:28.830 ","End":"03:36.215","Text":"In some sense, the grad behaves a bit like the derivative."},{"Start":"03:36.215 ","End":"03:41.340","Text":"You can see there\u0027s something with like f of b minus f of a,"},{"Start":"03:41.340 ","End":"03:45.720","Text":"and here we have f of something with b minus f of something with a,"},{"Start":"03:45.720 ","End":"03:48.265","Text":"and there is an analogy."},{"Start":"03:48.265 ","End":"03:51.260","Text":"To explain it, I\u0027ll do a little sketch."},{"Start":"03:51.260 ","End":"03:56.870","Text":"First of all, we have a curve c. Let\u0027s say the curve c,"},{"Start":"03:56.870 ","End":"04:02.004","Text":"just anything, and it goes from 1 point to another,"},{"Start":"04:02.004 ","End":"04:04.960","Text":"and I\u0027ll call this point p, well,"},{"Start":"04:04.960 ","End":"04:07.345","Text":"let\u0027s say its position vector is p,"},{"Start":"04:07.345 ","End":"04:11.440","Text":"and this is the point with position vector q,"},{"Start":"04:11.440 ","End":"04:13.645","Text":"and this is the curve r of"},{"Start":"04:13.645 ","End":"04:20.475","Text":"t. It could be in 2 dimensions or 3 dimensions or any dimension,"},{"Start":"04:20.475 ","End":"04:24.595","Text":"and it\u0027s given as a function of a parameter t,"},{"Start":"04:24.595 ","End":"04:28.585","Text":"where t goes from a to b,"},{"Start":"04:28.585 ","End":"04:32.440","Text":"and that means that p is just r of a,"},{"Start":"04:32.440 ","End":"04:37.390","Text":"and q is just r of b."},{"Start":"04:37.430 ","End":"04:40.230","Text":"The curve goes in this direction,"},{"Start":"04:40.230 ","End":"04:42.000","Text":"as a direction from p to q,"},{"Start":"04:42.000 ","End":"04:44.230","Text":"from a to b."},{"Start":"04:44.490 ","End":"04:52.545","Text":"It\u0027s also important for the theorem that this curve has to be smooth."},{"Start":"04:52.545 ","End":"04:59.390","Text":"What this is really saying is if we have some gradient,"},{"Start":"04:59.390 ","End":"05:03.300","Text":"this grad f might be some vector field"},{"Start":"05:03.300 ","End":"05:09.290","Text":"f. If we take the integral of a vector field over a smooth curve,"},{"Start":"05:09.290 ","End":"05:12.845","Text":"then all we do is take the value of"},{"Start":"05:12.845 ","End":"05:21.255","Text":"the potential function f at the point q."},{"Start":"05:21.255 ","End":"05:29.240","Text":"In other words, the end point minus the potential function at the start point."},{"Start":"05:29.390 ","End":"05:31.920","Text":"This is another way to say it,"},{"Start":"05:31.920 ","End":"05:39.220","Text":"that the integral of f.d r is little f of end point minus little f of start point."},{"Start":"05:40.010 ","End":"05:42.930","Text":"Let\u0027s give an example."},{"Start":"05:42.930 ","End":"05:46.230","Text":"Let\u0027s take a 3D example."},{"Start":"05:46.230 ","End":"05:51.270","Text":"Start out with the potential function f of x, y,"},{"Start":"05:51.270 ","End":"06:00.020","Text":"and z is equal to sine of pi x plus"},{"Start":"06:00.020 ","End":"06:09.840","Text":"cosine of pi y minus xyz."},{"Start":"06:09.840 ","End":"06:12.655","Text":"Changed my mind, I\u0027m going to switch the sine and the cosine."},{"Start":"06:12.655 ","End":"06:14.250","Text":"Okay, that\u0027s the function,"},{"Start":"06:14.250 ","End":"06:21.555","Text":"and now I need to give you a curve c. I\u0027m going to make c,"},{"Start":"06:21.555 ","End":"06:26.730","Text":"this will sound strange, any curve, well,"},{"Start":"06:26.730 ","End":"06:31.745","Text":"it has to be smooth or we just automatically assume that all curves are smooth,"},{"Start":"06:31.745 ","End":"06:38.735","Text":"from the point, let\u0027s say 1,"},{"Start":"06:38.735 ","End":"06:44.070","Text":"1/2, 2, it\u0027s in 3D, 2."}],"ID":10482},{"Watched":false,"Name":"The Gradient Theorem Part b","Duration":"12m ","ChapterTopicVideoID":10174,"CourseChapterTopicPlaylistID":82680,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10174.jpeg","UploadDate":"2017-08-21T14:46:53.4000000","DurationForVideoObject":"PT12M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.105","Text":"Go But just so you can get the idea,"},{"Start":"00:03.105 ","End":"00:04.935","Text":"I\u0027d like to do it the long way."},{"Start":"00:04.935 ","End":"00:08.435","Text":"Supposing I didn\u0027t have this theorem,"},{"Start":"00:08.435 ","End":"00:10.650","Text":"and suppose they said not any curve,"},{"Start":"00:10.650 ","End":"00:14.500","Text":"but the straight line from here to here."},{"Start":"00:15.050 ","End":"00:18.989","Text":"I\u0027ll do the same exercise using,"},{"Start":"00:18.989 ","End":"00:24.555","Text":"let\u0027s call it method 2, alternative solution."},{"Start":"00:24.555 ","End":"00:29.360","Text":"It\u0027s just really for educational purposes because we do have the theorem."},{"Start":"00:29.360 ","End":"00:31.700","Text":"If I take the straight line,"},{"Start":"00:31.700 ","End":"00:39.755","Text":"I\u0027ll take the line from 1, 1/2,"},{"Start":"00:39.755 ","End":"00:44.895","Text":"2 to 2, 1,"},{"Start":"00:44.895 ","End":"00:48.950","Text":"minus 1, and remember how we do that,"},{"Start":"00:48.950 ","End":"00:54.660","Text":"we just take the equation that r of"},{"Start":"00:54.660 ","End":"01:01.725","Text":"t is 1 minus t times the start point 1,"},{"Start":"01:01.725 ","End":"01:11.475","Text":"1/2, 2 plus t times the endpoint,"},{"Start":"01:11.475 ","End":"01:16.080","Text":"2, 1, minus 1,"},{"Start":"01:16.080 ","End":"01:20.120","Text":"and t goes from 0 to 1."},{"Start":"01:20.120 ","End":"01:23.170","Text":"When t is 0, we get the start point and t is 1,"},{"Start":"01:23.170 ","End":"01:24.955","Text":"we get the endpoint."},{"Start":"01:24.955 ","End":"01:27.670","Text":"In this case, if I write it out,"},{"Start":"01:27.670 ","End":"01:32.640","Text":"I get that r of t equals,"},{"Start":"01:32.640 ","End":"01:34.905","Text":"and if I do it coordinate-wise,"},{"Start":"01:34.905 ","End":"01:40.425","Text":"will get 1 minus t times 1 plus 2t,"},{"Start":"01:40.425 ","End":"01:47.025","Text":"that gives us 1 plus t. Second component,"},{"Start":"01:47.025 ","End":"01:52.020","Text":"1/2 of 1 minus t. It\u0027s 1/2 minus 1/2t"},{"Start":"01:52.020 ","End":"02:00.070","Text":"plus t is 1/2 plus 1/2t."},{"Start":"02:05.000 ","End":"02:11.625","Text":"The last component, twice 1 minus t is 2 minus 2t."},{"Start":"02:11.625 ","End":"02:13.970","Text":"From here I get another minus t,"},{"Start":"02:13.970 ","End":"02:16.890","Text":"so that\u0027s 2 minus 3t."},{"Start":"02:17.420 ","End":"02:21.420","Text":"Again, t goes from 0 to 1."},{"Start":"02:21.420 ","End":"02:23.150","Text":"Now let\u0027s compute the integral."},{"Start":"02:23.150 ","End":"02:28.655","Text":"This integral becomes the integral from 0 to 1."},{"Start":"02:28.655 ","End":"02:34.250","Text":"We don\u0027t have grad f and grad f,"},{"Start":"02:34.250 ","End":"02:37.700","Text":"if you remember, is just the derivative with respect to x,"},{"Start":"02:37.700 ","End":"02:38.810","Text":"derivative, respect to y,"},{"Start":"02:38.810 ","End":"02:40.910","Text":"derivative with respect to z."},{"Start":"02:40.910 ","End":"02:44.210","Text":"First, the derivative with respect to x, let\u0027s see."},{"Start":"02:44.210 ","End":"02:50.705","Text":"From here we get minus Pi sine Pi x."},{"Start":"02:50.705 ","End":"02:53.375","Text":"This is a constant as far as x goes,"},{"Start":"02:53.375 ","End":"02:57.365","Text":"and here we just get minus yz,"},{"Start":"02:57.365 ","End":"02:59.960","Text":"that\u0027s the derivative with respect to x."},{"Start":"02:59.960 ","End":"03:01.760","Text":"Now with respect to y,"},{"Start":"03:01.760 ","End":"03:04.950","Text":"we get Pi cosine Pi y."},{"Start":"03:05.600 ","End":"03:07.910","Text":"Again, y is a constant,"},{"Start":"03:07.910 ","End":"03:10.580","Text":"so we get minus xz."},{"Start":"03:10.580 ","End":"03:12.815","Text":"Finally, with respect to z,"},{"Start":"03:12.815 ","End":"03:17.750","Text":"this is nothing, and this is minus xy."},{"Start":"03:17.750 ","End":"03:21.620","Text":"All this dot with"},{"Start":"03:21.620 ","End":"03:29.390","Text":"dr and dr from r is here it is,"},{"Start":"03:29.390 ","End":"03:36.710","Text":"is just the derivative component-wise with respect to t. We get here 1,"},{"Start":"03:36.710 ","End":"03:45.700","Text":"here, 1/2, and here minus 3 and all this dt."},{"Start":"03:46.280 ","End":"03:50.645","Text":"Continuing, let\u0027s do the dot-product."},{"Start":"03:50.645 ","End":"03:54.515","Text":"We still have the integral from 0 to 1, the dot-product."},{"Start":"03:54.515 ","End":"03:58.755","Text":"We need 1 times the first component here,"},{"Start":"03:58.755 ","End":"04:05.420","Text":"so it\u0027s minus Pi sine Pi x minus yz."},{"Start":"04:05.420 ","End":"04:07.475","Text":"That\u0027s the first component."},{"Start":"04:07.475 ","End":"04:11.814","Text":"The second component is 1/2 times this,"},{"Start":"04:11.814 ","End":"04:16.605","Text":"so we get plus 1/2 Pi."},{"Start":"04:16.605 ","End":"04:25.150","Text":"I\u0027ll just write it as Pi over 2 cosine Pi y minus 1/2 xz."},{"Start":"04:26.870 ","End":"04:29.960","Text":"Then we have the last component,"},{"Start":"04:29.960 ","End":"04:32.900","Text":"which is minus xy times minus 3,"},{"Start":"04:32.900 ","End":"04:36.330","Text":"which is just plus 3 xy,"},{"Start":"04:36.330 ","End":"04:41.690","Text":"all this dt , and we need to substitute now x,"},{"Start":"04:41.690 ","End":"04:43.880","Text":"y, and z from here, that\u0027s the x,"},{"Start":"04:43.880 ","End":"04:47.495","Text":"that\u0027s the y, and that\u0027s the z from the curve."},{"Start":"04:47.495 ","End":"04:52.980","Text":"This gives us the integral from 0"},{"Start":"04:52.980 ","End":"04:59.310","Text":"to 1 of minus Pi sine,"},{"Start":"04:59.310 ","End":"05:02.025","Text":"x is 1 plus t,"},{"Start":"05:02.025 ","End":"05:07.845","Text":"so it\u0027s Pi times 1 plus t is Pi plus Pi"},{"Start":"05:07.845 ","End":"05:15.730","Text":"t minus yz would give us,"},{"Start":"05:16.430 ","End":"05:21.000","Text":"let me compute this at the side, so yz, well,"},{"Start":"05:21.000 ","End":"05:24.405","Text":"y is 1/2 of"},{"Start":"05:24.405 ","End":"05:32.140","Text":"1 plus t and z is 2 minus 3t."},{"Start":"05:32.540 ","End":"05:39.240","Text":"All right, this is t plus 1 and then 3t minus 2."},{"Start":"05:39.240 ","End":"05:42.765","Text":"I\u0027ll put a minus here,"},{"Start":"05:42.765 ","End":"05:45.560","Text":"and then I\u0027ll write a plus here."},{"Start":"05:45.560 ","End":"05:54.290","Text":"What I\u0027m saying is that this is the same as plus of minus yz and this is minus yz,"},{"Start":"05:54.290 ","End":"06:02.025","Text":"so multiplying out, we get 1/2,"},{"Start":"06:02.025 ","End":"06:08.520","Text":"3t squared plus 3t minus 2t is plus t minus 2."},{"Start":"06:08.520 ","End":"06:18.555","Text":"Write that here, 1/2 of 3 t squared plus t minus 2."},{"Start":"06:18.555 ","End":"06:20.910","Text":"Okay? Now the next bit,"},{"Start":"06:20.910 ","End":"06:26.820","Text":"Pi over 2 cosine Pi y. Pi y,"},{"Start":"06:26.820 ","End":"06:35.530","Text":"since y is this is Pi over 2 times,"},{"Start":"06:35.900 ","End":"06:38.240","Text":"well, multiplies by Pi,"},{"Start":"06:38.240 ","End":"06:41.220","Text":"we get Pi over 2 plus Pi over 2t."},{"Start":"06:44.050 ","End":"06:50.760","Text":"That\u0027s the cosine of Pi y minus 1/2 xz."},{"Start":"06:50.760 ","End":"06:56.765","Text":"I\u0027ll do plus 1/2 and I\u0027ll figure out what is minus xz at the side."},{"Start":"06:56.765 ","End":"06:59.060","Text":"Minus xz is minus."},{"Start":"06:59.060 ","End":"07:05.030","Text":"Now x is t plus 1 and z is 2 minus 3t,"},{"Start":"07:05.030 ","End":"07:12.050","Text":"so I,ll throw out the minus and then write it backwards as 3t minus 2."},{"Start":"07:12.050 ","End":"07:20.120","Text":"This is equal to 3t squared plus t minus 2."},{"Start":"07:20.120 ","End":"07:23.050","Text":"It\u0027s actually the same as what we had before."},{"Start":"07:23.050 ","End":"07:25.110","Text":"Yeah, just like here,"},{"Start":"07:25.110 ","End":"07:29.970","Text":"it\u0027s 3t squared plus t minus 2."},{"Start":"07:29.970 ","End":"07:32.685","Text":"That was the minus 1/2 xz,"},{"Start":"07:32.685 ","End":"07:36.640","Text":"the last one to go with 3 xy,"},{"Start":"07:40.520 ","End":"07:42.810","Text":"I\u0027ll do that also at the side,"},{"Start":"07:42.810 ","End":"07:44.820","Text":"3 xy is 3,"},{"Start":"07:44.820 ","End":"07:49.840","Text":"and then x is t plus 1,"},{"Start":"07:50.120 ","End":"07:54.150","Text":"and y is t plus 1 but over 2."},{"Start":"07:54.150 ","End":"07:58.335","Text":"I can write the over 2 here and then squared,"},{"Start":"07:58.335 ","End":"08:01.950","Text":"so plus 3 over 2,"},{"Start":"08:01.950 ","End":"08:04.680","Text":"t plus 1 squared,"},{"Start":"08:04.680 ","End":"08:09.400","Text":"just copied from here, and dt."},{"Start":"08:12.470 ","End":"08:18.650","Text":"Continuing, we can actually do the integral now."},{"Start":"08:18.650 ","End":"08:23.785","Text":"Now, the integral of minus sine is cosine."},{"Start":"08:23.785 ","End":"08:33.830","Text":"We start out with cosine of Pi plus Pi t. I would have written the pie here,"},{"Start":"08:33.830 ","End":"08:35.975","Text":"but the inner derivative is Pi,"},{"Start":"08:35.975 ","End":"08:38.060","Text":"and you can see that if you differentiate this,"},{"Start":"08:38.060 ","End":"08:39.950","Text":"you get from the cosine minus sine,"},{"Start":"08:39.950 ","End":"08:41.855","Text":"you get pi from here."},{"Start":"08:41.855 ","End":"08:44.910","Text":"This is just as is."},{"Start":"08:46.130 ","End":"08:51.150","Text":"Yeah, and I underline these 2 because they\u0027re actually the same."},{"Start":"08:51.150 ","End":"08:55.435","Text":"I can actually combine those 2."},{"Start":"08:55.435 ","End":"08:58.010","Text":"Instead of this plus 1/2,"},{"Start":"08:58.010 ","End":"09:00.515","Text":"I\u0027ll just get rid of this term altogether,"},{"Start":"09:00.515 ","End":"09:03.380","Text":"and instead of this 1/2,"},{"Start":"09:03.380 ","End":"09:06.515","Text":"I\u0027ll just make it 1 just as is."},{"Start":"09:06.515 ","End":"09:17.900","Text":"We get the integral of 3t squared is t cubed then plus 1/2 t squared then minus 2t."},{"Start":"09:17.900 ","End":"09:21.515","Text":"Going to use the same trick again as I did here."},{"Start":"09:21.515 ","End":"09:28.080","Text":"The integral of cosine is sine of the same thing."},{"Start":"09:28.300 ","End":"09:33.000","Text":"Normally, I would divide by the inner derivative,"},{"Start":"09:33.000 ","End":"09:34.330","Text":"divide by Pi over 2,"},{"Start":"09:34.330 ","End":"09:36.745","Text":"but it\u0027ll just get swallowed up with this Pi over 2."},{"Start":"09:36.745 ","End":"09:38.300","Text":"Again, if you differentiate this,"},{"Start":"09:38.300 ","End":"09:39.815","Text":"you\u0027ll see that you get that."},{"Start":"09:39.815 ","End":"09:44.465","Text":"Finally, the integral of this, because it\u0027s cubed."},{"Start":"09:44.465 ","End":"09:48.830","Text":"What I\u0027ll do is I\u0027ll take the t plus 1 cubed."},{"Start":"09:48.830 ","End":"09:51.200","Text":"Now I have to divide by 3,"},{"Start":"09:51.200 ","End":"09:54.380","Text":"and that will make this 3 over 2 just 1/2."},{"Start":"09:54.380 ","End":"09:55.730","Text":"The anti-derivative is 1,"},{"Start":"09:55.730 ","End":"09:58.090","Text":"so we don\u0027t need any further adjustment."},{"Start":"09:58.090 ","End":"10:00.540","Text":"Now I take this,"},{"Start":"10:00.540 ","End":"10:06.260","Text":"and I have to evaluate it between 0 and 1."},{"Start":"10:06.260 ","End":"10:08.375","Text":"What we get is,"},{"Start":"10:08.375 ","End":"10:10.760","Text":"let\u0027s plug in 1 first."},{"Start":"10:10.760 ","End":"10:12.530","Text":"This is cosine of 2 Pi,"},{"Start":"10:12.530 ","End":"10:16.615","Text":"which is cosine of 0, which is 1."},{"Start":"10:16.615 ","End":"10:20.550","Text":"I group these together."},{"Start":"10:20.550 ","End":"10:25.575","Text":"We get when t is 1,"},{"Start":"10:25.575 ","End":"10:32.835","Text":"1 plus 1/2 minus 2 is minus 1/2."},{"Start":"10:32.835 ","End":"10:36.490","Text":"When t is 1, that\u0027s sine of Pi,"},{"Start":"10:36.490 ","End":"10:39.770","Text":"which is 0, and t is 1,"},{"Start":"10:39.770 ","End":"10:43.685","Text":"2 cubed over 2 is 4."},{"Start":"10:43.685 ","End":"10:46.660","Text":"This is the t equals 1 bit."},{"Start":"10:46.660 ","End":"10:52.910","Text":"Then I need to subtract t equals 0 and t is 0."},{"Start":"10:52.910 ","End":"10:57.095","Text":"Cosine of pi is minus 1."},{"Start":"10:57.095 ","End":"11:07.425","Text":"That is 0, sine of Pi over 2 is 1,"},{"Start":"11:07.425 ","End":"11:15.700","Text":"and 0 plus 1 is 1 cubed is 1 over 2 is just 1/2."},{"Start":"11:15.700 ","End":"11:19.980","Text":"Let\u0027s see. 1 plus 4 is 5 minus 1/2."},{"Start":"11:19.980 ","End":"11:22.545","Text":"This is 4 1/2."},{"Start":"11:22.545 ","End":"11:28.020","Text":"This is minus 1 plus 1 is 0 plus 1/2 is 1/2,"},{"Start":"11:28.020 ","End":"11:35.700","Text":"and that means that the answer is 4 1/2 minus 1/2,"},{"Start":"11:35.700 ","End":"11:41.370","Text":"4, and 4 is what we got before."},{"Start":"11:41.370 ","End":"11:44.685","Text":"Also, let\u0027s just go check that."},{"Start":"11:44.685 ","End":"11:48.195","Text":"We got 4 also above,"},{"Start":"11:48.195 ","End":"11:52.320","Text":"and so that\u0027s what we expected."},{"Start":"11:52.320 ","End":"11:56.880","Text":"Here\u0027s an example to show that it really comes out."},{"Start":"11:57.370 ","End":"12:01.080","Text":"It\u0027s time for a break."}],"ID":10483},{"Watched":false,"Name":"The Gradient Theorem Part c","Duration":"3m 38s","ChapterTopicVideoID":10175,"CourseChapterTopicPlaylistID":82680,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10175.jpeg","UploadDate":"2020-02-26T06:20:21.4700000","DurationForVideoObject":"PT3M38S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"Here we are back from the example and"},{"Start":"00:02.820 ","End":"00:07.980","Text":"I just want to say some concluding words for this clip."},{"Start":"00:07.980 ","End":"00:12.705","Text":"What this actually mean or rather implies,"},{"Start":"00:12.705 ","End":"00:15.915","Text":"is that if we had 2 curves,"},{"Start":"00:15.915 ","End":"00:20.490","Text":"both going from the same point to the same point,"},{"Start":"00:20.490 ","End":"00:26.060","Text":"suppose this was curve C_1 and this was curve C_2,"},{"Start":"00:26.060 ","End":"00:34.185","Text":"then the line integral doesn\u0027t depend on the actual curve,"},{"Start":"00:34.185 ","End":"00:36.225","Text":"only at the start and end points."},{"Start":"00:36.225 ","End":"00:41.325","Text":"So we can actually write this as a formula or theorem,"},{"Start":"00:41.325 ","End":"00:47.490","Text":"that if we let F be the grad of f,"},{"Start":"00:47.490 ","End":"00:56.075","Text":"then this integral over path C_1 equals the integral over path C2,"},{"Start":"00:56.075 ","End":"01:06.030","Text":"and it actually works whenever F is conservative vector field."},{"Start":"01:06.530 ","End":"01:08.780","Text":"Well, it\u0027s the same thing."},{"Start":"01:08.780 ","End":"01:12.680","Text":"If it\u0027s equal to the grad of some scalar function f,"},{"Start":"01:12.680 ","End":"01:16.700","Text":"this property is called path independence."},{"Start":"01:16.700 ","End":"01:26.640","Text":"We say that the integral of F dr is path independent,"},{"Start":"01:26.640 ","End":"01:28.270","Text":"doesn\u0027t depend on C,"},{"Start":"01:28.270 ","End":"01:30.290","Text":"just on the end points."},{"Start":"01:30.290 ","End":"01:34.399","Text":"There is another concept I wanted to introduce,"},{"Start":"01:34.399 ","End":"01:36.770","Text":"the concept of a closed path."},{"Start":"01:36.770 ","End":"01:47.170","Text":"Suppose I took a point and then I had a path that went and came back to itself."},{"Start":"01:47.840 ","End":"01:54.395","Text":"If the start and end point are the same then this is called a closed path."},{"Start":"01:54.395 ","End":"01:57.710","Text":"Turns out there\u0027s a very important theorem that the"},{"Start":"01:57.710 ","End":"02:02.135","Text":"integral over a closed path C, and it has to be smoother,"},{"Start":"02:02.135 ","End":"02:04.625","Text":"of the same thing,"},{"Start":"02:04.625 ","End":"02:13.005","Text":"F.dr is always equal to 0."},{"Start":"02:13.005 ","End":"02:15.620","Text":"This is easily explained."},{"Start":"02:15.620 ","End":"02:19.220","Text":"In fact, you could look at this picture here, I could break it up,"},{"Start":"02:19.220 ","End":"02:23.000","Text":"I could take another point somewhere here and then"},{"Start":"02:23.000 ","End":"02:27.330","Text":"consider it as up to here and from here."},{"Start":"02:27.330 ","End":"02:32.805","Text":"If I look at this picture, this could be easier."},{"Start":"02:32.805 ","End":"02:38.840","Text":"If I consider the closed path to be from P to Q and back again,"},{"Start":"02:38.840 ","End":"02:41.905","Text":"I go along C_1 and return along C_2,"},{"Start":"02:41.905 ","End":"02:46.205","Text":"when I\u0027m coming back, I\u0027m going along minus C_2."},{"Start":"02:46.205 ","End":"02:48.545","Text":"Here is some curve C_1,"},{"Start":"02:48.545 ","End":"02:50.810","Text":"and if the curve from here to here was C_2,"},{"Start":"02:50.810 ","End":"02:52.805","Text":"this would be minus C_2."},{"Start":"02:52.805 ","End":"02:57.695","Text":"But since the integral of this equals the integral of this,"},{"Start":"02:57.695 ","End":"03:00.305","Text":"when I\u0027m going in a irregular direction."},{"Start":"03:00.305 ","End":"03:01.760","Text":"If I\u0027m going in the opposite direction,"},{"Start":"03:01.760 ","End":"03:03.080","Text":"it\u0027s going to have a minus sign,"},{"Start":"03:03.080 ","End":"03:04.580","Text":"it\u0027s going to cancel each other out."},{"Start":"03:04.580 ","End":"03:06.650","Text":"I don\u0027t want to give a formal proof."},{"Start":"03:06.650 ","End":"03:08.210","Text":"This is a very important theorem."},{"Start":"03:08.210 ","End":"03:14.035","Text":"Let me add if C is closed and smooth path."},{"Start":"03:14.035 ","End":"03:16.625","Text":"If C is a closed curve,"},{"Start":"03:16.625 ","End":"03:19.840","Text":"there\u0027s a special notation that tells us,"},{"Start":"03:19.840 ","End":"03:23.670","Text":"we usually write a little circle here."},{"Start":"03:23.720 ","End":"03:27.650","Text":"Now I\u0027m redundant because when I write a little circle,"},{"Start":"03:27.650 ","End":"03:31.520","Text":"it automatically means that if C is closed,"},{"Start":"03:31.520 ","End":"03:33.410","Text":"so I could strike this out."},{"Start":"03:33.410 ","End":"03:39.420","Text":"That\u0027s really all there is for this clip. Bye for now."}],"ID":10484},{"Watched":false,"Name":"Conservative Fields in 3D - Example","Duration":"15m 58s","ChapterTopicVideoID":27706,"CourseChapterTopicPlaylistID":82680,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/27706.jpeg","UploadDate":"2022-05-03T09:24:35.0070000","DurationForVideoObject":"PT15M58S","Description":null,"VideoComments":[],"Subtitles":[],"ID":29000}],"Thumbnail":null,"ID":82680},{"Name":"Vector Analysis","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Operators on 3D Vector Fields Part a","Duration":"13m 25s","ChapterTopicVideoID":10176,"CourseChapterTopicPlaylistID":82681,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10176.jpeg","UploadDate":"2017-08-21T14:38:02.0000000","DurationForVideoObject":"PT13M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.780","Text":"This clip will be the last section in the chapter on line integrals."},{"Start":"00:06.780 ","End":"00:11.820","Text":"Here I\u0027m going to talk about the concept of operators."},{"Start":"00:11.820 ","End":"00:15.270","Text":"We\u0027re mainly going to apply them to a 3D vector fields,"},{"Start":"00:15.270 ","End":"00:18.375","Text":"although it could be 2D also."},{"Start":"00:18.375 ","End":"00:22.290","Text":"The examples of vector fields are grad,"},{"Start":"00:22.290 ","End":"00:29.160","Text":"which we\u0027ve already discussed in 2D but now we\u0027re going to discuss it in 3D."},{"Start":"00:29.160 ","End":"00:34.800","Text":"Remember that\u0027s the one that\u0027s the upside-down triangle called nabla or Del,"},{"Start":"00:34.800 ","End":"00:37.520","Text":"and then there\u0027s going to be one called curl,"},{"Start":"00:37.520 ","End":"00:41.300","Text":"one called divergence or div for short, and there will be others."},{"Start":"00:41.300 ","End":"00:43.415","Text":"There\u0027ll be a Laplacian operator,"},{"Start":"00:43.415 ","End":"00:46.170","Text":"which will not be an upside-down triangle,"},{"Start":"00:46.170 ","End":"00:48.495","Text":"it will be a right-way-up triangle,"},{"Start":"00:48.495 ","End":"00:52.700","Text":"and at the end we\u0027ll revisit Green\u0027s theorem"},{"Start":"00:52.700 ","End":"01:02.075","Text":"and show the 2 vector forms of Green\u0027s theorem involving curl and divergence."},{"Start":"01:02.075 ","End":"01:04.190","Text":"That\u0027s the general idea."},{"Start":"01:04.190 ","End":"01:08.900","Text":"Let\u0027s just remind you what we have in 2D."},{"Start":"01:08.900 ","End":"01:11.400","Text":"If we had a function,"},{"Start":"01:11.800 ","End":"01:17.240","Text":"a scalar function, f in 2 variables,"},{"Start":"01:17.240 ","End":"01:41.800","Text":"then we defined the grad or gradient of f to be df/dx, df/dy."},{"Start":"01:41.800 ","End":"01:46.170","Text":"I would prefer to use the ij notation."},{"Start":"01:46.170 ","End":"01:59.170","Text":"This will be df/dx of standard basis vector i plus df/dy times j."},{"Start":"01:59.630 ","End":"02:04.545","Text":"Now this operator, this Del,"},{"Start":"02:04.545 ","End":"02:08.330","Text":"it\u0027s not a function and it\u0027s not something times f,"},{"Start":"02:08.330 ","End":"02:11.300","Text":"it\u0027s an operator, something we do to f."},{"Start":"02:11.300 ","End":"02:13.490","Text":"But symbolically,"},{"Start":"02:13.490 ","End":"02:19.590","Text":"we can think of it as being equal to, say just symbolic,"},{"Start":"02:19.590 ","End":"02:21.300","Text":"let\u0027s take the f out,"},{"Start":"02:21.300 ","End":"02:35.430","Text":"and say that it\u0027s d/dx i plus d/dy j,"},{"Start":"02:35.430 ","End":"02:38.480","Text":"not times f, but applied to f."},{"Start":"02:38.480 ","End":"02:40.370","Text":"This is an operator applied to f."},{"Start":"02:40.370 ","End":"02:45.230","Text":"But we treat it as if it\u0027s like a product like d/dx i of f,"},{"Start":"02:45.230 ","End":"02:46.805","Text":"so we put the f in here."},{"Start":"02:46.805 ","End":"02:49.480","Text":"But this thing on its own,"},{"Start":"02:49.480 ","End":"02:55.700","Text":"this part here is like the equivalent of the Del."},{"Start":"02:55.700 ","End":"02:58.780","Text":"Now this was in 2D,"},{"Start":"02:58.780 ","End":"03:03.985","Text":"and now I just want to extend it to 3D."},{"Start":"03:03.985 ","End":"03:07.995","Text":"If we had a function of 3 variables,"},{"Start":"03:07.995 ","End":"03:12.420","Text":"f of x, y, z in 3D,"},{"Start":"03:12.420 ","End":"03:22.505","Text":"then we would say that grad of f is equal to,"},{"Start":"03:22.505 ","End":"03:25.700","Text":"and I\u0027ll skip the middle step,"},{"Start":"03:25.700 ","End":"03:28.535","Text":"I\u0027ll just say it\u0027s this operator,"},{"Start":"03:28.535 ","End":"03:42.360","Text":"d/dx i plus d/dy j plus d/dz k applied to the function f,"},{"Start":"03:42.360 ","End":"03:47.040","Text":"and it gives us a vector field from a scale."}],"ID":10478},{"Watched":false,"Name":"Operators on 3D Vector Fields Part b","Duration":"7m 30s","ChapterTopicVideoID":10177,"CourseChapterTopicPlaylistID":82681,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10177.jpeg","UploadDate":"2020-02-26T06:20:54.9400000","DurationForVideoObject":"PT7M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:11.609","Text":"I want to show you the alternative notation or definition for curl and here it is."},{"Start":"00:11.609 ","End":"00:14.700","Text":"The curl of f is"},{"Start":"00:14.700 ","End":"00:22.665","Text":"this operator cross product with f and for those of you who have studied determinants,"},{"Start":"00:22.665 ","End":"00:26.040","Text":"the cross product can be defined this way."},{"Start":"00:26.040 ","End":"00:27.870","Text":"If you haven\u0027t studied determinants,"},{"Start":"00:27.870 ","End":"00:29.445","Text":"just ignore the last bit,"},{"Start":"00:29.445 ","End":"00:35.850","Text":"but this part still holds and let\u0027s just scroll back up a second so we can"},{"Start":"00:35.850 ","End":"00:45.160","Text":"see the definition of this operator, grad or dell."},{"Start":"00:45.160 ","End":"00:50.930","Text":"If you take the definition of the cross-product and you take"},{"Start":"00:50.930 ","End":"00:59.810","Text":"this operator and cross it with this vector field,"},{"Start":"00:59.810 ","End":"01:03.170","Text":"this cross with this according to the definition,"},{"Start":"01:03.170 ","End":"01:08.105","Text":"one of the definitions of cross-product component-wise, you\u0027ll get this."},{"Start":"01:08.105 ","End":"01:13.340","Text":"But it\u0027s neater if you know determinants because then you just have here i,"},{"Start":"01:13.340 ","End":"01:17.740","Text":"j, k partial derivative with respect to x, y, z,"},{"Start":"01:17.740 ","End":"01:19.850","Text":"and here the three functions P, Q,"},{"Start":"01:19.850 ","End":"01:25.350","Text":"and R. But if you don\u0027t know determinants,"},{"Start":"01:25.350 ","End":"01:30.215","Text":"it won\u0027t really hurt and note that now that we have this notation,"},{"Start":"01:30.215 ","End":"01:33.505","Text":"if I look at this formula,"},{"Start":"01:33.505 ","End":"01:42.110","Text":"it make sense in a different way because this now reads curl is"},{"Start":"01:42.110 ","End":"01:49.375","Text":"del cross and here we have"},{"Start":"01:49.375 ","End":"01:55.215","Text":"this del grad of f and"},{"Start":"01:55.215 ","End":"02:00.905","Text":"symbolically it makes sense because if you look at this bit,"},{"Start":"02:00.905 ","End":"02:08.105","Text":"we know that any vector cross product with itself is 0."},{"Start":"02:08.105 ","End":"02:11.985","Text":"This would give us 0 symbolically."},{"Start":"02:11.985 ","End":"02:18.650","Text":"This is another way of remembering that curl of grad of f is 0."},{"Start":"02:18.650 ","End":"02:23.200","Text":"It\u0027s because del cross del is symbolically 0."},{"Start":"02:23.200 ","End":"02:25.740","Text":"I need some room for the example,"},{"Start":"02:25.740 ","End":"02:33.090","Text":"so let me delete some stuff I don\u0027t need and see what else don\u0027t I need?"},{"Start":"02:33.090 ","End":"02:38.100","Text":"Don\u0027t need this, I don\u0027t need the 2D."},{"Start":"02:38.100 ","End":"02:43.920","Text":"I now move this stuff up a bit and as an example,"},{"Start":"02:43.920 ","End":"02:48.750","Text":"let\u0027s take the vector field f of x,"},{"Start":"02:48.750 ","End":"02:59.265","Text":"y, z to equal x squared y times i plus"},{"Start":"02:59.265 ","End":"03:05.810","Text":"xyz times j minus"},{"Start":"03:05.810 ","End":"03:13.070","Text":"x squared y squared times k and the question is,"},{"Start":"03:13.070 ","End":"03:18.065","Text":"is this vector field conservative?"},{"Start":"03:18.065 ","End":"03:27.045","Text":"Because this function is defined on all of space for any XYZ that is,"},{"Start":"03:27.045 ","End":"03:33.290","Text":"then we can say that we just have to check the curl and if it\u0027s 0,"},{"Start":"03:33.290 ","End":"03:36.950","Text":"it\u0027s conservative and otherwise not."},{"Start":"03:36.950 ","End":"03:38.855","Text":"I\u0027ll try with the determinant."},{"Start":"03:38.855 ","End":"03:40.670","Text":"But if you haven\u0027t learned determinants,"},{"Start":"03:40.670 ","End":"03:43.370","Text":"you can always fall back on this."},{"Start":"03:43.370 ","End":"03:51.370","Text":"Curl f and I\u0027ll write it with the new notation, curl of f,"},{"Start":"03:51.370 ","End":"03:54.330","Text":"I could have also written it here"},{"Start":"03:54.330 ","End":"04:01.835","Text":"curl f instead of writing the word curl whenever you like,"},{"Start":"04:01.835 ","End":"04:04.115","Text":"is going to equal,"},{"Start":"04:04.115 ","End":"04:06.740","Text":"using the determinant notation,"},{"Start":"04:06.740 ","End":"04:14.220","Text":"I\u0027ve got i, j and k vectors."},{"Start":"04:14.220 ","End":"04:18.665","Text":"In the next row I have partial derivative operators,"},{"Start":"04:18.665 ","End":"04:23.525","Text":"d by dx, d by dy,"},{"Start":"04:23.525 ","End":"04:28.070","Text":"d by dz and in the third line,"},{"Start":"04:28.070 ","End":"04:30.470","Text":"I have the functions p,"},{"Start":"04:30.470 ","End":"04:33.745","Text":"q and r. I mean this is my P, this is my Q,"},{"Start":"04:33.745 ","End":"04:38.450","Text":"this is the R. Here I have x squared"},{"Start":"04:38.450 ","End":"04:46.320","Text":"y. I have to leave enough space like it\u0027s small x,"},{"Start":"04:46.320 ","End":"04:52.050","Text":"y, z, and here x squared,"},{"Start":"04:52.050 ","End":"05:00.080","Text":"y squared and many ways to compute determinants also,"},{"Start":"05:00.080 ","End":"05:02.885","Text":"I\u0027ll use the method of the cofactors,"},{"Start":"05:02.885 ","End":"05:04.955","Text":"which is to say,"},{"Start":"05:04.955 ","End":"05:15.075","Text":"I\u0027ll choose the eye and multiply by its cofactor which is the 2-by-2 determinant of this."},{"Start":"05:15.075 ","End":"05:20.960","Text":"For i, I get d by dy of x squared y squared,"},{"Start":"05:20.960 ","End":"05:29.900","Text":"which is partial derivatives so it\u0027s 2x squared y."},{"Start":"05:29.900 ","End":"05:35.644","Text":"Then less this diagonal,"},{"Start":"05:35.644 ","End":"05:41.390","Text":"which is the derivative with respect to z of this thing which is x,"},{"Start":"05:41.390 ","End":"05:47.980","Text":"y and all this goes with i and then I\u0027ll do the j."