Exercises - Conservative Vector Fields
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[{"Name":"Exercises - Conservative Vector Fields","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Curves and Regions","Duration":"6m 43s","ChapterTopicVideoID":8594,"CourseChapterTopicPlaylistID":4965,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8594.jpeg","UploadDate":"2020-02-26T12:14:59.8900000","DurationForVideoObject":"PT6M43S","Description":null,"MetaTitle":"Curves and Regions: Video + Workbook | Proprep","MetaDescription":"Conservative Vector Fields - Exercises - Conservative Vector Fields. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/calculus-i%2c-ii-and-iii/conservative-vector-fields/exercises-_-conservative-vector-fields/vid8751","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.290","Text":"In this clip, I\u0027ll be talking about conservative vector fields."},{"Start":"00:04.290 ","End":"00:06.630","Text":"But before we get into it,"},{"Start":"00:06.630 ","End":"00:11.550","Text":"I want to get some definitions out of the way that"},{"Start":"00:11.550 ","End":"00:16.740","Text":"involve paths and regions, curves and domains."},{"Start":"00:16.740 ","End":"00:21.300","Text":"Let\u0027s start with curves,"},{"Start":"00:21.300 ","End":"00:24.360","Text":"and a similar concept,"},{"Start":"00:24.360 ","End":"00:28.210","Text":"almost the same, paths."},{"Start":"00:28.370 ","End":"00:32.580","Text":"The main difference is that a path is actually"},{"Start":"00:32.580 ","End":"00:38.185","Text":"a parameterized function that"},{"Start":"00:38.185 ","End":"00:43.560","Text":"describes a moving point from 1 place to another in terms of a parameter t,"},{"Start":"00:43.560 ","End":"00:48.320","Text":"and the curve is just the geometrical shape it traces out,"},{"Start":"00:48.320 ","End":"00:52.555","Text":"but we sometimes mix them up curves, paths."},{"Start":"00:52.555 ","End":"01:00.020","Text":"What I wanted to talk about was 2 concepts called simple and closed in regard to paths."},{"Start":"01:00.020 ","End":"01:05.945","Text":"Now, this would be an example of a simple path because it doesn\u0027t cross itself,"},{"Start":"01:05.945 ","End":"01:14.100","Text":"but if I drew it something like this and this."},{"Start":"01:14.100 ","End":"01:19.490","Text":"We\u0027ll talk about paths and also it has to have an arrow on it."},{"Start":"01:19.490 ","End":"01:23.180","Text":"Although it suppose it could change direction in the middle,"},{"Start":"01:23.180 ","End":"01:25.985","Text":"but usually we have a direction."},{"Start":"01:25.985 ","End":"01:30.695","Text":"This is not simple and this is simple."},{"Start":"01:30.695 ","End":"01:33.600","Text":"Also if it starts and ends at the same point,"},{"Start":"01:33.600 ","End":"01:41.680","Text":"so if it goes like this and it has the same start and endpoint, then it\u0027s closed."},{"Start":"01:41.680 ","End":"01:45.350","Text":"Of course, it could be closed but not simple like,"},{"Start":"01:45.350 ","End":"01:47.610","Text":"I don\u0027t know, figure 8."},{"Start":"01:48.130 ","End":"01:56.915","Text":"Maybe this is the start point and it goes in this way along here and here."},{"Start":"01:56.915 ","End":"01:58.565","Text":"So this would be closed,"},{"Start":"01:58.565 ","End":"02:01.315","Text":"but it\u0027s not simple."},{"Start":"02:01.315 ","End":"02:10.410","Text":"The top 2 would be simple and the other 2 are not,"},{"Start":"02:10.410 ","End":"02:15.360","Text":"whereas these 2 are"},{"Start":"02:15.360 ","End":"02:21.380","Text":"both closed and these 2 are not."},{"Start":"02:21.380 ","End":"02:24.035","Text":"So this is simple and closed."},{"Start":"02:24.035 ","End":"02:28.129","Text":"Okay, that\u0027s as far as paths and curves,"},{"Start":"02:28.129 ","End":"02:32.165","Text":"2D and 3D, or any number of dimensions."},{"Start":"02:32.165 ","End":"02:38.900","Text":"Let\u0027s now talk about regions which are"},{"Start":"02:38.900 ","End":"02:42.290","Text":"closely related to domains and there"},{"Start":"02:42.290 ","End":"02:46.800","Text":"is some confusion and overlap between the 2 concepts."