},{"Start":"05:47.980 ","End":"05:53.990","Text":"The j goes with a minus there alternate plus, minus,"},{"Start":"05:53.990 ","End":"06:01.395","Text":"plus and then I look at the 2-by-2 determinant this, this, this, this."},{"Start":"06:01.395 ","End":"06:06.350","Text":"I\u0027ve got d by dx of x squared y squared which is"},{"Start":"06:06.350 ","End":"06:12.905","Text":"2xy squared minus the other product."},{"Start":"06:12.905 ","End":"06:15.050","Text":"Really a product operator,"},{"Start":"06:15.050 ","End":"06:20.495","Text":"this applied to this that would be 0 because there is no z"},{"Start":"06:20.495 ","End":"06:30.115","Text":"here so minus 0 and all this is j and for the k,"},{"Start":"06:30.115 ","End":"06:32.810","Text":"you get this, this minus this,"},{"Start":"06:32.810 ","End":"06:36.560","Text":"this so derivative with respect to x of x,"},{"Start":"06:36.560 ","End":"06:41.160","Text":"y, z is yz,"},{"Start":"06:43.520 ","End":"06:51.770","Text":"and minus derivative with respect to y of this is just"},{"Start":"06:51.770 ","End":"06:59.270","Text":"x squared and this is k. Now,"},{"Start":"06:59.270 ","End":"07:06.180","Text":"this is definitely not equal to 0,"},{"Start":"07:06.180 ","End":"07:08.090","Text":"when I say not equal to 0,"},{"Start":"07:08.090 ","End":"07:11.750","Text":"I mean it could be 0 for some values of x and y."},{"Start":"07:11.750 ","End":"07:15.380","Text":"In fact, if x, y, and z are all 0, this is going to be 0."},{"Start":"07:15.380 ","End":"07:18.200","Text":"I mean it\u0027s not the 0 function,"},{"Start":"07:18.200 ","End":"07:21.815","Text":"it\u0027s not 0 for all x, y, and z."},{"Start":"07:21.815 ","End":"07:24.260","Text":"Didn\u0027t everything cancel out."},{"Start":"07:24.260 ","End":"07:31.050","Text":"The answer is no, not conservative."}],"ID":10479},{"Watched":false,"Name":"Operators on 3D Vector Fields Part c","Duration":"12m 14s","ChapterTopicVideoID":10178,"CourseChapterTopicPlaylistID":82681,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10178.jpeg","UploadDate":"2020-02-26T06:21:13.7300000","DurationForVideoObject":"PT12M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.785","Text":"Continuing after curl, we\u0027re going to learn about divergence or div."},{"Start":"00:04.785 ","End":"00:07.260","Text":"Let me just clear the board a bit."},{"Start":"00:07.260 ","End":"00:10.875","Text":"Just put this out of the way here."},{"Start":"00:10.875 ","End":"00:14.385","Text":"You might remember that besides the cross product,"},{"Start":"00:14.385 ","End":"00:17.970","Text":"there\u0027s also a dot product which we learned even before the cross-product."},{"Start":"00:17.970 ","End":"00:22.185","Text":"I\u0027m going to give an analogous definition for divergence."},{"Start":"00:22.185 ","End":"00:26.970","Text":"I\u0027ll write it in 2 forms and here it is."},{"Start":"00:26.970 ","End":"00:30.600","Text":"The divergence, is this operator,"},{"Start":"00:30.600 ","End":"00:36.255","Text":"the Del operator, which is this part here."},{"Start":"00:36.255 ","End":"00:40.365","Text":"Dot product with a vector field F,"},{"Start":"00:40.365 ","End":"00:45.634","Text":"it\u0027s going to give us a scalar because the dot product gives a scalar."},{"Start":"00:45.634 ","End":"00:53.530","Text":"Let me write it out in full just like we did with curl and here it is."},{"Start":"00:53.530 ","End":"00:59.375","Text":"This is what we get when we take this Del operator."},{"Start":"00:59.375 ","End":"01:01.990","Text":"Remember this bit here,"},{"Start":"01:01.990 ","End":"01:07.950","Text":"up to here, this part is the Del operator."},{"Start":"01:07.950 ","End":"01:12.605","Text":"If we dot product it with F,"},{"Start":"01:12.605 ","End":"01:16.835","Text":"which is this, we just multiply component-wise and add."},{"Start":"01:16.835 ","End":"01:21.675","Text":"In other words, this d by dxp,"},{"Start":"01:21.675 ","End":"01:26.880","Text":"which is this d by dyq, d by dzr."},{"Start":"01:26.880 ","End":"01:29.370","Text":"It\u0027s not exactly multiplication,"},{"Start":"01:29.370 ","End":"01:32.990","Text":"it\u0027s applying the operator d over dz to r,"},{"Start":"01:32.990 ","End":"01:34.730","Text":"for example, to get this."},{"Start":"01:34.730 ","End":"01:41.765","Text":"For consistency, let\u0027s also write this part is P with respect to x,"},{"Start":"01:41.765 ","End":"01:44.014","Text":"Q with respect to y,"},{"Start":"01:44.014 ","End":"01:49.585","Text":"Z with respect to r. Just so it\u0027s more consistent with what I did with curl."},{"Start":"01:49.585 ","End":"01:52.990","Text":"Let\u0027s give an example straight away."},{"Start":"01:53.660 ","End":"02:04.245","Text":"Let me take the vector function F as x squared y"},{"Start":"02:04.245 ","End":"02:09.420","Text":"times i plus xyz times"},{"Start":"02:09.420 ","End":"02:15.555","Text":"j minus x squared y squared k,"},{"Start":"02:15.555 ","End":"02:26.485","Text":"this are my pq and r. The div of F will just be,"},{"Start":"02:26.485 ","End":"02:30.650","Text":"you just take the derivative of the first,"},{"Start":"02:30.650 ","End":"02:32.180","Text":"this is p, q and r,"},{"Start":"02:32.180 ","End":"02:35.420","Text":"derivative of P with respect to x."},{"Start":"02:35.420 ","End":"02:40.760","Text":"This would be 2xy and"},{"Start":"02:40.760 ","End":"02:47.960","Text":"then the derivative of the second with respect to y, that\u0027s just xz."},{"Start":"02:47.960 ","End":"02:53.820","Text":"Then the derivative of the third with respect to z,"},{"Start":"02:53.820 ","End":"02:57.610","Text":"but there is no z here, so it\u0027s 0."},{"Start":"02:58.580 ","End":"03:05.070","Text":"This is basically the answer because I can forget about the 0."},{"Start":"03:05.080 ","End":"03:12.365","Text":"Now I\u0027d like to introduce another formula or theorem that relates the div and curl."},{"Start":"03:12.365 ","End":"03:22.430","Text":"It says that the div of curl of a vector field is equal to 0."},{"Start":"03:22.430 ","End":"03:24.065","Text":"This is the scalar 0."},{"Start":"03:24.065 ","End":"03:27.050","Text":"Remember the curl is a vector,"},{"Start":"03:27.050 ","End":"03:30.190","Text":"but the div gives us a scalar."},{"Start":"03:30.190 ","End":"03:34.660","Text":"Just like here, there\u0027s no ijk in it."},{"Start":"03:35.090 ","End":"03:39.140","Text":"I\u0027ll give an example of it and then I\u0027ll tell you why this"},{"Start":"03:39.140 ","End":"03:43.620","Text":"makes sense in terms of this del operator."},{"Start":"03:43.760 ","End":"03:52.625","Text":"For the example, I\u0027ll take the vector field yz squared"},{"Start":"03:52.625 ","End":"04:00.890","Text":"in the i direction plus xy in the j direction,"},{"Start":"04:00.890 ","End":"04:07.235","Text":"and yz in the k direction."},{"Start":"04:07.235 ","End":"04:09.320","Text":"Let\u0027s see. Let\u0027s first of all,"},{"Start":"04:09.320 ","End":"04:16.610","Text":"compute the curl of F. The curl of F is using"},{"Start":"04:16.610 ","End":"04:25.680","Text":"this formula is equal to."},{"Start":"04:25.680 ","End":"04:28.910","Text":"We already did 1 example with the determinant method."},{"Start":"04:28.910 ","End":"04:32.900","Text":"Let me do it with the more boring definition."},{"Start":"04:32.900 ","End":"04:35.720","Text":"This is well P,"},{"Start":"04:35.720 ","End":"04:38.285","Text":"Q and R, just to help us remind us."},{"Start":"04:38.285 ","End":"04:42.890","Text":"Using this formula, the derivative of r with respect to"},{"Start":"04:42.890 ","End":"04:49.270","Text":"y is z minus derivative of Q with respect to z."},{"Start":"04:49.270 ","End":"04:51.520","Text":"Well, that\u0027s 0 because there is no z here."},{"Start":"04:51.520 ","End":"04:55.150","Text":"That\u0027s all we have as far as i."},{"Start":"04:55.150 ","End":"04:57.640","Text":"Now, the j component."},{"Start":"04:57.640 ","End":"05:07.690","Text":"The j component P with respect to z is 2yz and r with respect to x is nothing."},{"Start":"05:07.690 ","End":"05:15.770","Text":"It\u0027s just 2yz in the j direction and finally the r,"},{"Start":"05:15.770 ","End":"05:22.855","Text":"it\u0027s Q with respect to x minus P with respect to y. Q with respect to x is y,"},{"Start":"05:22.855 ","End":"05:31.080","Text":"P with respect to y Is z squared."},{"Start":"05:31.080 ","End":"05:38.595","Text":"The subtraction gives us y minus z squared"},{"Start":"05:38.595 ","End":"05:47.345","Text":"k. That\u0027s curl of F and now we need to take the div of that."},{"Start":"05:47.345 ","End":"05:55.205","Text":"Div, which is Del dot"},{"Start":"05:55.205 ","End":"06:01.770","Text":"of the above cross F of the above,"},{"Start":"06:02.380 ","End":"06:05.660","Text":"is just, I take this with"},{"Start":"06:05.660 ","End":"06:14.300","Text":"respect to x is 0."},{"Start":"06:14.300 ","End":"06:17.760","Text":"This with respect to y,"},{"Start":"06:17.770 ","End":"06:23.810","Text":"that is just 2z."},{"Start":"06:23.810 ","End":"06:25.880","Text":"The reason I put little bar in the z,"},{"Start":"06:25.880 ","End":"06:30.500","Text":"people think it\u0027s unusual is to distinguish it from 2, so to 2z."},{"Start":"06:30.500 ","End":"06:40.070","Text":"Then this with respect to z is just minus 2z and if we compute this,"},{"Start":"06:40.070 ","End":"06:43.500","Text":"well, it\u0027s just 0 as promised."},{"Start":"06:43.670 ","End":"06:48.695","Text":"Now I just want to show you why this equation makes sense."},{"Start":"06:48.695 ","End":"06:53.105","Text":"If we write the div and the curl in terms of the Del or nabla operator,"},{"Start":"06:53.105 ","End":"06:57.995","Text":"this is what it looks like here we have Del cross and then a Del dot."},{"Start":"06:57.995 ","End":"07:01.790","Text":"This reminds me of 1 of the formulas we learned"},{"Start":"07:01.790 ","End":"07:07.835","Text":"and this is it when we learned dot and cross products that there was a relationship."},{"Start":"07:07.835 ","End":"07:16.515","Text":"Basically this which is I\u0027m just copying from here is Del dot,"},{"Start":"07:16.515 ","End":"07:22.310","Text":"Del cross F and even though this is symbolic,"},{"Start":"07:22.310 ","End":"07:25.655","Text":"we\u0027ll just take it from here and see what this would say."},{"Start":"07:25.655 ","End":"07:29.030","Text":"This would say that this is Del cross,"},{"Start":"07:29.030 ","End":"07:36.045","Text":"Del dot F. F"},{"Start":"07:36.045 ","End":"07:40.950","Text":"is a vector and Del cross Del is 0."},{"Start":"07:40.950 ","End":"07:46.580","Text":"We get a 0 dot with something, it\u0027s just 0."},{"Start":"07:49.500 ","End":"07:54.185","Text":"That\u0027s proof. But anyway,"},{"Start":"07:54.185 ","End":"07:57.225","Text":"div of curl is 0."},{"Start":"07:57.225 ","End":"08:06.365","Text":"Okay, we\u0027ve covered curl and div and before that grad."},{"Start":"08:06.365 ","End":"08:11.620","Text":"I\u0027m now going to introduce another operator, the Laplace operator."},{"Start":"08:11.620 ","End":"08:19.400","Text":"Let me just clear the board and we\u0027ll introduce the 4th after grad,"},{"Start":"08:19.400 ","End":"08:23.015","Text":"curl and div and I\u0027ll just write it here."},{"Start":"08:23.015 ","End":"08:30.470","Text":"We have the Laplace operator named after the French mathematician Laplace."},{"Start":"08:30.470 ","End":"08:34.040","Text":"There\u0027s different ways of writing the Laplace operator."},{"Start":"08:34.040 ","End":"08:36.785","Text":"In any event it applies to a scalar function,"},{"Start":"08:36.785 ","End":"08:38.210","Text":"let\u0027s say of x, y,"},{"Start":"08:38.210 ","End":"08:40.339","Text":"and z are though it could be in any dimension."},{"Start":"08:40.339 ","End":"08:43.300","Text":"Sometimes it\u0027s written like a triangle,"},{"Start":"08:43.300 ","End":"08:46.490","Text":"sometimes it\u0027s written as this,"},{"Start":"08:46.490 ","End":"08:48.575","Text":"and this is the notation I like."},{"Start":"08:48.575 ","End":"08:50.570","Text":"Besides the Laplace operator,"},{"Start":"08:50.570 ","End":"08:56.510","Text":"it\u0027s also sometimes called the Del squared operator for"},{"Start":"08:56.510 ","End":"09:05.760","Text":"obvious reasons and it\u0027s actually defined as Del dot Del F. In other words,"},{"Start":"09:05.760 ","End":"09:08.510","Text":"this is grad F when we take the div of that."},{"Start":"09:08.510 ","End":"09:12.980","Text":"I could actually say that this is the div of"},{"Start":"09:12.980 ","End":"09:21.010","Text":"the grad of f. Although you don\u0027t usually write grad in letters,"},{"Start":"09:21.010 ","End":"09:29.150","Text":"I\u0027ll just write it as div of grad f. We can actually compute this in a more closed form,"},{"Start":"09:29.150 ","End":"09:33.295","Text":"something more along the lines of partial derivatives."},{"Start":"09:33.295 ","End":"09:36.885","Text":"This is equal to div of,"},{"Start":"09:36.885 ","End":"09:42.430","Text":"now this means derivative of f with respect to xi,"},{"Start":"09:42.430 ","End":"09:46.170","Text":"plus derivative of f with respect to yj,"},{"Start":"09:46.170 ","End":"09:49.390","Text":"plus derivative of f with respect to zk."},{"Start":"09:50.410 ","End":"09:55.970","Text":"Now the div means take the derivative of the first bit, like this is P,"},{"Start":"09:55.970 ","End":"09:58.400","Text":"this is Q, and this is R,"},{"Start":"09:58.400 ","End":"10:01.265","Text":"we take a derivative of P with respect to x."},{"Start":"10:01.265 ","End":"10:04.075","Text":"We just get fxx,"},{"Start":"10:04.075 ","End":"10:09.465","Text":"second derivative and then here we get fyy,"},{"Start":"10:09.465 ","End":"10:11.985","Text":"from here we get fzz."},{"Start":"10:11.985 ","End":"10:14.130","Text":"This is a scalar, no i,"},{"Start":"10:14.130 ","End":"10:15.585","Text":"j\u0027s and k\u0027s here."},{"Start":"10:15.585 ","End":"10:21.750","Text":"It\u0027s just the sum of the non-mixed second derivatives."},{"Start":"10:22.460 ","End":"10:30.170","Text":"Example, suppose I take the scalar function f of 3 variables,"},{"Start":"10:30.170 ","End":"10:37.025","Text":"x, y, z to be x squared plus y squared plus z squared."},{"Start":"10:37.025 ","End":"10:43.205","Text":"The Laplacian Del squared of f is equal to"},{"Start":"10:43.205 ","End":"10:51.005","Text":"the derivative of this with respect to x is just 2x and the second derivative is 2."},{"Start":"10:51.005 ","End":"10:53.659","Text":"Now, with respect to y,"},{"Start":"10:53.659 ","End":"10:56.645","Text":"we\u0027ll get the first derivative is 2y,"},{"Start":"10:56.645 ","End":"11:01.670","Text":"because this and this is constants and then it\u0027s 2 and then clearly plus 2 again,"},{"Start":"11:01.670 ","End":"11:05.475","Text":"it\u0027s 6 constant, function 6."},{"Start":"11:05.475 ","End":"11:10.455","Text":"Another example, suppose I take f of x, y,"},{"Start":"11:10.455 ","End":"11:17.375","Text":"z equal xy squared z cubed."},{"Start":"11:17.375 ","End":"11:25.730","Text":"Then the Laplacian Del squared of f is second derivative with respect to x."},{"Start":"11:25.730 ","End":"11:28.490","Text":"The first derivative is just y squared and z cubed."},{"Start":"11:28.490 ","End":"11:30.530","Text":"Second derivative is 0,"},{"Start":"11:30.530 ","End":"11:37.475","Text":"0 plus second derivative with respect to y."},{"Start":"11:37.475 ","End":"11:40.760","Text":"Well basically from here I get 2y and then 2,"},{"Start":"11:40.760 ","End":"11:43.950","Text":"so it\u0027s just 2xz cubed."},{"Start":"11:44.260 ","End":"11:49.430","Text":"Then with respect to z, this is basically the constant from here."},{"Start":"11:49.430 ","End":"11:54.020","Text":"First derivative is 3z squared and then 6z,"},{"Start":"11:54.020 ","End":"11:57.545","Text":"I get xy squared 6z,"},{"Start":"11:57.545 ","End":"12:02.455","Text":"or 6xy squared z."},{"Start":"12:02.455 ","End":"12:05.870","Text":"Very straightforward, you don\u0027t even have to do intermediate steps,"},{"Start":"12:05.870 ","End":"12:11.340","Text":"just second derivative with respect to each of the 3 variables and add them all."},{"Start":"12:12.040 ","End":"12:15.510","Text":"It\u0027s time for a break."}],"ID":10480},{"Watched":false,"Name":"Operators on 3D Vector Fields Part d","Duration":"8m 40s","ChapterTopicVideoID":10179,"CourseChapterTopicPlaylistID":82681,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10179.jpeg","UploadDate":"2020-02-26T06:20:42.5370000","DurationForVideoObject":"PT8M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.890","Text":"That\u0027s all 4 operators grad,"},{"Start":"00:04.890 ","End":"00:10.185","Text":"curl, div and del squared or the laplace operator."},{"Start":"00:10.185 ","End":"00:12.030","Text":"As I promised in the beginning,"},{"Start":"00:12.030 ","End":"00:15.990","Text":"we\u0027re going to return to Green\u0027s theorem and we\u0027re going to give 2 vector forms,"},{"Start":"00:15.990 ","End":"00:19.830","Text":"1 using curl and 1 using divergence."},{"Start":"00:19.830 ","End":"00:23.415","Text":"Let me just write a little title here."},{"Start":"00:23.415 ","End":"00:26.175","Text":"Vector forms of Green\u0027s theorem,"},{"Start":"00:26.175 ","End":"00:27.630","Text":"going to be 1 and 2."},{"Start":"00:27.630 ","End":"00:32.335","Text":"The first 1 is going to use curl and here it is."},{"Start":"00:32.335 ","End":"00:34.280","Text":"This is what it looks like."},{"Start":"00:34.280 ","End":"00:35.330","Text":"I\u0027ll explain in a moment."},{"Start":"00:35.330 ","End":"00:41.670","Text":"Let me just first show you the second 1 using div and this is it."},{"Start":"00:41.670 ","End":"00:44.625","Text":"I\u0027m not going to do worked examples."},{"Start":"00:44.625 ","End":"00:48.530","Text":"I would just like you to have these formulas for future reference."},{"Start":"00:48.530 ","End":"00:50.045","Text":"I will explain though,"},{"Start":"00:50.045 ","End":"00:53.825","Text":"why they are forms of Green\u0027s theorem."},{"Start":"00:53.825 ","End":"00:56.120","Text":"I don\u0027t want to scroll down,"},{"Start":"00:56.120 ","End":"00:58.070","Text":"I need to keep these definitions,"},{"Start":"00:58.070 ","End":"01:03.425","Text":"let me just erase this stuff and I\u0027ll move these up a bit."},{"Start":"01:03.425 ","End":"01:05.690","Text":"I\u0027ll explain this 1."},{"Start":"01:05.690 ","End":"01:09.950","Text":"What this means is that,"},{"Start":"01:09.950 ","End":"01:13.310","Text":"well, this applies to a 2-dimensional function f,"},{"Start":"01:13.310 ","End":"01:15.890","Text":"but we did this trick before of taking"},{"Start":"01:15.890 ","End":"01:21.455","Text":"a 2-dimensional vector field and making it a 3-dimensional vector field."},{"Start":"01:21.455 ","End":"01:29.465","Text":"Really what I\u0027m assuming is that f, well, let\u0027s just say this."},{"Start":"01:29.465 ","End":"01:31.430","Text":"Instead of looking at it as f of x,"},{"Start":"01:31.430 ","End":"01:33.380","Text":"y is p of x,"},{"Start":"01:33.380 ","End":"01:39.080","Text":"y i plus Q of x,"},{"Start":"01:39.080 ","End":"01:43.850","Text":"y j. I\u0027m going to use the same letter again,"},{"Start":"01:43.850 ","End":"01:47.390","Text":"even though it\u0027s not really mathematically proper, but there\u0027s no confusion."},{"Start":"01:47.390 ","End":"01:52.205","Text":"I\u0027m going to look at f as a function of 3 variables; x, y, and z,"},{"Start":"01:52.205 ","End":"01:57.885","Text":"that just ignores the variable z and doesn\u0027t have a k component."},{"Start":"01:57.885 ","End":"02:02.395","Text":"I can look at it as this,"},{"Start":"02:02.395 ","End":"02:09.175","Text":"which I just copy pasted and 0 in the k direction."},{"Start":"02:09.175 ","End":"02:13.960","Text":"I\u0027m using the same letter f because there won\u0027t be any confusion."},{"Start":"02:14.120 ","End":"02:22.830","Text":"The left-hand side is the line integral in 2D of the 2-dimensional f,"},{"Start":"02:22.830 ","End":"02:27.319","Text":"and the right-hand side,"},{"Start":"02:27.319 ","End":"02:31.295","Text":"it\u0027s a double integral of a scalar with respect to area."},{"Start":"02:31.295 ","End":"02:35.930","Text":"Now all I have to do is show you that this bit here is what we"},{"Start":"02:35.930 ","End":"02:40.970","Text":"classically had as Green\u0027s theorem. Now let\u0027s see."},{"Start":"02:40.970 ","End":"02:44.000","Text":"The curl of f,"},{"Start":"02:44.000 ","End":"02:47.100","Text":"I actually don\u0027t need it all."},{"Start":"02:47.100 ","End":"02:50.100","Text":"I don\u0027t actually care what the i component is."},{"Start":"02:50.100 ","End":"02:52.070","Text":"I\u0027ll just put a question mark here."},{"Start":"02:52.070 ","End":"02:59.000","Text":"It\u0027s something i plus something I don\u0027t care j,"},{"Start":"02:59.000 ","End":"03:02.345","Text":"but I do care about the last component."},{"Start":"03:02.345 ","End":"03:06.440","Text":"If I look here, it\u0027s dQ by dx."},{"Start":"03:06.440 ","End":"03:07.745","Text":"Let\u0027s write it in this form."},{"Start":"03:07.745 ","End":"03:11.655","Text":"dQ by dx minus"},{"Start":"03:11.655 ","End":"03:18.110","Text":"dp by dy k. Now,"},{"Start":"03:18.110 ","End":"03:21.965","Text":"why don\u0027t I care about this question mark and this question mark,"},{"Start":"03:21.965 ","End":"03:24.290","Text":"because I\u0027m going to do the dot product with"},{"Start":"03:24.290 ","End":"03:31.195","Text":"k. I just show you in general and perhaps it\u0027s easier with angular bracket."},{"Start":"03:31.195 ","End":"03:36.395","Text":"Suppose I have in general a, b, c,"},{"Start":"03:36.395 ","End":"03:39.530","Text":"and then I dot product it with k. Well, in this form,"},{"Start":"03:39.530 ","End":"03:45.700","Text":"k is 0i, 0j plus 1k."},{"Start":"03:45.700 ","End":"03:47.970","Text":"If I do the dot product, it\u0027s this with this,"},{"Start":"03:47.970 ","End":"03:50.595","Text":"this with this, this with this, and we add them,"},{"Start":"03:50.595 ","End":"03:55.110","Text":"it\u0027s just equal to c. I don\u0027t care about a and b,"},{"Start":"03:55.110 ","End":"03:56.805","Text":"that\u0027s the question mark."},{"Start":"03:56.805 ","End":"04:00.580","Text":"All I get is the c,"},{"Start":"04:00.590 ","End":"04:09.225","Text":"this curl f dot with k is just this thing here,"},{"Start":"04:09.225 ","End":"04:17.955","Text":"is dq by dx minus dp by dy."},{"Start":"04:17.955 ","End":"04:23.020","Text":"Now, if you remember Green\u0027s theorem,"},{"Start":"04:23.020 ","End":"04:24.145","Text":"the way we presented it,"},{"Start":"04:24.145 ","End":"04:26.520","Text":"I\u0027ll show you again, and here it is."},{"Start":"04:26.520 ","End":"04:27.990","Text":"I just squeezed it in here."},{"Start":"04:27.990 ","End":"04:29.470","Text":"This is the Green\u0027s theorem,"},{"Start":"04:29.470 ","End":"04:31.970","Text":"the way I presented it earlier."},{"Start":"04:33.950 ","End":"04:39.790","Text":"I mean the Pdx plus Qdy is exactly F.dr."},{"Start":"04:40.490 ","End":"04:43.390","Text":"Well, I\u0027ll explain that in any event,"},{"Start":"04:43.390 ","End":"04:52.560","Text":"because F.dr, F is just P,"},{"Start":"04:53.210 ","End":"04:57.850","Text":"Q, R of a components,"},{"Start":"04:57.850 ","End":"05:06.560","Text":"and dr is just dx,"},{"Start":"05:07.160 ","End":"05:15.270","Text":"the i component plus dy in the j direction and dz."},{"Start":"05:15.270 ","End":"05:18.920","Text":"I should have written in an angular brackets,"},{"Start":"05:18.920 ","End":"05:20.380","Text":"and perhaps still will."},{"Start":"05:20.380 ","End":"05:22.820","Text":"Let me change this to angular."},{"Start":"05:23.660 ","End":"05:29.010","Text":"Here it is P,Q,R but R is 0."},{"Start":"05:29.010 ","End":"05:33.555","Text":"Remember, we took R as being 0 here."},{"Start":"05:33.555 ","End":"05:36.110","Text":"If we just expand this,"},{"Start":"05:36.110 ","End":"05:40.725","Text":"we get exactly Pdx plus Qdy,"},{"Start":"05:40.725 ","End":"05:43.050","Text":"which is what\u0027s written here."},{"Start":"05:43.050 ","End":"05:45.635","Text":"That\u0027s the left-hand side is that,"},{"Start":"05:45.635 ","End":"05:48.710","Text":"and the right-hand side is this,"},{"Start":"05:48.710 ","End":"05:50.750","Text":"so everything works out."},{"Start":"05:50.750 ","End":"05:56.505","Text":"This is 1 of the 2 vector forms of Green\u0027s theorem."},{"Start":"05:56.505 ","End":"05:59.130","Text":"Let me go over to the other 1 now."},{"Start":"05:59.130 ","End":"06:00.360","Text":"That\u0027s the 1 with the curl,"},{"Start":"06:00.360 ","End":"06:03.040","Text":"this is the 1 with div."},{"Start":"06:03.230 ","End":"06:08.540","Text":"This 1is a little more difficult to demonstrate and I won\u0027t."},{"Start":"06:08.540 ","End":"06:11.015","Text":"I\u0027ll just illustrate the concepts,"},{"Start":"06:11.015 ","End":"06:15.060","Text":"especially what this normal vector is."},{"Start":"06:15.490 ","End":"06:25.120","Text":"Well, if we take a parameterized form of the curve C in 2D where r of t is,"},{"Start":"06:25.120 ","End":"06:26.875","Text":"let me use the angular brackets,"},{"Start":"06:26.875 ","End":"06:29.050","Text":"is x of t,"},{"Start":"06:29.050 ","End":"06:35.095","Text":"y of t then the normal vector n"},{"Start":"06:35.095 ","End":"06:42.250","Text":"is a vector which is perpendicular to the direction of the curve,"},{"Start":"06:42.250 ","End":"06:45.460","Text":"to the tangent and I think we\u0027ve seen this before."},{"Start":"06:45.460 ","End":"06:52.420","Text":"It\u0027s given by y prime of"},{"Start":"06:52.420 ","End":"07:01.550","Text":"t over the magnitude of r prime of t,"},{"Start":"07:03.200 ","End":"07:09.420","Text":"minus x prime of t over same thing,"},{"Start":"07:09.420 ","End":"07:13.030","Text":"r prime of t magnitudes."},{"Start":"07:13.190 ","End":"07:19.340","Text":"Roughly, you could see that this dot with this is 0 because it\u0027s the same denominator."},{"Start":"07:19.340 ","End":"07:23.405","Text":"We get xy prime minus yx prime, it\u0027s perpendicular."},{"Start":"07:23.405 ","End":"07:28.160","Text":"The reason this is a unit vector is if I take its magnitude,"},{"Start":"07:28.160 ","End":"07:32.240","Text":"then this thing squared plus this thing squared is"},{"Start":"07:32.240 ","End":"07:37.770","Text":"exactly the magnitude of r prime squared."},{"Start":"07:37.790 ","End":"07:40.050","Text":"That\u0027s what the magnitude is."},{"Start":"07:40.050 ","End":"07:43.535","Text":"It\u0027s the square root of the sum of the squares of the 2 components."},{"Start":"07:43.535 ","End":"07:45.230","Text":"I\u0027m not going to get too much into it."},{"Start":"07:45.230 ","End":"07:47.300","Text":"I will give you a picture here."},{"Start":"07:47.300 ","End":"07:55.905","Text":"Here\u0027s the picture and basically this is the curve C,"},{"Start":"07:55.905 ","End":"07:59.510","Text":"that\u0027s parametrized and it\u0027s going in this direction."},{"Start":"07:59.510 ","End":"08:06.615","Text":"At any given point, n is the normal to the tangent or to the curve."},{"Start":"08:06.615 ","End":"08:12.510","Text":"What we\u0027re saying is that the vector field dot with the normal, that will be a scalar."},{"Start":"08:12.510 ","End":"08:15.005","Text":"If I take the integral of that along the curve,"},{"Start":"08:15.005 ","End":"08:18.379","Text":"I\u0027ll get the integral over D,"},{"Start":"08:18.379 ","End":"08:21.110","Text":"which is the inside of the curve,"},{"Start":"08:21.110 ","End":"08:25.475","Text":"of the divergence of the vector field F,"},{"Start":"08:25.475 ","End":"08:29.380","Text":"and I just leave it at that."},{"Start":"08:29.380 ","End":"08:32.850","Text":"It\u0027s just something for reference for the future."},{"Start":"08:32.920 ","End":"08:37.040","Text":"Actually, we\u0027re done with this section."},{"Start":"08:37.040 ","End":"08:41.010","Text":"Actually, we\u0027re done with line integrals."}],"ID":10481}],"Thumbnail":null,"ID":82681},{"Name":"Exercises","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1 Part a","Duration":"4m 17s","ChapterTopicVideoID":8709,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8709.jpeg","UploadDate":"2017-02-13T05:29:00.2070000","DurationForVideoObject":"PT4M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.455","Text":"In this exercise, we have to compute this type 1 line integral along the curve C,"},{"Start":"00:07.455 ","End":"00:10.650","Text":"and C is given in parametric form."},{"Start":"00:10.650 ","End":"00:16.275","Text":"There are several formulas of variations of the formula we can use."},{"Start":"00:16.275 ","End":"00:25.630","Text":"The simplest is that ds is just the square root of dx squared plus dy squared."},{"Start":"00:25.630 ","End":"00:28.640","Text":"But that\u0027s a bit awkward to work with,"},{"Start":"00:28.640 ","End":"00:35.420","Text":"and so usually, we have dx over dt squared,"},{"Start":"00:35.420 ","End":"00:43.500","Text":"plus dy over dt squared, dt."},{"Start":"00:43.500 ","End":"00:48.400","Text":"It\u0027s like the dt squared square root cancels out with the dt."},{"Start":"00:48.920 ","End":"00:51.630","Text":"This is the Leibniz form."},{"Start":"00:51.630 ","End":"00:54.880","Text":"Those who don\u0027t like the dx, dt notation,"},{"Start":"00:54.880 ","End":"00:59.805","Text":"could always use the formula x-prime of t"},{"Start":"00:59.805 ","End":"01:10.450","Text":"squared plus y-prime of t squared dt."},{"Start":"01:11.300 ","End":"01:13.470","Text":"Not to be too confusing,"},{"Start":"01:13.470 ","End":"01:14.700","Text":"let me stick with 1 of them."},{"Start":"01:14.700 ","End":"01:19.170","Text":"Let me choose the last 1. I\u0027ll move it here."},{"Start":"01:19.170 ","End":"01:21.260","Text":"After we\u0027ve computed ds,"},{"Start":"01:21.260 ","End":"01:25.760","Text":"we also have to remember that this integral becomes an integral with"},{"Start":"01:25.760 ","End":"01:32.700","Text":"respect to t. We take it that t goes from 0 to 2Pi."},{"Start":"01:32.700 ","End":"01:37.190","Text":"We also substitute where we have x and y,"},{"Start":"01:37.190 ","End":"01:39.230","Text":"we don\u0027t have a y, we just have an x."},{"Start":"01:39.230 ","End":"01:43.805","Text":"But in this case, we have 1 minus x squared is cosine"},{"Start":"01:43.805 ","End":"01:49.905","Text":"squared t. Then I have to put ds in."},{"Start":"01:49.905 ","End":"01:52.590","Text":"Let\u0027s compute ds and then get back here."},{"Start":"01:52.590 ","End":"01:54.420","Text":"This has to be continued."},{"Start":"01:54.420 ","End":"01:57.270","Text":"This is the square root."},{"Start":"01:57.270 ","End":"02:07.680","Text":"Now, the derivative of x with respect to t is minus sine t, and that\u0027s squared."},{"Start":"02:07.680 ","End":"02:16.480","Text":"The derivative of y with respect to t is cosine t, also squared, dt."},{"Start":"02:16.480 ","End":"02:21.415","Text":"Now, sine squared plus cosine squared is 1."},{"Start":"02:21.415 ","End":"02:27.270","Text":"This comes out to be just 1dt, which is dt."},{"Start":"02:27.270 ","End":"02:31.720","Text":"Ds comes out to be dt."},{"Start":"02:32.290 ","End":"02:37.520","Text":"Now, we can use some trigonometrical identities to solve this."},{"Start":"02:37.520 ","End":"02:48.090","Text":"First of all, it\u0027s equal to the integral 1 minus cosine squared is sine squared t dt."},{"Start":"02:48.090 ","End":"02:51.770","Text":"Then as another trigonometrical identity,"},{"Start":"02:51.770 ","End":"02:58.800","Text":"that sine squared is 1/2 of 1 minus cosine twice the angle,"},{"Start":"02:58.800 ","End":"03:02.350","Text":"in this case, 2t dt."},{"Start":"03:02.360 ","End":"03:06.310","Text":"Now, we can actually do the integral."},{"Start":"03:06.620 ","End":"03:10.290","Text":"Let\u0027s leave the 1/2 outside."},{"Start":"03:10.290 ","End":"03:16.580","Text":"What we have is the integral of 1 is t. The"},{"Start":"03:16.580 ","End":"03:22.790","Text":"integral of cosine 2t is not quite sine 2t,"},{"Start":"03:22.790 ","End":"03:25.710","Text":"we have to divide by the 2."},{"Start":"03:26.120 ","End":"03:33.480","Text":"All this has to be taken from 0 to 2Pi."},{"Start":"03:33.480 ","End":"03:36.940","Text":"What does this come out to be?"},{"Start":"03:37.580 ","End":"03:40.410","Text":"I\u0027ll leave the 1/2 here."},{"Start":"03:40.410 ","End":"03:43.260","Text":"Now, if I put in 2Pi,"},{"Start":"03:43.260 ","End":"03:51.825","Text":"I get 2Pi minus sine of 4Pi,"},{"Start":"03:51.825 ","End":"03:54.030","Text":"it\u0027s a multiple of 2Pi,"},{"Start":"03:54.030 ","End":"03:56.190","Text":"so it\u0027s like 0."},{"Start":"03:56.190 ","End":"04:00.165","Text":"This would be 0. That\u0027s for the 2Pi,"},{"Start":"04:00.165 ","End":"04:07.910","Text":"and now for 0, I just get 0 minus sine of 0 is 0."},{"Start":"04:07.910 ","End":"04:11.575","Text":"Altogether, 1/2 of 2Pi,"},{"Start":"04:11.575 ","End":"04:15.030","Text":"and the answer is just Pi."},{"Start":"04:15.030 ","End":"04:17.980","Text":"We are done."}],"ID":8808},{"Watched":false,"Name":"Exercise 1 Part b","Duration":"13m 50s","ChapterTopicVideoID":8710,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8710.jpeg","UploadDate":"2017-02-13T05:32:00.6470000","DurationForVideoObject":"PT13M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.490","Text":"In this exercise, we have to compute the following line integral,"},{"Start":"00:05.490 ","End":"00:10.440","Text":"where the curve C is given in parametric form, x,"},{"Start":"00:10.440 ","End":"00:16.540","Text":"y in terms of t and the range where t goes from."},{"Start":"00:16.730 ","End":"00:22.350","Text":"There\u0027s only really one basic formula that we have to use."},{"Start":"00:22.350 ","End":"00:27.090","Text":"That is that ds,"},{"Start":"00:27.090 ","End":"00:29.760","Text":"one of the variations of the formula,"},{"Start":"00:29.760 ","End":"00:37.230","Text":"is the square root of x prime as a function of t squared,"},{"Start":"00:37.230 ","End":"00:44.940","Text":"plus y prime as a function of t squared, dt."},{"Start":"00:44.940 ","End":"00:49.130","Text":"Probably, best to put an extra set of brackets here to avoid confusion."},{"Start":"00:49.130 ","End":"00:52.260","Text":"The whole thing here is squared."},{"Start":"00:53.030 ","End":"00:57.295","Text":"Let\u0027s compute this, and then we\u0027ll get back here."},{"Start":"00:57.295 ","End":"01:06.105","Text":"This is equal to the derivative of x"},{"Start":"01:06.105 ","End":"01:11.310","Text":"is 1 minus cosine t squared."},{"Start":"01:11.310 ","End":"01:19.710","Text":"Then the derivative of y is just sine t. 1 disappears,"},{"Start":"01:19.710 ","End":"01:21.900","Text":"it\u0027s a constant and minus cosine."},{"Start":"01:21.900 ","End":"01:25.515","Text":"That\u0027s sine t squared."},{"Start":"01:25.515 ","End":"01:29.530","Text":"Square root of that, dt."},{"Start":"01:30.230 ","End":"01:33.005","Text":"If we simplify this,"},{"Start":"01:33.005 ","End":"01:36.545","Text":"we get the square root of,"},{"Start":"01:36.545 ","End":"01:39.725","Text":"it\u0027s just simple algebra here."},{"Start":"01:39.725 ","End":"01:46.650","Text":"I have 1 minus 2 cosine t plus cosine squared t. I will just write the side,"},{"Start":"01:46.650 ","End":"01:51.645","Text":"1 minus 2 cosine t plus cosine squared t,"},{"Start":"01:51.645 ","End":"01:56.600","Text":"plus sine squared t. The cosine squared and sine squared is 1,"},{"Start":"01:56.600 ","End":"01:58.730","Text":"together with the 1 gives me 2."},{"Start":"01:58.730 ","End":"02:03.320","Text":"I have twice, 2 minus 2 cosine t,"},{"Start":"02:03.320 ","End":"02:09.110","Text":"which I can write as 2 times 1 minus cosine t. That\u0027s ds."},{"Start":"02:09.110 ","End":"02:10.670","Text":"Now getting back to the integral,"},{"Start":"02:10.670 ","End":"02:12.590","Text":"we\u0027ll replace the curve,"},{"Start":"02:12.590 ","End":"02:19.665","Text":"with the range of t. T goes from 0 to Pi."},{"Start":"02:19.665 ","End":"02:22.425","Text":"Then I replace the x,"},{"Start":"02:22.425 ","End":"02:24.570","Text":"as the x from the curve."},{"Start":"02:24.570 ","End":"02:33.010","Text":"That t minus sine t. Then the ds,"},{"Start":"02:35.600 ","End":"02:38.430","Text":"sorry, I forgot the dt here,"},{"Start":"02:38.430 ","End":"02:42.780","Text":"which is the square root of twice,"},{"Start":"02:42.780 ","End":"02:45.490","Text":"1 minus cosine of t, dt."},{"Start":"02:45.490 ","End":"02:49.445","Text":"But I\u0027m going to use a trigonometrical formula here,"},{"Start":"02:49.445 ","End":"02:57.785","Text":"that 1 minus cosine t is 2 sine squared t over 2."},{"Start":"02:57.785 ","End":"02:59.480","Text":"It\u0027s just a variation of one of"},{"Start":"02:59.480 ","End":"03:02.960","Text":"the standard trigonometric formulas after switching sides a bit."},{"Start":"03:02.960 ","End":"03:06.930","Text":"I have here 2 times 2,"},{"Start":"03:06.930 ","End":"03:13.000","Text":"times sine squared t over 2."},{"Start":"03:13.000 ","End":"03:18.770","Text":"Now, what I have under the square root sign is 4 sine squared t over 2."},{"Start":"03:18.770 ","End":"03:24.620","Text":"The square root would be 2 sine t over 2."},{"Start":"03:24.620 ","End":"03:27.440","Text":"But that would normally be an absolute value,"},{"Start":"03:27.440 ","End":"03:29.825","Text":"and you take the square root of something squared."},{"Start":"03:29.825 ","End":"03:34.355","Text":"But when t goes from 0 to Pi,"},{"Start":"03:34.355 ","End":"03:40.680","Text":"t over 2 goes from 0 to Pi over 2."},{"Start":"03:40.680 ","End":"03:42.510","Text":"The sine is positive,"},{"Start":"03:42.510 ","End":"03:45.380","Text":"because that\u0027s the whole first quadrant,"},{"Start":"03:45.380 ","End":"03:46.910","Text":"or at least non-negative."},{"Start":"03:46.910 ","End":"03:48.995","Text":"I don\u0027t need the absolute value."},{"Start":"03:48.995 ","End":"03:53.630","Text":"What I get is the integral from 0 to Pi."},{"Start":"03:53.630 ","End":"03:55.685","Text":"This whole thing now is just,"},{"Start":"03:55.685 ","End":"03:59.429","Text":"this without the absolute value."},{"Start":"04:00.970 ","End":"04:04.970","Text":"If I multiply out this with this,"},{"Start":"04:04.970 ","End":"04:12.785","Text":"we just get twice t minus sine t,"},{"Start":"04:12.785 ","End":"04:17.330","Text":"sine of t over 2, dt."},{"Start":"04:17.330 ","End":"04:23.270","Text":"I\u0027m going to split it up into 2 integrals from the minus."},{"Start":"04:23.270 ","End":"04:25.160","Text":"We get, on the one hand,"},{"Start":"04:25.160 ","End":"04:32.735","Text":"the integral from 0 to Pi of 2t,"},{"Start":"04:32.735 ","End":"04:37.595","Text":"sine of t over 2, dt."},{"Start":"04:37.595 ","End":"04:48.680","Text":"Then we have minus the integral from 0 to Pi of 2 sine t,"},{"Start":"04:48.680 ","End":"04:53.490","Text":"sine t over 2, dt."},{"Start":"04:53.500 ","End":"04:55.970","Text":"I\u0027ll do each one separately,"},{"Start":"04:55.970 ","End":"04:57.685","Text":"and then we\u0027ll do a subtraction."},{"Start":"04:57.685 ","End":"05:00.485","Text":"I just give them names."},{"Start":"05:00.485 ","End":"05:04.950","Text":"I\u0027ll call this one asterisk,"},{"Start":"05:04.950 ","End":"05:11.235","Text":"and I\u0027ll call the second integral double asterisk."},{"Start":"05:11.235 ","End":"05:16.680","Text":"We just have an integration problem now."},{"Start":"05:17.090 ","End":"05:20.340","Text":"That\u0027s all I really need now."},{"Start":"05:20.340 ","End":"05:24.525","Text":"I think I\u0027ll work on a split page."},{"Start":"05:24.525 ","End":"05:34.085","Text":"Also, here I\u0027ll compute the integral of 2t sine t over 2,"},{"Start":"05:34.085 ","End":"05:42.090","Text":"dt, from 0 to Pi."},{"Start":"05:42.090 ","End":"05:52.140","Text":"Then, afterwards, I\u0027ll do later the 0 to Pi of 2 sine t,"},{"Start":"05:52.140 ","End":"05:55.540","Text":"sine t over 2, dt."},{"Start":"05:55.880 ","End":"06:00.815","Text":"At the end we we\u0027ll subtract this one minus this one."},{"Start":"06:00.815 ","End":"06:05.960","Text":"For this one, we\u0027re going to do integration by parts."},{"Start":"06:05.960 ","End":"06:11.645","Text":"To remind you, the integration by parts says the integral of udv,"},{"Start":"06:11.645 ","End":"06:17.930","Text":"is uv, minus the integral of vdu."},{"Start":"06:17.930 ","End":"06:26.880","Text":"I\u0027m going to let the 2t be u."},{"Start":"06:26.880 ","End":"06:36.615","Text":"This one will be the v. The 2 quantities I\u0027m missing are du and v. Well,"},{"Start":"06:36.615 ","End":"06:39.960","Text":"du is simply 2dt,"},{"Start":"06:39.960 ","End":"06:47.555","Text":"and v is the antiderivative of sine t over 2,"},{"Start":"06:47.555 ","End":"06:50.765","Text":"which if you think about it,"},{"Start":"06:50.765 ","End":"06:55.920","Text":"the antiderivative of sine is minus cosine,"},{"Start":"06:58.660 ","End":"07:03.050","Text":"of t over 2, but we have to divide by the 1/2."},{"Start":"07:03.050 ","End":"07:06.375","Text":"Dividing by a 1/2 is this."},{"Start":"07:06.375 ","End":"07:08.270","Text":"If you\u0027re not sure,"},{"Start":"07:08.270 ","End":"07:10.670","Text":"then just differentiate this,"},{"Start":"07:10.670 ","End":"07:13.670","Text":"and you\u0027ll see that you get that."},{"Start":"07:13.670 ","End":"07:17.955","Text":"Now, we\u0027ve got, from here,"},{"Start":"07:17.955 ","End":"07:22.260","Text":"uv, let\u0027s see, u times v,"},{"Start":"07:22.260 ","End":"07:26.190","Text":"that would be minus because of the minus,"},{"Start":"07:26.190 ","End":"07:34.560","Text":"and then 2 times 2 is 4t cosine t over 2."},{"Start":"07:34.560 ","End":"07:37.410","Text":"But we\u0027re doing a definite integral."},{"Start":"07:37.410 ","End":"07:41.030","Text":"We have to take this between the limits."},{"Start":"07:41.030 ","End":"07:43.415","Text":"Maybe I\u0027ll put brackets here,"},{"Start":"07:43.415 ","End":"07:49.030","Text":"between the limits 0 to Pi."},{"Start":"07:49.030 ","End":"07:52.915","Text":"Then we have a minus vdu."},{"Start":"07:52.915 ","End":"07:57.815","Text":"We have minus the integral from 0 to Pi."},{"Start":"07:57.815 ","End":"08:02.875","Text":"Let\u0027s see this with this."},{"Start":"08:02.875 ","End":"08:05.714","Text":"You know what? The minus,"},{"Start":"08:05.714 ","End":"08:07.845","Text":"I\u0027ll just make this a plus."},{"Start":"08:07.845 ","End":"08:12.390","Text":"Then I have again 2 times 2 is 4,"},{"Start":"08:12.390 ","End":"08:24.090","Text":"and cosine of t over 2, dt."},{"Start":"08:24.090 ","End":"08:27.680","Text":"Here I have a substitution to do."},{"Start":"08:27.680 ","End":"08:30.590","Text":"Now when we put in 0 for t,"},{"Start":"08:30.590 ","End":"08:33.305","Text":"we\u0027re going to get 0 because of this t here."},{"Start":"08:33.305 ","End":"08:36.660","Text":"We just have to substitute the Pi."},{"Start":"08:36.660 ","End":"08:43.145","Text":"A cosine of Pi over 2 is cosine of 90 degrees,"},{"Start":"08:43.145 ","End":"08:45.965","Text":"is 0. That\u0027s also 0."},{"Start":"08:45.965 ","End":"08:48.640","Text":"Everything here is 0."},{"Start":"08:48.640 ","End":"08:51.905","Text":"Then I just have this other integral."},{"Start":"08:51.905 ","End":"08:59.055","Text":"The integral of cosine is almost sine t over 2."},{"Start":"08:59.055 ","End":"09:02.415","Text":"But because of the 1/2 I have to divide by a 1/2,"},{"Start":"09:02.415 ","End":"09:05.040","Text":"which is like multiplying by 2."},{"Start":"09:05.040 ","End":"09:08.385","Text":"It\u0027s 8 sine t over 2."},{"Start":"09:08.385 ","End":"09:12.670","Text":"Again, differentiate this if you\u0027re not sure and you\u0027ll see that you get this."},{"Start":"09:12.670 ","End":"09:18.840","Text":"This I have to take from 0 to Pi."},{"Start":"09:20.440 ","End":"09:22.894","Text":"What we have here,"},{"Start":"09:22.894 ","End":"09:24.740","Text":"forgetting the 8 for the moment,"},{"Start":"09:24.740 ","End":"09:28.449","Text":"is we have sine Pi over 2 minus sine 0."},{"Start":"09:28.449 ","End":"09:30.950","Text":"Sine Pi over 2 is 1."},{"Start":"09:30.950 ","End":"09:32.570","Text":"Sine of 90 degrees,"},{"Start":"09:32.570 ","End":"09:34.565","Text":"so we just get 1."},{"Start":"09:34.565 ","End":"09:42.060","Text":"This whole thing comes out to be just the 8 from here,"},{"Start":"09:42.060 ","End":"09:45.690","Text":"because this is 0, and we said this minus this is 1."},{"Start":"09:45.690 ","End":"09:50.715","Text":"That\u0027s 8. That finishes the asterisk."},{"Start":"09:50.715 ","End":"09:54.400","Text":"Now let\u0027s go to do the double asterisk."},{"Start":"09:54.400 ","End":"09:59.110","Text":"Here, again, I\u0027m going to use a trigonometric formula."},{"Start":"09:59.110 ","End":"10:03.110","Text":"There\u0027s a trigonometric formula for,"},{"Start":"10:03.220 ","End":"10:09.200","Text":"I meant to say sine Alpha, sine Beta."},{"Start":"10:09.200 ","End":"10:13.475","Text":"This is equal to, it\u0027s 1/2 something."},{"Start":"10:13.475 ","End":"10:15.170","Text":"Instead of the 1/2,"},{"Start":"10:15.170 ","End":"10:17.500","Text":"I\u0027ll put the 2 here."},{"Start":"10:17.500 ","End":"10:19.785","Text":"What this is equal to,"},{"Start":"10:19.785 ","End":"10:25.345","Text":"is cosine of Alpha minus Beta."},{"Start":"10:25.345 ","End":"10:32.780","Text":"Take away the cosine of Alpha plus Beta."},{"Start":"10:32.780 ","End":"10:37.755","Text":"Here we\u0027ll use it with Alpha being t,"},{"Start":"10:37.755 ","End":"10:41.295","Text":"and Beta will be t over 2."},{"Start":"10:41.295 ","End":"10:43.980","Text":"What we get here,"},{"Start":"10:43.980 ","End":"10:51.660","Text":"is the integral from 0 to Pi of cosine."},{"Start":"10:51.660 ","End":"10:56.800","Text":"Now Alpha minus beta is just t over 2."},{"Start":"10:58.190 ","End":"11:01.470","Text":"Then we have a minus,"},{"Start":"11:01.470 ","End":"11:04.755","Text":"cosine Alpha plus Beta,"},{"Start":"11:04.755 ","End":"11:07.275","Text":"is t plus t over 2."},{"Start":"11:07.275 ","End":"11:11.500","Text":"It\u0027s 3t over 2."},{"Start":"11:11.810 ","End":"11:15.240","Text":"This dt."},{"Start":"11:15.240 ","End":"11:19.060","Text":"We split it up into 2 separate bits."},{"Start":"11:19.060 ","End":"11:24.270","Text":"Now the integral of cosine generally is sine."},{"Start":"11:26.420 ","End":"11:29.100","Text":"Because it\u0027s not t, it\u0027s t over 2,"},{"Start":"11:29.100 ","End":"11:30.675","Text":"I have to divide by a 1/2,"},{"Start":"11:30.675 ","End":"11:33.300","Text":"which is like multiplying by 2."},{"Start":"11:33.300 ","End":"11:35.285","Text":"Similarly for the other one,"},{"Start":"11:35.285 ","End":"11:36.830","Text":"I would normally say,"},{"Start":"11:36.830 ","End":"11:39.200","Text":"if it was just cosine of t,"},{"Start":"11:39.200 ","End":"11:42.560","Text":"it would be sine of whatever it is."},{"Start":"11:42.560 ","End":"11:46.700","Text":"But because it\u0027s not t it\u0027s, times 3 over 2,"},{"Start":"11:46.700 ","End":"11:48.995","Text":"I have to divide by 3 over 2,"},{"Start":"11:48.995 ","End":"11:52.475","Text":"which is like multiplying by 2/3."},{"Start":"11:52.475 ","End":"11:54.125","Text":"This is what I get,"},{"Start":"11:54.125 ","End":"11:58.575","Text":"and this has to be evaluated from 0 to Pi."},{"Start":"11:58.575 ","End":"12:00.585","Text":"Let\u0027s see what we get."},{"Start":"12:00.585 ","End":"12:03.135","Text":"If we put in Pi,"},{"Start":"12:03.135 ","End":"12:10.454","Text":"we get sine of Pi over 2 is 1, so this is 2."},{"Start":"12:10.454 ","End":"12:20.470","Text":"3Pi over 2 is 270 degrees."},{"Start":"12:21.710 ","End":"12:27.585","Text":"Sine of that is minus 1."},{"Start":"12:27.585 ","End":"12:32.790","Text":"It\u0027s going to be plus 2/3,"},{"Start":"12:32.790 ","End":"12:36.090","Text":"because it\u0027s minus 2/3 times minus 1."},{"Start":"12:36.090 ","End":"12:38.910","Text":"That\u0027s the part for Pi."},{"Start":"12:38.910 ","End":"12:41.930","Text":"Now I have to take away the part to 0."},{"Start":"12:41.930 ","End":"12:45.200","Text":"Well, when t is 0,"},{"Start":"12:45.200 ","End":"12:47.330","Text":"the sine of t is also 0."},{"Start":"12:47.330 ","End":"12:50.340","Text":"It\u0027s just 0 minus 0."},{"Start":"12:50.340 ","End":"12:56.745","Text":"In short, what we get is 2 and 2/3."},{"Start":"12:56.745 ","End":"13:03.350","Text":"That\u0027s the answer for the double asterisk part here."},{"Start":"13:03.350 ","End":"13:06.325","Text":"Now I just have to subtract."},{"Start":"13:06.325 ","End":"13:09.665","Text":"Finally our line integral,"},{"Start":"13:09.665 ","End":"13:12.305","Text":"which is this minus this,"},{"Start":"13:12.305 ","End":"13:17.205","Text":"becomes 8 minus 2 and 2/3,"},{"Start":"13:17.205 ","End":"13:19.995","Text":"which is 5 and a 1/3."},{"Start":"13:19.995 ","End":"13:21.500","Text":"You can leave it like that,"},{"Start":"13:21.500 ","End":"13:23.390","Text":"or if you like improper fractions,"},{"Start":"13:23.390 ","End":"13:26.695","Text":"that\u0027s 16 over 3."},{"Start":"13:26.695 ","End":"13:29.550","Text":"I\u0027ll just highlight it."},{"Start":"13:29.550 ","End":"13:33.030","Text":"We are done. I want to apologize that,"},{"Start":"13:33.030 ","End":"13:35.640","Text":"it was such a nuisance some,"},{"Start":"13:35.640 ","End":"13:37.665","Text":"such a length integral."},{"Start":"13:37.665 ","End":"13:46.714","Text":"We did more integration than we talk about the line integral concept."},{"Start":"13:46.714 ","End":"13:49.950","Text":"Anyway, it\u0027s solved."}],"ID":8809},{"Watched":false,"Name":"Exercise 1 Part c","Duration":"3m 53s","ChapterTopicVideoID":8711,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8711.jpeg","UploadDate":"2017-02-13T05:32:52.9870000","DurationForVideoObject":"PT3M53S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.329","Text":"In this exercise, we have to compute this line integral over the curve,"},{"Start":"00:06.329 ","End":"00:09.670","Text":"but we\u0027re not given the curve in parametric form,"},{"Start":"00:09.670 ","End":"00:11.370","Text":"we\u0027re just given a description,"},{"Start":"00:11.370 ","End":"00:15.570","Text":"and it\u0027s the line segment joining this point to this point."},{"Start":"00:15.570 ","End":"00:21.119","Text":"The first thing I want to do is to describe C in parametric form,"},{"Start":"00:21.119 ","End":"00:25.830","Text":"and there are several formulas that are all similar,"},{"Start":"00:25.830 ","End":"00:27.435","Text":"I\u0027ll use one of the forms."},{"Start":"00:27.435 ","End":"00:31.380","Text":"One of the forms says that the x and the y,"},{"Start":"00:31.380 ","End":"00:34.244","Text":"I use the angular brackets for vectors,"},{"Start":"00:34.244 ","End":"00:40.544","Text":"is equal to 1 minus t times the start point,"},{"Start":"00:40.544 ","End":"00:49.690","Text":"which is 0,0 plus t times the end point which is 1,2."},{"Start":"00:50.750 ","End":"00:55.620","Text":"If I expand this out, I\u0027ll just get, well,"},{"Start":"00:55.620 ","End":"01:03.740","Text":"these are both 0s, so I get just t,"},{"Start":"01:03.740 ","End":"01:08.015","Text":"2t, which if I write it out,"},{"Start":"01:08.015 ","End":"01:18.245","Text":"tells me basically that x is equal to t and y equals 2t,"},{"Start":"01:18.245 ","End":"01:24.350","Text":"oh and I forgot to say this always goes from 0-1, the parameter."},{"Start":"01:24.350 ","End":"01:29.745","Text":"Here also t goes from 0-1."},{"Start":"01:29.745 ","End":"01:32.855","Text":"That\u0027s the parametric form of the curve."},{"Start":"01:32.855 ","End":"01:35.810","Text":"Now what we do is we do the substitution,"},{"Start":"01:35.810 ","End":"01:38.360","Text":"there\u0027s a long formula and a short formula."},{"Start":"01:38.360 ","End":"01:42.860","Text":"I just use the short formula for just the ds part,"},{"Start":"01:42.860 ","End":"01:52.310","Text":"which is the square root of it\u0027s x prime with respect to t squared,"},{"Start":"01:52.310 ","End":"01:56.715","Text":"and y prime squared,"},{"Start":"01:56.715 ","End":"01:59.010","Text":"and all this is dt,"},{"Start":"01:59.010 ","End":"02:00.420","Text":"and as for the rest of it,"},{"Start":"02:00.420 ","End":"02:03.630","Text":"you just set the integral along the curve,"},{"Start":"02:03.630 ","End":"02:07.530","Text":"you put the integral from the limits of t,"},{"Start":"02:07.530 ","End":"02:14.015","Text":"so that would be 0-1, and then you replace x and y according to the formula."},{"Start":"02:14.015 ","End":"02:21.770","Text":"X plus y would be t plus 2t, and then ds."},{"Start":"02:21.770 ","End":"02:24.825","Text":"Well, let\u0027s compute what ds is in our case."},{"Start":"02:24.825 ","End":"02:34.200","Text":"In our case, x prime is 1 and y prime with respect to t is 2,"},{"Start":"02:34.200 ","End":"02:37.589","Text":"so we get 1 squared, plus 2 squared,"},{"Start":"02:37.589 ","End":"02:43.840","Text":"square root, and that comes out to be root 5."},{"Start":"02:44.060 ","End":"02:51.780","Text":"So ds is root 5 dt."},{"Start":"02:51.780 ","End":"02:53.740","Text":"Now I can take constants in front."},{"Start":"02:53.740 ","End":"02:56.485","Text":"First of all, t plus 2t is 3t,"},{"Start":"02:56.485 ","End":"03:01.925","Text":"so the 3 I can put in front and the square root of 5,"},{"Start":"03:01.925 ","End":"03:10.290","Text":"and all we\u0027re left with is the integral of t-dt from 0-1."},{"Start":"03:10.290 ","End":"03:16.555","Text":"Let\u0027s see The integral of t. I\u0027ll just do this bit at the side."},{"Start":"03:16.555 ","End":"03:21.100","Text":"The integral from 0-1 of t-dt is"},{"Start":"03:21.100 ","End":"03:27.300","Text":"1.5t squared, taken from 0-1."},{"Start":"03:27.300 ","End":"03:30.000","Text":"At 0 I get nothing, and I plug in 1,"},{"Start":"03:30.000 ","End":"03:35.250","Text":"I get 1.5, so what I get is 3 root 5,"},{"Start":"03:35.250 ","End":"03:41.830","Text":"and this comes out to be 1.5,"},{"Start":"03:42.050 ","End":"03:44.840","Text":"just rewrite it slightly."},{"Start":"03:44.840 ","End":"03:50.435","Text":"Looks nicer if I write it as 3 over 2 square root of 5,"},{"Start":"03:50.435 ","End":"03:53.580","Text":"and that\u0027s the answer."}],"ID":8810},{"Watched":false,"Name":"Exercise 1 Part d","Duration":"10m 57s","ChapterTopicVideoID":8712,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8712.jpeg","UploadDate":"2017-02-13T05:35:15.0770000","DurationForVideoObject":"PT10M57S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.625","Text":"In this exercise, we\u0027re given a line integral of type 1 to compute."},{"Start":"00:05.625 ","End":"00:11.925","Text":"But the curve C is described as the perimeter of a triangle."},{"Start":"00:11.925 ","End":"00:14.355","Text":"Let\u0027s sketch the triangle."},{"Start":"00:14.355 ","End":"00:16.110","Text":"O is the origin,"},{"Start":"00:16.110 ","End":"00:18.465","Text":"that\u0027s this point O."},{"Start":"00:18.465 ","End":"00:22.140","Text":"A is the point, 0, 1,"},{"Start":"00:22.140 ","End":"00:24.000","Text":"which means that y is 1,"},{"Start":"00:24.000 ","End":"00:26.985","Text":"so that would be somewhere, let\u0027s say here."},{"Start":"00:26.985 ","End":"00:29.985","Text":"B is 1, 0,"},{"Start":"00:29.985 ","End":"00:33.150","Text":"so that would be somewhere here."},{"Start":"00:33.150 ","End":"00:39.450","Text":"Then OAB, usually we take it in order."},{"Start":"00:39.450 ","End":"00:48.885","Text":"From O to A and then from A to B."},{"Start":"00:48.885 ","End":"00:53.310","Text":"Then finally from B back to O."},{"Start":"00:53.310 ","End":"00:57.525","Text":"It\u0027s a closed path."},{"Start":"00:57.525 ","End":"01:00.960","Text":"What we have to do is,"},{"Start":"01:00.960 ","End":"01:03.580","Text":"since this is defined piecewise,"},{"Start":"01:03.580 ","End":"01:08.880","Text":"is just take this and break it up into 3 separate integrals."},{"Start":"01:08.900 ","End":"01:14.215","Text":"This integral of x plus y squared,"},{"Start":"01:14.215 ","End":"01:16.565","Text":"ds over the curve C,"},{"Start":"01:16.565 ","End":"01:18.050","Text":"I\u0027ll just break it up."},{"Start":"01:18.050 ","End":"01:23.720","Text":"I\u0027ll first of all take the integral of the same thing along OA."},{"Start":"01:23.720 ","End":"01:30.215","Text":"Then I\u0027ll take the integral of the whatever it is along AB,"},{"Start":"01:30.215 ","End":"01:37.930","Text":"and then the integral of whatever it is along BO."},{"Start":"01:39.710 ","End":"01:43.860","Text":"I\u0027m going to need to parametrize each of these segments,"},{"Start":"01:43.860 ","End":"01:48.910","Text":"so I\u0027ll just remind you of the formula for a parametrized segment."},{"Start":"01:48.910 ","End":"01:50.530","Text":"It\u0027s actually more than 1."},{"Start":"01:50.530 ","End":"01:54.680","Text":"Previously, I used the formula that x,"},{"Start":"01:54.680 ","End":"02:00.580","Text":"y is equal to 1 minus t times the first point."},{"Start":"02:00.580 ","End":"02:04.864","Text":"Let\u0027s say, the first was x naught, y naught,"},{"Start":"02:04.864 ","End":"02:08.235","Text":"and then t times the second point,"},{"Start":"02:08.235 ","End":"02:12.165","Text":"whatever that is, let\u0027s call it x1, y1."},{"Start":"02:12.165 ","End":"02:16.230","Text":"We\u0027re going to use this 3 times once,"},{"Start":"02:16.340 ","End":"02:18.855","Text":"well, with each of the 3 sides."},{"Start":"02:18.855 ","End":"02:22.070","Text":"There is a variation of this which is equivalent,"},{"Start":"02:22.070 ","End":"02:24.140","Text":"which is sometimes I find more useful,"},{"Start":"02:24.140 ","End":"02:26.570","Text":"is just to say that this is x naught,"},{"Start":"02:26.570 ","End":"02:32.445","Text":"y naught plus t times the difference of the 2,"},{"Start":"02:32.445 ","End":"02:35.265","Text":"x1 minus x naught,"},{"Start":"02:35.265 ","End":"02:37.930","Text":"y1 minus y naught."},{"Start":"02:37.930 ","End":"02:39.440","Text":"Sometimes I use this form,"},{"Start":"02:39.440 ","End":"02:40.530","Text":"sometimes I use this form."},{"Start":"02:40.530 ","End":"02:43.085","Text":"I may even repeat the x, y."},{"Start":"02:43.085 ","End":"02:48.930","Text":"Let\u0027s work with the second form in this exercise."},{"Start":"02:49.310 ","End":"02:53.810","Text":"I can now parametrize each of them."},{"Start":"02:53.940 ","End":"02:57.460","Text":"I forgot to add that in each of these cases,"},{"Start":"02:57.460 ","End":"03:00.310","Text":"t goes from 0 to 1."},{"Start":"03:00.310 ","End":"03:05.280","Text":"Let\u0027s parametrize first of all, OA."},{"Start":"03:05.280 ","End":"03:12.630","Text":"For OA, what I have is that O is this point working with this formula."},{"Start":"03:12.630 ","End":"03:14.070","Text":"You know what, I\u0027ll erase this one,"},{"Start":"03:14.070 ","End":"03:16.275","Text":"so I don\u0027t get confused."},{"Start":"03:16.275 ","End":"03:20.130","Text":"Let\u0027s see. What I get here, is x naught,"},{"Start":"03:20.130 ","End":"03:22.920","Text":"y naught is the O,"},{"Start":"03:22.920 ","End":"03:27.420","Text":"and x1, y1 is A."},{"Start":"03:27.420 ","End":"03:32.280","Text":"Let me just rewrite these coordinates, 0,"},{"Start":"03:32.280 ","End":"03:38.820","Text":"0, 0, 1, and 1, 0."},{"Start":"03:38.820 ","End":"03:47.540","Text":"Here we get that x, y is 0,"},{"Start":"03:47.540 ","End":"03:52.325","Text":"0 plus t times the difference,"},{"Start":"03:52.325 ","End":"03:57.715","Text":"this minus this, which is 0, 1."},{"Start":"03:57.715 ","End":"04:01.680","Text":"That just tells us that,"},{"Start":"04:01.680 ","End":"04:12.165","Text":"I can write it that x equals 0 and y equals t,"},{"Start":"04:12.165 ","End":"04:16.170","Text":"t goes from 0 to 1."},{"Start":"04:16.170 ","End":"04:21.675","Text":"That\u0027s the parametrization for OA."},{"Start":"04:21.675 ","End":"04:26.160","Text":"Next. Next one is AB."},{"Start":"04:26.160 ","End":"04:32.340","Text":"AB, x, y is the first point which is A,"},{"Start":"04:32.340 ","End":"04:40.675","Text":"which is 0,1 plus t times the difference B minus A."},{"Start":"04:40.675 ","End":"04:47.190","Text":"B minus A is 1 minus 0,"},{"Start":"04:47.190 ","End":"04:49.364","Text":"and then 0 minus 1,"},{"Start":"04:49.364 ","End":"04:53.135","Text":"so it\u0027s 1 minus 1,"},{"Start":"04:53.135 ","End":"05:02.965","Text":"which means that x equals 0 plus 1t, x equals t,"},{"Start":"05:02.965 ","End":"05:13.605","Text":"and y equals 1 minus t. I take this minus t times this,"},{"Start":"05:13.605 ","End":"05:18.880","Text":"and again, t between 0 and 1."},{"Start":"05:18.950 ","End":"05:23.460","Text":"Lastly BO. Where x,"},{"Start":"05:23.460 ","End":"05:27.190","Text":"y is going to be the point for B"},{"Start":"05:27.610 ","End":"05:34.850","Text":"plus t times the difference of O minus B,"},{"Start":"05:34.850 ","End":"05:39.060","Text":"which is minus 1, 0."},{"Start":"05:39.060 ","End":"05:43.145","Text":"What we get is that x is equal to"},{"Start":"05:43.145 ","End":"05:51.990","Text":"1 minus t and y equals just 0,"},{"Start":"05:51.990 ","End":"05:57.465","Text":"and still t goes as always from 0 to 1."},{"Start":"05:57.465 ","End":"05:59.500","Text":"Before I write the integrals,"},{"Start":"05:59.500 ","End":"06:03.140","Text":"I want to write another formula for what ds is equal to."},{"Start":"06:03.140 ","End":"06:10.025","Text":"In general, ds is going to be the square root of x"},{"Start":"06:10.025 ","End":"06:18.955","Text":"prime with respect to t plus y prime squared dt."},{"Start":"06:18.955 ","End":"06:23.140","Text":"Let\u0027s do ds for each of these 3 cases."},{"Start":"06:23.140 ","End":"06:26.960","Text":"In this case, I will get that ds is"},{"Start":"06:26.960 ","End":"06:32.285","Text":"the square root of this derivative squared plus this derivative squared,"},{"Start":"06:32.285 ","End":"06:36.845","Text":"0 squared plus 1 squared is 1."},{"Start":"06:36.845 ","End":"06:41.870","Text":"It\u0027s just 1dt. In here,"},{"Start":"06:41.870 ","End":"06:43.670","Text":"if I differentiate this, I get 1,"},{"Start":"06:43.670 ","End":"06:47.360","Text":"I differentiate this, I get minus 1."},{"Start":"06:47.360 ","End":"06:51.185","Text":"The square root of this squared plus this squared will be square root of 2,"},{"Start":"06:51.185 ","End":"06:54.570","Text":"so ds is square root of 2dt."},{"Start":"06:54.620 ","End":"06:58.640","Text":"Here, the derivative of this is minus 1,"},{"Start":"06:58.640 ","End":"06:59.840","Text":"of this is 0."},{"Start":"06:59.840 ","End":"07:02.990","Text":"Taking the square root of the sum of squares, again,"},{"Start":"07:02.990 ","End":"07:09.820","Text":"I get 1, so here also ds is equal to dt."},{"Start":"07:11.570 ","End":"07:16.335","Text":"Now, I\u0027m going to write each of the 3 integrals."},{"Start":"07:16.335 ","End":"07:20.640","Text":"The first integral, the one that belongs to"},{"Start":"07:20.640 ","End":"07:26.180","Text":"OA is going to be the integral where t goes from 0 to."},{"Start":"07:26.180 ","End":"07:29.700","Text":"Well, they\u0027re all going to go from 0 to 1."},{"Start":"07:30.470 ","End":"07:34.744","Text":"Just have to substitute x plus y squared,"},{"Start":"07:34.744 ","End":"07:41.254","Text":"and here x plus y squared is 0 plus t squared."},{"Start":"07:41.254 ","End":"07:43.760","Text":"This is going to be t squared."},{"Start":"07:43.760 ","End":"07:49.770","Text":"Ds is 1dt, so this is just dt."},{"Start":"07:50.210 ","End":"07:52.590","Text":"We\u0027ll see what this equals in a moment."},{"Start":"07:52.590 ","End":"07:55.000","Text":"I\u0027ll work on them in parallel maybe."},{"Start":"07:55.000 ","End":"08:00.125","Text":"The AB integral is going to be the integral from 0 to 1."},{"Start":"08:00.125 ","End":"08:02.060","Text":"In fact, they\u0027re all, as I said,"},{"Start":"08:02.060 ","End":"08:05.640","Text":"going to be the integral from 0 to 1."},{"Start":"08:05.640 ","End":"08:07.910","Text":"Now, x plus y squared,"},{"Start":"08:07.910 ","End":"08:09.710","Text":"in the second case,"},{"Start":"08:09.710 ","End":"08:16.760","Text":"x plus y squared is t plus 1 minus t squared."},{"Start":"08:16.760 ","End":"08:21.030","Text":"This is 1 minus 2t plus t squared,"},{"Start":"08:21.030 ","End":"08:27.410","Text":"so altogether I\u0027ll get 1 minus t plus t squared."},{"Start":"08:27.410 ","End":"08:31.375","Text":"Then I need the ds, which is root 2."},{"Start":"08:31.375 ","End":"08:35.745","Text":"I can put the root 2 in front."},{"Start":"08:35.745 ","End":"08:41.505","Text":"Here I put the root 2 and here I put the dt from here."},{"Start":"08:41.505 ","End":"08:44.940","Text":"Let\u0027s see what the last integral is."},{"Start":"08:44.940 ","End":"08:48.510","Text":"I need x plus y squared,"},{"Start":"08:48.510 ","End":"08:50.730","Text":"x is this, y is 0,"},{"Start":"08:50.730 ","End":"08:59.035","Text":"so it\u0027s just the 1 minus t and ds is dt."},{"Start":"08:59.035 ","End":"09:03.140","Text":"That\u0027s 3 integrals to compute."},{"Start":"09:03.140 ","End":"09:11.670","Text":"Let\u0027s see, the first 1 would be 1/3t cubed from 0 to 1."},{"Start":"09:11.670 ","End":"09:13.710","Text":"Plug in 0 is nothing, plug in 1,"},{"Start":"09:13.710 ","End":"09:20.280","Text":"it\u0027s a 1/3, so this is just 1/3. Let\u0027s see."},{"Start":"09:20.280 ","End":"09:25.095","Text":"This one will be the square root of 2,"},{"Start":"09:25.095 ","End":"09:29.175","Text":"and then t minus 1/2t"},{"Start":"09:29.175 ","End":"09:36.285","Text":"squared plus 1/3t cubed from 0 to 1,"},{"Start":"09:36.285 ","End":"09:40.050","Text":"0 gives 0, so I just have to plug in 1."},{"Start":"09:40.050 ","End":"09:44.055","Text":"1 minus a1/2 is a 1/2,"},{"Start":"09:44.055 ","End":"09:47.025","Text":"plus a 1/3 is 5/6."},{"Start":"09:47.025 ","End":"09:54.840","Text":"I get 5/6 root 2. Let\u0027s see."},{"Start":"09:54.840 ","End":"10:03.455","Text":"This is t minus 1/2t squared from 0 to 1,"},{"Start":"10:03.455 ","End":"10:05.090","Text":"0 doesn\u0027t give anything,"},{"Start":"10:05.090 ","End":"10:06.725","Text":"could just plug in 1,"},{"Start":"10:06.725 ","End":"10:11.840","Text":"it\u0027s 1 minus 1/2."},{"Start":"10:11.840 ","End":"10:15.360","Text":"This comes out to be 1/2."},{"Start":"10:15.360 ","End":"10:20.190","Text":"Finally, I have to add the 3."},{"Start":"10:20.190 ","End":"10:26.400","Text":"The integral along C of whatever it was,"},{"Start":"10:26.400 ","End":"10:28.860","Text":"is just this plus this plus this,"},{"Start":"10:28.860 ","End":"10:36.090","Text":"I get 1/3 plus 5/6 root"},{"Start":"10:36.090 ","End":"10:39.760","Text":"2 plus 1/2,"},{"Start":"10:39.760 ","End":"10:43.695","Text":"and could simplify it a bit,"},{"Start":"10:43.695 ","End":"10:46.155","Text":"1/2 plus a 1/3 is 5/6."},{"Start":"10:46.155 ","End":"10:54.720","Text":"I can take the 5/6 then outside the brackets and write it as 1 plus root 2."},{"Start":"10:54.720 ","End":"10:58.870","Text":"I\u0027ll leave the answer in this form. We are done."}],"ID":8811},{"Watched":false,"Name":"Exercise 2 Part a","Duration":"3m 18s","ChapterTopicVideoID":8713,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8713.jpeg","UploadDate":"2017-02-13T05:35:59.4330000","DurationForVideoObject":"PT3M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.055","Text":"In this exercise, we have a type 1 line integral."},{"Start":"00:05.055 ","End":"00:07.560","Text":"This time it\u0027s in 3-dimensions."},{"Start":"00:07.560 ","End":"00:11.010","Text":"Previously, we had some in 2-dimensions."},{"Start":"00:11.010 ","End":"00:16.830","Text":"The main difference in the formula is that in 2-dimensions,"},{"Start":"00:16.830 ","End":"00:21.870","Text":"one of the formulas we had for ds was the square root of x"},{"Start":"00:21.870 ","End":"00:29.970","Text":"prime with respect to t squared plus y prime with respect to t squared."},{"Start":"00:29.970 ","End":"00:35.190","Text":"Then we had dt. The difference is in 3D that we just extended a"},{"Start":"00:35.190 ","End":"00:41.465","Text":"bit and add also a z prime squared before we put the dt."},{"Start":"00:41.465 ","End":"00:43.940","Text":"In 2-dimensions is just these 2,"},{"Start":"00:43.940 ","End":"00:45.515","Text":"in 3-dimensions is these 3,"},{"Start":"00:45.515 ","End":"00:48.845","Text":"other than that, it\u0027s the same idea."