},{"Start":"02:46.880 ","End":"02:51.090","Text":"Mostly I\u0027ll be talking about these in 2D,"},{"Start":"02:51.090 ","End":"02:54.340","Text":"gets a bit complicated when we move to 3D."},{"Start":"02:54.340 ","End":"03:00.295","Text":"A domain is often the domain of a function, we mean."},{"Start":"03:00.295 ","End":"03:03.415","Text":"So for example, if I took the function,"},{"Start":"03:03.415 ","End":"03:08.740","Text":"say the square root of 9 minus x"},{"Start":"03:08.740 ","End":"03:14.140","Text":"squared minus y squared as a function in 2 variables,"},{"Start":"03:14.140 ","End":"03:18.264","Text":"then its domain would be where"},{"Start":"03:18.264 ","End":"03:24.205","Text":"the 9 minus x squared minus y squared is bigger or equal to 0,"},{"Start":"03:24.205 ","End":"03:31.580","Text":"and this will give us x squared plus y squared less than or equal to 9."},{"Start":"03:31.580 ","End":"03:34.460","Text":"That would be the domain of the functional,"},{"Start":"03:34.460 ","End":"03:38.410","Text":"I\u0027ll call it d, and if we illustrate that,"},{"Start":"03:38.410 ","End":"03:43.920","Text":"we\u0027d get a circle of radius 3 centered at the origin,"},{"Start":"03:43.920 ","End":"03:49.880","Text":"and the shaded bit is this and it includes the circle cause of the less than or equal to."},{"Start":"03:49.880 ","End":"03:53.030","Text":"If I had taken some other function, for example,"},{"Start":"03:53.030 ","End":"03:57.139","Text":"the natural log of the same expression,"},{"Start":"03:57.139 ","End":"04:00.095","Text":"9 minus x squared minus y squared,"},{"Start":"04:00.095 ","End":"04:03.800","Text":"then it would look like this, almost,"},{"Start":"04:03.800 ","End":"04:07.230","Text":"but the less than or equal to would be less than,"},{"Start":"04:07.230 ","End":"04:11.285","Text":"and then it wouldn\u0027t include the boundary which is the circle."},{"Start":"04:11.285 ","End":"04:17.780","Text":"This will be useful in a moment when I talk about open and closed regions."},{"Start":"04:17.780 ","End":"04:22.445","Text":"A domain is just often the domain of definition part and"},{"Start":"04:22.445 ","End":"04:27.665","Text":"sometimes just a region meaning part of the plane or part of 3D space."},{"Start":"04:27.665 ","End":"04:30.335","Text":"I\u0027m not going to get very formal about it."},{"Start":"04:30.335 ","End":"04:32.570","Text":"I\u0027ll typically use the letter D,"},{"Start":"04:32.570 ","End":"04:34.130","Text":"whether it\u0027s region or domain,"},{"Start":"04:34.130 ","End":"04:36.535","Text":"or I could use R, I suppose."},{"Start":"04:36.535 ","End":"04:41.679","Text":"The circle itself, excluding the interior,"},{"Start":"04:41.679 ","End":"04:45.215","Text":"in this case, is called the boundary."},{"Start":"04:45.215 ","End":"04:49.290","Text":"Here\u0027s a picture that explains it better,"},{"Start":"04:49.290 ","End":"04:52.610","Text":"and I noticed they use the letter R here,"},{"Start":"04:52.610 ","End":"04:54.110","Text":"could have used the letter D,"},{"Start":"04:54.110 ","End":"04:56.640","Text":"so we\u0027re going to use both."},{"Start":"04:57.220 ","End":"05:01.235","Text":"Now, if we take away the boundary,"},{"Start":"05:01.235 ","End":"05:05.254","Text":"then we have what we call an open region."},{"Start":"05:05.254 ","End":"05:11.030","Text":"Here\u0027s an illustration of the open region corresponding to this picture."},{"Start":"05:11.030 ","End":"05:12.920","Text":"Just strip away the boundary,"},{"Start":"05:12.920 ","End":"05:16.760","Text":"just like here if I took the natural log and I got"},{"Start":"05:16.760 ","End":"05:23.420","Text":"to x squared plus y squared less than 9."},{"Start":"05:23.420 ","End":"05:27.645","Text":"Then you would indicate it with dotted lines as here,"},{"Start":"05:27.645 ","End":"05:30.670","Text":"the same circle with dotted lines."},{"Start":"05:30.670 ","End":"05:35.515","Text":"That\u0027s the concept of boundary open region."