},{"Start":"00:48.845 ","End":"00:52.370","Text":"The integral along the curve,"},{"Start":"00:52.370 ","End":"00:55.235","Text":"we replace by the integral of the parameter,"},{"Start":"00:55.235 ","End":"00:57.485","Text":"which in this case is t,"},{"Start":"00:57.485 ","End":"01:01.630","Text":"and it goes from 0 to Pi."},{"Start":"01:01.630 ","End":"01:08.260","Text":"Then we substitute each of the xyz according to the parametric equation."},{"Start":"01:08.260 ","End":"01:14.485","Text":"We have x squared is cosine squared t,"},{"Start":"01:14.485 ","End":"01:19.195","Text":"y squared is sine squared t,"},{"Start":"01:19.195 ","End":"01:23.960","Text":"and z squared is just t squared."},{"Start":"01:23.960 ","End":"01:26.790","Text":"Then I need the ds."},{"Start":"01:26.790 ","End":"01:29.559","Text":"Let\u0027s just compute that over here."},{"Start":"01:29.559 ","End":"01:35.550","Text":"This is going to equal x prime"},{"Start":"01:35.550 ","End":"01:42.975","Text":"is minus sine t squared,"},{"Start":"01:42.975 ","End":"01:48.010","Text":"y prime is cosine t,"},{"Start":"01:48.380 ","End":"01:51.135","Text":"and I want that squared,"},{"Start":"01:51.135 ","End":"01:54.750","Text":"and z prime is 1,"},{"Start":"01:54.750 ","End":"01:56.670","Text":"so I need 1 squared,"},{"Start":"01:56.670 ","End":"01:58.905","Text":"and I need the square root of that,"},{"Start":"01:58.905 ","End":"02:01.150","Text":"and at the end, the dt."},{"Start":"02:01.150 ","End":"02:06.095","Text":"Now, sine squared plus cosine squared is 1,"},{"Start":"02:06.095 ","End":"02:07.970","Text":"1 plus 1 is 2."},{"Start":"02:07.970 ","End":"02:11.300","Text":"So we have the square root of 2dt,"},{"Start":"02:11.300 ","End":"02:14.290","Text":"which I put here."},{"Start":"02:14.290 ","End":"02:17.790","Text":"I can put the dt here,"},{"Start":"02:17.790 ","End":"02:26.330","Text":"and the square root of 2 I\u0027d rather put in front right here instead of here."},{"Start":"02:26.330 ","End":"02:31.025","Text":"Once again, we see cosine squared plus sine squared,"},{"Start":"02:31.025 ","End":"02:34.125","Text":"and this is equal to 1."},{"Start":"02:34.125 ","End":"02:38.780","Text":"When I do the integral I have the square root of 2,"},{"Start":"02:38.780 ","End":"02:48.755","Text":"the integral of 1 is just t. The integral of t squared is 1/3 t cubed."},{"Start":"02:48.755 ","End":"02:54.360","Text":"All this needs to go from 0 to Pi."},{"Start":"02:55.100 ","End":"02:58.380","Text":"When t is 0, we get nothing."},{"Start":"02:58.380 ","End":"03:00.930","Text":"We just have to plug in Pi."},{"Start":"03:00.930 ","End":"03:11.755","Text":"We get the square root of 2 of Pi plus 1/3 Pi cubed."},{"Start":"03:11.755 ","End":"03:14.390","Text":"There\u0027s not really much to simplify."},{"Start":"03:14.390 ","End":"03:18.390","Text":"I\u0027ll just leave the answer like this and we\u0027re done."}],"ID":8812},{"Watched":false,"Name":"Exercise 2 Part b","Duration":"6m 22s","ChapterTopicVideoID":8714,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8714.jpeg","UploadDate":"2017-02-13T05:37:16.9570000","DurationForVideoObject":"PT6M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.285","Text":"Here we have a type 1 lightning to grow."},{"Start":"00:03.285 ","End":"00:06.420","Text":"This time it\u0027s in 3D because we have an x,"},{"Start":"00:06.420 ","End":"00:08.235","Text":"a y, and a z."},{"Start":"00:08.235 ","End":"00:10.725","Text":"It\u0027s given in parametric form."},{"Start":"00:10.725 ","End":"00:13.575","Text":"When we have it in parametric form,"},{"Start":"00:13.575 ","End":"00:21.140","Text":"1 of the formulas I use for the ds is there several variations of"},{"Start":"00:21.140 ","End":"00:29.285","Text":"formulas I use this 1 that ds is the square root of x prime."},{"Start":"00:29.285 ","End":"00:31.580","Text":"Sometimes I write x prime of t,"},{"Start":"00:31.580 ","End":"00:35.330","Text":"usually I just abbreviate it to x prime squared."},{"Start":"00:35.330 ","End":"00:44.040","Text":"We assume x, y, and z are functions of t along the curve plus y prime squared."},{"Start":"00:44.040 ","End":"00:46.070","Text":"If it was 2D, we\u0027d stop here,"},{"Start":"00:46.070 ","End":"00:47.750","Text":"but we are in 3D,"},{"Start":"00:47.750 ","End":"00:50.375","Text":"so we also have z prime squared,"},{"Start":"00:50.375 ","End":"00:53.430","Text":"and then it\u0027s dt."},{"Start":"00:53.430 ","End":"00:55.320","Text":"I will compute this in a moment,"},{"Start":"00:55.320 ","End":"00:59.060","Text":"and we have to remember that the line"},{"Start":"00:59.060 ","End":"01:05.570","Text":"integral is replaced by a simple 1 variable integral,"},{"Start":"01:05.570 ","End":"01:07.744","Text":"which is the parameter t,"},{"Start":"01:07.744 ","End":"01:13.420","Text":"which goes from 0-3."},{"Start":"01:13.420 ","End":"01:19.895","Text":"Then I need to compute what is x cubed plus 3z that are along the curve."},{"Start":"01:19.895 ","End":"01:24.245","Text":"Let me also do that at the side."},{"Start":"01:24.245 ","End":"01:26.440","Text":"We can just do it in here,"},{"Start":"01:26.440 ","End":"01:32.595","Text":"x cubed is t cubed,"},{"Start":"01:32.595 ","End":"01:38.240","Text":"and 3z is 3 times 1 third,"},{"Start":"01:38.240 ","End":"01:40.070","Text":"3 times 1/3 is 1,"},{"Start":"01:40.070 ","End":"01:43.200","Text":"so plus t cubed."},{"Start":"01:43.200 ","End":"01:46.965","Text":"Then we need the ds,"},{"Start":"01:46.965 ","End":"01:51.380","Text":"so we need the square root of,"},{"Start":"01:51.380 ","End":"01:57.635","Text":"let\u0027s see, x prime is 1 squared."},{"Start":"01:57.635 ","End":"02:03.890","Text":"Y prime is derivative of t squared is 2t,"},{"Start":"02:03.890 ","End":"02:11.330","Text":"so it\u0027s 2 over root 2t squared,"},{"Start":"02:11.330 ","End":"02:18.890","Text":"and then z prime is 3 times t squared,"},{"Start":"02:18.890 ","End":"02:23.540","Text":"but times 1/3 just comes out t squared,"},{"Start":"02:23.540 ","End":"02:28.980","Text":"also squared, and then dt."},{"Start":"02:30.400 ","End":"02:36.965","Text":"Now, I\u0027ll do this square root thing at the side."},{"Start":"02:36.965 ","End":"02:44.655","Text":"What I get is the square root of 1 squared is 1."},{"Start":"02:44.655 ","End":"02:50.880","Text":"This thing squared, 2 over root 2 squared is 4 over 2, which is 2,"},{"Start":"02:50.880 ","End":"02:54.255","Text":"so that\u0027s 2t squared,"},{"Start":"02:54.255 ","End":"03:00.600","Text":"and t squared squared is t^4."},{"Start":"03:00.600 ","End":"03:08.360","Text":"This is exactly 1 plus t squared squared."},{"Start":"03:08.360 ","End":"03:10.040","Text":"You could have made a substitution,"},{"Start":"03:10.040 ","End":"03:13.400","Text":"u equals t squared or something."},{"Start":"03:13.400 ","End":"03:15.095","Text":"Anyway, if you square this,"},{"Start":"03:15.095 ","End":"03:22.025","Text":"you\u0027ll see this 1 squared is this twice this times this plus the last 1 squared."},{"Start":"03:22.025 ","End":"03:24.590","Text":"When I take the square root,"},{"Start":"03:24.590 ","End":"03:30.860","Text":"it\u0027s actually the absolute value"},{"Start":"03:30.860 ","End":"03:38.890","Text":"of just 1 plus t squared."},{"Start":"03:39.350 ","End":"03:44.910","Text":"But the absolute value this thing is always non-negative,"},{"Start":"03:44.910 ","End":"03:47.750","Text":"so I don\u0027t need the absolute value."},{"Start":"03:47.750 ","End":"03:51.995","Text":"It just comes out to be 1 plus t squared."},{"Start":"03:51.995 ","End":"03:55.880","Text":"We get the integral from 0-3."},{"Start":"03:55.880 ","End":"03:59.810","Text":"Now this plus this is just 2t cubed,"},{"Start":"03:59.810 ","End":"04:07.110","Text":"and from here I get 1 plus t squared dt."},{"Start":"04:09.230 ","End":"04:13.335","Text":"I\u0027ll break it up into 2 integrals,"},{"Start":"04:13.335 ","End":"04:15.060","Text":"or I can just do it as 1."},{"Start":"04:15.060 ","End":"04:16.545","Text":"Let\u0027s do it as 1 then."},{"Start":"04:16.545 ","End":"04:19.125","Text":"It\u0027s the integral from 0-3."},{"Start":"04:19.125 ","End":"04:23.055","Text":"This with this is 2t cubed,"},{"Start":"04:23.055 ","End":"04:30.210","Text":"and this with this is 2t^5."},{"Start":"04:30.210 ","End":"04:34.820","Text":"All this 0-3dt."},{"Start":"04:34.820 ","End":"04:40.380","Text":"Now to the integral raised the power of 1 is 4, divide by 4."},{"Start":"04:41.400 ","End":"04:46.335","Text":"1.5t^4, here, I raise it to 6 and divide by 6,"},{"Start":"04:46.335 ","End":"04:56.135","Text":"so it\u0027s 1/3t^6, and all this has to be evaluated from 0-3."},{"Start":"04:56.135 ","End":"04:59.720","Text":"Now, 0 when we substitute gives nothing,"},{"Start":"04:59.720 ","End":"05:03.230","Text":"so we just have to substitute the 3. What do we get?"},{"Start":"05:03.230 ","End":"05:08.030","Text":"1.5, 3^4,"},{"Start":"05:08.030 ","End":"05:17.900","Text":"plus 1 third times 3^6."},{"Start":"05:17.900 ","End":"05:23.965","Text":"Well, I could cancel the 1/3with 1 of the threes and make it just 5,"},{"Start":"05:23.965 ","End":"05:32.670","Text":"and then I could take 3^4 outside the brackets, 3^4."},{"Start":"05:32.670 ","End":"05:34.050","Text":"What am I left with?"},{"Start":"05:34.050 ","End":"05:40.120","Text":"1.5 plus 3. That\u0027s 3.5."},{"Start":"05:41.960 ","End":"05:49.125","Text":"So it\u0027s 81, and this is 7 over 2"},{"Start":"05:49.125 ","End":"05:54.100","Text":"times 7 over 2,"},{"Start":"05:54.980 ","End":"06:02.770","Text":"567 over 2."},{"Start":"06:02.770 ","End":"06:06.620","Text":"I could leave it like that or well, yeah,"},{"Start":"06:06.620 ","End":"06:15.080","Text":"I could do the division and you get something like 283.5 and write that down 283.5,"},{"Start":"06:15.080 ","End":"06:17.059","Text":"whichever one you prefer."},{"Start":"06:17.059 ","End":"06:21.690","Text":"I\u0027ll stick with this and we are done."}],"ID":8813},{"Watched":false,"Name":"Exercise 3","Duration":"11m 3s","ChapterTopicVideoID":8715,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8715.jpeg","UploadDate":"2017-02-13T05:40:02.4870000","DurationForVideoObject":"PT11M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.145","Text":"In this exercise, we need to compute the length of the following curve,"},{"Start":"00:05.145 ","End":"00:11.400","Text":"which is given an implicit form with x and y so we\u0027re working in 2 dimensions."},{"Start":"00:11.400 ","End":"00:15.570","Text":"Now, suppose I restrict x and y to the first quadrant,"},{"Start":"00:15.570 ","End":"00:19.935","Text":"you could try plugging in a few points for example, if x is 0,"},{"Start":"00:19.935 ","End":"00:26.490","Text":"we get that y is 1 and if we let y is 0,"},{"Start":"00:26.490 ","End":"00:31.285","Text":"then we get that x is 1 and if you try a few more points,"},{"Start":"00:31.285 ","End":"00:38.810","Text":"basically what we get is some curve like this and that\u0027s just for the first quadrant."},{"Start":"00:38.810 ","End":"00:42.560","Text":"Now notice if I replace x by minus x,"},{"Start":"00:42.560 ","End":"00:45.200","Text":"because it\u0027s to the power of 2/3,"},{"Start":"00:45.200 ","End":"00:47.315","Text":"it\u0027s squared and then the cube root."},{"Start":"00:47.315 ","End":"00:52.880","Text":"But I could replace x by minus x and I could also replace y by minus y,"},{"Start":"00:52.880 ","End":"00:58.445","Text":"so really I get a symmetry with respect to both the x- and the y-axis."},{"Start":"00:58.445 ","End":"01:01.130","Text":"If I draw in the points, minus 1,"},{"Start":"01:01.130 ","End":"01:04.615","Text":"0, and 0, minus 1,"},{"Start":"01:04.615 ","End":"01:07.140","Text":"we get a star shape,"},{"Start":"01:07.140 ","End":"01:14.145","Text":"something like this and this is the curve."},{"Start":"01:14.145 ","End":"01:18.420","Text":"Because of the total symmetry,"},{"Start":"01:18.420 ","End":"01:22.700","Text":"what we can do is just compute this part in"},{"Start":"01:22.700 ","End":"01:27.490","Text":"the first quadrant and then multiply the final answer by 4."},{"Start":"01:27.490 ","End":"01:31.970","Text":"I\u0027m just making a note to myself to remember to take this length"},{"Start":"01:31.970 ","End":"01:37.175","Text":"and then multiply it by 4."},{"Start":"01:37.175 ","End":"01:39.775","Text":"Now, length of curve,"},{"Start":"01:39.775 ","End":"01:41.530","Text":"if this was the curve,"},{"Start":"01:41.530 ","End":"01:48.035","Text":"let\u0027s say this is the curve C. Suppose I had it in parametric form or any other form,"},{"Start":"01:48.035 ","End":"01:58.330","Text":"the length is the integral along the curve of just 1 ds."},{"Start":"01:58.330 ","End":"02:02.770","Text":"What I\u0027d like to do first is try and write C in parametric form."},{"Start":"02:02.770 ","End":"02:04.135","Text":"I\u0027m talking about this,"},{"Start":"02:04.135 ","End":"02:12.690","Text":"x to the 2/3 plus y to the 2/3 equals 1."},{"Start":"02:12.690 ","End":"02:15.530","Text":"From experience,"},{"Start":"02:15.530 ","End":"02:21.295","Text":"what I know is that when I have something squared plus something squared equals 1,"},{"Start":"02:21.295 ","End":"02:23.965","Text":"often a trigonometric substitution works."},{"Start":"02:23.965 ","End":"02:28.074","Text":"I could write this as the cube root of x"},{"Start":"02:28.074 ","End":"02:35.430","Text":"squared plus the cube root of y squared equals 1."},{"Start":"02:35.430 ","End":"02:38.535","Text":"What I\u0027d like to do is let"},{"Start":"02:38.535 ","End":"02:44.745","Text":"the cube root of x equal cosine t"},{"Start":"02:44.745 ","End":"02:50.335","Text":"and the cube root of y to be sine t,"},{"Start":"02:50.335 ","End":"02:53.900","Text":"because cosine squared plus sine squared is 1."},{"Start":"02:53.900 ","End":"02:57.810","Text":"From experience, this substitution works."},{"Start":"03:01.280 ","End":"03:05.105","Text":"First of all, I want to write this differently,"},{"Start":"03:05.105 ","End":"03:13.140","Text":"this is the same as saying that x equals cosine cube t and"},{"Start":"03:13.140 ","End":"03:22.860","Text":"y equals sine cube t. Then it satisfies this equation,"},{"Start":"03:22.860 ","End":"03:27.680","Text":"to the power of 3/2 gives me cosine squared plus sine squared is 1."},{"Start":"03:27.680 ","End":"03:31.625","Text":"I also have to figure out where t goes from in 2."},{"Start":"03:31.625 ","End":"03:34.960","Text":"Notice that when t is 0,"},{"Start":"03:34.960 ","End":"03:38.550","Text":"I get cosine 0 is 1,"},{"Start":"03:38.550 ","End":"03:42.150","Text":"so x is 1 cubed and y is 0 cubed,"},{"Start":"03:42.150 ","End":"03:44.075","Text":"that gives me this point."},{"Start":"03:44.075 ","End":"03:46.985","Text":"That\u0027s where t equals 0."},{"Start":"03:46.985 ","End":"03:52.830","Text":"If I let t equals 90 degrees or Pi/2,"},{"Start":"03:52.830 ","End":"03:58.050","Text":"cosine of Pi/2 is 0 and sine Pi/2 is 1,"},{"Start":"03:58.050 ","End":"04:03.640","Text":"so this corresponds to t equals Pi/2 and we\u0027re"},{"Start":"04:03.640 ","End":"04:09.810","Text":"traveling along the curve from 0-Pi/2."},{"Start":"04:09.810 ","End":"04:14.570","Text":"What we get is the integral"},{"Start":"04:14.570 ","End":"04:21.560","Text":"where t goes from 0-Pi/2."},{"Start":"04:21.560 ","End":"04:23.645","Text":"We\u0027re still missing the ds."},{"Start":"04:23.645 ","End":"04:26.240","Text":"When we have a parametric form,"},{"Start":"04:26.240 ","End":"04:31.770","Text":"we can use the formula that ds is the square root"},{"Start":"04:31.770 ","End":"04:39.815","Text":"of x prime squared plus y prime squared dt,"},{"Start":"04:39.815 ","End":"04:45.875","Text":"where the prime means derivative with respect to t. Now,"},{"Start":"04:45.875 ","End":"04:47.585","Text":"if I look at this,"},{"Start":"04:47.585 ","End":"04:50.900","Text":"I\u0027ll make the computation."},{"Start":"04:50.900 ","End":"04:53.480","Text":"In our case, x prime"},{"Start":"04:53.480 ","End":"05:02.365","Text":"is 3 cosine squared t,"},{"Start":"05:02.365 ","End":"05:07.680","Text":"it\u0027s like something cubed so it\u0027s 3 times that something squared but times the inner"},{"Start":"05:07.680 ","End":"05:15.765","Text":"derivative which is minus sine t. All this is x prime."},{"Start":"05:15.765 ","End":"05:18.390","Text":"Let me just write that x prime is this,"},{"Start":"05:18.390 ","End":"05:22.445","Text":"y prime is very similar."},{"Start":"05:22.445 ","End":"05:25.385","Text":"Just here, I have sine squared t,"},{"Start":"05:25.385 ","End":"05:26.810","Text":"and here I don\u0027t have the minus."},{"Start":"05:26.810 ","End":"05:31.630","Text":"Derivative of sine is just cosine t. Now,"},{"Start":"05:31.630 ","End":"05:39.260","Text":"I can plug it in here and get that ds equals the square root of, now,"},{"Start":"05:39.260 ","End":"05:44.270","Text":"x prime squared is 3"},{"Start":"05:44.270 ","End":"05:53.080","Text":"squared times cosine to the 4th t,"},{"Start":"05:53.080 ","End":"05:54.725","Text":"and the minus doesn\u0027t matter,"},{"Start":"05:54.725 ","End":"06:00.560","Text":"sine squared t plus,"},{"Start":"06:00.560 ","End":"06:03.600","Text":"let\u0027s make this a bit longer."},{"Start":"06:04.040 ","End":"06:12.329","Text":"Similar thing here, 3 squared sine to the 4th t,"},{"Start":"06:12.329 ","End":"06:22.820","Text":"cosine squared t. What we can do is we can take the common stuff out."},{"Start":"06:22.820 ","End":"06:25.310","Text":"We have 3 squared, cosine squared,"},{"Start":"06:25.310 ","End":"06:28.675","Text":"sine squared in common."},{"Start":"06:28.675 ","End":"06:34.850","Text":"I can write this as 3 squared cosine squared sine"},{"Start":"06:34.850 ","End":"06:41.140","Text":"squared t. What we\u0027re left with is,"},{"Start":"06:41.140 ","End":"06:44.735","Text":"here we\u0027re just left with a cosine squared."},{"Start":"06:44.735 ","End":"06:49.020","Text":"Here we\u0027re left with a sine squared."},{"Start":"06:50.210 ","End":"06:52.710","Text":"This here is 1,"},{"Start":"06:52.710 ","End":"06:55.510","Text":"square root of 1 is 1."},{"Start":"06:55.640 ","End":"07:00.310","Text":"This bit, the square root is just"},{"Start":"07:00.310 ","End":"07:07.260","Text":"3 cosine t sine t. Normally,"},{"Start":"07:07.260 ","End":"07:11.520","Text":"I\u0027d say absolute value but we\u0027re working in the first quadrant,"},{"Start":"07:11.520 ","End":"07:14.010","Text":"so all these things are positive and this,"},{"Start":"07:14.010 ","End":"07:16.230","Text":"as I said, is times 1."},{"Start":"07:16.230 ","End":"07:19.169","Text":"Getting back to here,"},{"Start":"07:19.169 ","End":"07:28.120","Text":"we get the integral from 0-Pi/2 of 3."},{"Start":"07:28.120 ","End":"07:29.560","Text":"I\u0027ll just change the order."},{"Start":"07:29.560 ","End":"07:32.615","Text":"I\u0027d like to write the sine before the cosine,"},{"Start":"07:32.615 ","End":"07:38.185","Text":"dt because ds, yeah."},{"Start":"07:38.185 ","End":"07:45.210","Text":"I should have written dt, dt, dt, yeah."},{"Start":"07:45.210 ","End":"07:47.320","Text":"Now, the trigonometric formula,"},{"Start":"07:47.320 ","End":"07:53.709","Text":"that if I had a 2 sine t,"},{"Start":"07:53.709 ","End":"07:57.055","Text":"usually written as Alpha in the books, but doesn\u0027t matter,"},{"Start":"07:57.055 ","End":"08:03.635","Text":"2 sine t, cosine t is sine of 2t."},{"Start":"08:03.635 ","End":"08:07.225","Text":"You usually see it. This is on the right, this is on the left."},{"Start":"08:07.225 ","End":"08:10.250","Text":"I don\u0027t have a 2, I have a 3."},{"Start":"08:10.250 ","End":"08:14.755","Text":"What I can do is I can force it to be a 2."},{"Start":"08:14.755 ","End":"08:21.500","Text":"If I make this a 2, then I have to compensate and put a 3/2 here,"},{"Start":"08:21.500 ","End":"08:24.095","Text":"and then everything will be okay."},{"Start":"08:24.095 ","End":"08:30.055","Text":"I have 3/2 times the integral from 0-Pi/2,"},{"Start":"08:30.055 ","End":"08:39.900","Text":"2 sine t cosine t is sine of 2t dt."},{"Start":"08:39.900 ","End":"08:45.095","Text":"The integral of sine is roughly minus cosine."},{"Start":"08:45.095 ","End":"08:50.015","Text":"What I get is 3/2 and"},{"Start":"08:50.015 ","End":"08:58.200","Text":"then I start off with minus cosine of 2t."},{"Start":"08:58.200 ","End":"09:01.975","Text":"But because it\u0027s not t, it\u0027s 2t,"},{"Start":"09:01.975 ","End":"09:06.050","Text":"I have to also put in a 1/2 because if I was to differentiate,"},{"Start":"09:06.050 ","End":"09:07.910","Text":"I\u0027d get times 2."},{"Start":"09:07.910 ","End":"09:12.150","Text":"This has to be taken from 0-Pi/2."},{"Start":"09:15.080 ","End":"09:17.900","Text":"Let\u0027s see what we get."},{"Start":"09:17.900 ","End":"09:21.095","Text":"One thing I like to do is instead of a minus,"},{"Start":"09:21.095 ","End":"09:30.890","Text":"I like to get rid of the minus and then switch these 2 around and what I have"},{"Start":"09:30.890 ","End":"09:35.480","Text":"pulling the 1/2 in front is I have 3/4 and then I"},{"Start":"09:35.480 ","End":"09:42.170","Text":"have cosine of 2t."},{"Start":"09:42.170 ","End":"09:45.150","Text":"But, in reverse order,"},{"Start":"09:45.850 ","End":"09:49.130","Text":"Pi/2-0, that\u0027s what got rid of the minus,"},{"Start":"09:49.130 ","End":"09:51.920","Text":"and the 1/2, I threw in with the 3/2."},{"Start":"09:51.920 ","End":"09:55.830","Text":"Let\u0027s see. If I have 3/4."},{"Start":"09:55.830 ","End":"09:58.720","Text":"Now. If I plug in 0,"},{"Start":"09:58.720 ","End":"10:01.930","Text":"I get cosine of,"},{"Start":"10:01.930 ","End":"10:06.085","Text":"twice 0 with cosine of 0 is 1."},{"Start":"10:06.085 ","End":"10:10.015","Text":"If I plug in Pi/2,"},{"Start":"10:10.015 ","End":"10:12.960","Text":"twice Pi/2 is Pi,"},{"Start":"10:12.960 ","End":"10:20.415","Text":"and cosine of Pi is minus 1."},{"Start":"10:20.415 ","End":"10:22.260","Text":"Let\u0027s see what do I get altogether?"},{"Start":"10:22.260 ","End":"10:24.440","Text":"1 minus minus 1 is 2,"},{"Start":"10:24.440 ","End":"10:28.490","Text":"2 cancels with the 4 partially,"},{"Start":"10:28.490 ","End":"10:32.135","Text":"I make this 3/2."},{"Start":"10:32.135 ","End":"10:35.540","Text":"But that\u0027s not the answer because if you remember,"},{"Start":"10:35.540 ","End":"10:39.610","Text":"we had to multiply this answer by 4."},{"Start":"10:39.610 ","End":"10:42.740","Text":"See, we had this is only a 1/4 of it,"},{"Start":"10:42.740 ","End":"10:45.820","Text":"so what I need for"},{"Start":"10:45.820 ","End":"10:53.320","Text":"the answer is 3/2 times 4."},{"Start":"10:53.320 ","End":"10:56.840","Text":"In other words, let\u0027s see what this comes out to,"},{"Start":"10:56.840 ","End":"11:04.320","Text":"6 and that\u0027s the answer I\u0027m going to highlight. Now we\u0027re done."}],"ID":8814},{"Watched":false,"Name":"Exercise 4","Duration":"4m 39s","ChapterTopicVideoID":8716,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8716.jpeg","UploadDate":"2017-02-13T05:41:04.5400000","DurationForVideoObject":"PT4M39S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.380","Text":"In this exercise, you have a question from physics or engineering,"},{"Start":"00:04.380 ","End":"00:10.810","Text":"but don\u0027t worry, I\u0027ll give you all the formulas you need. It\u0027s about a coil."},{"Start":"00:11.180 ","End":"00:15.629","Text":"I can recognize that this is in the shape of a helix."},{"Start":"00:15.629 ","End":"00:21.690","Text":"It\u0027s made of wire and we neglect the thickness of the wire."},{"Start":"00:21.690 ","End":"00:24.465","Text":"It\u0027s just like a line,"},{"Start":"00:24.465 ","End":"00:28.020","Text":"a curve given parametric form as x, y,"},{"Start":"00:28.020 ","End":"00:33.000","Text":"and z in terms of t and we\u0027re given the range t goes from zero to Pi."},{"Start":"00:33.000 ","End":"00:36.195","Text":"We\u0027re also given a density function."},{"Start":"00:36.195 ","End":"00:39.075","Text":"This is actually something called linear density,"},{"Start":"00:39.075 ","End":"00:47.085","Text":"not the usual density of mass per unit of volume but mass per unit length."},{"Start":"00:47.085 ","End":"00:48.420","Text":"Anyway, never mind that."},{"Start":"00:48.420 ","End":"00:53.280","Text":"This is a density function and it\u0027s proportional to z,"},{"Start":"00:53.280 ","End":"00:54.720","Text":"it\u0027s some constant times z,"},{"Start":"00:54.720 ","End":"00:59.825","Text":"the positive constant, and we have to compute the mass of the coil."},{"Start":"00:59.825 ","End":"01:04.940","Text":"In general, we want the formula for the mass given the density."},{"Start":"01:04.940 ","End":"01:14.060","Text":"In general, the mass is the integral along the parametrized curve,"},{"Start":"01:14.060 ","End":"01:19.245","Text":"c of Delta of x,"},{"Start":"01:19.245 ","End":"01:25.685","Text":"y, and z, ds like type 1 line integral."},{"Start":"01:25.685 ","End":"01:28.100","Text":"But in our case,"},{"Start":"01:28.100 ","End":"01:35.970","Text":"we know that delta is just kz."},{"Start":"01:35.970 ","End":"01:40.905","Text":"What we\u0027re missing still is ds and in 3 dimensions,"},{"Start":"01:40.905 ","End":"01:43.590","Text":"the formula for ds,"},{"Start":"01:43.590 ","End":"01:48.680","Text":"1 of the variations of the several ways of writing the formula."},{"Start":"01:48.680 ","End":"01:54.640","Text":"One way is just to write it as x prime squared,"},{"Start":"01:54.640 ","End":"01:58.110","Text":"plus y prime squared,"},{"Start":"01:58.110 ","End":"02:02.150","Text":"plus z prime squared dt,"},{"Start":"02:02.150 ","End":"02:05.760","Text":"where the prime means derivative with respect to t. Sometimes,"},{"Start":"02:05.760 ","End":"02:07.790","Text":"you write it as parentheses t,"},{"Start":"02:07.790 ","End":"02:09.920","Text":"I was just saving space."},{"Start":"02:09.920 ","End":"02:17.945","Text":"We could actually compute this here is quite easy to do because x prime"},{"Start":"02:17.945 ","End":"02:26.540","Text":"is minus sine t. Here I have minus sine t. Let me just write the derivatives."},{"Start":"02:26.540 ","End":"02:33.480","Text":"Y prime is cosine t and z prime is just 2."},{"Start":"02:33.480 ","End":"02:43.915","Text":"I need to put each of these squared and add them and then take the square root dt."},{"Start":"02:43.915 ","End":"02:48.430","Text":"Sine squared plus cosine squared is 1,"},{"Start":"02:48.430 ","End":"02:51.090","Text":"and 2 squared is 4."},{"Start":"02:51.090 ","End":"02:58.180","Text":"All together, I have the square root of 1 plus 4 is 5, square roots of 5dt."},{"Start":"03:01.370 ","End":"03:07.290","Text":"If we plug everything in and we plug in the Delta is kz,"},{"Start":"03:08.440 ","End":"03:11.314","Text":"we get the integral,"},{"Start":"03:11.314 ","End":"03:15.245","Text":"becomes the integral of 1 variable which is the parameter,"},{"Start":"03:15.245 ","End":"03:18.545","Text":"and it goes from 0 to Pi."},{"Start":"03:18.545 ","End":"03:22.620","Text":"Delta is just kz."},{"Start":"03:23.320 ","End":"03:33.170","Text":"But instead of z, we write what z is, which is 2t."},{"Start":"03:33.170 ","End":"03:36.730","Text":"It\u0027s k times 2t,"},{"Start":"03:36.730 ","End":"03:39.585","Text":"and then we need the ds,"},{"Start":"03:39.585 ","End":"03:43.180","Text":"which is root 5dt."},{"Start":"03:46.390 ","End":"03:49.940","Text":"I\u0027ll take some of the constants in front."},{"Start":"03:49.940 ","End":"03:53.270","Text":"I\u0027ll take out root 5,"},{"Start":"03:53.270 ","End":"03:56.360","Text":"I\u0027ll take out k, I was going to take the 2"},{"Start":"03:56.360 ","End":"04:00.190","Text":"out and then I realized that\u0027s useful for me to leave the 2 in."},{"Start":"04:00.190 ","End":"04:04.100","Text":"The reason I\u0027m leaving the 2 in is because the integral"},{"Start":"04:04.100 ","End":"04:08.585","Text":"of 2t comes out to be an t squared."},{"Start":"04:08.585 ","End":"04:13.140","Text":"What we get here is root 5k."},{"Start":"04:13.140 ","End":"04:19.080","Text":"Then we have t squared from zero to Pi."},{"Start":"04:19.080 ","End":"04:21.870","Text":"If I plugin Pi, I get Pi squared,"},{"Start":"04:21.870 ","End":"04:24.300","Text":"I plugin 0, I get just 0."},{"Start":"04:24.300 ","End":"04:26.700","Text":"Altogether, this is Pi squared."},{"Start":"04:26.700 ","End":"04:37.920","Text":"The answer would be root 5k times Pi squared and that\u0027s the answer. That\u0027s all there is."}],"ID":8815},{"Watched":false,"Name":"Exercise 5 Part a","Duration":"7m ","ChapterTopicVideoID":8717,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8717.jpeg","UploadDate":"2017-02-13T05:42:39.0700000","DurationForVideoObject":"PT7M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:09.300","Text":"In this exercise, we have a line integral of type 2 over a parameterized curve C,"},{"Start":"00:09.300 ","End":"00:11.835","Text":"given it\u0027s in 2 dimensions,"},{"Start":"00:11.835 ","End":"00:15.030","Text":"x and y in terms of t and here\u0027s the range of"},{"Start":"00:15.030 ","End":"00:18.705","Text":"t so we just have to basically substitute everything into"},{"Start":"00:18.705 ","End":"00:25.855","Text":"language of t. The integral becomes the integral t goes from 0 to Pi over 2."},{"Start":"00:25.855 ","End":"00:29.765","Text":"I\u0027ll emphasize and write t equals 0 up to"},{"Start":"00:29.765 ","End":"00:34.070","Text":"Pi over 2 and then everything has to be substituted."},{"Start":"00:34.070 ","End":"00:42.285","Text":"We have here 2x we take along the curve so x is cosine t,"},{"Start":"00:42.285 ","End":"00:47.610","Text":"y is sine t and then I need dx."},{"Start":"00:47.610 ","End":"00:51.745","Text":"Let me just do dx and dy on this side."},{"Start":"00:51.745 ","End":"00:54.500","Text":"If I take the d of both sides,"},{"Start":"00:54.500 ","End":"00:59.585","Text":"we have 1dx which is dx and the other side,"},{"Start":"00:59.585 ","End":"01:06.200","Text":"I get minus sine t dt and"},{"Start":"01:06.200 ","End":"01:14.500","Text":"the dy becomes 1dy equals cosine t dt."},{"Start":"01:18.650 ","End":"01:21.880","Text":"Yeah, put the brackets there."},{"Start":"01:22.640 ","End":"01:26.285","Text":"What I get here,"},{"Start":"01:26.285 ","End":"01:33.270","Text":"dx is minus sine t and then dt."},{"Start":"01:38.210 ","End":"01:40.410","Text":"Later I\u0027m going to take the dt,"},{"Start":"01:40.410 ","End":"01:43.580","Text":"I\u0027m going to get a dt here and later I\u0027ll take it outside the brackets."},{"Start":"01:43.580 ","End":"01:49.445","Text":"Meanwhile, x squared is cosine squared"},{"Start":"01:49.445 ","End":"01:56.000","Text":"t and y squared"},{"Start":"01:56.000 ","End":"02:01.190","Text":"is sine squared t and here we have dy,"},{"Start":"02:01.190 ","End":"02:10.770","Text":"which is cosine t dt and we have a dt here and a dt here,"},{"Start":"02:10.770 ","End":"02:15.365","Text":"let\u0027s combine everything and put a single dt at the end."},{"Start":"02:15.365 ","End":"02:18.305","Text":"From 0 to Pi over 2,"},{"Start":"02:18.305 ","End":"02:21.270","Text":"let me start opening the brackets."},{"Start":"02:21.280 ","End":"02:23.435","Text":"What do we have here?"},{"Start":"02:23.435 ","End":"02:29.585","Text":"We have minus 2 cosine t,"},{"Start":"02:29.585 ","End":"02:36.135","Text":"sine squared t and from the second 1,"},{"Start":"02:36.135 ","End":"02:44.090","Text":"I have cosine squared cosine that\u0027s cosine cube t and then sine squared"},{"Start":"02:44.090 ","End":"02:52.920","Text":"times cosine is just sine squared t cosine t and all this dt."},{"Start":"02:53.560 ","End":"02:59.510","Text":"Notice that cosine times sine squared is the same as sine squared times"},{"Start":"02:59.510 ","End":"03:06.730","Text":"cosine so this could be combined and we just have minus 1 of these."},{"Start":"03:06.730 ","End":"03:14.415","Text":"We can also take the cosine outside the brackets so let\u0027s see,"},{"Start":"03:14.415 ","End":"03:17.080","Text":"from 0 to Pi over 2."},{"Start":"03:17.080 ","End":"03:21.160","Text":"I\u0027m taking cosine t outside the brackets."},{"Start":"03:21.160 ","End":"03:25.505","Text":"Now, then we said these 2 combine,"},{"Start":"03:25.505 ","End":"03:33.590","Text":"we get minus 2 sine squared plus sine squared is just minus sine squared t and from here,"},{"Start":"03:33.590 ","End":"03:43.500","Text":"I get just cosine squared t dt."},{"Start":"03:43.500 ","End":"03:45.390","Text":"There\u0027s more than 1 way to do this."},{"Start":"03:45.390 ","End":"03:47.625","Text":"I\u0027m going to suggest using"},{"Start":"03:47.625 ","End":"03:53.510","Text":"a trigonometric identity to convert sine squared into cosine squared because"},{"Start":"03:53.510 ","End":"04:04.040","Text":"sine squared t is 1 minus cosine squared t. If I do the computation,"},{"Start":"04:04.040 ","End":"04:10.370","Text":"we\u0027ll get cosine squared minus 1 plus cosine squared."},{"Start":"04:10.370 ","End":"04:20.855","Text":"In other words, I\u0027ll get the integral same limits of and you know what?"},{"Start":"04:20.855 ","End":"04:21.995","Text":"I changed my mind."},{"Start":"04:21.995 ","End":"04:25.820","Text":"I\u0027m going to convert the cosine into sine so I\u0027m"},{"Start":"04:25.820 ","End":"04:30.859","Text":"erasing this and I\u0027m going to write the cosine squared"},{"Start":"04:30.859 ","End":"04:37.370","Text":"as 1 minus sine squared t. The reason I prefer to stay with sine and not with"},{"Start":"04:37.370 ","End":"04:44.900","Text":"cosine is because I have the derivative of sine here and then I can use a substitution."},{"Start":"04:47.780 ","End":"04:50.070","Text":"I\u0027ll do this part first,"},{"Start":"04:50.070 ","End":"04:59.875","Text":"this is 1 minus twice sine squared t and then I\u0027ll take the cosine t dt"},{"Start":"04:59.875 ","End":"05:09.790","Text":"and now I\u0027ll do the substitution I mentioned by letting u"},{"Start":"05:09.790 ","End":"05:19.585","Text":"equal sine t and then 1du"},{"Start":"05:19.585 ","End":"05:27.470","Text":"equals cosine t dt and so this integral"},{"Start":"05:27.490 ","End":"05:38.390","Text":"becomes the integral of 1 minus 2u squared."},{"Start":"05:38.390 ","End":"05:43.250","Text":"The cosine t dt is du but that\u0027s not all because I also have to"},{"Start":"05:43.250 ","End":"05:49.475","Text":"substitute the limits unless I want to do it with an indefinite and return to t,"},{"Start":"05:49.475 ","End":"05:55.880","Text":"I\u0027ll rather stay in the land of u and so I say that when t is 0,"},{"Start":"05:55.880 ","End":"06:00.090","Text":"u equals and when t equals Pi over 2,"},{"Start":"06:00.090 ","End":"06:02.160","Text":"we\u0027ll see what u equals."},{"Start":"06:02.160 ","End":"06:06.419","Text":"Sine t, so sine of 0 is 0,"},{"Start":"06:06.419 ","End":"06:14.210","Text":"sine of Pi over 2 is 1 and so this is the integral from 0 to 1."},{"Start":"06:14.210 ","End":"06:19.435","Text":"This is very straightforward."},{"Start":"06:19.435 ","End":"06:21.755","Text":"This integral of 1 is u."},{"Start":"06:21.755 ","End":"06:23.960","Text":"The integral of this,"},{"Start":"06:23.960 ","End":"06:33.635","Text":"raise it to u cubed and divide by the 3 so 2/3 u cubed."},{"Start":"06:33.635 ","End":"06:37.370","Text":"This I want to take from 0 to"},{"Start":"06:37.370 ","End":"06:45.610","Text":"1 and if I plug in 0 for u,"},{"Start":"06:45.610 ","End":"06:48.725","Text":"I don\u0027t get anything, so I just need to plug in the 1,"},{"Start":"06:48.725 ","End":"06:53.030","Text":"so I get 1 minus 2/3,"},{"Start":"06:53.030 ","End":"06:59.760","Text":"which is 1/3 and that\u0027s the answer."}],"ID":8816},{"Watched":false,"Name":"Exercise 5 Part b","Duration":"2m 43s","ChapterTopicVideoID":8718,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8718.jpeg","UploadDate":"2017-02-13T05:43:16.8870000","DurationForVideoObject":"PT2M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"Here we have this type 2 line integral."},{"Start":"00:03.720 ","End":"00:07.110","Text":"It\u0027s in 2-dimensions, just x, and y."},{"Start":"00:07.110 ","End":"00:11.970","Text":"The line integral over a parametrized curve C,"},{"Start":"00:11.970 ","End":"00:15.720","Text":"where we have x and y in terms of t and we have the range for t."},{"Start":"00:15.720 ","End":"00:19.530","Text":"We just translate everything into t."},{"Start":"00:19.530 ","End":"00:26.710","Text":"So this integral becomes the integral where t goes from 0-1."},{"Start":"00:27.290 ","End":"00:32.489","Text":"Just write the t for emphasis and then I just substitute everything."},{"Start":"00:32.489 ","End":"00:35.610","Text":"So 2x is 2t,"},{"Start":"00:35.610 ","End":"00:38.940","Text":"y is t squared."},{"Start":"00:38.940 ","End":"00:41.545","Text":"Now I need dx."},{"Start":"00:41.545 ","End":"00:44.760","Text":"Let\u0027s do those at the side."},{"Start":"00:44.950 ","End":"00:48.170","Text":"From this x equals t,"},{"Start":"00:48.170 ","End":"00:49.970","Text":"I take the d of both sides."},{"Start":"00:49.970 ","End":"00:55.220","Text":"So I have dx equals dt and for y,"},{"Start":"00:55.220 ","End":"00:58.640","Text":"derivative of y is 1 with respect to y."},{"Start":"00:58.640 ","End":"01:03.900","Text":"So it\u0027s 1dy or just dy and here derivative is 2tdt."},{"Start":"01:05.170 ","End":"01:08.265","Text":"Now I\u0027ve got those as well."},{"Start":"01:08.265 ","End":"01:14.780","Text":"Dx is dt and I\u0027m going to have another one with a dt"},{"Start":"01:14.780 ","End":"01:18.385","Text":"and later I\u0027ll just take the dt outside the brackets."},{"Start":"01:18.385 ","End":"01:27.960","Text":"Let\u0027s see, x squared is t squared minus y is t squared."},{"Start":"01:27.960 ","End":"01:30.460","Text":"That\u0027s lucky."},{"Start":"01:30.770 ","End":"01:33.130","Text":"I\u0027ll write dy anyway,"},{"Start":"01:33.130 ","End":"01:37.880","Text":"even though I already see this is going to be zero but I\u0027ll write dy as 2tdt."},{"Start":"01:40.100 ","End":"01:44.200","Text":"We got lucky here because t squared minus t squared is 0."},{"Start":"01:44.200 ","End":"01:51.020","Text":"This whole thing can just be removed and we just have this integral to do."},{"Start":"01:51.020 ","End":"01:54.960","Text":"The integral of 2t is t squared,"},{"Start":"01:54.960 ","End":"01:59.125","Text":"integral of t squared is 1/3t cubed."},{"Start":"01:59.125 ","End":"02:11.129","Text":"I need to take this from 0-1 and when I plug in 1,"},{"Start":"02:11.129 ","End":"02:15.150","Text":"I get 1 squared plus 1,"},{"Start":"02:15.150 ","End":"02:26.295","Text":"I\u0027ll write it 1 squared plus 1/3 times 1 cubed minus 0 squared plus 1/3 0 cubed."},{"Start":"02:26.295 ","End":"02:31.840","Text":"This is 0 and this is just 1 plus 1/3,"},{"Start":"02:33.950 ","End":"02:42.400","Text":"1 plus a 1/3 is 1 and a 1/3 or 4/3 and that\u0027s the answer."}],"ID":8817},{"Watched":false,"Name":"Exercise 6 Part a","Duration":"4m 20s","ChapterTopicVideoID":8719,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8719.jpeg","UploadDate":"2017-02-13T05:44:22.8200000","DurationForVideoObject":"PT4M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.840","Text":"In this exercise, we want to compute this type 2 line integral"},{"Start":"00:06.840 ","End":"00:15.690","Text":"where the C here is a path from 0,0 to 2,4."},{"Start":"00:15.690 ","End":"00:20.130","Text":"Of course, this depends on the path C. In part A,"},{"Start":"00:20.130 ","End":"00:24.620","Text":"we\u0027ll take the straight line y equals 2x,"},{"Start":"00:24.620 ","End":"00:30.195","Text":"this will look something like this and that will be"},{"Start":"00:30.195 ","End":"00:38.785","Text":"the C. In the following clip, we\u0027ll go and do it along the parabola y equals x squared."},{"Start":"00:38.785 ","End":"00:40.770","Text":"This is our path C,"},{"Start":"00:40.770 ","End":"00:43.545","Text":"it goes from 0,0 to 2,4."},{"Start":"00:43.545 ","End":"00:48.445","Text":"What we want to do is parameterize this path."},{"Start":"00:48.445 ","End":"00:51.710","Text":"When y is given as a function of x,"},{"Start":"00:51.710 ","End":"00:56.820","Text":"then the easiest thing to do is to let x equals t,"},{"Start":"00:57.290 ","End":"01:01.980","Text":"and then y is just the same function,"},{"Start":"01:01.980 ","End":"01:03.570","Text":"instead of x you put t,"},{"Start":"01:03.570 ","End":"01:06.970","Text":"we\u0027ll have y equals 2t."},{"Start":"01:07.550 ","End":"01:11.360","Text":"Since we know the beginning and the endpoints,"},{"Start":"01:11.360 ","End":"01:14.710","Text":"you see that x goes from 0,"},{"Start":"01:14.710 ","End":"01:19.770","Text":"this 0 here for the x up to 2."},{"Start":"01:19.770 ","End":"01:23.700","Text":"So x goes between 0 and 2."},{"Start":"01:23.700 ","End":"01:34.040","Text":"Or let\u0027s replace x by t. Now we need to convert this to an integral in terms of t."},{"Start":"01:34.040 ","End":"01:40.850","Text":"Notice that from here, we can get that"},{"Start":"01:40.850 ","End":"01:51.305","Text":"dx is equal to dt and dy is equal to 2dt."},{"Start":"01:51.305 ","End":"01:56.305","Text":"A simple differentiation, derivative here is 1, here it\u0027s 2,"},{"Start":"01:56.305 ","End":"02:00.725","Text":"this is 1dt which is just dt,"},{"Start":"02:00.725 ","End":"02:05.160","Text":"1dx equals 1dt, 1dy equals 2dt."},{"Start":"02:05.870 ","End":"02:08.445","Text":"We just substitute in here,"},{"Start":"02:08.445 ","End":"02:14.210","Text":"instead of the curve we put the value of the parameter t."},{"Start":"02:14.210 ","End":"02:20.840","Text":"In other words we have t going from 0-2,"},{"Start":"02:21.170 ","End":"02:26.340","Text":"and then we plug in y is 2t."},{"Start":"02:26.340 ","End":"02:35.219","Text":"We have 2t and then dx is dt plus x squared"},{"Start":"02:35.219 ","End":"02:43.650","Text":"is t squared and dy is 2dt."},{"Start":"02:43.650 ","End":"02:46.695","Text":"I just want to bring the dt out,"},{"Start":"02:46.695 ","End":"02:48.555","Text":"it\u0027s just 1 integral,"},{"Start":"02:48.555 ","End":"02:51.950","Text":"just 1 time dt. What do we have here?"},{"Start":"02:51.950 ","End":"02:55.915","Text":"We have 2t plus 2t squared,"},{"Start":"02:55.915 ","End":"03:04.045","Text":"dt from 0-2."},{"Start":"03:04.045 ","End":"03:11.460","Text":"Straightforward integral, 2t gives me t squared, 2t squared gives,"},{"Start":"03:11.460 ","End":"03:16.215","Text":"see, it raises it to 1 to the power of 3 and divide by 3,"},{"Start":"03:16.215 ","End":"03:23.170","Text":"so 2/3t cubed this from 0-2."},{"Start":"03:23.890 ","End":"03:27.350","Text":"When we substitute t equals 0,"},{"Start":"03:27.350 ","End":"03:29.630","Text":"both of these are 0, forget about that."},{"Start":"03:29.630 ","End":"03:31.490","Text":"We just need to substitute 2,"},{"Start":"03:31.490 ","End":"03:36.990","Text":"so we get 2 squared is 4 plus 2/3,"},{"Start":"03:36.990 ","End":"03:39.615","Text":"2 cubed is 8."},{"Start":"03:39.615 ","End":"03:42.080","Text":"Let\u0027s see if we can simplify this."},{"Start":"03:42.080 ","End":"03:48.225","Text":"If I put it over 3, I\u0027ll get what?"},{"Start":"03:48.225 ","End":"03:54.735","Text":"12 over 3 plus 16 over 3,"},{"Start":"03:54.735 ","End":"04:03.015","Text":"that\u0027s 28 over 3 and that\u0027s the answer which I\u0027ll highlight."},{"Start":"04:03.015 ","End":"04:07.190","Text":"If you prefer mixed fractions,"},{"Start":"04:07.190 ","End":"04:09.320","Text":"then we could write this also as,"},{"Start":"04:09.320 ","End":"04:12.530","Text":"3 in 27 goes 9 remainder 1,"},{"Start":"04:12.530 ","End":"04:14.210","Text":"9 and 1/3 anyway."},{"Start":"04:14.210 ","End":"04:15.500","Text":"That\u0027s all there is to it."},{"Start":"04:15.500 ","End":"04:17.645","Text":"Now, on to part b,"},{"Start":"04:17.645 ","End":"04:20.760","Text":"in the next clip that is."}],"ID":8818},{"Watched":false,"Name":"Exercise 6 Part b","Duration":"2m 48s","ChapterTopicVideoID":8720,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8720.jpeg","UploadDate":"2017-02-13T05:44:57.5670000","DurationForVideoObject":"PT2M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.660","Text":"We just finished part A and we want to move on to part B."},{"Start":"00:03.660 ","End":"00:05.310","Text":"Let me erase what we don\u0027t need,"},{"Start":"00:05.310 ","End":"00:07.410","Text":"first of all, this stuff."},{"Start":"00:07.410 ","End":"00:15.750","Text":"The curve is going to be different and the function of t will be different."},{"Start":"00:15.750 ","End":"00:21.990","Text":"What we\u0027ll get in fact is y equals t squared"},{"Start":"00:21.990 ","End":"00:31.600","Text":"and dy therefore will be 2t dt."},{"Start":"00:31.600 ","End":"00:36.200","Text":"This part still stays and the sketch is different."},{"Start":"00:36.200 ","End":"00:38.870","Text":"It\u0027s a bit of a parabola."},{"Start":"00:38.870 ","End":"00:41.465","Text":"It doesn\u0027t have to be accurate."},{"Start":"00:41.465 ","End":"00:45.395","Text":"C, doesn\u0027t need a sketch at all really,"},{"Start":"00:45.395 ","End":"00:52.565","Text":"and now we\u0027ll go and do it with these equations."},{"Start":"00:52.565 ","End":"00:57.440","Text":"We get the integral from 0-2,"},{"Start":"00:57.440 ","End":"01:01.340","Text":"so it must be 4, y,"},{"Start":"01:01.340 ","End":"01:05.645","Text":"this time is t squared and dx is"},{"Start":"01:05.645 ","End":"01:13.830","Text":"dt and x squared is t squared,"},{"Start":"01:14.140 ","End":"01:25.780","Text":"and dy is 2t dt."},{"Start":"01:25.780 ","End":"01:30.450","Text":"Let\u0027s put it all with a single dt."},{"Start":"01:30.450 ","End":"01:39.250","Text":"What I have here is t squared plus 2t cubed dt."},{"Start":"01:40.070 ","End":"01:47.205","Text":"We get from here 1/3 t cubed, from here,"},{"Start":"01:47.205 ","End":"01:52.635","Text":"2t^4 over 4, 2 over 4 is easier to write as a half,"},{"Start":"01:52.635 ","End":"01:56.820","Text":"all this from 0-2."},{"Start":"01:56.820 ","End":"02:01.530","Text":"Now, 0 when we plug it in gives nothing so I only need to put in the 2."},{"Start":"02:01.530 ","End":"02:06.030","Text":"Here 2 cubed is 8 over 3,"},{"Start":"02:06.030 ","End":"02:17.040","Text":"and here t^4 is"},{"Start":"02:17.040 ","End":"02:24.400","Text":"16 over 2 is 8."},{"Start":"02:24.830 ","End":"02:29.625","Text":"This is 8 plus 2 and 2/3,"},{"Start":"02:29.625 ","End":"02:39.860","Text":"so I can write it as 10 and 2/3 and this is the answer as a mixed number."},{"Start":"02:39.860 ","End":"02:45.380","Text":"But if you prefer an improper fraction you can also write it as 32 over 3."},{"Start":"02:45.380 ","End":"02:48.240","Text":"Anyway, we are done."}],"ID":8819},{"Watched":false,"Name":"Exercise 7 Part a","Duration":"5m 40s","ChapterTopicVideoID":8696,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8696.jpeg","UploadDate":"2017-02-13T05:01:43.5470000","DurationForVideoObject":"PT5M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.805","Text":"In this exercise, we have to compute a line integral of type 2,"},{"Start":"00:05.805 ","End":"00:09.735","Text":"and it\u0027s in 2 dimensions is just x and y."},{"Start":"00:09.735 ","End":"00:13.740","Text":"Along each of the following curves, it\u0027s going to be 4 of them,"},{"Start":"00:13.740 ","End":"00:16.935","Text":"and I\u0027ll do them 1 at a time."},{"Start":"00:16.935 ","End":"00:20.400","Text":"This is not really precise notation,"},{"Start":"00:20.400 ","End":"00:25.020","Text":"what we really mean is the integral along a curve c,"},{"Start":"00:25.020 ","End":"00:28.095","Text":"and c will take us from here to here,"},{"Start":"00:28.095 ","End":"00:30.645","Text":"will make it as a parametrized curve,"},{"Start":"00:30.645 ","End":"00:35.850","Text":"and will be 4 different possibilities for each of these subsections."},{"Start":"00:35.850 ","End":"00:39.905","Text":"Let\u0027s start with a, y squared equals x."},{"Start":"00:39.905 ","End":"00:44.850","Text":"First of all, notice that these points, 1,"},{"Start":"00:44.850 ","End":"00:48.050","Text":"1, and 4, 2 are really on the curve."},{"Start":"00:48.050 ","End":"00:52.849","Text":"In fact, it\u0027s easier to write it as x equals y squared."},{"Start":"00:52.849 ","End":"00:59.490","Text":"You can see that 1 squared is 1 and 2 squared is 4,"},{"Start":"01:00.380 ","End":"01:05.410","Text":"so the parameter will go from 1-2."},{"Start":"01:05.900 ","End":"01:08.205","Text":"When x is y squared,"},{"Start":"01:08.205 ","End":"01:13.800","Text":"easiest thing to do is to take y as t. What we\u0027ll do is,"},{"Start":"01:13.800 ","End":"01:19.500","Text":"we\u0027ll describe c as x equals,"},{"Start":"01:19.500 ","End":"01:20.985","Text":"I\u0027ll write that in a moment,"},{"Start":"01:20.985 ","End":"01:26.060","Text":"y equals t. Since x is y squared,"},{"Start":"01:26.060 ","End":"01:32.270","Text":"then x is t squared and y equals t. The parameter goes from,"},{"Start":"01:32.270 ","End":"01:34.635","Text":"since y is the parameter t,"},{"Start":"01:34.635 ","End":"01:39.780","Text":"from 1-2, so I\u0027ll write it like that."},{"Start":"01:39.780 ","End":"01:44.630","Text":"Then in b, c, and d we\u0027ll have a different curve at each time,"},{"Start":"01:44.630 ","End":"01:47.105","Text":"but it will be the same expression,"},{"Start":"01:47.105 ","End":"01:49.440","Text":"line integral along c,"},{"Start":"01:50.150 ","End":"01:53.930","Text":"and each of them will take us from here to here."},{"Start":"01:53.930 ","End":"01:57.310","Text":"Let\u0027s get started with this 1 then."},{"Start":"01:57.310 ","End":"02:00.650","Text":"Now, this integral becomes, because it\u0027s parametrized,"},{"Start":"02:00.650 ","End":"02:04.430","Text":"we just take the upper and lower limits for the parameter,"},{"Start":"02:04.430 ","End":"02:06.185","Text":"so it\u0027s from 1-2."},{"Start":"02:06.185 ","End":"02:10.255","Text":"Maybe I\u0027ll emphasize it by putting t equals 1-2,"},{"Start":"02:10.255 ","End":"02:12.585","Text":"then we just substitute,"},{"Start":"02:12.585 ","End":"02:15.810","Text":"x is t squared,"},{"Start":"02:15.810 ","End":"02:22.215","Text":"y is t, and then we need dx."},{"Start":"02:22.215 ","End":"02:28.175","Text":"From here, if I differentiate I\u0027ll get 1 dx,"},{"Start":"02:28.175 ","End":"02:32.320","Text":"the d of both of them on here, I\u0027ll have 2tdt."},{"Start":"02:33.620 ","End":"02:36.060","Text":"As for y, well,"},{"Start":"02:36.060 ","End":"02:39.960","Text":"y equals t, so dy equals dt."},{"Start":"02:39.960 ","End":"02:44.350","Text":"Here I need dx, which is 2tdt,"},{"Start":"02:44.930 ","End":"02:51.014","Text":"so 2tdt plus y,"},{"Start":"02:51.014 ","End":"02:54.570","Text":"which is t minus x,"},{"Start":"02:54.570 ","End":"02:56.385","Text":"which is t squared,"},{"Start":"02:56.385 ","End":"03:00.405","Text":"and dy is just dt."},{"Start":"03:00.405 ","End":"03:10.095","Text":"What we want do is just write this as 1 single dt of integral from 1-2,"},{"Start":"03:10.095 ","End":"03:13.955","Text":"and we\u0027ll see what we can take out."},{"Start":"03:13.955 ","End":"03:20.160","Text":"Multiplying 2t times t squared is 2t cubed."},{"Start":"03:20.160 ","End":"03:24.630","Text":"Here, I have plus 2t squared,"},{"Start":"03:24.630 ","End":"03:29.060","Text":"but the 2t squared from here, minus the t squared"},{"Start":"03:29.060 ","End":"03:33.880","Text":"from here, will just leave us with 1t squared."},{"Start":"03:33.880 ","End":"03:38.880","Text":"Then what we\u0027re left with is still plus t,"},{"Start":"03:38.880 ","End":"03:41.950","Text":"and then all this is dt."},{"Start":"03:42.680 ","End":"03:49.320","Text":"Straightforward integral, 2t cubed becomes t^4,"},{"Start":"03:49.320 ","End":"03:53.640","Text":"divide by 4, 2 over 4 is a half."},{"Start":"03:53.640 ","End":"03:57.955","Text":"Next 1, I need t cubed divide by 3."},{"Start":"03:57.955 ","End":"04:02.915","Text":"Here I need t squared, and I divide by 2,"},{"Start":"04:02.915 ","End":"04:08.460","Text":"so this is what we have, and evaluate it from 1-2."},{"Start":"04:08.840 ","End":"04:11.520","Text":"Let\u0027s see what we get."},{"Start":"04:11.520 ","End":"04:13.950","Text":"If we put in 2,"},{"Start":"04:13.950 ","End":"04:20.265","Text":"we have 1/2 times 2^4 is"},{"Start":"04:20.265 ","End":"04:30.345","Text":"16 plus 1/3 times 8 plus 1/2 times 4,"},{"Start":"04:30.345 ","End":"04:32.475","Text":"that\u0027s for the 2."},{"Start":"04:32.475 ","End":"04:39.945","Text":"For the 1, we just get a 1/2 plus a 1/3 plus a 1/2."},{"Start":"04:39.945 ","End":"04:43.650","Text":"What does this come out to be?"},{"Start":"04:43.650 ","End":"04:54.350","Text":"The first 1, we\u0027ll get 8 plus 2 from here,"},{"Start":"04:54.350 ","End":"05:02.280","Text":"that\u0027s altogether 10, 8/3 is 2 2/3."},{"Start":"05:02.280 ","End":"05:04.370","Text":"Altogether, I\u0027ll do it in mixed numbers,"},{"Start":"05:04.370 ","End":"05:08.070","Text":"12 and 2/3 for the first 1,"},{"Start":"05:09.310 ","End":"05:15.090","Text":"1/2 and a 1/2 is 1, 1 1/3."},{"Start":"05:15.090 ","End":"05:23.390","Text":"Let\u0027s see, that comes out to be 11 and 1/3, could leave the answer like that,"},{"Start":"05:23.390 ","End":"05:27.170","Text":"or we could write it as an improper fraction,"},{"Start":"05:27.170 ","End":"05:33.780","Text":"11 times 3 plus 1 is 34, 34/3."},{"Start":"05:33.780 ","End":"05:34.920","Text":"Either 1 of these,"},{"Start":"05:34.920 ","End":"05:36.465","Text":"I\u0027ll just go with this 1,"},{"Start":"05:36.465 ","End":"05:40.260","Text":"that\u0027s the answer. We\u0027re done for part a."}],"ID":8795},{"Watched":false,"Name":"Exercise 7 Part b","Duration":"5m 28s","ChapterTopicVideoID":8697,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8697.jpeg","UploadDate":"2017-02-13T05:03:45.8470000","DurationForVideoObject":"PT5M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.050","Text":"That was part a. Now, let\u0027s get on to part b. I erased what I don\u0027t need."},{"Start":"00:07.050 ","End":"00:09.960","Text":"I don\u0027t need this."},{"Start":"00:09.960 ","End":"00:12.120","Text":"We\u0027re still going to have x as a function of t,"},{"Start":"00:12.120 ","End":"00:13.830","Text":"y as a function of t,"},{"Start":"00:13.830 ","End":"00:17.070","Text":"and t is going to go from something to something."},{"Start":"00:17.070 ","End":"00:22.110","Text":"I can already tell you that t is going from"},{"Start":"00:22.110 ","End":"00:28.880","Text":"0-1, because I\u0027m going to use the formula for a line segment joining 2 points."},{"Start":"00:28.880 ","End":"00:32.540","Text":"I\u0027ll remind you, in general, that if I have 2 points,"},{"Start":"00:32.540 ","End":"00:34.265","Text":"let say x naught,"},{"Start":"00:34.265 ","End":"00:39.200","Text":"y naught, and x_1, y_1,"},{"Start":"00:39.200 ","End":"00:41.555","Text":"which in our case will turn out,"},{"Start":"00:41.555 ","End":"00:42.890","Text":"I mean they are 1,"},{"Start":"00:42.890 ","End":"00:45.455","Text":"1, and 4, 2,"},{"Start":"00:45.455 ","End":"00:47.990","Text":"but I\u0027ll write the general formula, then"},{"Start":"00:47.990 ","End":"00:52.789","Text":"the parametric curve is given by the several formulas."},{"Start":"00:52.789 ","End":"01:02.780","Text":"One way is to say that x equals the first point plus t"},{"Start":"01:02.780 ","End":"01:11.860","Text":"times the difference between the axes x_1 minus x_0,"},{"Start":"01:11.860 ","End":"01:22.714","Text":"and y is the first y plus t times the difference, second y minus the first y,"},{"Start":"01:22.714 ","End":"01:30.040","Text":"in both values of the parameter is from 0-1."},{"Start":"01:30.920 ","End":"01:39.945","Text":"In our case, what we\u0027ll get is x"},{"Start":"01:39.945 ","End":"01:48.900","Text":"equals the first point is 1, and the difference,"},{"Start":"01:48.900 ","End":"01:57.630","Text":"4 minus 1 is 3 times t, so it\u0027s 1 plus 3t,"},{"Start":"01:57.630 ","End":"02:02.590","Text":"and y is equal to,"},{"Start":"02:03.230 ","End":"02:12.435","Text":"it\u0027s the first y, which is 1, plus the difference 2 minus 1 is 1 times"},{"Start":"02:12.435 ","End":"02:17.160","Text":"t. What I get is here,"},{"Start":"02:17.160 ","End":"02:24.625","Text":"1 plus 3t and here 1 plus t. We can check,"},{"Start":"02:24.625 ","End":"02:29.194","Text":"when t is 0, we get the point 1, 1,"},{"Start":"02:29.194 ","End":"02:33.124","Text":"which is fine/ When t is 1,"},{"Start":"02:33.124 ","End":"02:36.290","Text":"we get 1 plus 3 is 4."},{"Start":"02:36.290 ","End":"02:39.020","Text":"Here, 1 plus 1 is 2, 4,"},{"Start":"02:39.020 ","End":"02:44.499","Text":"2. Here, this works."},{"Start":"02:45.560 ","End":"02:49.545","Text":"I know for later on, I\u0027ll need dx and dy."},{"Start":"02:49.545 ","End":"02:55.235","Text":"Here dx, the derivative of this is just 3dt, and"},{"Start":"02:55.235 ","End":"03:03.250","Text":"dy will just equal 1dt or simply dt."},{"Start":"03:03.250 ","End":"03:06.965","Text":"I\u0027ve got everything I need now to do this."},{"Start":"03:06.965 ","End":"03:11.970","Text":"This time the parameter goes from 0-1."},{"Start":"03:11.970 ","End":"03:18.375","Text":"That\u0027s the t. x is 1 plus 3t,"},{"Start":"03:18.375 ","End":"03:22.335","Text":"y is 1 plus t,"},{"Start":"03:22.335 ","End":"03:24.915","Text":"all of this is dx,"},{"Start":"03:24.915 ","End":"03:34.980","Text":"3dt and the second one y is 1 plus t minus x. I\u0027ll just make them both minuses,"},{"Start":"03:34.980 ","End":"03:39.615","Text":"minus 1 minus 3t and dy."},{"Start":"03:39.615 ","End":"03:43.060","Text":"Here it is, it\u0027s just dt."},{"Start":"03:43.250 ","End":"03:51.655","Text":"I want to write this as just one integral from 0-1."},{"Start":"03:51.655 ","End":"03:55.175","Text":"I could have emphasized that this is t equals,"},{"Start":"03:55.175 ","End":"03:57.725","Text":"well, we know it\u0027s t. Let\u0027s see."},{"Start":"03:57.725 ","End":"04:00.665","Text":"I want to collect together everything."},{"Start":"04:00.665 ","End":"04:06.450","Text":"Let\u0027s collect the terms with t. Now remember, this is a 3 here."},{"Start":"04:06.450 ","End":"04:10.440","Text":"Well, 3t plus t is 4t,"},{"Start":"04:10.440 ","End":"04:15.705","Text":"4t times 3 is 12t."},{"Start":"04:15.705 ","End":"04:19.100","Text":"I have a 12t from here."},{"Start":"04:19.100 ","End":"04:21.395","Text":"Maybe I\u0027ll just do each one separately."},{"Start":"04:21.395 ","End":"04:26.255","Text":"12t and here 1 plus 1 is 2, times 3 is 6."},{"Start":"04:26.255 ","End":"04:28.310","Text":"This is 12t plus 6."},{"Start":"04:28.310 ","End":"04:33.195","Text":"The second bit is 1 minus 1 is nothing,"},{"Start":"04:33.195 ","End":"04:37.710","Text":"t minus 3t is minus 2t."},{"Start":"04:37.960 ","End":"04:41.545","Text":"Combining these, what do I get?"},{"Start":"04:41.545 ","End":"04:52.045","Text":"10t plus 6dt. This integral of 10t,"},{"Start":"04:52.045 ","End":"04:57.350","Text":"I raise the power to t squared divide by the 2,"},{"Start":"04:57.350 ","End":"05:00.065","Text":"so that\u0027s just 5, constant."},{"Start":"05:00.065 ","End":"05:04.795","Text":"So it gives me 6t from 0-1,"},{"Start":"05:04.795 ","End":"05:12.320","Text":"and if I put in 0, I don\u0027t get anything."},{"Start":"05:12.320 ","End":"05:17.690","Text":"If I put in the 1, I just get 5 plus 6."},{"Start":"05:17.690 ","End":"05:25.020","Text":"This is 11, and this is the answer to part b."},{"Start":"05:25.020 ","End":"05:28.360","Text":"On the next clip, we\u0027ll do part c."}],"ID":8796},{"Watched":false,"Name":"Exercise 7 Part c","Duration":"8m 23s","ChapterTopicVideoID":8698,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8698.jpeg","UploadDate":"2017-02-13T05:05:48.8530000","DurationForVideoObject":"PT8M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.080 ","End":"00:09.840","Text":"Now, we come to part c. I\u0027ve included a sketch where the points 1,"},{"Start":"00:09.840 ","End":"00:12.240","Text":"1, and 4, 2 are."},{"Start":"00:12.240 ","End":"00:15.150","Text":"In the previous part b,"},{"Start":"00:15.150 ","End":"00:16.410","Text":"we took a line segment."},{"Start":"00:16.410 ","End":"00:17.760","Text":"It would be like I joined 1,"},{"Start":"00:17.760 ","End":"00:19.815","Text":"1 to 4, 2."},{"Start":"00:19.815 ","End":"00:24.960","Text":"In part c, it\u0027s like I make a stopover at the point 1, 2,"},{"Start":"00:24.960 ","End":"00:26.355","Text":"which is let see,"},{"Start":"00:26.355 ","End":"00:29.700","Text":"just on the same horizontal line as this,"},{"Start":"00:29.700 ","End":"00:32.850","Text":"on the same vertical line as this."},{"Start":"00:32.850 ","End":"00:35.310","Text":"I go from here to here,"},{"Start":"00:35.310 ","End":"00:37.605","Text":"and then from here to here."},{"Start":"00:37.605 ","End":"00:42.325","Text":"Vertical then horizontal and not directly."},{"Start":"00:42.325 ","End":"00:47.370","Text":"We\u0027ll get a different answer then in part b, probably."},{"Start":"00:47.370 ","End":"00:52.010","Text":"Let\u0027s see if we can parametrize each of these 2."},{"Start":"00:52.010 ","End":"00:55.610","Text":"What I\u0027m going to say is that this whole curve is c,"},{"Start":"00:55.610 ","End":"01:00.080","Text":"but this might be the curve C_1 and this might be the curve C_2."},{"Start":"01:00.080 ","End":"01:05.630","Text":"What we\u0027re going to do is break it up to the integral over C_1 of whatever it is,"},{"Start":"01:05.630 ","End":"01:10.965","Text":"plus the integral along C_2 of the same thing as here."},{"Start":"01:10.965 ","End":"01:14.345","Text":"Let\u0027s parametrize each one of them."},{"Start":"01:14.345 ","End":"01:21.065","Text":"Now, the easiest way to do it is to parametrize C_1 as follows."},{"Start":"01:21.065 ","End":"01:26.480","Text":"We just keep the same x and move with y,"},{"Start":"01:26.480 ","End":"01:28.070","Text":"let me just maybe mark some points here,"},{"Start":"01:28.070 ","End":"01:30.890","Text":"this is 1, this is 2,"},{"Start":"01:30.890 ","End":"01:35.230","Text":"this is 1, and this is 4."},{"Start":"01:35.230 ","End":"01:42.940","Text":"For C_1, I could take that x is the constant one,"},{"Start":"01:42.940 ","End":"01:46.515","Text":"and y equals t,"},{"Start":"01:46.515 ","End":"01:53.255","Text":"which is just the value of the y here, goes from 1 to 2."},{"Start":"01:53.255 ","End":"01:57.995","Text":"I can say t goes from 1 to 2."},{"Start":"01:57.995 ","End":"01:59.735","Text":"For the second part,"},{"Start":"01:59.735 ","End":"02:05.240","Text":"the horizontal part, x will be the one that\u0027s moving."},{"Start":"02:05.240 ","End":"02:10.970","Text":"X will be the t. Y will be constantly 2."},{"Start":"02:10.970 ","End":"02:13.055","Text":"The value of x, which is t,"},{"Start":"02:13.055 ","End":"02:15.890","Text":"will go from 1 to 4."},{"Start":"02:17.680 ","End":"02:21.980","Text":"It\u0027ll be a different t because it\u0027s different curve."},{"Start":"02:21.980 ","End":"02:25.100","Text":"Let\u0027s see if we can translate this."},{"Start":"02:25.100 ","End":"02:26.990","Text":"Well, let\u0027s do each one separately."},{"Start":"02:26.990 ","End":"02:30.975","Text":"The integral over C_1."},{"Start":"02:30.975 ","End":"02:33.855","Text":"Let\u0027s do that one first."},{"Start":"02:33.855 ","End":"02:40.115","Text":"Maybe I\u0027ll call this one asterisk and the other one double asterisk."},{"Start":"02:40.115 ","End":"02:43.685","Text":"Let\u0027s, first of all, do the asterisk."},{"Start":"02:43.685 ","End":"02:46.085","Text":"That\u0027s this one."},{"Start":"02:46.085 ","End":"02:54.710","Text":"We got the integral from t equals 1 to 2."},{"Start":"02:54.710 ","End":"03:02.990","Text":"X plus y is 1 plus t. We\u0027ll need dx and dy."},{"Start":"03:02.990 ","End":"03:04.520","Text":"Well, x is a constant,"},{"Start":"03:04.520 ","End":"03:08.615","Text":"so dx is just the derivative of this 0 dt,"},{"Start":"03:08.615 ","End":"03:17.240","Text":"which is just 0, and dy is equal to dt."},{"Start":"03:17.240 ","End":"03:20.120","Text":"For the first one, you might as well do the second one already."},{"Start":"03:20.120 ","End":"03:22.220","Text":"Well, I\u0027m differentiating here."},{"Start":"03:22.220 ","End":"03:30.860","Text":"Dx will equal dt and dy will equal 0,"},{"Start":"03:30.860 ","End":"03:34.889","Text":"0 dt, but it\u0027s 0."},{"Start":"03:36.320 ","End":"03:42.705","Text":"When we get here, we get 1 plus t. I\u0027ll write it,"},{"Start":"03:42.705 ","End":"03:51.900","Text":"0 dt that I\u0027m just omitting it so you can see where I\u0027m coming from plus y minus x,"},{"Start":"03:51.900 ","End":"03:56.805","Text":"which is at t minus 1."},{"Start":"03:56.805 ","End":"04:03.990","Text":"Dy along this curve is equal to just dt."},{"Start":"04:08.990 ","End":"04:12.935","Text":"I\u0027ll continue on the next line."},{"Start":"04:12.935 ","End":"04:16.175","Text":"This will equal, this is 0."},{"Start":"04:16.175 ","End":"04:25.590","Text":"It\u0027s just the integral from 1 to 2 of t minus 1 dt."},{"Start":"04:26.980 ","End":"04:33.590","Text":"This equals 1.5 t"},{"Start":"04:33.590 ","End":"04:39.030","Text":"squared minus t from 1 to 2."},{"Start":"04:39.100 ","End":"04:43.415","Text":"What we get, we plug in 2."},{"Start":"04:43.415 ","End":"04:48.000","Text":"We get 1/2 of t squared is 1/2 of 4,"},{"Start":"04:48.000 ","End":"04:52.780","Text":"is 2 minus 2 is 0."},{"Start":"04:53.030 ","End":"04:59.880","Text":"That it\u0027s 2 minus 2."},{"Start":"04:59.880 ","End":"05:02.520","Text":"I have to subtract."},{"Start":"05:02.520 ","End":"05:04.770","Text":"What happens when I plug in 1,"},{"Start":"05:04.770 ","End":"05:09.150","Text":"which is 1/2 minus 1?"},{"Start":"05:09.150 ","End":"05:12.675","Text":"This is 0 minus 1/2,"},{"Start":"05:12.675 ","End":"05:14.670","Text":"so this is 1/2."},{"Start":"05:14.670 ","End":"05:19.280","Text":"That\u0027s the first one and now the second one,"},{"Start":"05:19.280 ","End":"05:27.340","Text":"the double asterisk is the integral this time it\u0027s from"},{"Start":"05:27.340 ","End":"05:36.590","Text":"1 to 4 and we have the same thing,"},{"Start":"05:36.590 ","End":"05:39.590","Text":"x plus y, but it\u0027s different."},{"Start":"05:39.590 ","End":"05:43.890","Text":"X plus y here is t plus 2."},{"Start":"05:44.980 ","End":"05:48.530","Text":"Let me just say this is the asterisk and"},{"Start":"05:48.530 ","End":"05:51.380","Text":"this is the double asterisk so we\u0027re reading off here,"},{"Start":"05:51.380 ","End":"05:54.245","Text":"x plus y, t plus 2."},{"Start":"05:54.245 ","End":"05:59.960","Text":"Then dx is, where is it?"},{"Start":"05:59.960 ","End":"06:03.935","Text":"Here, dt plus."},{"Start":"06:03.935 ","End":"06:06.755","Text":"Then we need y minus x,"},{"Start":"06:06.755 ","End":"06:12.080","Text":"which is 2 minus"},{"Start":"06:12.080 ","End":"06:24.200","Text":"t. That is dy,"},{"Start":"06:24.200 ","End":"06:27.900","Text":"which is 0 dt."},{"Start":"06:28.850 ","End":"06:31.430","Text":"The second part is nothing."},{"Start":"06:31.430 ","End":"06:34.330","Text":"We just get the integral,"},{"Start":"06:34.330 ","End":"06:41.590","Text":"write it again, 1 to 4 of t plus 2 dt."},{"Start":"06:43.780 ","End":"06:50.180","Text":"This is equal to 1.5 t"},{"Start":"06:50.180 ","End":"06:56.300","Text":"squared plus 2t from 1 to 4."},{"Start":"06:56.300 ","End":"06:59.285","Text":"Let\u0027s see. When I plug in 4,"},{"Start":"06:59.285 ","End":"07:04.910","Text":"I get 4 squared over 2 is 8,"},{"Start":"07:04.910 ","End":"07:11.475","Text":"2 times 4 is 8 minus,"},{"Start":"07:11.475 ","End":"07:17.805","Text":"I plug in 1, I get 1.5 plus 2."},{"Start":"07:17.805 ","End":"07:26.535","Text":"What do I get? 16 minus 2.5 is 13 and 1/2."},{"Start":"07:26.535 ","End":"07:28.830","Text":"For those who like mixed numbers,"},{"Start":"07:28.830 ","End":"07:35.800","Text":"and if you like improper fractions, that\u0027s 27/2."},{"Start":"07:37.430 ","End":"07:42.920","Text":"Let\u0027s see, this is the answer for the asterisk."},{"Start":"07:42.920 ","End":"07:46.415","Text":"This is the answer for the double asterisk."