},{"Start":"05:35.515 ","End":"05:39.220","Text":"I want to talk about the word connected"},{"Start":"05:39.220 ","End":"05:45.980","Text":"and I have a picture here that shows the concept of connected."},{"Start":"05:47.010 ","End":"05:50.755","Text":"Here\u0027s an example of something that\u0027s not connected."},{"Start":"05:50.755 ","End":"05:52.825","Text":"It\u0027s in 2 separate bits,"},{"Start":"05:52.825 ","End":"05:57.115","Text":"and I would even remove the word,"},{"Start":"05:57.115 ","End":"05:58.840","Text":"just let me remove this word."},{"Start":"05:58.840 ","End":"06:01.660","Text":"This is not connected and this is connected."},{"Start":"06:01.660 ","End":"06:04.850","Text":"Now, there\u0027s also a concept of simply connected,"},{"Start":"06:04.850 ","End":"06:06.935","Text":"which means doesn\u0027t have holes."},{"Start":"06:06.935 ","End":"06:11.525","Text":"So this is connected because you can get from any point to any point,"},{"Start":"06:11.525 ","End":"06:13.775","Text":"but it\u0027s not simply connected."},{"Start":"06:13.775 ","End":"06:16.640","Text":"This 1 is simply not connected,"},{"Start":"06:16.640 ","End":"06:19.925","Text":"and this is simply connected."},{"Start":"06:19.925 ","End":"06:23.890","Text":"If we strip away the boundaries then, they\u0027re also open."},{"Start":"06:23.890 ","End":"06:27.620","Text":"Well, usually with the boundary it\u0027s called closed, so actually,"},{"Start":"06:27.620 ","End":"06:32.495","Text":"all these 3 are examples of closed regions,"},{"Start":"06:32.495 ","End":"06:36.774","Text":"whether connected, not connected, or simply connected."},{"Start":"06:36.774 ","End":"06:38.400","Text":"That will do for now."},{"Start":"06:38.400 ","End":"06:44.230","Text":"Now, let\u0027s get back to conservative vector fields."}],"ID":8751},{"Watched":false,"Name":"Conservative Vector Fields - 2D","Duration":"5m 36s","ChapterTopicVideoID":8595,"CourseChapterTopicPlaylistID":4965,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.500 ","End":"00:06.435","Text":"Let\u0027s get back to conservative vector fields and I\u0027ll just clear the board."},{"Start":"00:06.435 ","End":"00:08.790","Text":"Now until I say otherwise,"},{"Start":"00:08.790 ","End":"00:10.290","Text":"we\u0027ll be working in 2D,"},{"Start":"00:10.290 ","End":"00:13.305","Text":"it\u0027s a lot more complex in 3D."},{"Start":"00:13.305 ","End":"00:15.915","Text":"Just to refresh your memory,"},{"Start":"00:15.915 ","End":"00:20.630","Text":"conservative, if we have a vector field F, well,"},{"Start":"00:20.630 ","End":"00:23.720","Text":"this will be in 2 dimensions of x and y,"},{"Start":"00:23.720 ","End":"00:31.110","Text":"it\u0027s conservative if it\u0027s equal to the grad of some scalar function f of x, y."},{"Start":"00:31.110 ","End":"00:35.940","Text":"If you\u0027ve forgotten what grad is,"},{"Start":"00:35.940 ","End":"00:42.425","Text":"then this means the vector derivative of f with respect to x,"},{"Start":"00:42.425 ","End":"00:46.440","Text":"the derivative of f with respect to y."},{"Start":"00:46.550 ","End":"00:55.010","Text":"The important property of a conservative vector field is that the line integral,"},{"Start":"00:55.010 ","End":"00:58.880","Text":"which is the integral over some curve of"},{"Start":"00:58.880 ","End":"01:07.400","Text":"F.dr is not dependent on the path c,"},{"Start":"01:07.400 ","End":"01:09.745","Text":"just on the endpoints."},{"Start":"01:09.745 ","End":"01:13.965","Text":"We use the term path independent to say that."},{"Start":"01:13.965 ","End":"01:17.900","Text":"1 of the things that this implies also that\u0027s useful is that"},{"Start":"01:17.900 ","End":"01:22.780","Text":"the path along a closed curve c which we write with a little circle here,"},{"Start":"01:22.780 ","End":"01:26.775","Text":"is always equal to 0,"},{"Start":"01:26.775 ","End":"01:29.260","Text":"if we\u0027re path independent."},{"Start":"01:29.540 ","End":"01:33.785","Text":"That\u0027s just a quick refresher."},{"Start":"01:33.785 ","End":"01:38.900","Text":"Now, suppose we have the vector field F,"},{"Start":"01:38.900 ","End":"01:41.480","Text":"and suppose it\u0027s equal to,"},{"Start":"01:41.480 ","End":"01:48.505","Text":"and I\u0027ll break it up into components, P,Q."