},{"Start":"07:46.415 ","End":"07:48.560","Text":"I have to add them together."},{"Start":"07:48.560 ","End":"07:53.475","Text":"I\u0027ve got, maybe I\u0027ll use the 13.5,"},{"Start":"07:53.475 ","End":"07:56.190","Text":"doesn\u0027t matter, this plus this."},{"Start":"07:56.190 ","End":"08:02.250","Text":"13.5 plus 1.5 is equal"},{"Start":"08:02.250 ","End":"08:09.765","Text":"to 14 or if you did it as 28 over 2, it would still be 14."},{"Start":"08:09.765 ","End":"08:18.245","Text":"This is the answer for the integral along this broken path."},{"Start":"08:18.245 ","End":"08:22.800","Text":"That concludes part c. We still have part d to do."}],"ID":8797},{"Watched":false,"Name":"Exercise 7 Part d","Duration":"6m 7s","ChapterTopicVideoID":8699,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8699.jpeg","UploadDate":"2017-02-13T05:07:17.1570000","DurationForVideoObject":"PT6M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.410 ","End":"00:04.875","Text":"Now we come to part D of this exercise."},{"Start":"00:04.875 ","End":"00:11.175","Text":"In part D, in some ways it\u0027s easier because we already have it in parameterized form."},{"Start":"00:11.175 ","End":"00:15.360","Text":"In all the others we had to figure out parametric representation."},{"Start":"00:15.360 ","End":"00:19.560","Text":"Let\u0027s just check that when t is 0 and t is 1,"},{"Start":"00:19.560 ","End":"00:22.380","Text":"we really do get our 2 points. Let\u0027s see."},{"Start":"00:22.380 ","End":"00:24.300","Text":"When t is 0,"},{"Start":"00:24.300 ","End":"00:26.460","Text":"then what is x, y equal?"},{"Start":"00:26.460 ","End":"00:30.495","Text":"X when it\u0027s 0 this is 1,"},{"Start":"00:30.495 ","End":"00:32.760","Text":"and when t is 0,"},{"Start":"00:32.760 ","End":"00:35.385","Text":"y is 1 and that\u0027s okay."},{"Start":"00:35.385 ","End":"00:41.860","Text":"When t is 1, then we get 2 plus 1 plus 1."},{"Start":"00:41.930 ","End":"00:45.810","Text":"X, y is 4,"},{"Start":"00:45.810 ","End":"00:51.180","Text":"2 plus 1 plus 1. What else?"},{"Start":"00:51.180 ","End":"00:54.780","Text":"1 squared plus 1 is 2,"},{"Start":"00:54.780 ","End":"00:58.080","Text":"and that\u0027s also okay."},{"Start":"00:58.080 ","End":"01:01.620","Text":"We have from 1,1-4,2."},{"Start":"01:01.620 ","End":"01:06.165","Text":"What I will still need is the dx and the dy."},{"Start":"01:06.165 ","End":"01:08.760","Text":"I\u0027ll just write them under here."},{"Start":"01:08.760 ","End":"01:17.420","Text":"Dx is the derivative of this is 4t plus 1 and that\u0027s dt,"},{"Start":"01:17.420 ","End":"01:22.680","Text":"and dy is just 2tdt."},{"Start":"01:22.730 ","End":"01:25.710","Text":"Now we have everything we need."},{"Start":"01:25.710 ","End":"01:33.065","Text":"We write this as an integral according to parameter t from 0-1."},{"Start":"01:33.065 ","End":"01:40.065","Text":"I\u0027ll even stress that this is t, goes from 0-1."},{"Start":"01:40.065 ","End":"01:46.120","Text":"X plus y, I\u0027m just reading it off here,"},{"Start":"01:46.220 ","End":"01:53.310","Text":"x plus y is 2t squared plus t plus 1 is x,"},{"Start":"01:53.310 ","End":"01:57.330","Text":"and then y is t squared plus 1,"},{"Start":"01:57.330 ","End":"02:02.950","Text":"and dx is 4t plus 1dt."},{"Start":"02:03.530 ","End":"02:05.690","Text":"That\u0027s just the first part."},{"Start":"02:05.690 ","End":"02:08.630","Text":"Now the second part, y minus x,"},{"Start":"02:08.630 ","End":"02:12.620","Text":"t squared plus 1 minus all of these,"},{"Start":"02:12.620 ","End":"02:13.970","Text":"let\u0027s put them all with a minus,"},{"Start":"02:13.970 ","End":"02:18.815","Text":"minus 2t squared, minus t, minus 1."},{"Start":"02:18.815 ","End":"02:28.610","Text":"Here we have a dy which is 2tdt."},{"Start":"02:28.610 ","End":"02:33.365","Text":"You just have to do a bit of algebra here to collect like terms."},{"Start":"02:33.365 ","End":"02:35.975","Text":"We\u0027ll get the integral from 0-1."},{"Start":"02:35.975 ","End":"02:38.905","Text":"Let\u0027s see what we can do."},{"Start":"02:38.905 ","End":"02:43.510","Text":"Perhaps first of all simplify this."},{"Start":"02:44.180 ","End":"02:54.780","Text":"I\u0027ll just rewrite this as 3t squared plus t plus 2."},{"Start":"02:54.780 ","End":"02:59.210","Text":"This 1 we can collect like terms. What do we get?"},{"Start":"02:59.210 ","End":"03:07.620","Text":"T squared minus 2t squared is minus t squared,"},{"Start":"03:07.870 ","End":"03:13.885","Text":"and then the minus t and the plus 1, minus 1 cancels."},{"Start":"03:13.885 ","End":"03:17.265","Text":"Let\u0027s start multiplying out."},{"Start":"03:17.265 ","End":"03:21.875","Text":"I\u0027ll begin by multiplying 4t by all of these."},{"Start":"03:21.875 ","End":"03:31.620","Text":"I\u0027ve got 12t cubed plus 4t squared plus 8t, then plus 1."},{"Start":"03:31.620 ","End":"03:33.915","Text":"I just have to write this again,"},{"Start":"03:33.915 ","End":"03:38.590","Text":"plus 3t squared plus t plus 2."},{"Start":"03:38.590 ","End":"03:40.580","Text":"That\u0027s this times this."},{"Start":"03:40.580 ","End":"03:43.670","Text":"Now here I just have to multiply by 2t."},{"Start":"03:43.670 ","End":"03:45.215","Text":"Everything\u0027s going to be negative."},{"Start":"03:45.215 ","End":"03:49.865","Text":"It\u0027s minus 2t times t squared is 2t cubed,"},{"Start":"03:49.865 ","End":"03:59.640","Text":"and then minus 2t squared and all of this dt."},{"Start":"03:59.640 ","End":"04:04.155","Text":"Let\u0027s collect first of all the terms with t cubed, I have 1 here,"},{"Start":"04:04.155 ","End":"04:06.165","Text":"and I have 1 here,"},{"Start":"04:06.165 ","End":"04:10.800","Text":"so 12 minus 10t cubed."},{"Start":"04:10.800 ","End":"04:12.990","Text":"Next, let\u0027s go for t squared,"},{"Start":"04:12.990 ","End":"04:16.070","Text":"I have here, and I have here,"},{"Start":"04:16.070 ","End":"04:18.355","Text":"and I have here."},{"Start":"04:18.355 ","End":"04:27.220","Text":"That will give me 4 plus 3 minus 2, that\u0027s 5t squared."},{"Start":"04:27.220 ","End":"04:29.810","Text":"There\u0027s not much else."},{"Start":"04:29.810 ","End":"04:33.845","Text":"I have a t here and a t here,"},{"Start":"04:33.845 ","End":"04:39.109","Text":"8 plus 1 is 9t, and a constant,"},{"Start":"04:39.109 ","End":"04:42.080","Text":"I just have the 2dt"},{"Start":"04:42.080 ","End":"04:49.580","Text":"from 0-1."},{"Start":"04:49.580 ","End":"04:52.520","Text":"Next, just do the integration,"},{"Start":"04:52.520 ","End":"04:57.260","Text":"10 over 4 is like 5"},{"Start":"04:57.260 ","End":"05:06.810","Text":"over 2t^4,5 over 3t cubed,"},{"Start":"05:06.810 ","End":"05:12.630","Text":"9 over 2t squared and 2t."},{"Start":"05:12.630 ","End":"05:16.380","Text":"This from 0-1."},{"Start":"05:16.380 ","End":"05:19.615","Text":"The 0 doesn\u0027t give us anything, It\u0027s all 0."},{"Start":"05:19.615 ","End":"05:21.745","Text":"Just have to plug in the 1."},{"Start":"05:21.745 ","End":"05:24.760","Text":"I have a fraction, 5 over 2,"},{"Start":"05:24.760 ","End":"05:26.685","Text":"plus 5 over 3,"},{"Start":"05:26.685 ","End":"05:31.125","Text":"plus 9 over 2, plus 2."},{"Start":"05:31.125 ","End":"05:37.100","Text":"Let\u0027s see. If I take the 5 over 2 and 9 over 2,"},{"Start":"05:37.100 ","End":"05:39.725","Text":"I get 14 over 2, which is 7."},{"Start":"05:39.725 ","End":"05:43.410","Text":"7 plus 2 is 9,"},{"Start":"05:44.830 ","End":"05:48.980","Text":"and 5/3 is 1 and 2/3."},{"Start":"05:48.980 ","End":"05:52.680","Text":"I make that 9/2,10 and 2/3."},{"Start":"05:53.560 ","End":"05:55.850","Text":"Or if you prefer,"},{"Start":"05:55.850 ","End":"05:59.135","Text":"this could be written as 32/3."},{"Start":"05:59.135 ","End":"06:01.520","Text":"I\u0027ll choose this 1,"},{"Start":"06:01.520 ","End":"06:03.545","Text":"and we are done."},{"Start":"06:03.545 ","End":"06:07.260","Text":"That\u0027s part D, so that finishes this exercise."}],"ID":8798},{"Watched":false,"Name":"Exercise 8","Duration":"15m 13s","ChapterTopicVideoID":8700,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8700.jpeg","UploadDate":"2017-02-13T05:11:21.8430000","DurationForVideoObject":"PT15M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.370","Text":"In this exercise, we need to compute the type 2 line integral along the path C,"},{"Start":"00:08.370 ","End":"00:10.380","Text":"where C is described as in the picture."},{"Start":"00:10.380 ","End":"00:12.930","Text":"I guess I need to put an extra arrow here."},{"Start":"00:12.930 ","End":"00:20.310","Text":"C is this whole thing but obviously we\u0027re going to be breaking it up into 3 parts."},{"Start":"00:20.310 ","End":"00:23.595","Text":"This will be maybe C_1,"},{"Start":"00:23.595 ","End":"00:25.665","Text":"this will be C_2,"},{"Start":"00:25.665 ","End":"00:27.180","Text":"and this will be C_3."},{"Start":"00:27.180 ","End":"00:29.550","Text":"We can\u0027t do it all at once."},{"Start":"00:29.550 ","End":"00:33.585","Text":"We want a parametric form of each of them."},{"Start":"00:33.585 ","End":"00:36.105","Text":"Let\u0027s see."},{"Start":"00:36.105 ","End":"00:37.230","Text":"Let\u0027s mark some of the points."},{"Start":"00:37.230 ","End":"00:43.590","Text":"This is the origin so that\u0027s 00."},{"Start":"00:43.590 ","End":"00:45.665","Text":"Let me start with the more difficult one,"},{"Start":"00:45.665 ","End":"00:53.145","Text":"let\u0027s start with C_3. I\u0027ll mark it."},{"Start":"00:53.145 ","End":"00:59.040","Text":"This is 1, 2 and the origin is, I\u0027ll write it here."},{"Start":"00:59.040 ","End":"01:06.255","Text":"C_3 goes from 1, 2-0, 0."},{"Start":"01:06.255 ","End":"01:11.195","Text":"One way of parametrizing a line from 2 points is"},{"Start":"01:11.195 ","End":"01:16.970","Text":"just to say that what x equals and y equals,"},{"Start":"01:16.970 ","End":"01:22.019","Text":"we take the component from the first point,"},{"Start":"01:22.019 ","End":"01:24.150","Text":"in this case it\u0027s 1."},{"Start":"01:24.150 ","End":"01:30.980","Text":"Then we take t times the difference from the second minus the first."},{"Start":"01:30.980 ","End":"01:35.585","Text":"it\u0027s 0 minus 1 and the same thing for y."},{"Start":"01:35.585 ","End":"01:37.220","Text":"We take the 1st point,"},{"Start":"01:37.220 ","End":"01:43.500","Text":"which is 2 and put t times 0 minus 2."},{"Start":"01:44.360 ","End":"01:47.330","Text":"When we do it with this method,"},{"Start":"01:47.330 ","End":"01:51.360","Text":"it\u0027s always that t goes from 0-1."},{"Start":"01:51.560 ","End":"01:54.185","Text":"We just simplify this."},{"Start":"01:54.185 ","End":"01:56.180","Text":"What does this come out to be?"},{"Start":"01:56.180 ","End":"02:03.210","Text":"This comes out as x equals 1"},{"Start":"02:03.210 ","End":"02:12.370","Text":"minus t and y equals 2 minus 2t."},{"Start":"02:12.680 ","End":"02:17.700","Text":"You can always check by plugging in 0."},{"Start":"02:17.700 ","End":"02:20.450","Text":"When t is 0 we get 1, 2,"},{"Start":"02:20.450 ","End":"02:22.955","Text":"fine and when t is 1,"},{"Start":"02:22.955 ","End":"02:24.350","Text":"we get 0, 0."},{"Start":"02:24.350 ","End":"02:26.700","Text":"Yeah, that\u0027s all fine."},{"Start":"02:27.340 ","End":"02:29.900","Text":"That would be C_3."},{"Start":"02:29.900 ","End":"02:36.175","Text":"I might as well do all the curves first."},{"Start":"02:36.175 ","End":"02:39.300","Text":"I Should have been writing in the solution area."},{"Start":"02:39.300 ","End":"02:43.660","Text":"Well, forgive me for that. Let\u0027s go for C_1."},{"Start":"02:44.650 ","End":"02:49.640","Text":"Now when it\u0027s horizontal or vertical line,"},{"Start":"02:49.640 ","End":"02:53.825","Text":"we don\u0027t have to do it with this method because if it\u0027s horizontal,"},{"Start":"02:53.825 ","End":"02:56.269","Text":"then one of the variables is constant."},{"Start":"02:56.269 ","End":"03:04.670","Text":"In this case, we see that the y is always 0 and it\u0027s just x going from 0-1."},{"Start":"03:04.670 ","End":"03:08.870","Text":"Instead of saying x goes from 0-1,"},{"Start":"03:08.870 ","End":"03:14.465","Text":"I say x equals t and t goes from 0-1."},{"Start":"03:14.465 ","End":"03:23.795","Text":"As for C_2, then the x stays at"},{"Start":"03:23.795 ","End":"03:32.400","Text":"1 and y moves from 0-2."},{"Start":"03:32.400 ","End":"03:39.625","Text":"I let y equals t and let t go from 0-2."},{"Start":"03:39.625 ","End":"03:43.280","Text":"I find this is just a bit easier when I have a vertical or horizontal line."},{"Start":"03:43.280 ","End":"03:47.740","Text":"Of course you could do it the same system as we did here."},{"Start":"03:47.740 ","End":"03:52.610","Text":"Now what we\u0027re going to do is say that the integral over C,"},{"Start":"03:52.610 ","End":"03:56.030","Text":"this one here, we break it up into 3 pieces."},{"Start":"03:56.030 ","End":"04:01.710","Text":"The integral over C_1 plus the"},{"Start":"04:01.710 ","End":"04:10.480","Text":"integral along C_2 plus the integral along C_3 of whatever it is here."},{"Start":"04:11.540 ","End":"04:14.345","Text":"Let\u0027s do them one by one."},{"Start":"04:14.345 ","End":"04:20.070","Text":"The first one, the integral of C_1,"},{"Start":"04:21.040 ","End":"04:23.750","Text":"well, maybe I\u0027ll copy it."},{"Start":"04:23.750 ","End":"04:34.180","Text":"X squared ydx plus xdy is equal to,"},{"Start":"04:34.180 ","End":"04:39.090","Text":"we have C_1 we\u0027re talking about,"},{"Start":"04:39.090 ","End":"04:41.685","Text":"t goes from 0 to 1."},{"Start":"04:41.685 ","End":"04:44.885","Text":"We put 0, 1 if you want to emphasize it,"},{"Start":"04:44.885 ","End":"04:49.390","Text":"like t equals and then we start substituting."},{"Start":"04:49.390 ","End":"04:53.205","Text":"We\u0027re going to need dx and dy."},{"Start":"04:53.205 ","End":"04:56.635","Text":"Let me add those here."},{"Start":"04:56.635 ","End":"05:03.660","Text":"If x is t, then I have that dx is equal to"},{"Start":"05:03.660 ","End":"05:11.780","Text":"dt and dy is equal to 0 dt which is just 0."},{"Start":"05:11.780 ","End":"05:15.180","Text":"I might as well continue for all of them."},{"Start":"05:16.460 ","End":"05:20.040","Text":"Maybe I\u0027ll write them outside."},{"Start":"05:20.040 ","End":"05:22.340","Text":"It was getting a bit crowded."},{"Start":"05:22.340 ","End":"05:29.590","Text":"Here, we\u0027ll get that dx is also 0 because the derivative of 1 is 0,"},{"Start":"05:29.590 ","End":"05:34.490","Text":"0dt and dy will equal dt."},{"Start":"05:35.120 ","End":"05:38.050","Text":"For the curve C_3,"},{"Start":"05:38.050 ","End":"05:39.605","Text":"I\u0027ll do it from here."},{"Start":"05:39.605 ","End":"05:44.840","Text":"We have that dx is equal to minus 1dt,"},{"Start":"05:44.840 ","End":"05:52.345","Text":"which is just minus dt and dy is minus 2dt."},{"Start":"05:52.345 ","End":"05:58.035","Text":"I think we\u0027re all set up for doing this integral."},{"Start":"05:58.035 ","End":"06:02.230","Text":"Back here, we\u0027re doing number 1."},{"Start":"06:07.550 ","End":"06:14.680","Text":"Y is 0, so x squared y dx is 0,"},{"Start":"06:15.200 ","End":"06:19.590","Text":"and dy is also 0."},{"Start":"06:19.590 ","End":"06:25.180","Text":"Basically we get 0."},{"Start":"06:25.970 ","End":"06:30.965","Text":"I\u0027ll write it a bit more in full because I just want to say 0."},{"Start":"06:30.965 ","End":"06:40.500","Text":"The x squared y is 0 because y is 0 and that the dx is just dt."},{"Start":"06:42.080 ","End":"06:50.270","Text":"Then we have that x is equal to t,"},{"Start":"06:50.270 ","End":"06:58.880","Text":"but dy is 0 or even 0dt."},{"Start":"06:58.880 ","End":"07:02.280","Text":"I could write the dt here, I suppose."},{"Start":"07:02.280 ","End":"07:07.115","Text":"It doesn\u0027t hurt. Altogether this is 0."},{"Start":"07:07.115 ","End":"07:12.045","Text":"It\u0027s just the integral of 0, which is 0."},{"Start":"07:12.045 ","End":"07:14.280","Text":"That\u0027s the first one done."},{"Start":"07:14.280 ","End":"07:19.620","Text":"Now let\u0027s go to the next one."},{"Start":"07:19.620 ","End":"07:25.430","Text":"The integral of C_2 of the same thing. Just copy it."},{"Start":"07:25.430 ","End":"07:32.850","Text":"X squared ydx plus xdy and that will equal the integral."},{"Start":"07:32.850 ","End":"07:40.595","Text":"For C_2 I have from 0-2 dt."},{"Start":"07:40.595 ","End":"07:46.765","Text":"Let\u0027s see, x squared y. X squared y is 1 squared t,"},{"Start":"07:46.765 ","End":"07:50.560","Text":"which is t. Then dx."},{"Start":"07:50.560 ","End":"07:56.680","Text":"But dx is 0dt,"},{"Start":"07:56.680 ","End":"08:01.585","Text":"and the other 1 xdy, x is 1,"},{"Start":"08:01.585 ","End":"08:07.420","Text":"dy is dt, 1dt."},{"Start":"08:07.420 ","End":"08:10.840","Text":"Altogether, it\u0027s the integral of just 1dt."},{"Start":"08:10.840 ","End":"08:18.040","Text":"The integral of 1 is just t taken from 0-2,"},{"Start":"08:18.040 ","End":"08:21.985","Text":"just 2 minus 0, which is 2."},{"Start":"08:21.985 ","End":"08:28.420","Text":"Maybe I should highlight the subtotals, C_1, C_2, C_3."},{"Start":"08:28.420 ","End":"08:32.065","Text":"That\u0027s C_1, that\u0027s C_2,"},{"Start":"08:32.065 ","End":"08:34.030","Text":"at the end we have to add all 3 of them up."},{"Start":"08:34.030 ","End":"08:36.055","Text":"Now, we have 1 more."},{"Start":"08:36.055 ","End":"08:42.820","Text":"We have the integral along C_3 of the same thing,"},{"Start":"08:42.820 ","End":"08:46.120","Text":"x squared ydx plus xdy."},{"Start":"08:46.120 ","End":"08:52.554","Text":"For C_3, we have,"},{"Start":"08:52.554 ","End":"08:58.095","Text":"let\u0027s see, there it is, we just see it."},{"Start":"08:58.095 ","End":"09:03.075","Text":"X squared y is going to be"},{"Start":"09:03.075 ","End":"09:09.610","Text":"1 minus t squared times y,"},{"Start":"09:09.610 ","End":"09:11.755","Text":"which is 2 minus 2t."},{"Start":"09:11.755 ","End":"09:19.400","Text":"I need to write that this goes from 0-1."},{"Start":"09:19.710 ","End":"09:23.470","Text":"Then I need the dx,"},{"Start":"09:23.470 ","End":"09:26.750","Text":"which is minus dt."},{"Start":"09:27.210 ","End":"09:30.175","Text":"Then we need the xdy,"},{"Start":"09:30.175 ","End":"09:41.810","Text":"x is 1 minus t. Then dy is minus 2dt."},{"Start":"09:44.700 ","End":"09:48.620","Text":"Let\u0027s see if we can simplify this."},{"Start":"09:57.090 ","End":"10:05.200","Text":"What we have here is 2 minus t. I can take a 2 out."},{"Start":"10:05.200 ","End":"10:08.095","Text":"I\u0027m going to continue on the next line."},{"Start":"10:08.095 ","End":"10:10.255","Text":"I can take a 2 out."},{"Start":"10:10.255 ","End":"10:13.825","Text":"I\u0027ve got the integral from 0-1,"},{"Start":"10:13.825 ","End":"10:25.375","Text":"twice 1 minus t. Then 1 minus t with 1 minus t squared will be 1 minus t cubed."},{"Start":"10:25.375 ","End":"10:31.460","Text":"I have twice 1 minus t cubed,"},{"Start":"10:31.980 ","End":"10:35.260","Text":"and it\u0027s a minus."},{"Start":"10:35.260 ","End":"10:44.410","Text":"Minus this, dt, and put brackets here because you have to put everything dt,"},{"Start":"10:44.410 ","End":"10:50.845","Text":"and here I have minus"},{"Start":"10:50.845 ","End":"10:58.100","Text":"2 plus 2t, and that dt."},{"Start":"10:59.760 ","End":"11:02.859","Text":"I\u0027ll compute this using the formula."},{"Start":"11:02.859 ","End":"11:03.985","Text":"Write it at the side."},{"Start":"11:03.985 ","End":"11:09.130","Text":"That a minus b cubed is a cubed minus 3a"},{"Start":"11:09.130 ","End":"11:15.685","Text":"squared b plus 3ab squared minus b cubed."},{"Start":"11:15.685 ","End":"11:24.380","Text":"So we will get the integral from 0-1 of minus 2,"},{"Start":"11:27.900 ","End":"11:37.000","Text":"1 minus 3t plus 3t squared minus t"},{"Start":"11:37.000 ","End":"11:47.440","Text":"cubed minus 2 plus 2t dt."},{"Start":"11:47.440 ","End":"11:54.820","Text":"We can now compute this in our heads."},{"Start":"11:54.820 ","End":"12:01.355","Text":"Just collect together minus 2,"},{"Start":"12:01.355 ","End":"12:05.880","Text":"with minus 2, will give me minus 4."},{"Start":"12:05.880 ","End":"12:12.080","Text":"It\u0027s from 0-1. We have here,"},{"Start":"12:12.080 ","End":"12:19.590","Text":"plus 6t plus 2t, is plus 8t."},{"Start":"12:19.590 ","End":"12:25.150","Text":"Then t squared, I\u0027ve got minus 6 of those"},{"Start":"12:25.150 ","End":"12:32.830","Text":"and plus 2t cubed, all this dt."},{"Start":"12:33.540 ","End":"12:36.835","Text":"Let\u0027s see what we get."},{"Start":"12:36.835 ","End":"12:45.205","Text":"We get minus 4t plus 8t squared over 2,"},{"Start":"12:45.205 ","End":"12:50.185","Text":"makes it 4t squared here,"},{"Start":"12:50.185 ","End":"12:53.305","Text":"minus 6 over 3t cubed,"},{"Start":"12:53.305 ","End":"12:58.430","Text":"minus 2t cubed, and here,"},{"Start":"13:00.060 ","End":"13:04.360","Text":"t^4 over 4 times 2"},{"Start":"13:04.360 ","End":"13:13.240","Text":"is 1/2t^4, from 0-1."},{"Start":"13:13.240 ","End":"13:15.520","Text":"I plug in 0, everything 0."},{"Start":"13:15.520 ","End":"13:19.450","Text":"So I just need to plug in the 1,"},{"Start":"13:19.450 ","End":"13:25.510","Text":"so I get minus 4 plus 4,"},{"Start":"13:25.510 ","End":"13:29.875","Text":"minus 2 plus a 1/2,"},{"Start":"13:29.875 ","End":"13:35.725","Text":"altogether, that is minus 1.5."},{"Start":"13:35.725 ","End":"13:38.740","Text":"I\u0027ll highlight this."},{"Start":"13:38.740 ","End":"13:42.940","Text":"Now, I go back here."},{"Start":"13:42.940 ","End":"13:53.785","Text":"I can write this as 0 plus 2 plus minus 1 and 1/2,"},{"Start":"13:53.785 ","End":"13:58.370","Text":"and that is just equal to 1/2."},{"Start":"13:58.470 ","End":"14:01.030","Text":"That is the answer."},{"Start":"14:01.030 ","End":"14:06.760","Text":"But I Just like to say something else as an alternative method for doing the integral."},{"Start":"14:06.760 ","End":"14:09.985","Text":"We could have done the integral,"},{"Start":"14:09.985 ","End":"14:16.345","Text":"I\u0027ll just take this part of 1 minus t, all cubed."},{"Start":"14:16.345 ","End":"14:19.480","Text":"Let\u0027s just say we\u0027re taking the indefinite integral."},{"Start":"14:19.480 ","End":"14:21.145","Text":"What we could\u0027ve done,"},{"Start":"14:21.145 ","End":"14:23.965","Text":"is because it\u0027s a linear function of t,"},{"Start":"14:23.965 ","End":"14:28.585","Text":"treated as if 1 minus t was just a variable,"},{"Start":"14:28.585 ","End":"14:35.965","Text":"and we could have said this is 1 minus t^4 over 4."},{"Start":"14:35.965 ","End":"14:39.430","Text":"Then because it isn\u0027t t,"},{"Start":"14:39.430 ","End":"14:40.690","Text":"it\u0027s 1 minus t,"},{"Start":"14:40.690 ","End":"14:43.300","Text":"the inner derivative is minus 1,"},{"Start":"14:43.300 ","End":"14:46.615","Text":"we could have divided by minus 1."},{"Start":"14:46.615 ","End":"14:51.340","Text":"Dividing by minus 1 is like multiplying by minus 1,"},{"Start":"14:51.340 ","End":"14:55.070","Text":"and we could have just put a minus here."},{"Start":"14:56.190 ","End":"14:59.335","Text":"I just wanted to mention that from this point,"},{"Start":"14:59.335 ","End":"15:06.355","Text":"you could have used this instead of using the formula for the cube of a binomial,"},{"Start":"15:06.355 ","End":"15:12.740","Text":"would have got the same answer. We\u0027re done."}],"ID":8799},{"Watched":false,"Name":"Exercise 9","Duration":"10m 16s","ChapterTopicVideoID":8701,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8701.jpeg","UploadDate":"2017-02-13T05:14:05.1600000","DurationForVideoObject":"PT10M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.520","Text":"In this exercise, we\u0027re asked to compute the line integral of type 2 here,"},{"Start":"00:08.520 ","End":"00:12.900","Text":"and C is the closed path,"},{"Start":"00:12.900 ","End":"00:14.535","Text":"as in the picture."},{"Start":"00:14.535 ","End":"00:16.875","Text":"Maybe we should mark the points."},{"Start":"00:16.875 ","End":"00:21.495","Text":"This one, this one, and this one."},{"Start":"00:21.495 ","End":"00:25.065","Text":"The origin is obviously 0,0."},{"Start":"00:25.065 ","End":"00:27.000","Text":"Here, we\u0027re given that y is 1,"},{"Start":"00:27.000 ","End":"00:31.060","Text":"so this is the point 0,1."},{"Start":"00:31.160 ","End":"00:35.055","Text":"If y equals x squared and y equals 1,"},{"Start":"00:35.055 ","End":"00:38.895","Text":"then this has got to be the point 1,1."},{"Start":"00:38.895 ","End":"00:43.200","Text":"We go in a circular way."},{"Start":"00:43.200 ","End":"00:49.520","Text":"All this is C, the counterclockwise path."},{"Start":"00:49.520 ","End":"00:51.320","Text":"But to do the integral,"},{"Start":"00:51.320 ","End":"00:54.430","Text":"we want to break it up into 3 parts."},{"Start":"00:54.430 ","End":"00:58.470","Text":"Let\u0027s call this bit C_1,"},{"Start":"00:58.470 ","End":"01:00.300","Text":"and then we go along here,"},{"Start":"01:00.300 ","End":"01:04.665","Text":"that\u0027s C_2, and here, C_3."},{"Start":"01:04.665 ","End":"01:09.050","Text":"Then what we\u0027ll do is instead of computing the integral over"},{"Start":"01:09.050 ","End":"01:13.085","Text":"the whole of C of this thing, whatever it is,"},{"Start":"01:13.085 ","End":"01:16.215","Text":"we\u0027ll break it up into the integral over"},{"Start":"01:16.215 ","End":"01:22.220","Text":"C_1 plus the integral over C_2 of whatever it is,"},{"Start":"01:22.220 ","End":"01:27.515","Text":"plus the integral over C_3, 3 separate bits."},{"Start":"01:27.515 ","End":"01:34.500","Text":"We\u0027d like to parametrize each of the 3 separate pieces."},{"Start":"01:34.500 ","End":"01:38.915","Text":"I\u0027ll do it over here even though it says solution here."},{"Start":"01:38.915 ","End":"01:43.375","Text":"Let\u0027s take C_1. Now, C_1,"},{"Start":"01:43.375 ","End":"01:47.480","Text":"we\u0027re going from here to here along the curve y equals x squared,"},{"Start":"01:47.480 ","End":"01:48.920","Text":"and y is a function of x,"},{"Start":"01:48.920 ","End":"01:51.200","Text":"we just let x be the parameter t."},{"Start":"01:51.200 ","End":"01:54.405","Text":"We\u0027ll say x equals t,"},{"Start":"01:54.405 ","End":"01:58.385","Text":"and y which is x squared is now t squared."},{"Start":"01:58.385 ","End":"02:03.495","Text":"If you look at it, the x goes from 0 to 1,"},{"Start":"02:03.495 ","End":"02:08.580","Text":"so that\u0027s t goes from 0 to 1."},{"Start":"02:08.580 ","End":"02:10.740","Text":"That\u0027s C_1."},{"Start":"02:10.740 ","End":"02:13.860","Text":"Well, let\u0027s see, C_2."},{"Start":"02:13.860 ","End":"02:16.970","Text":"When it\u0027s a horizontal or vertical line,"},{"Start":"02:16.970 ","End":"02:22.580","Text":"I don\u0027t use the usual formula for a line segment between 2 points."},{"Start":"02:22.580 ","End":"02:27.440","Text":"If it\u0027s horizontal, then one of them is constant."},{"Start":"02:27.440 ","End":"02:29.570","Text":"I mean, the y is constant."},{"Start":"02:29.570 ","End":"02:32.330","Text":"We know here that y equals 1."},{"Start":"02:32.330 ","End":"02:39.150","Text":"But x goes from"},{"Start":"02:40.160 ","End":"02:43.560","Text":"1 down to 0,"},{"Start":"02:43.560 ","End":"02:52.305","Text":"so I can\u0027t let x equals t. I\u0027ll do a take 2 on the last bit."},{"Start":"02:52.305 ","End":"02:58.200","Text":"Here, I use the formula that x is the x of"},{"Start":"02:58.200 ","End":"03:06.830","Text":"the start point which is 1 plus t times the difference in the x\u0027s,"},{"Start":"03:06.830 ","End":"03:12.175","Text":"the destination minus the source, 0 minus 1,"},{"Start":"03:12.175 ","End":"03:16.590","Text":"and same thing for y but"},{"Start":"03:16.590 ","End":"03:23.490","Text":"y is just equal to 1 so we can take a shortcut here and just say y equals 1."},{"Start":"03:24.010 ","End":"03:33.585","Text":"I can rewrite this as just 1 minus t,"},{"Start":"03:33.585 ","End":"03:36.555","Text":"and that will be simpler."},{"Start":"03:36.555 ","End":"03:41.160","Text":"Then for C_3, something similar,"},{"Start":"03:41.160 ","End":"03:48.090","Text":"I forgot to say, t goes from 0 to 1."},{"Start":"03:48.090 ","End":"03:51.930","Text":"For C_3, similar idea."},{"Start":"03:51.930 ","End":"03:54.350","Text":"Only here, the x,"},{"Start":"03:54.350 ","End":"04:01.305","Text":"we don\u0027t have to mess with because it\u0027s always 0 along the y axis,"},{"Start":"04:01.305 ","End":"04:05.979","Text":"and y will equal the"},{"Start":"04:06.020 ","End":"04:13.880","Text":"source y which is 1 plus t times the end minus the start y,"},{"Start":"04:13.880 ","End":"04:17.405","Text":"also 0 minus 1."},{"Start":"04:17.405 ","End":"04:21.630","Text":"Again, t goes from 0 to 1."},{"Start":"04:21.630 ","End":"04:23.550","Text":"I\u0027ll save a line."},{"Start":"04:23.550 ","End":"04:25.770","Text":"This comes out to be 1 minus t,"},{"Start":"04:25.770 ","End":"04:31.185","Text":"I\u0027ll just cross this out and put that y is"},{"Start":"04:31.185 ","End":"04:40.090","Text":"1 minus t. Let\u0027s see what we get."},{"Start":"04:40.250 ","End":"04:43.710","Text":"Let\u0027s start with the C_1 part."},{"Start":"04:43.710 ","End":"04:46.410","Text":"For this part, we have the integral,"},{"Start":"04:46.410 ","End":"04:49.215","Text":"t goes from 0 to 1."},{"Start":"04:49.215 ","End":"04:54.190","Text":"We want x minus y squared which is"},{"Start":"04:54.190 ","End":"05:00.950","Text":"t minus y squared is t squared squared which is t^4."},{"Start":"05:00.950 ","End":"05:03.635","Text":"We don\u0027t have dx and dy,"},{"Start":"05:03.635 ","End":"05:05.885","Text":"so let\u0027s fill those in."},{"Start":"05:05.885 ","End":"05:08.435","Text":"dx is just dt,"},{"Start":"05:08.435 ","End":"05:13.200","Text":"and dy would be 2tdt."},{"Start":"05:15.130 ","End":"05:18.745","Text":"Here, I need dx so that\u0027s dt,"},{"Start":"05:18.745 ","End":"05:23.830","Text":"and then I need dy and just copy 2tdt."},{"Start":"05:28.010 ","End":"05:32.660","Text":"Let\u0027s see. I just combine this."},{"Start":"05:32.660 ","End":"05:35.060","Text":"I\u0027ll continue on the same line."},{"Start":"05:35.060 ","End":"05:38.695","Text":"The integral from 0 to 1."},{"Start":"05:38.695 ","End":"05:41.460","Text":"I\u0027ll just combine it all dt."},{"Start":"05:41.460 ","End":"05:44.940","Text":"I have t and another 2t is"},{"Start":"05:44.940 ","End":"05:51.830","Text":"3t minus t^4 dt."},{"Start":"05:51.830 ","End":"05:54.660","Text":"I\u0027ll just keep going on the same line."},{"Start":"05:56.890 ","End":"06:02.465","Text":"This is 3 over 2t squared."},{"Start":"06:02.465 ","End":"06:07.225","Text":"This is 1/5 t^5."},{"Start":"06:07.225 ","End":"06:11.070","Text":"This has to be taken from 0 to 1."},{"Start":"06:11.070 ","End":"06:14.789","Text":"I just need the 1 because 0 gives 0,"},{"Start":"06:14.789 ","End":"06:23.140","Text":"so it\u0027s the fraction 3/2 minus 1/5."},{"Start":"06:23.140 ","End":"06:25.255","Text":"If we put it all over 10,"},{"Start":"06:25.255 ","End":"06:28.030","Text":"that will be what?"},{"Start":"06:28.030 ","End":"06:34.505","Text":"3 times 5 is 15 minus 2 over 10."},{"Start":"06:34.505 ","End":"06:37.270","Text":"This is 13 over 10."},{"Start":"06:37.270 ","End":"06:41.630","Text":"That\u0027s C_1."},{"Start":"06:41.630 ","End":"06:48.000","Text":"Now C_2."},{"Start":"06:48.000 ","End":"06:50.650","Text":"C_2, again, I\u0027ll need the dx and dy."},{"Start":"06:50.780 ","End":"06:57.650","Text":"Here, dx, the derivative of this is minus 1dt,"},{"Start":"06:57.650 ","End":"06:59.750","Text":"so it\u0027s just minus dt."},{"Start":"06:59.750 ","End":"07:06.005","Text":"dy is just 0."},{"Start":"07:06.005 ","End":"07:08.690","Text":"We\u0027ll put it as 0dt."},{"Start":"07:08.690 ","End":"07:11.900","Text":"I might as well continue already with the dx\u0027s."},{"Start":"07:11.900 ","End":"07:15.905","Text":"Here, dx is just 0."},{"Start":"07:15.905 ","End":"07:18.500","Text":"We\u0027ll, write it as 0dt,"},{"Start":"07:18.500 ","End":"07:22.940","Text":"and dy is also because of the minus 1,"},{"Start":"07:22.940 ","End":"07:25.070","Text":"I\u0027ve got minus dt."},{"Start":"07:25.070 ","End":"07:28.205","Text":"Now, we\u0027ll continue over here."},{"Start":"07:28.205 ","End":"07:32.375","Text":"Let\u0027s see, C_2."},{"Start":"07:32.375 ","End":"07:36.110","Text":"All of them are going to be from 0 to 1."},{"Start":"07:36.110 ","End":"07:43.265","Text":"In fact, I might as well just work on the 3 of them simultaneously or partially, 0 to 1."},{"Start":"07:43.265 ","End":"07:45.305","Text":"Now, let\u0027s see. For C_2,"},{"Start":"07:45.305 ","End":"07:48.395","Text":"I need x minus y squared."},{"Start":"07:48.395 ","End":"07:57.225","Text":"It\u0027s x is 1 minus t minus y squared minus 1 squared."},{"Start":"07:57.225 ","End":"08:02.385","Text":"All this is dx which is minus dt."},{"Start":"08:02.385 ","End":"08:08.490","Text":"Then I need plus dy plus 0dt."},{"Start":"08:08.490 ","End":"08:10.895","Text":"If I combine all this,"},{"Start":"08:10.895 ","End":"08:13.879","Text":"1 minus 1 squared is just nothing."},{"Start":"08:13.879 ","End":"08:19.595","Text":"I\u0027ve just got minus t and minus dt so it\u0027s tdt."},{"Start":"08:19.595 ","End":"08:22.085","Text":"This part gives nothing."},{"Start":"08:22.085 ","End":"08:29.849","Text":"This comes out to be 1/2 t squared from 0 to 1,"},{"Start":"08:29.849 ","End":"08:33.820","Text":"and this is just a 1/2."},{"Start":"08:34.250 ","End":"08:38.270","Text":"The last one, let\u0027s see."},{"Start":"08:38.270 ","End":"08:42.995","Text":"Again, I need the x minus y squared,"},{"Start":"08:42.995 ","End":"08:49.095","Text":"so x minus y squared is just 0"},{"Start":"08:49.095 ","End":"08:55.685","Text":"minus 1 minus t squared."},{"Start":"08:55.685 ","End":"09:00.990","Text":"Then dx which is 0dt."},{"Start":"09:00.990 ","End":"09:04.890","Text":"I didn\u0027t have to bother with this but anyway."},{"Start":"09:04.890 ","End":"09:10.795","Text":"Then plus dy, dy is minus dt which is minus dt."},{"Start":"09:10.795 ","End":"09:19.050","Text":"It\u0027s just the integral from 0 to 1 of minus dt or minus 1dt."},{"Start":"09:19.050 ","End":"09:25.420","Text":"This is minus t from 0 to 1,"},{"Start":"09:25.520 ","End":"09:30.910","Text":"and this comes out to be just minus 1."},{"Start":"09:31.180 ","End":"09:34.535","Text":"Now, I can go back here."},{"Start":"09:34.535 ","End":"09:37.760","Text":"This part is the C_1 part."},{"Start":"09:37.760 ","End":"09:39.905","Text":"Here, I have C_2,"},{"Start":"09:39.905 ","End":"09:42.840","Text":"here, I have C_3."},{"Start":"09:45.920 ","End":"09:48.814","Text":"You know what? Maybe I\u0027ll do it in decimal."},{"Start":"09:48.814 ","End":"09:52.530","Text":"This will be 1.3"},{"Start":"09:54.320 ","End":"10:01.060","Text":"plus 0.5 minus 1."},{"Start":"10:01.060 ","End":"10:02.910","Text":"What do I get?"},{"Start":"10:02.910 ","End":"10:07.435","Text":"1.8 minus 1 is 0.8."},{"Start":"10:07.435 ","End":"10:09.499","Text":"Really, we should do it as a fraction,"},{"Start":"10:09.499 ","End":"10:13.970","Text":"so I\u0027ll write it as 4/5,"},{"Start":"10:13.970 ","End":"10:16.530","Text":"and that\u0027s the answer."}],"ID":8800},{"Watched":false,"Name":"Exercise 10 Part a","Duration":"9m 1s","ChapterTopicVideoID":8702,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8702.jpeg","UploadDate":"2017-02-13T05:16:29.1130000","DurationForVideoObject":"PT9M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.290 ","End":"00:04.710","Text":"This exercise is made up of 3 parts."},{"Start":"00:04.710 ","End":"00:06.285","Text":"But in each part,"},{"Start":"00:06.285 ","End":"00:14.220","Text":"we\u0027re given a function f of 3 variables and it\u0027s a vector function."},{"Start":"00:14.220 ","End":"00:18.150","Text":"These are the i, j, and k components."},{"Start":"00:18.150 ","End":"00:22.590","Text":"It\u0027s in 3D and we have to compute the line"},{"Start":"00:22.590 ","End":"00:31.340","Text":"integral F.dr from this point to this point along each of 3 different paths."},{"Start":"00:31.340 ","End":"00:35.100","Text":"Just want to point out that, strictly speaking,"},{"Start":"00:35.100 ","End":"00:41.350","Text":"should really be writing vectors with an arrow or a bar on top."},{"Start":"00:41.350 ","End":"00:43.760","Text":"When it\u0027s in bold phase like this,"},{"Start":"00:43.760 ","End":"00:45.770","Text":"then we understand that it\u0027s a vector."},{"Start":"00:45.770 ","End":"00:52.320","Text":"It\u0027s another way of doing it and actually dr is also a vector."},{"Start":"00:52.320 ","End":"00:58.485","Text":"In fact, dr is equal to,"},{"Start":"00:58.485 ","End":"01:01.620","Text":"if I do it in the i, j, k form,"},{"Start":"01:01.620 ","End":"01:08.085","Text":"is just dx times i plus"},{"Start":"01:08.085 ","End":"01:15.420","Text":"dy times j plus dz times k,"},{"Start":"01:15.420 ","End":"01:18.050","Text":"or if you want it in the angular bracket form,"},{"Start":"01:18.050 ","End":"01:24.560","Text":"it\u0027s dx, dy, dz."},{"Start":"01:24.560 ","End":"01:25.790","Text":"For all 3 parts,"},{"Start":"01:25.790 ","End":"01:29.780","Text":"we can actually rewrite this integral because we have an i,"},{"Start":"01:29.780 ","End":"01:33.635","Text":"j, k here, dot with an i, j, k here."},{"Start":"01:33.635 ","End":"01:37.530","Text":"What we get, really and this,"},{"Start":"01:37.530 ","End":"01:44.930","Text":"as I say for all 3 cases is the integral along some path from here to here,"},{"Start":"01:44.930 ","End":"01:49.380","Text":"we can just informally write from 0,"},{"Start":"01:49.380 ","End":"01:51.890","Text":"0, 0 to 1, 1, 1."},{"Start":"01:51.890 ","End":"01:54.695","Text":"Although of course this depends on the path,"},{"Start":"01:54.695 ","End":"02:00.950","Text":"but we just sometimes informally write like this of this dx."},{"Start":"02:00.950 ","End":"02:10.760","Text":"So 3x squared minus 6yz dx plus the next component,"},{"Start":"02:10.760 ","End":"02:24.410","Text":"the y component, 2y plus 3xz dy."},{"Start":"02:24.410 ","End":"02:26.435","Text":"Just made some more room here."},{"Start":"02:26.435 ","End":"02:31.370","Text":"Plus the last 1 is 1 minus"},{"Start":"02:31.370 ","End":"02:38.370","Text":"4xyz squared times dz."},{"Start":"02:39.130 ","End":"02:42.530","Text":"This we\u0027ll compute in 3 different ways and in"},{"Start":"02:42.530 ","End":"02:45.560","Text":"each case we\u0027ll get a different parameterized path."},{"Start":"02:45.560 ","End":"02:53.130","Text":"This will be the integral along a path c and differently for a,"},{"Start":"02:53.130 ","End":"02:55.850","Text":"b, and c. Let\u0027s start with part a."},{"Start":"02:55.850 ","End":"03:00.305","Text":"Now in part a, we already have the parametric representation."},{"Start":"03:00.305 ","End":"03:05.270","Text":"We have that the curve c is given by x equals t,"},{"Start":"03:05.270 ","End":"03:07.685","Text":"y equals t squared,"},{"Start":"03:07.685 ","End":"03:10.130","Text":"z equals t cubed."},{"Start":"03:10.130 ","End":"03:12.409","Text":"But there is something missing,"},{"Start":"03:12.409 ","End":"03:15.520","Text":"we have to know where t goes from and to."},{"Start":"03:15.520 ","End":"03:18.665","Text":"T goes from something to something."},{"Start":"03:18.665 ","End":"03:21.335","Text":"They\u0027ve left that out or have they?"},{"Start":"03:21.335 ","End":"03:28.070","Text":"Well, we can easily deduce this because we know that we start from the point 0,"},{"Start":"03:28.070 ","End":"03:33.620","Text":"0, 0 so what value of t could I possibly put to get 0, 0, 0?"},{"Start":"03:33.620 ","End":"03:36.760","Text":"I think it\u0027s clear that that\u0027s 0."},{"Start":"03:36.760 ","End":"03:40.340","Text":"Likewise, to get the point 1, 1,"},{"Start":"03:40.340 ","End":"03:44.045","Text":"1, if I let t equals 1, that will do it."},{"Start":"03:44.045 ","End":"03:48.890","Text":"Now we have the path parameterized and we have the range of values"},{"Start":"03:48.890 ","End":"03:53.900","Text":"for t. Now we can convert this integral."},{"Start":"03:53.900 ","End":"03:56.780","Text":"The integral along the curve c,"},{"Start":"03:56.780 ","End":"04:04.770","Text":"becomes just the integral of the parameter t from 0 to 1."},{"Start":"04:05.510 ","End":"04:10.100","Text":"Then we just have to interpret this because we have x, y,"},{"Start":"04:10.100 ","End":"04:15.185","Text":"and z in terms of t. What we don\u0027t have,"},{"Start":"04:15.185 ","End":"04:20.010","Text":"we don\u0027t have dx, dy, and dz."},{"Start":"04:20.540 ","End":"04:22.920","Text":"I could write these at the side."},{"Start":"04:22.920 ","End":"04:25.470","Text":"Let\u0027s see we want to know what dx equals,"},{"Start":"04:25.470 ","End":"04:30.740","Text":"what dy equals, and what dz equals."},{"Start":"04:30.740 ","End":"04:36.245","Text":"If x equals t, then just dx equals dt. No problem."},{"Start":"04:36.245 ","End":"04:38.254","Text":"Y equals t squared,"},{"Start":"04:38.254 ","End":"04:43.460","Text":"take the derivative of d of each of these."},{"Start":"04:43.460 ","End":"04:51.200","Text":"I\u0027ve got 1dy or dy equals 2t dt."},{"Start":"04:51.200 ","End":"04:55.290","Text":"Here dz is just 3t squared,"},{"Start":"04:55.290 ","End":"04:58.980","Text":"derivative of t cubed dt."},{"Start":"04:58.980 ","End":"05:05.820","Text":"I think now we\u0027re ready to substitute into this expression or this expression."},{"Start":"05:05.860 ","End":"05:08.270","Text":"This 1 I mean."},{"Start":"05:08.270 ","End":"05:18.930","Text":"3x squared is 3t squared and I need a bracket minus 6yz."},{"Start":"05:19.430 ","End":"05:30.170","Text":"Y times z is t^5 so I have minus 6t^5 and then dx is dt,"},{"Start":"05:30.170 ","End":"05:33.525","Text":"and that\u0027s just the first part."},{"Start":"05:33.525 ","End":"05:43.070","Text":"Then continuing 2y is 2t squared plus 3xz,"},{"Start":"05:43.140 ","End":"05:49.600","Text":"3 and xz is t^4."},{"Start":"05:49.600 ","End":"05:53.750","Text":"Here we have dy, which is 2tdt."},{"Start":"05:55.910 ","End":"06:05.925","Text":"The last piece, the third component is 1 minus 4xyz squared."},{"Start":"06:05.925 ","End":"06:09.905","Text":"Now xyz squared is t,"},{"Start":"06:09.905 ","End":"06:13.955","Text":"t squared and t^6."},{"Start":"06:13.955 ","End":"06:18.935","Text":"Let\u0027s see, do the arithmetic t times t squared is t cubed,"},{"Start":"06:18.935 ","End":"06:26.955","Text":"times t^6 that should be t^9."},{"Start":"06:26.955 ","End":"06:33.340","Text":"Then dz is 3t squared dt."},{"Start":"06:33.340 ","End":"06:37.040","Text":"Now we just have an integral in t,"},{"Start":"06:37.040 ","End":"06:42.480","Text":"the first thing I\u0027ll do is just tidy it up."},{"Start":"06:42.480 ","End":"06:46.305","Text":"You want to have something dt."},{"Start":"06:46.305 ","End":"06:51.885","Text":"First step, I\u0027ll just open up brackets here,"},{"Start":"06:51.885 ","End":"06:58.920","Text":"3t squared minus 6t^5 and here multiplying by 2t."},{"Start":"06:58.920 ","End":"07:05.890","Text":"So it\u0027s 4t cubed plus 6t^5."},{"Start":"07:06.500 ","End":"07:11.175","Text":"Then this 1, 3t squared times each of these."},{"Start":"07:11.175 ","End":"07:13.635","Text":"So 3t squared"},{"Start":"07:13.635 ","End":"07:22.890","Text":"minus 12t^11 dt."},{"Start":"07:22.890 ","End":"07:26.375","Text":"Let\u0027s see, do I have any like terms to collect?"},{"Start":"07:26.375 ","End":"07:28.920","Text":"Well, yes."},{"Start":"07:29.000 ","End":"07:35.120","Text":"I have a 3t squared and I have another 3t squared."},{"Start":"07:35.120 ","End":"07:40.280","Text":"That will be like 6t squared and what else can I combine?"},{"Start":"07:40.280 ","End":"07:44.330","Text":"I have a t^5 and a t^5,"},{"Start":"07:44.330 ","End":"07:46.550","Text":"these just cancel each other out."},{"Start":"07:46.550 ","End":"07:49.560","Text":"It\u0027s a plus 6 and minus 6."},{"Start":"07:51.730 ","End":"07:56.300","Text":"I\u0027ll also cross these 2 out just so I don\u0027t count them twice because"},{"Start":"07:56.300 ","End":"08:01.440","Text":"I\u0027ve already got it here in the 6t squared."},{"Start":"08:01.630 ","End":"08:05.700","Text":"The integral is here,"},{"Start":"08:05.700 ","End":"08:08.625","Text":"I get 6 over 3t cubed,"},{"Start":"08:08.625 ","End":"08:13.530","Text":"6 over 3 is 2t cubed."},{"Start":"08:13.530 ","End":"08:17.865","Text":"From here 4t^4 over 4,"},{"Start":"08:17.865 ","End":"08:22.170","Text":"so that\u0027s just t^4."},{"Start":"08:22.170 ","End":"08:27.979","Text":"Then finally t^12 over 12."},{"Start":"08:27.979 ","End":"08:30.200","Text":"But the 12 with the 12 cancels,"},{"Start":"08:30.200 ","End":"08:34.715","Text":"so it\u0027s just minus t^12."},{"Start":"08:34.715 ","End":"08:39.065","Text":"All this taken from 0-1."},{"Start":"08:39.065 ","End":"08:42.530","Text":"When I put in 0, I get nothing,"},{"Start":"08:42.530 ","End":"08:45.425","Text":"so I just have to plug in 1."},{"Start":"08:45.425 ","End":"08:55.540","Text":"All we get is 2 plus 1 minus 1 and so the answer to part A is 2."},{"Start":"08:55.540 ","End":"09:01.990","Text":"I\u0027ll now highlight it and then we\u0027ll move on to part B."}],"ID":8801},{"Watched":false,"Name":"Exercise 10 Part b","Duration":"12m 29s","ChapterTopicVideoID":8703,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8703.jpeg","UploadDate":"2017-02-13T05:19:42.5870000","DurationForVideoObject":"PT12M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.515","Text":"Here we are in part B. I just left some of the stuff from part A that we can reuse."},{"Start":"00:07.515 ","End":"00:14.640","Text":"The path C this time is still the path from 0,0,0 to 1,1,1."},{"Start":"00:14.640 ","End":"00:17.760","Text":"But here we\u0027re going in 3 straight lines."},{"Start":"00:17.760 ","End":"00:20.685","Text":"We\u0027re starting from here and making a stop over"},{"Start":"00:20.685 ","End":"00:24.105","Text":"here and here until we finally end up where we want to go."},{"Start":"00:24.105 ","End":"00:26.715","Text":"That\u0027s like 3 straight lines."},{"Start":"00:26.715 ","End":"00:28.470","Text":"I\u0027m not going to draw it in 3D."},{"Start":"00:28.470 ","End":"00:32.280","Text":"I\u0027ll just represent the curve C of the picture."},{"Start":"00:32.280 ","End":"00:35.715","Text":"We go from this point to some other point,"},{"Start":"00:35.715 ","End":"00:37.710","Text":"and then to some other point,"},{"Start":"00:37.710 ","End":"00:42.630","Text":"and then finally to our destination."},{"Start":"00:42.630 ","End":"00:47.565","Text":"This would be the point 0,0,0,"},{"Start":"00:47.565 ","End":"00:56.670","Text":"this is the point 0,0,1,"},{"Start":"00:56.670 ","End":"01:00.600","Text":"then the point 0,1,1,"},{"Start":"01:00.600 ","End":"01:03.810","Text":"and then we end up at 1,1,1."},{"Start":"01:03.810 ","End":"01:09.595","Text":"This whole thing will be the path C,"},{"Start":"01:09.595 ","End":"01:14.665","Text":"but it\u0027s going to be made up of 3 separate paths."},{"Start":"01:14.665 ","End":"01:18.430","Text":"We\u0027ll call the first portion C_1,"},{"Start":"01:18.430 ","End":"01:21.645","Text":"the second portion C_2,"},{"Start":"01:21.645 ","End":"01:24.960","Text":"and the third portion C_3."},{"Start":"01:24.960 ","End":"01:30.965","Text":"What we do when we have a piecewise path is we just take the sum."},{"Start":"01:30.965 ","End":"01:36.695","Text":"The integral over C of all this stuff, blah, blah, blah,"},{"Start":"01:36.695 ","End":"01:41.075","Text":"is equal to the integral over C_1 of whatever it is,"},{"Start":"01:41.075 ","End":"01:44.570","Text":"plus the integral over C_2 of whatever it is,"},{"Start":"01:44.570 ","End":"01:48.690","Text":"plus the integral over C_3 of whatever it is."},{"Start":"01:48.690 ","End":"01:54.590","Text":"We make 3 separate calculations and then we add the 3 results together."},{"Start":"01:54.590 ","End":"01:58.310","Text":"Let\u0027s start with the path C_1."},{"Start":"01:58.310 ","End":"02:01.420","Text":"Now I need to parameterize it."},{"Start":"02:01.420 ","End":"02:05.495","Text":"I\u0027m going to use a shortcut because if you look at it,"},{"Start":"02:05.495 ","End":"02:11.390","Text":"you see that the x and the y are the same for the start and end."},{"Start":"02:11.390 ","End":"02:15.530","Text":"0,0 and 0,0 are what x and y are."},{"Start":"02:15.530 ","End":"02:19.315","Text":"Only the z is changing from 0 to 1."},{"Start":"02:19.315 ","End":"02:24.515","Text":"When you get a simple case like this where only 1 of them is moving,"},{"Start":"02:24.515 ","End":"02:30.410","Text":"then we can easily say that x equals 0,"},{"Start":"02:30.410 ","End":"02:34.940","Text":"y equals 0, and z is the only one that depends on the"},{"Start":"02:34.940 ","End":"02:41.180","Text":"parameter t. We\u0027re going from 0 to 1,"},{"Start":"02:41.180 ","End":"02:45.710","Text":"and so we can say that z equals t and"},{"Start":"02:45.710 ","End":"02:53.280","Text":"then t goes from 0 to 1."},{"Start":"02:55.130 ","End":"02:58.020","Text":"We know we\u0027re going to need dx,"},{"Start":"02:58.020 ","End":"03:01.800","Text":"dy, and dz, so let\u0027s just write those."},{"Start":"03:01.800 ","End":"03:06.285","Text":"dx is just also 0,"},{"Start":"03:06.285 ","End":"03:10.970","Text":"I\u0027d like to write it as 0 dt to remind myself there is a dt there."},{"Start":"03:10.970 ","End":"03:15.860","Text":"Similarly, y is just a derivative of 0 is 0 dt,"},{"Start":"03:15.860 ","End":"03:18.080","Text":"and z equals t,"},{"Start":"03:18.080 ","End":"03:21.620","Text":"so dz equals dt."},{"Start":"03:21.620 ","End":"03:24.530","Text":"That\u0027s fairly straightforward."},{"Start":"03:24.530 ","End":"03:28.280","Text":"The integral will be,"},{"Start":"03:28.280 ","End":"03:34.355","Text":"we take the parameter range that t goes from 0 to 1,"},{"Start":"03:34.355 ","End":"03:36.140","Text":"and then we do this expression."},{"Start":"03:36.140 ","End":"03:45.505","Text":"But of course, since dx and dy are both 0,"},{"Start":"03:45.505 ","End":"03:48.409","Text":"I don\u0027t need this because that\u0027s 0."},{"Start":"03:48.409 ","End":"03:51.745","Text":"I don\u0027t need this, that\u0027s 0."},{"Start":"03:51.745 ","End":"03:55.275","Text":"I only need the last bit then."},{"Start":"03:55.275 ","End":"04:02.910","Text":"I have 1 minus 4xyz squared."},{"Start":"04:02.910 ","End":"04:10.845","Text":"But xyz squared is 0 already,"},{"Start":"04:10.845 ","End":"04:13.710","Text":"so it\u0027s just 1 minus 0,"},{"Start":"04:13.710 ","End":"04:17.220","Text":"I\u0027m going to leave the 0 there to show you that I haven\u0027t forgotten it,"},{"Start":"04:17.220 ","End":"04:20.560","Text":"dz which is dt."},{"Start":"04:22.040 ","End":"04:25.540","Text":"This is now very straightforward."},{"Start":"04:25.540 ","End":"04:30.980","Text":"The integral of 1 from 0 to 1."},{"Start":"04:30.980 ","End":"04:32.710","Text":"Whenever you have the integral of 1,"},{"Start":"04:32.710 ","End":"04:34.960","Text":"it\u0027s always the upper minus the lower,"},{"Start":"04:34.960 ","End":"04:36.760","Text":"it comes out to be just 1."},{"Start":"04:36.760 ","End":"04:40.510","Text":"If you want, you can write the integral of 1 is t. When t is 1,"},{"Start":"04:40.510 ","End":"04:42.590","Text":"it\u0027s 1, when t is 0, it\u0027s 0."},{"Start":"04:42.590 ","End":"04:46.610","Text":"It\u0027s so simple, I\u0027m just writing the answer 1."},{"Start":"04:46.610 ","End":"04:50.540","Text":"Maybe I\u0027ll just write it below here,"},{"Start":"04:50.540 ","End":"04:53.005","Text":"I write that this is 1."},{"Start":"04:53.005 ","End":"04:55.470","Text":"Then we\u0027ll write what this is and this is,"},{"Start":"04:55.470 ","End":"04:57.370","Text":"and then we\u0027ll do the addition at the end."},{"Start":"04:57.370 ","End":"05:00.650","Text":"Let\u0027s move on to C_2 now."},{"Start":"05:01.140 ","End":"05:05.380","Text":"Want to keep this in view. Let\u0027s see."},{"Start":"05:05.380 ","End":"05:14.320","Text":"For C_2, I\u0027m also going to use a shortcut because when I look at C_2,"},{"Start":"05:14.320 ","End":"05:22.510","Text":"you see that the x and the z don\u0027t change."},{"Start":"05:22.730 ","End":"05:29.250","Text":"I\u0027ve got 1 here,"},{"Start":"05:29.250 ","End":"05:32.325","Text":"and I have 1 here,"},{"Start":"05:32.325 ","End":"05:34.650","Text":"and I have 0 here,"},{"Start":"05:34.650 ","End":"05:37.900","Text":"and I have 0 here."},{"Start":"05:38.210 ","End":"05:41.670","Text":"Only the middle bit, the y bit,"},{"Start":"05:41.670 ","End":"05:44.460","Text":"is changing from 0 to 1."},{"Start":"05:44.460 ","End":"05:46.010","Text":"Just like we did before,"},{"Start":"05:46.010 ","End":"05:50.610","Text":"we don\u0027t need to crank out all this formula that"},{"Start":"05:50.610 ","End":"05:55.460","Text":"we used to do with the first point plus t times the second minus the first and all that,"},{"Start":"05:55.460 ","End":"05:57.860","Text":"we can just straight away say,"},{"Start":"05:57.860 ","End":"06:01.715","Text":"x is constantly 0,"},{"Start":"06:01.715 ","End":"06:05.730","Text":"z is constantly 1,"},{"Start":"06:05.730 ","End":"06:08.460","Text":"and only the y is changing."},{"Start":"06:08.460 ","End":"06:14.260","Text":"So it\u0027s like t, but t goes from 0 to 1."},{"Start":"06:14.260 ","End":"06:17.480","Text":"This is a fairly standard shortcut."},{"Start":"06:17.480 ","End":"06:19.730","Text":"This is because if we do it in 3D,"},{"Start":"06:19.730 ","End":"06:21.500","Text":"we\u0027re going parallel to the axes."},{"Start":"06:21.500 ","End":"06:24.335","Text":"Here we\u0027re going up in the z direction,"},{"Start":"06:24.335 ","End":"06:27.335","Text":"here we\u0027re going in the y direction,"},{"Start":"06:27.335 ","End":"06:29.845","Text":"parallel to the y-axis."},{"Start":"06:29.845 ","End":"06:33.850","Text":"Of course, the last bit we\u0027re going to be moving in the x direction anyway,"},{"Start":"06:33.850 ","End":"06:35.735","Text":"let\u0027s not get ahead of ourselves."},{"Start":"06:35.735 ","End":"06:39.800","Text":"As before, we know we\u0027re going to need dx."},{"Start":"06:39.800 ","End":"06:42.050","Text":"Did I miss a d here?"},{"Start":"06:42.050 ","End":"06:46.875","Text":"Sorry. Yeah, too late now, but never mind."},{"Start":"06:46.875 ","End":"06:49.905","Text":"Here also we\u0027ll need dx,"},{"Start":"06:49.905 ","End":"06:53.880","Text":"we\u0027ll need dy, and we\u0027ll need dz."},{"Start":"06:53.880 ","End":"06:59.145","Text":"Let\u0027s see, again, x and z are constants."},{"Start":"06:59.145 ","End":"07:02.250","Text":"These are both going to be 0,"},{"Start":"07:02.250 ","End":"07:05.150","Text":"this is also going to be 0."},{"Start":"07:05.150 ","End":"07:07.970","Text":"Only the y is moving and y is t,"},{"Start":"07:07.970 ","End":"07:10.980","Text":"so dy is dt."},{"Start":"07:11.690 ","End":"07:16.270","Text":"In the case of the integral,"},{"Start":"07:18.530 ","End":"07:20.760","Text":"the dx is still 0,"},{"Start":"07:20.760 ","End":"07:24.830","Text":"dy no longer, and the dz will be 0."},{"Start":"07:24.830 ","End":"07:28.865","Text":"So we just need the middle bit, the dy."},{"Start":"07:28.865 ","End":"07:39.840","Text":"We get the integral from 0 to 1 of 2y plus 3xz."},{"Start":"07:40.090 ","End":"07:49.620","Text":"2y is 2t plus 3xz."},{"Start":"07:49.620 ","End":"07:51.570","Text":"Well, x is 0,"},{"Start":"07:51.570 ","End":"07:58.750","Text":"so that makes this 0 dt."},{"Start":"07:59.150 ","End":"08:03.000","Text":"The integral of 2t is t squared,"},{"Start":"08:03.000 ","End":"08:06.495","Text":"so it\u0027s t squared from 0 to 1."},{"Start":"08:06.495 ","End":"08:12.760","Text":"It\u0027s just 1 squared minus 0 squared is just 1."},{"Start":"08:12.760 ","End":"08:15.300","Text":"That also came out as 1,"},{"Start":"08:15.300 ","End":"08:19.800","Text":"and I\u0027ll write that here as 1."},{"Start":"08:19.800 ","End":"08:27.080","Text":"That\u0027s 2 down and 1 to go, just C_3."},{"Start":"08:27.080 ","End":"08:31.150","Text":"I just copied the formula from here"},{"Start":"08:31.150 ","End":"08:35.450","Text":"to here because I\u0027m going to scroll now and I\u0027ll lose it."},{"Start":"08:35.550 ","End":"08:40.575","Text":"What we need to do really is keep this here."},{"Start":"08:40.575 ","End":"08:42.840","Text":"I\u0027ve got what I need."},{"Start":"08:42.840 ","End":"08:46.845","Text":"This time we need to parameterize C_3,"},{"Start":"08:46.845 ","End":"08:49.845","Text":"which is this bit here."},{"Start":"08:49.845 ","End":"08:52.675","Text":"In this bit here,"},{"Start":"08:52.675 ","End":"08:54.190","Text":"if I look at the numbers,"},{"Start":"08:54.190 ","End":"08:58.210","Text":"notice that the y and z are not changing,"},{"Start":"08:58.210 ","End":"09:03.050","Text":"I have 1 and 1, and here I have 1 and 1."},{"Start":"09:03.050 ","End":"09:08.075","Text":"Only the x is changing from 0 to 1,"},{"Start":"09:08.075 ","End":"09:14.704","Text":"so we don\u0027t need to crank out the whole formula for segment between 2 points."},{"Start":"09:14.704 ","End":"09:24.065","Text":"We can easily see that the curve C_3 can be parameterized by y is 1,"},{"Start":"09:24.065 ","End":"09:27.455","Text":"z is 1, that\u0027s the 1 and the 1 here,"},{"Start":"09:27.455 ","End":"09:29.530","Text":"and only x is moving,"},{"Start":"09:29.530 ","End":"09:31.650","Text":"so it goes from 0 to 1,"},{"Start":"09:31.650 ","End":"09:39.240","Text":"so I\u0027ll write it as x equals t and we\u0027ll let t go from 0 to 1."},{"Start":"09:39.240 ","End":"09:43.060","Text":"As before, we need dx, dy, and dz."},{"Start":"09:43.160 ","End":"09:48.000","Text":"dx from here is just dt,"},{"Start":"09:48.000 ","End":"09:52.890","Text":"and dy, that y is a constant,"},{"Start":"09:52.890 ","End":"09:56.285","Text":"so dy is just 0dt,"},{"Start":"09:56.285 ","End":"09:59.250","Text":"and dz is also,"},{"Start":"09:59.250 ","End":"10:03.360","Text":"z is a constant, it\u0027s also 0dt, it\u0027s just 0."},{"Start":"10:03.360 ","End":"10:13.880","Text":"Basically what we get is that this part is going to be 0 and this part is going to be 0."},{"Start":"10:13.880 ","End":"10:17.820","Text":"So I\u0027ll only need the integral of the first bit."},{"Start":"10:19.250 ","End":"10:26.640","Text":"What we will get is the integral from 0 to 1,"},{"Start":"10:26.640 ","End":"10:34.170","Text":"that\u0027s for t, of we just need 3x squared minus 6yz."},{"Start":"10:34.170 ","End":"10:40.425","Text":"Let\u0027s see, 3x squared is 3t squared"},{"Start":"10:40.425 ","End":"10:49.840","Text":"minus 6 times y is 1 and z is 1."},{"Start":"11:04.130 ","End":"11:08.025","Text":"We need dx, of course,"},{"Start":"11:08.025 ","End":"11:13.080","Text":"and dx is equal to dt."},{"Start":"11:13.080 ","End":"11:16.185","Text":"I don\u0027t know why I said dz before, sorry,"},{"Start":"11:16.185 ","End":"11:22.380","Text":"dx which is dt."},{"Start":"11:22.380 ","End":"11:24.330","Text":"I can just even do it over here,"},{"Start":"11:24.330 ","End":"11:26.680","Text":"I don\u0027t want to lose this bit."},{"Start":"11:27.590 ","End":"11:34.620","Text":"For 3t squared I get 3t cubed over 3 is just t cubed,"},{"Start":"11:34.620 ","End":"11:38.620","Text":"6 just gives me 6t."},{"Start":"11:38.720 ","End":"11:43.365","Text":"Then this from 0 to 1, 0 gives nothing,"},{"Start":"11:43.365 ","End":"11:47.820","Text":"1 gives me 1 minus 6,"},{"Start":"11:47.820 ","End":"11:51.070","Text":"so that\u0027s minus 5."},{"Start":"11:51.380 ","End":"11:59.290","Text":"That is the integral over the third part is minus 5."},{"Start":"11:59.290 ","End":"12:02.230","Text":"If I add all these 3 together,"},{"Start":"12:02.230 ","End":"12:06.820","Text":"I get 1 plus 1."},{"Start":"12:06.820 ","End":"12:14.070","Text":"Well, I\u0027ll just write it, it\u0027s 1 plus 1 plus negative 5,"},{"Start":"12:14.070 ","End":"12:16.574","Text":"and that\u0027s 2 minus 5,"},{"Start":"12:16.574 ","End":"12:19.485","Text":"so that\u0027s minus 3."},{"Start":"12:19.485 ","End":"12:24.015","Text":"That\u0027s the answer to part B."},{"Start":"12:24.015 ","End":"12:28.090","Text":"Next clip, we\u0027ll do part C."}],"ID":8802},{"Watched":false,"Name":"Exercise 11 Part a","Duration":"7m 44s","ChapterTopicVideoID":8704,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8704.jpeg","UploadDate":"2017-02-13T05:21:20.7870000","DurationForVideoObject":"PT7M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.970","Text":"In this exercise, we have to compute a type 2 line integral."},{"Start":"00:05.970 ","End":"00:10.230","Text":"It\u0027s in 2 variables in the plane x,y."},{"Start":"00:10.230 ","End":"00:14.280","Text":"But it\u0027s not written in the usual customary"},{"Start":"00:14.280 ","End":"00:17.580","Text":"form that you\u0027re familiar with."},{"Start":"00:17.580 ","End":"00:20.440","Text":"We\u0027re going to rewrite it a bit."},{"Start":"00:21.590 ","End":"00:25.220","Text":"We usually write vectors with an arrow,"},{"Start":"00:25.220 ","End":"00:28.860","Text":"so let me add arrows everywhere."},{"Start":"00:29.630 ","End":"00:32.625","Text":"Here this is a vector,"},{"Start":"00:32.625 ","End":"00:35.460","Text":"F is a vector,"},{"Start":"00:35.460 ","End":"00:39.220","Text":"r is a vector."},{"Start":"00:42.470 ","End":"00:48.235","Text":"What does this mean when we write r in this form?"},{"Start":"00:48.235 ","End":"00:50.525","Text":"Just straightforward."},{"Start":"00:50.525 ","End":"00:57.260","Text":"We have that r which is like x,y,"},{"Start":"00:57.260 ","End":"01:16.020","Text":"r is just x,y, and dr is just dx, dy."},{"Start":"01:16.020 ","End":"01:18.790","Text":"So the integral of"},{"Start":"01:18.790 ","End":"01:27.320","Text":"F.dr could be written in the more familiar form,"},{"Start":"01:27.320 ","End":"01:29.555","Text":"I\u0027ll use this particular F,"},{"Start":"01:29.555 ","End":"01:35.420","Text":"it\u0027s the dot product of this vector with this vector,"},{"Start":"01:35.420 ","End":"01:38.765","Text":"it\u0027s just integral."},{"Start":"01:38.765 ","End":"01:46.455","Text":"X squared y cubed goes with dx plus,"},{"Start":"01:46.455 ","End":"01:49.620","Text":"and then this with dy,"},{"Start":"01:49.620 ","End":"01:54.200","Text":"it\u0027s negative, I\u0027ll just change that plus to a minus,"},{"Start":"01:54.200 ","End":"02:02.670","Text":"and then we have y square root of x times dy."},{"Start":"02:02.670 ","End":"02:04.680","Text":"This is a more familiar form,"},{"Start":"02:04.680 ","End":"02:06.989","Text":"and as for the curve C,"},{"Start":"02:06.989 ","End":"02:08.955","Text":"r also defines that."},{"Start":"02:08.955 ","End":"02:13.265","Text":"We can write our curve C in parametrized form."},{"Start":"02:13.265 ","End":"02:17.960","Text":"I could write it as x equals, and y equals."},{"Start":"02:17.960 ","End":"02:21.250","Text":"This vector is x,"},{"Start":"02:21.250 ","End":"02:25.515","Text":"y, so component y is x is t squared,"},{"Start":"02:25.515 ","End":"02:28.095","Text":"y is minus t cubed,"},{"Start":"02:28.095 ","End":"02:33.780","Text":"and we have the parameter range where it goes from 0-1."},{"Start":"02:33.780 ","End":"02:39.560","Text":"Now we can even write this in terms of t"},{"Start":"02:39.560 ","End":"02:41.285","Text":"and get a regular integral,"},{"Start":"02:41.285 ","End":"02:45.995","Text":"we get the integral from 0-1,"},{"Start":"02:45.995 ","End":"02:53.490","Text":"which I sometimes emphasize it\u0027s t. Just take it 1 at a time,"},{"Start":"02:53.490 ","End":"02:57.135","Text":"x, I look up is t squared."},{"Start":"02:57.135 ","End":"03:08.010","Text":"T squared, squared, y cubed is minus t cubed, cubed."},{"Start":"03:08.010 ","End":"03:11.175","Text":"All this is dx."},{"Start":"03:11.175 ","End":"03:14.850","Text":"We still don\u0027t have dx,"},{"Start":"03:14.850 ","End":"03:18.000","Text":"dy for our particular case."},{"Start":"03:18.000 ","End":"03:19.230","Text":"Let me just write that,"},{"Start":"03:19.230 ","End":"03:23.100","Text":"dx equals and dy equals."},{"Start":"03:23.100 ","End":"03:26.350","Text":"This will be 2tdt,"},{"Start":"03:27.230 ","End":"03:33.690","Text":"and this will be minus 3t squared dt."},{"Start":"03:33.690 ","End":"03:36.165","Text":"I can get back here now,"},{"Start":"03:36.165 ","End":"03:39.285","Text":"and write times 2tdt."},{"Start":"03:39.285 ","End":"03:43.665","Text":"That was the dx which is here."},{"Start":"03:43.665 ","End":"03:46.200","Text":"That\u0027s just the first part,"},{"Start":"03:46.200 ","End":"03:48.330","Text":"now we have a minus."},{"Start":"03:48.330 ","End":"03:54.400","Text":"We need y which is minus t cubed,"},{"Start":"03:56.090 ","End":"04:00.624","Text":"and square root of x."},{"Start":"04:00.624 ","End":"04:05.795","Text":"Just like to go over a small important technical point."},{"Start":"04:05.795 ","End":"04:12.020","Text":"The square root of x is the square root of t squared."},{"Start":"04:12.020 ","End":"04:17.870","Text":"Now, the square root of t squared is not just automatically t,"},{"Start":"04:17.870 ","End":"04:21.050","Text":"it\u0027s the absolute value of t if you remember."},{"Start":"04:21.050 ","End":"04:25.294","Text":"But because our t is in the non-negative range,"},{"Start":"04:25.294 ","End":"04:30.170","Text":"in our case it is equal to t. Getting back here,"},{"Start":"04:30.170 ","End":"04:33.380","Text":"the square root of x is square root of t squared,"},{"Start":"04:33.380 ","End":"04:38.900","Text":"in our case is t. Finally the dy from here"},{"Start":"04:38.900 ","End":"04:44.220","Text":"is minus 3t squared dt."},{"Start":"04:44.890 ","End":"04:48.865","Text":"Now we want to simplify this a bit."},{"Start":"04:48.865 ","End":"04:54.870","Text":"It\u0027s the integral, it\u0027s still from 0-1. Let\u0027s see."},{"Start":"04:54.870 ","End":"05:00.265","Text":"The first 1 is an exercise is an exponents."},{"Start":"05:00.265 ","End":"05:08.100","Text":"Certainly we have 2, in fact, even we have a minus 2,"},{"Start":"05:08.660 ","End":"05:14.940","Text":"everything else is powers of t. This would be t^4."},{"Start":"05:14.940 ","End":"05:18.720","Text":"Here we have a t^4, here t cubed,"},{"Start":"05:18.720 ","End":"05:21.120","Text":"cubed is t^9,"},{"Start":"05:21.120 ","End":"05:26.475","Text":"and here we\u0027ll have just a t^1."},{"Start":"05:26.475 ","End":"05:30.485","Text":"If I take 4 plus 9 plus 1,"},{"Start":"05:30.485 ","End":"05:33.175","Text":"that makes it 14."},{"Start":"05:33.175 ","End":"05:36.915","Text":"So t to the power of 14."},{"Start":"05:36.915 ","End":"05:43.040","Text":"Here, I have 3 minuses, minus, minus, minus."},{"Start":"05:43.040 ","End":"05:47.445","Text":"It\u0027s going to be minus constant,"},{"Start":"05:47.445 ","End":"05:49.935","Text":"the numbers I have are 3."},{"Start":"05:49.935 ","End":"05:55.290","Text":"Everything else is just powers of t, t cubed,"},{"Start":"05:55.290 ","End":"05:58.860","Text":"t to the 1, t to the 2,"},{"Start":"05:58.860 ","End":"06:02.805","Text":"3 and 1 and 2 is 6."},{"Start":"06:02.805 ","End":"06:12.120","Text":"It\u0027s t to the 6 and dt."},{"Start":"06:12.120 ","End":"06:15.100","Text":"Let\u0027s do this integral."},{"Start":"06:15.680 ","End":"06:20.730","Text":"Raise the power by 1 is 15 and divide by that."},{"Start":"06:20.730 ","End":"06:25.330","Text":"I\u0027ve got minus 2/15t^15,"},{"Start":"06:26.900 ","End":"06:35.010","Text":"and here it\u0027s 7 divide by the 7 minus 3/7t^7."},{"Start":"06:35.010 ","End":"06:43.890","Text":"Evaluate this from 0-1."},{"Start":"06:43.890 ","End":"06:46.380","Text":"At 0, I don\u0027t get anything."},{"Start":"06:46.380 ","End":"06:54.060","Text":"At 1, we just get the fraction minus 2/15 minus 3/7,"},{"Start":"06:54.060 ","End":"06:56.150","Text":"why don\u0027t I just put a bracket and make"},{"Start":"06:56.150 ","End":"07:02.550","Text":"that a plus 3/7, exercise in fractions."},{"Start":"07:03.460 ","End":"07:10.605","Text":"A common denominator has to be 15 times 7, nothing less."},{"Start":"07:10.605 ","End":"07:15.300","Text":"That\u0027s 105, 15 into a 105 goes 7 times,"},{"Start":"07:15.300 ","End":"07:18.765","Text":"7 times 2 is 14,"},{"Start":"07:18.765 ","End":"07:22.230","Text":"7 into 105 goes 15 times,"},{"Start":"07:22.230 ","End":"07:26.865","Text":"15 times 3 is 45."},{"Start":"07:26.865 ","End":"07:34.560","Text":"I\u0027ve kept the negative here, 14 and 45 is 59."},{"Start":"07:34.560 ","End":"07:39.580","Text":"The final answer is minus 59/105."},{"Start":"07:39.940 ","End":"07:43.340","Text":"I like to highlight it,"},{"Start":"07:43.340 ","End":"07:45.810","Text":"and we are done."}],"ID":8803},{"Watched":false,"Name":"Exercise 11 Part b","Duration":"9m 17s","ChapterTopicVideoID":8705,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8705.jpeg","UploadDate":"2017-02-13T05:23:32.8100000","DurationForVideoObject":"PT9M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:09.075","Text":"In this exercise, we have a line integral of type 2 over a curve C,"},{"Start":"00:09.075 ","End":"00:11.415","Text":"function f it\u0027s in 3 dimensions."},{"Start":"00:11.415 ","End":"00:13.470","Text":"We have x, y, and z."},{"Start":"00:13.470 ","End":"00:19.230","Text":"Actually, I think vector should be written with an arrow on top."},{"Start":"00:19.230 ","End":"00:21.135","Text":"F is a vector,"},{"Start":"00:21.135 ","End":"00:25.320","Text":"and this r is a vector. Let me explain."},{"Start":"00:25.320 ","End":"00:27.660","Text":"This is not the standard notation,"},{"Start":"00:27.660 ","End":"00:30.840","Text":"r or r of t,"},{"Start":"00:30.840 ","End":"00:33.660","Text":"but in general, r is the vector."},{"Start":"00:33.660 ","End":"00:36.270","Text":"I use the bracket form of a vector."},{"Start":"00:36.270 ","End":"00:39.375","Text":"It\u0027s x, y, z,"},{"Start":"00:39.375 ","End":"00:48.790","Text":"and dr also a vector is just dx, dy, dz."},{"Start":"00:49.040 ","End":"00:54.160","Text":"Now we can interpret this dot-product F.dr."},{"Start":"00:54.160 ","End":"01:02.315","Text":"It\u0027s this vector, dot with this vector and we just multiply component-wise and add."},{"Start":"01:02.315 ","End":"01:06.290","Text":"What we have is the integral along the curve"},{"Start":"01:06.290 ","End":"01:13.915","Text":"C of sine x times dx,"},{"Start":"01:13.915 ","End":"01:17.480","Text":"plus cosine y from here,"},{"Start":"01:17.480 ","End":"01:25.640","Text":"times dy, plus 1/3 component, xz times dz."},{"Start":"01:26.540 ","End":"01:29.335","Text":"What is this curve C?"},{"Start":"01:29.335 ","End":"01:32.314","Text":"Well, it\u0027s given in parametric form."},{"Start":"01:32.314 ","End":"01:38.405","Text":"C is just, if I write it with the curly bracket,"},{"Start":"01:38.405 ","End":"01:42.500","Text":"this is x, y, z as functions of t,"},{"Start":"01:42.500 ","End":"01:45.900","Text":"but it\u0027s x of t,"},{"Start":"01:45.900 ","End":"01:52.040","Text":"I don\u0027t bother to write the x of t is just t cubed, y of t,"},{"Start":"01:52.040 ","End":"01:55.055","Text":"or just y is minus t squared,"},{"Start":"01:55.055 ","End":"01:58.085","Text":"and z of t is just t,"},{"Start":"01:58.085 ","End":"02:00.950","Text":"and of course we need to know where the parameter goes from and to,"},{"Start":"02:00.950 ","End":"02:04.825","Text":"and it\u0027s just written here from 0-1."},{"Start":"02:04.825 ","End":"02:08.510","Text":"Now we can convert this to a regular integral in terms of"},{"Start":"02:08.510 ","End":"02:14.915","Text":"the parameter t. We have t goes from 0-1."},{"Start":"02:14.915 ","End":"02:17.660","Text":"I would like to emphasize the parameter,"},{"Start":"02:17.660 ","End":"02:20.945","Text":"and then we just translate everything."},{"Start":"02:20.945 ","End":"02:28.260","Text":"I see x, then I see that x is t cubed along the curve,"},{"Start":"02:28.260 ","End":"02:34.055","Text":"so I have sine of t cubed and then dx,"},{"Start":"02:34.055 ","End":"02:36.595","Text":"that we don\u0027t have."},{"Start":"02:36.595 ","End":"02:38.490","Text":"Let\u0027s just write those,"},{"Start":"02:38.490 ","End":"02:40.580","Text":"let\u0027s see what dx equals,"},{"Start":"02:40.580 ","End":"02:44.635","Text":"what dy equals and dz equals."},{"Start":"02:44.635 ","End":"02:47.150","Text":"Basically just the derivative of this times dt."},{"Start":"02:47.150 ","End":"02:52.850","Text":"3t squared dt minus"},{"Start":"02:52.850 ","End":"02:59.700","Text":"2t dt and just 1 dt."},{"Start":"02:59.700 ","End":"03:10.125","Text":"Getting back here, now I can write that dx is 3t squared dt plus cosine."},{"Start":"03:10.125 ","End":"03:15.190","Text":"I look up y, y is minus t squared,"},{"Start":"03:15.350 ","End":"03:24.810","Text":"and dy is minus 2t dt,"},{"Start":"03:24.810 ","End":"03:26.925","Text":"and the third component,"},{"Start":"03:26.925 ","End":"03:33.585","Text":"xz is t cubed times t,"},{"Start":"03:33.585 ","End":"03:37.095","Text":"and dz is just dt."},{"Start":"03:37.095 ","End":"03:40.905","Text":"We\u0027re going to simplify this expression a bit,"},{"Start":"03:40.905 ","End":"03:47.180","Text":"and I\u0027d like to just write this with a single dt."},{"Start":"03:47.180 ","End":"03:57.500","Text":"I\u0027ve got the integral from 0-1 of sine of t cubed times 3t"},{"Start":"03:57.500 ","End":"04:03.215","Text":"squared plus cosine of minus"},{"Start":"04:03.215 ","End":"04:09.120","Text":"t squared times minus 2t plus,"},{"Start":"04:09.120 ","End":"04:11.685","Text":"and this I can write as t^4,"},{"Start":"04:11.685 ","End":"04:15.090","Text":"and all this is dt."},{"Start":"04:15.090 ","End":"04:18.705","Text":"Now I have 3 integrals to do."},{"Start":"04:18.705 ","End":"04:23.945","Text":"These they look difficult to actually simpler than you think,"},{"Start":"04:23.945 ","End":"04:26.674","Text":"they were cooked up to be simple."},{"Start":"04:26.674 ","End":"04:29.600","Text":"Let me do an aside here."},{"Start":"04:29.600 ","End":"04:36.995","Text":"If I have a function sine of something with x in it,"},{"Start":"04:36.995 ","End":"04:39.275","Text":"and I take its derivative,"},{"Start":"04:39.275 ","End":"04:43.250","Text":"what I would get would be cosine of that thing."},{"Start":"04:43.250 ","End":"04:46.820","Text":"But then I\u0027d also have to multiply by the inner derivative,"},{"Start":"04:46.820 ","End":"04:50.255","Text":"which is say, box prime."},{"Start":"04:50.255 ","End":"04:59.465","Text":"I can do this backwards and say if I have the integral of cosine of some function of x,"},{"Start":"04:59.465 ","End":"05:03.125","Text":"and I have the derivative of that thing alongside,"},{"Start":"05:03.125 ","End":"05:08.150","Text":"then this is just sine of this."},{"Start":"05:08.150 ","End":"05:10.550","Text":"If we\u0027re doing indefinite integrals,"},{"Start":"05:10.550 ","End":"05:14.335","Text":"would have to add a plus c for definite integrals, we wouldn\u0027t need it."},{"Start":"05:14.335 ","End":"05:21.070","Text":"Similarly, the integral, if we have the sine of something,"},{"Start":"05:21.070 ","End":"05:25.005","Text":"maybe I better put this in brackets,"},{"Start":"05:25.005 ","End":"05:31.930","Text":"sine of this, this, this."},{"Start":"05:36.410 ","End":"05:41.380","Text":"I have also the derivative alongside,"},{"Start":"05:41.380 ","End":"05:44.460","Text":"and maybe it\u0027s dx here,"},{"Start":"05:44.460 ","End":"05:49.025","Text":"say the box is a function of x,"},{"Start":"05:49.025 ","End":"05:55.740","Text":"then the integral of sine is minus cosine of whatever this was,"},{"Start":"05:55.740 ","End":"05:58.380","Text":"and if we\u0027re doing indefinite integral we would"},{"Start":"05:58.380 ","End":"06:01.625","Text":"add plus c. Now why I\u0027m mentioning all this?"},{"Start":"06:01.625 ","End":"06:08.689","Text":"Because I happened to notice that if I take t cubed to be the box,"},{"Start":"06:08.689 ","End":"06:13.570","Text":"then this 3t squared is exactly box prime."},{"Start":"06:13.570 ","End":"06:17.795","Text":"The question was cooked up to come out easy and similarly,"},{"Start":"06:17.795 ","End":"06:22.830","Text":"if minus t squared is what I call box,"},{"Start":"06:22.830 ","End":"06:27.845","Text":"then minus 2t is the derivative of that."},{"Start":"06:27.845 ","End":"06:30.965","Text":"Here I worked with x, but it could have worked just as well with"},{"Start":"06:30.965 ","End":"06:36.705","Text":"t, doesn\u0027t really matter."},{"Start":"06:36.705 ","End":"06:40.070","Text":"What I can say is that this integral using"},{"Start":"06:40.070 ","End":"06:46.470","Text":"this scheme comes out to be using the one for sine."},{"Start":"06:46.580 ","End":"06:50.985","Text":"Then I get minus cosine,"},{"Start":"06:50.985 ","End":"06:57.070","Text":"so here I have minus cosine of t cubed,"},{"Start":"06:57.070 ","End":"07:01.730","Text":"and I don\u0027t need the plus c because it\u0027s going to be a definite integral."},{"Start":"07:01.730 ","End":"07:07.720","Text":"The next bit for the cosine, I\u0027m using these 2."},{"Start":"07:07.720 ","End":"07:11.130","Text":"This was just a way to show you how I got to this."},{"Start":"07:14.530 ","End":"07:20.370","Text":"The integral of cosine is just sine,"},{"Start":"07:20.370 ","End":"07:26.285","Text":"so I have sine of minus t squared,"},{"Start":"07:26.285 ","End":"07:29.420","Text":"and the last one is just a polynomial,"},{"Start":"07:29.420 ","End":"07:33.140","Text":"so it\u0027s 1/5 t^5."},{"Start":"07:33.140 ","End":"07:39.630","Text":"Now all this has to be evaluated between 0 and 1."},{"Start":"07:40.160 ","End":"07:42.935","Text":"Let\u0027s see what we get."},{"Start":"07:42.935 ","End":"07:46.800","Text":"First of all, we want to substitute 1,"},{"Start":"07:48.830 ","End":"07:51.260","Text":"1 cubed is 1,"},{"Start":"07:51.260 ","End":"07:56.360","Text":"so we get minus cosine of 1."},{"Start":"07:56.360 ","End":"08:01.400","Text":"Then we have plus sine of minus 1."},{"Start":"08:01.400 ","End":"08:03.590","Text":"But sine is an odd function."},{"Start":"08:03.590 ","End":"08:06.220","Text":"The sine of a minus I can bring the minus in front,"},{"Start":"08:06.220 ","End":"08:09.080","Text":"so this is minus sine 1."},{"Start":"08:09.080 ","End":"08:11.530","Text":"Make a note that that\u0027s an odd function."},{"Start":"08:11.530 ","End":"08:13.955","Text":"I can take the minus in front,"},{"Start":"08:13.955 ","End":"08:22.080","Text":"and here I get plus 1/5,"},{"Start":"08:22.080 ","End":"08:25.055","Text":"and all this is the 1 part."},{"Start":"08:25.055 ","End":"08:28.325","Text":"Now I need to subtract the 0 part."},{"Start":"08:28.325 ","End":"08:35.525","Text":"I get minus cosine 0 and cosine 0 is 1."},{"Start":"08:35.525 ","End":"08:42.920","Text":"That\u0027s minus 1, and then sine of 0 is 0,"},{"Start":"08:42.920 ","End":"08:49.155","Text":"so I have just 0 and here also 0."},{"Start":"08:49.155 ","End":"08:52.475","Text":"If I add it all up together,"},{"Start":"08:52.475 ","End":"08:57.160","Text":"let\u0027s see the numbers without the trigonometric stuff."},{"Start":"08:57.160 ","End":"09:04.470","Text":"Fifth minus minus 1 is 1 and 1/5 or 6/5 whichever,"},{"Start":"09:04.470 ","End":"09:12.805","Text":"and then minus cosine 1 minus sine 1,"},{"Start":"09:12.805 ","End":"09:17.050","Text":"and this is the answer to the question. We\u0027re done."}],"ID":8804},{"Watched":false,"Name":"Exercise 12 Part a","Duration":"9m 40s","ChapterTopicVideoID":8706,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8706.jpeg","UploadDate":"2017-02-13T05:25:54.3100000","DurationForVideoObject":"PT9M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.540","Text":"This exercise is really from physics"},{"Start":"00:03.540 ","End":"00:07.815","Text":"but don\u0027t worry, I\u0027ll give you all the formulas that you need."},{"Start":"00:07.815 ","End":"00:12.045","Text":"We have to compute the work done by a force field,"},{"Start":"00:12.045 ","End":"00:14.205","Text":"which is given as follows."},{"Start":"00:14.205 ","End":"00:18.285","Text":"On a particle which moves along the parabola,"},{"Start":"00:18.285 ","End":"00:21.420","Text":"y equals x squared from this point to this point."},{"Start":"00:21.420 ","End":"00:25.515","Text":"Let\u0027s just worry about part a first and we\u0027ll get to part b."},{"Start":"00:25.515 ","End":"00:32.385","Text":"Now, I brought in the formula or the theory that we need for doing this problem."},{"Start":"00:32.385 ","End":"00:35.630","Text":"I found this on the Internet and I copied it."},{"Start":"00:35.630 ","End":"00:41.470","Text":"It says that as a particle moves through a force field along the path C,"},{"Start":"00:41.470 ","End":"00:44.900","Text":"the work done by the force is the line integral,"},{"Start":"00:44.900 ","End":"00:47.495","Text":"and that\u0027s the chapter we\u0027re on, line integrals."},{"Start":"00:47.495 ","End":"00:50.995","Text":"Given by W, W is the work,"},{"Start":"00:50.995 ","End":"00:53.385","Text":"F is the force field,"},{"Start":"00:53.385 ","End":"00:57.825","Text":"and what we don\u0027t have here is C,"},{"Start":"00:57.825 ","End":"01:02.185","Text":"and I need to explain again what is dr."},{"Start":"01:02.185 ","End":"01:04.665","Text":"I\u0027d like to start with C,"},{"Start":"01:04.665 ","End":"01:09.635","Text":"and C is this path along the parabola from minus 2,4 to 1,1."},{"Start":"01:09.635 ","End":"01:10.955","Text":"We don\u0027t need a sketch,"},{"Start":"01:10.955 ","End":"01:13.340","Text":"but it might just help."},{"Start":"01:13.340 ","End":"01:16.625","Text":"Here\u0027s a pair of axis,"},{"Start":"01:16.625 ","End":"01:18.950","Text":"and let\u0027s get 1,1."},{"Start":"01:18.950 ","End":"01:21.200","Text":"Let\u0027s say this is 1 unit,"},{"Start":"01:21.200 ","End":"01:24.340","Text":"1 unit somewhere around here,"},{"Start":"01:24.340 ","End":"01:28.995","Text":"and minus 2,4, let\u0027s say it\u0027s about 2."},{"Start":"01:28.995 ","End":"01:31.755","Text":"It doesn\u0027t really matter, somewhere here."},{"Start":"01:31.755 ","End":"01:35.035","Text":"Certainly, the parabola goes through the origin."},{"Start":"01:35.035 ","End":"01:38.190","Text":"We get something like this."},{"Start":"01:38.190 ","End":"01:41.495","Text":"Over here, we get something like this."},{"Start":"01:41.495 ","End":"01:44.765","Text":"It really doesn\u0027t have to be exact"},{"Start":"01:44.765 ","End":"01:47.285","Text":"but you don\u0027t need a sketch at all."},{"Start":"01:47.285 ","End":"01:55.310","Text":"The point is that the x goes from minus 2 to 1."},{"Start":"01:55.310 ","End":"02:00.240","Text":"The curve, just this part, I\u0027ll highlight it,"},{"Start":"02:00.240 ","End":"02:07.275","Text":"we just want this path and going in this direction."},{"Start":"02:07.275 ","End":"02:09.680","Text":"That\u0027s going to be our curve C,"},{"Start":"02:09.680 ","End":"02:12.575","Text":"just from here to here."},{"Start":"02:12.575 ","End":"02:17.260","Text":"Now, I would like to have C as parametrized curve."},{"Start":"02:17.260 ","End":"02:22.270","Text":"This is easy to do because we have y as a function of x,"},{"Start":"02:22.270 ","End":"02:26.730","Text":"and we have where x goes from and to."},{"Start":"02:26.730 ","End":"02:35.389","Text":"We just let x be t. Here\u0027s some curly braces and then x equals t,"},{"Start":"02:35.389 ","End":"02:40.325","Text":"and then y being x squared is just t squared."},{"Start":"02:40.325 ","End":"02:43.220","Text":"The range of t, which is x,"},{"Start":"02:43.220 ","End":"02:49.395","Text":"is from minus 2 up to 1."},{"Start":"02:49.395 ","End":"02:51.930","Text":"That\u0027s the parametrized, the"},{"Start":"02:51.930 ","End":"03:00.630","Text":"C. I know that we\u0027ll also need later dx and dy,"},{"Start":"03:00.630 ","End":"03:03.105","Text":"and we always do."},{"Start":"03:03.105 ","End":"03:09.450","Text":"From here, I can get that dx is equal to dt."},{"Start":"03:09.450 ","End":"03:13.680","Text":"From here, I can see that dy j,"},{"Start":"03:13.680 ","End":"03:17.380","Text":"ust the derivative 2t squared, which is 2tdt."},{"Start":"03:18.530 ","End":"03:21.840","Text":"That\u0027s the curve C. Now,"},{"Start":"03:21.840 ","End":"03:25.120","Text":"what\u0027s this r and dr?"},{"Start":"03:25.910 ","End":"03:33.080","Text":"Well, the vector r is the position vector of a point along"},{"Start":"03:33.080 ","End":"03:36.545","Text":"our curve C. It\u0027s"},{"Start":"03:36.545 ","End":"03:45.680","Text":"just x times the vector i plus y times vector j,"},{"Start":"03:45.680 ","End":"03:48.950","Text":"where x and y are the x and y of a point on the curve."},{"Start":"03:48.950 ","End":"03:53.020","Text":"We could also write it in the angular brackets notation."},{"Start":"03:53.020 ","End":"03:55.710","Text":"You could write it x, y,"},{"Start":"03:55.710 ","End":"03:57.600","Text":"but since f is given with the i,"},{"Start":"03:57.600 ","End":"03:59.580","Text":"j, I\u0027m sticking with the i,"},{"Start":"03:59.580 ","End":"04:02.005","Text":"j and I\u0027ll erase this."},{"Start":"04:02.005 ","End":"04:05.370","Text":"In general, I deliberately like to sometimes use the i,"},{"Start":"04:05.370 ","End":"04:08.255","Text":"j notation and sometimes the brackets"},{"Start":"04:08.255 ","End":"04:13.025","Text":"notation for vectors because they\u0027re both used in the literature."},{"Start":"04:13.025 ","End":"04:16.625","Text":"You should know both. That\u0027s that,"},{"Start":"04:16.625 ","End":"04:20.250","Text":"and as I was saying, we also need to know what is dr."},{"Start":"04:20.990 ","End":"04:24.240","Text":"This is also a vector,"},{"Start":"04:24.240 ","End":"04:26.505","Text":"and that\u0027s just dx,"},{"Start":"04:26.505 ","End":"04:30.920","Text":"I\u0027ll write it as dx,dy or dx times vector i,"},{"Start":"04:30.920 ","End":"04:35.260","Text":"plus dy times vector j."},{"Start":"04:35.260 ","End":"04:38.620","Text":"Here, I have a dot-product."},{"Start":"04:38.720 ","End":"04:45.270","Text":"What we have if I do this dot product, the f part,"},{"Start":"04:45.270 ","End":"04:48.310","Text":"I can take from here,"},{"Start":"04:48.890 ","End":"04:55.035","Text":"and the dr, which is this one here,"},{"Start":"04:55.035 ","End":"04:57.855","Text":"I can take from here."},{"Start":"04:57.855 ","End":"05:04.220","Text":"The dot-product is just we take the i components and multiply them,"},{"Start":"05:04.220 ","End":"05:06.935","Text":"the j components and multiply them, and then add."},{"Start":"05:06.935 ","End":"05:08.905","Text":"I\u0027ll write that here."},{"Start":"05:08.905 ","End":"05:14.045","Text":"That W is just the integral along the curve."},{"Start":"05:14.045 ","End":"05:17.690","Text":"That\u0027s the type 2 line integral."},{"Start":"05:18.960 ","End":"05:26.230","Text":"We need x cubed y times dx"},{"Start":"05:26.230 ","End":"05:34.770","Text":"plus then x minus y times dy."},{"Start":"05:34.770 ","End":"05:38.940","Text":"This is a more familiar form of the line integral."},{"Start":"05:38.940 ","End":"05:42.410","Text":"Now, we use the standard methods since C is"},{"Start":"05:42.410 ","End":"05:46.700","Text":"parametrized and we have the complete parametrization,"},{"Start":"05:46.700 ","End":"05:52.430","Text":"with x and y is in the range of t. We convert this to a regular"},{"Start":"05:52.430 ","End":"06:00.569","Text":"integral in terms of t. We say that t goes from minus 2 to 1,"},{"Start":"06:00.680 ","End":"06:04.355","Text":"and then we substitute everything."},{"Start":"06:04.355 ","End":"06:08.480","Text":"The x and the y here are the x and the y along the curve,"},{"Start":"06:08.480 ","End":"06:10.595","Text":"and that\u0027s this here."},{"Start":"06:10.595 ","End":"06:17.690","Text":"So x cubed y is t cubed times t squared."},{"Start":"06:17.690 ","End":"06:24.760","Text":"Then dx, here it is, dt,"},{"Start":"06:24.760 ","End":"06:28.200","Text":"plus x minus y,"},{"Start":"06:28.200 ","End":"06:30.869","Text":"which is t minus t squared,"},{"Start":"06:30.869 ","End":"06:35.620","Text":"and dy is here, 2tdt."},{"Start":"06:36.830 ","End":"06:43.880","Text":"Now, I\u0027ll just arrange this so that I have just everything dt."},{"Start":"06:43.880 ","End":"06:46.685","Text":"Let\u0027s open the bracket and let\u0027s see what we have."},{"Start":"06:46.685 ","End":"06:50.305","Text":"From here, we get t^5."},{"Start":"06:50.305 ","End":"06:56.010","Text":"This times this gives me plus 2t squared,"},{"Start":"06:56.010 ","End":"07:01.665","Text":"and then minus 2t cubed dt,"},{"Start":"07:01.665 ","End":"07:04.740","Text":"and everything is completely routine."},{"Start":"07:04.740 ","End":"07:09.100","Text":"Here, we get t^6 over 6."},{"Start":"07:10.400 ","End":"07:16.950","Text":"Here, I have 2t cubed over 3,"},{"Start":"07:16.950 ","End":"07:19.650","Text":"so it\u0027s 2/3t cubed."},{"Start":"07:19.650 ","End":"07:21.600","Text":"Here, I have minus,"},{"Start":"07:21.600 ","End":"07:24.375","Text":"it\u0027s t^4, and here,"},{"Start":"07:24.375 ","End":"07:26.610","Text":"minus 2 over 4."},{"Start":"07:26.610 ","End":"07:30.530","Text":"So 2/4 is a 1/2, and all this,"},{"Start":"07:30.530 ","End":"07:34.445","Text":"we substitute from minus 2 to 1 this,"},{"Start":"07:34.445 ","End":"07:36.930","Text":"and then this and subtract."},{"Start":"07:38.330 ","End":"07:41.655","Text":"Let\u0027s see, if I plug in 1,"},{"Start":"07:41.655 ","End":"07:48.345","Text":"I get 1/6 plus 2/3 minus 1/2."},{"Start":"07:48.345 ","End":"07:53.590","Text":"If I plug in minus 2, let\u0027s see."},{"Start":"07:53.590 ","End":"07:55.490","Text":"It\u0027s an even power,"},{"Start":"07:55.490 ","End":"07:56.675","Text":"so it\u0027s like 2^6,"},{"Start":"07:56.675 ","End":"08:03.850","Text":"which is 64/6, I\u0027ll write it as 32/3."},{"Start":"08:03.850 ","End":"08:08.285","Text":"Next, it\u0027s going to be negative because it\u0027s an odd power."},{"Start":"08:08.285 ","End":"08:12.500","Text":"We have minus 8 times 2/3."},{"Start":"08:12.500 ","End":"08:21.610","Text":"That minus 8 times 2/3, 16/3."},{"Start":"08:21.610 ","End":"08:24.030","Text":"Then here, the minus 2_4,"},{"Start":"08:24.030 ","End":"08:26.625","Text":"so it\u0027s plus 16,"},{"Start":"08:26.625 ","End":"08:28.455","Text":"but it\u0027s minus a 1/2."},{"Start":"08:28.455 ","End":"08:31.545","Text":"So it\u0027s minus 8."},{"Start":"08:31.545 ","End":"08:38.775","Text":"Let\u0027s see now, 1/6 and 2/3 is 5/6, minus 1/2,"},{"Start":"08:38.775 ","End":"08:45.990","Text":"which is 3/6, so altogether 2/6, this is 1/3."},{"Start":"08:45.990 ","End":"08:48.270","Text":"As for the other one, this minus,"},{"Start":"08:48.270 ","End":"08:55.860","Text":"this is just 16/3."},{"Start":"08:55.860 ","End":"09:04.450","Text":"16/3 is 5 and 1/3, minus 8."},{"Start":"09:04.970 ","End":"09:08.685","Text":"I\u0027ll just write it as 5 and 1/3 minus 8,"},{"Start":"09:08.685 ","End":"09:10.485","Text":"I\u0027ll compute it later."},{"Start":"09:10.485 ","End":"09:19.485","Text":"Let\u0027s see, 1/3 minus 5/3 is just minus 5,"},{"Start":"09:19.485 ","End":"09:22.995","Text":"and then plus 8."},{"Start":"09:22.995 ","End":"09:26.560","Text":"I make the answer as 3."},{"Start":"09:26.560 ","End":"09:29.015","Text":"This is the answer to part a."},{"Start":"09:29.015 ","End":"09:36.485","Text":"This is the work performed by the force on the particle that travels along the curve."},{"Start":"09:36.485 ","End":"09:40.830","Text":"That concludes part a and part b in the next clip."}],"ID":8805},{"Watched":false,"Name":"Exercise 12 Part b","Duration":"2m 14s","ChapterTopicVideoID":8707,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8707.jpeg","UploadDate":"2017-02-13T05:26:29.1630000","DurationForVideoObject":"PT2M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.585","Text":"Now in part b, they\u0027re asking us,"},{"Start":"00:03.585 ","End":"00:10.545","Text":"how would the answer change if the particle moved from this point,"},{"Start":"00:10.545 ","End":"00:16.890","Text":"this is the 1,1 to minus 2,4,"},{"Start":"00:16.890 ","End":"00:20.910","Text":"implying it\u0027s along the same curve."},{"Start":"00:20.910 ","End":"00:25.050","Text":"In fact, if we go in the opposite direction,"},{"Start":"00:25.050 ","End":"00:27.210","Text":"there is a name for this curve."},{"Start":"00:27.210 ","End":"00:30.600","Text":"We call this curve minus c,"},{"Start":"00:30.600 ","End":"00:34.480","Text":"same thing but to the opposite direction."},{"Start":"00:35.780 ","End":"00:40.395","Text":"In fact, there\u0027s a theorem that relates to that."},{"Start":"00:40.395 ","End":"00:46.010","Text":"That in general, if we take the integral along"},{"Start":"00:46.010 ","End":"00:54.200","Text":"the opposite curve of some line integral of type 2 F.dr in this case,"},{"Start":"00:54.200 ","End":"01:02.380","Text":"that this is just equal to minus the regular direction F.dr."},{"Start":"01:05.050 ","End":"01:10.610","Text":"Another way of expressing this is that suppose we give them names."},{"Start":"01:10.610 ","End":"01:15.140","Text":"Maybe this is the point M and this is the point N,"},{"Start":"01:15.140 ","End":"01:17.374","Text":"just pick a pair of letters."},{"Start":"01:17.374 ","End":"01:27.125","Text":"As you could say that the integral from M to N, I won\u0027t repeat it, of the same thing"},{"Start":"01:27.125 ","End":"01:32.700","Text":"is equal to minus the"},{"Start":"01:32.700 ","End":"01:40.665","Text":"integral from N to M. Or perhaps I meant backwards."},{"Start":"01:40.665 ","End":"01:46.055","Text":"The 1 I want in terms of the 1 I have, but anyway, in both cases,"},{"Start":"01:46.055 ","End":"01:51.950","Text":"we have the extra minus here if we reverse the curve or reverse the start and end points."},{"Start":"01:51.950 ","End":"01:55.940","Text":"In our case, the answer would just make it a minus."},{"Start":"01:55.940 ","End":"02:01.580","Text":"In part b, the answer would just become then minus 3."},{"Start":"02:01.580 ","End":"02:02.885","Text":"Let\u0027s do it with 3."},{"Start":"02:02.885 ","End":"02:06.320","Text":"This is for part a in the regular direction and this is"},{"Start":"02:06.320 ","End":"02:10.955","Text":"the answer for part b. I guess I\u0027ll highlight this 1 also."},{"Start":"02:10.955 ","End":"02:13.600","Text":"That concludes part b."}],"ID":8806},{"Watched":false,"Name":"Exercise 13","Duration":"6m 35s","ChapterTopicVideoID":8708,"CourseChapterTopicPlaylistID":82682,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8708.jpeg","UploadDate":"2017-02-13T05:28:03.0430000","DurationForVideoObject":"PT6M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.660","Text":"Here we have a question from physics involving the work done by"},{"Start":"00:03.660 ","End":"00:07.650","Text":"a force field on a particle which moves along a path."},{"Start":"00:07.650 ","End":"00:09.930","Text":"We\u0027ve had 1 of these before,"},{"Start":"00:09.930 ","End":"00:11.460","Text":"but in case you haven\u0027t,"},{"Start":"00:11.460 ","End":"00:14.910","Text":"I brought the formula again."},{"Start":"00:14.910 ","End":"00:19.485","Text":"Essentially, all it says here is that to calculate the work,"},{"Start":"00:19.485 ","End":"00:21.870","Text":"you just have to compute the line integral,"},{"Start":"00:21.870 ","End":"00:28.140","Text":"the type 2 line integral of F dot product with dr."},{"Start":"00:28.140 ","End":"00:35.665","Text":"In general, we use the vector r to be an abbreviation for"},{"Start":"00:35.665 ","End":"00:45.065","Text":"x in the i direction plus y in the j direction,"},{"Start":"00:45.065 ","End":"00:51.485","Text":"plus z in the k direction."},{"Start":"00:51.485 ","End":"00:54.320","Text":"If you\u0027re using brackets notation,"},{"Start":"00:54.320 ","End":"00:59.240","Text":"you would just write x, y, z,"},{"Start":"00:59.240 ","End":"01:05.165","Text":"and in general also as r could be a function of t,"},{"Start":"01:05.165 ","End":"01:10.114","Text":"which case we\u0027d have r of t is x of t, y of t,"},{"Start":"01:10.114 ","End":"01:13.620","Text":"z of t, and I\u0027m going to use the i,"},{"Start":"01:13.620 ","End":"01:15.810","Text":"j, k so I\u0027ll erase this,"},{"Start":"01:15.810 ","End":"01:20.195","Text":"and I can rewrite this now in our case because we have F,"},{"Start":"01:20.195 ","End":"01:24.200","Text":"which is given as this."},{"Start":"01:25.280 ","End":"01:30.374","Text":"I didn\u0027t say what is dr. Well, in general,"},{"Start":"01:30.374 ","End":"01:34.890","Text":"dr is just dx in"},{"Start":"01:34.890 ","End":"01:45.795","Text":"the i direction plus dyj plus dzk,"},{"Start":"01:45.795 ","End":"01:51.260","Text":"and so now I do have this which I can replace by this."},{"Start":"01:51.260 ","End":"01:55.190","Text":"Now I need the dot product of this with this."},{"Start":"01:55.190 ","End":"01:56.810","Text":"You know how to do dot-product,"},{"Start":"01:56.810 ","End":"02:00.050","Text":"we just multiply the i components,"},{"Start":"02:00.050 ","End":"02:04.245","Text":"the j components, and the k components and add them."},{"Start":"02:04.245 ","End":"02:10.850","Text":"We would get that w is the integral along C,"},{"Start":"02:10.850 ","End":"02:16.435","Text":"and we still haven\u0027t quite said what is C. We\u0027ll get to that."},{"Start":"02:16.435 ","End":"02:24.090","Text":"Let\u0027s see, we have yzdx,"},{"Start":"02:24.090 ","End":"02:25.650","Text":"for the first component."},{"Start":"02:25.650 ","End":"02:27.000","Text":"For the second component,"},{"Start":"02:27.000 ","End":"02:37.080","Text":"I have xz with dy and xy with dz."},{"Start":"02:37.720 ","End":"02:40.520","Text":"This is a more familiar form."},{"Start":"02:40.520 ","End":"02:46.355","Text":"Now, the curve C or the path is just what\u0027s given by the definition of"},{"Start":"02:46.355 ","End":"02:55.160","Text":"r. We could write it in parametric form that the path C is given by,"},{"Start":"02:55.160 ","End":"02:57.470","Text":"let\u0027s see, x equals,"},{"Start":"02:57.470 ","End":"03:00.200","Text":"y equals, z equals."},{"Start":"03:00.200 ","End":"03:04.249","Text":"X is just t, the first component,"},{"Start":"03:04.249 ","End":"03:09.905","Text":"y is t squared and z is t cubed,"},{"Start":"03:09.905 ","End":"03:12.110","Text":"and we also need to know what the parameter"},{"Start":"03:12.110 ","End":"03:14.765","Text":"goes from and to and that\u0027s what\u0027s written here."},{"Start":"03:14.765 ","End":"03:17.810","Text":"0 less than or equal to t,"},{"Start":"03:17.810 ","End":"03:20.430","Text":"less than or equal to 1."},{"Start":"03:20.770 ","End":"03:23.780","Text":"We need to know what dx,"},{"Start":"03:23.780 ","End":"03:26.704","Text":"dy, and dz in our case."},{"Start":"03:26.704 ","End":"03:28.850","Text":"Well, if x is t,"},{"Start":"03:28.850 ","End":"03:31.730","Text":"then dx is just dt."},{"Start":"03:31.730 ","End":"03:38.900","Text":"Here, dy is just the derivative of this dt so 2t, dt,"},{"Start":"03:38.900 ","End":"03:46.405","Text":"and dz is 3t squared dt."},{"Start":"03:46.405 ","End":"03:50.060","Text":"I\u0027ve got everything, the x, y,"},{"Start":"03:50.060 ","End":"03:52.490","Text":"and z here are all the x, y,"},{"Start":"03:52.490 ","End":"03:55.850","Text":"and z along the curve so that\u0027s these."},{"Start":"03:55.850 ","End":"04:01.520","Text":"Just substitute everything and instead of integral along the curve,"},{"Start":"04:01.520 ","End":"04:08.550","Text":"we just get a regular integral for t going from 0 to 1, 0 to 1."},{"Start":"04:08.550 ","End":"04:13.930","Text":"Let\u0027s see, yz is t squared,"},{"Start":"04:13.930 ","End":"04:18.780","Text":"t cubed and dx is dt."},{"Start":"04:18.780 ","End":"04:22.185","Text":"The next bit, x is t,"},{"Start":"04:22.185 ","End":"04:29.820","Text":"z is t cubed and dy is 2tdt."},{"Start":"04:29.820 ","End":"04:37.080","Text":"What\u0027s here? Lastly, xy is t,"},{"Start":"04:37.080 ","End":"04:42.135","Text":"t squared from here and here and the dz from here"},{"Start":"04:42.135 ","End":"04:48.210","Text":"is 3t squared dt."},{"Start":"04:48.210 ","End":"04:50.360","Text":"Just want to simplify this,"},{"Start":"04:50.360 ","End":"04:55.520","Text":"just get it as a single dt so integral from 0 to 1."},{"Start":"04:55.520 ","End":"04:57.455","Text":"Now, what do we have here?"},{"Start":"04:57.455 ","End":"05:01.450","Text":"T squared times t cubed is t^5."},{"Start":"05:01.450 ","End":"05:04.440","Text":"From here we have t,"},{"Start":"05:04.440 ","End":"05:06.180","Text":"t cubed and t,"},{"Start":"05:06.180 ","End":"05:08.445","Text":"give me t^5 with the 2."},{"Start":"05:08.445 ","End":"05:13.410","Text":"That\u0027s 2t^5 and this bit,"},{"Start":"05:13.410 ","End":"05:17.630","Text":"I also get t^5 because it\u0027s 1 plus 2 plus 2."},{"Start":"05:17.630 ","End":"05:24.940","Text":"But there\u0027s a 3 here, 3t^5 dt."},{"Start":"05:24.940 ","End":"05:28.080","Text":"They\u0027re all t^5 so I could just add them up."},{"Start":"05:28.080 ","End":"05:30.885","Text":"1 plus 2 plus 3 is 6."},{"Start":"05:30.885 ","End":"05:37.580","Text":"In fact, I can take the 6 in front of the integral so I get 6 times the integral"},{"Start":"05:37.580 ","End":"05:41.525","Text":"from 0 to 1"},{"Start":"05:41.525 ","End":"05:49.140","Text":"of t^5 dt and I\u0027ve changed my mind,"},{"Start":"05:49.140 ","End":"05:52.905","Text":"I\u0027m going to put the 6 back in here, you\u0027ll see why."},{"Start":"05:52.905 ","End":"05:55.290","Text":"The reason I put the 6 back here,"},{"Start":"05:55.290 ","End":"05:57.775","Text":"is I realized that 6t^5,"},{"Start":"05:57.775 ","End":"06:02.675","Text":"the integral of that is exactly t^6"},{"Start":"06:02.675 ","End":"06:09.950","Text":"because you raise the power by 1 and divide or if you just look at it the other way,"},{"Start":"06:09.950 ","End":"06:13.685","Text":"if I differentiate t^6 I get 6t^5."},{"Start":"06:13.685 ","End":"06:18.740","Text":"This is t^6, taken from 0 to 1."},{"Start":"06:18.740 ","End":"06:21.770","Text":"When I put in 0, I get nothing."},{"Start":"06:21.770 ","End":"06:27.630","Text":"When I put in 1, I just get 1 so the answer is nice and simple."},{"Start":"06:27.630 ","End":"06:35.670","Text":"It\u0027s just 1 unit of work which I shall highlight and that\u0027s all there is to it."}],"ID":8807}],"Thumbnail":null,"ID":82682}]

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