},{"Start":"01:48.505 ","End":"01:51.090","Text":"F of x, y is P of x,"},{"Start":"01:51.090 ","End":"01:53.310","Text":"y, Q of x, y."},{"Start":"01:53.310 ","End":"01:59.090","Text":"Now, because it\u0027s the grad of some big function f,"},{"Start":"01:59.090 ","End":"02:06.980","Text":"this is really equal to above f_x, f_y."},{"Start":"02:06.980 ","End":"02:12.890","Text":"Some books prefer the notation partial derivative of f with respect to x,"},{"Start":"02:12.890 ","End":"02:15.230","Text":"partial derivative of f with respect to y."},{"Start":"02:15.230 ","End":"02:24.610","Text":"Notice now that if I differentiate p partially with respect to y,"},{"Start":"02:24.860 ","End":"02:32.525","Text":"what this equals to is the derivative of this with respect to y,"},{"Start":"02:32.525 ","End":"02:36.655","Text":"which sometimes we write it as just f_xy,"},{"Start":"02:36.655 ","End":"02:39.860","Text":"meaning partial derivative of f,"},{"Start":"02:39.860 ","End":"02:42.770","Text":"second derivative with respect to x and then y."},{"Start":"02:42.770 ","End":"02:50.450","Text":"If I take this Q and take its partial derivative with respect to x,"},{"Start":"02:50.450 ","End":"02:56.825","Text":"then this is equal to because Q is f_y, this is f_yx."},{"Start":"02:56.825 ","End":"03:05.940","Text":"But there\u0027s a theorem that these 2 are equal, P_y equals Q_x."},{"Start":"03:05.940 ","End":"03:09.165","Text":"I\u0027ll write that again here, it\u0027s very important,"},{"Start":"03:09.165 ","End":"03:16.240","Text":"so P_y equals Q_x."},{"Start":"03:16.370 ","End":"03:24.455","Text":"That\u0027s an important property of conservative vector fields in 2D."},{"Start":"03:24.455 ","End":"03:29.450","Text":"I just want to emphasize again that what I said here is true."},{"Start":"03:29.450 ","End":"03:33.120","Text":"I\u0027ll just write it if the vector field F,"},{"Start":"03:33.120 ","End":"03:39.045","Text":"which is P,Q is conservative."},{"Start":"03:39.045 ","End":"03:44.330","Text":"Conservative implies this."},{"Start":"03:44.330 ","End":"03:46.670","Text":"I\u0027ll just erase this arrow."},{"Start":"03:46.670 ","End":"03:49.250","Text":"Now, I\u0027d like to ask a question,"},{"Start":"03:49.250 ","End":"03:51.470","Text":"is the reverse also true?"},{"Start":"03:51.470 ","End":"03:56.930","Text":"In other words, they have a vector field F with components P and Q."},{"Start":"03:56.930 ","End":"04:02.120","Text":"Suppose that the partial derivatives like this are equal,"},{"Start":"04:02.120 ","End":"04:07.820","Text":"can I say that the vector field F is conservative?"},{"Start":"04:07.820 ","End":"04:10.490","Text":"If the answer is yes,"},{"Start":"04:10.490 ","End":"04:14.884","Text":"how do we go about finding little f, the potential function?"},{"Start":"04:14.884 ","End":"04:17.975","Text":"The answer is yes, but conditionally."},{"Start":"04:17.975 ","End":"04:24.395","Text":"If F, which is a vector field with components P and Q,"},{"Start":"04:24.395 ","End":"04:27.980","Text":"and it\u0027s defined on"},{"Start":"04:27.980 ","End":"04:36.975","Text":"a simply-connected and open region D,"},{"Start":"04:36.975 ","End":"04:43.440","Text":"and the above dP by dy,"},{"Start":"04:43.440 ","End":"04:48.260","Text":"in other words partial derivative equals to partial derivative of Q with respect to x,"},{"Start":"04:48.260 ","End":"04:55.385","Text":"then we can say that F is conservative."},{"Start":"04:55.385 ","End":"05:00.290","Text":"In other words, the reverse is true provided the vector field is"},{"Start":"05:00.290 ","End":"05:06.500","Text":"defined on a region which is simply-connected and open."},{"Start":"05:06.500 ","End":"05:09.050","Text":"1 example would be the whole plane."},{"Start":"05:09.050 ","End":"05:15.090","Text":"The whole plane is considered to be simply-connected, there\u0027s no holes,"},{"Start":"05:15.090 ","End":"05:17.400","Text":"and open, it\u0027s got no boundaries,"},{"Start":"05:17.400 ","End":"05:22.505","Text":"and we saw other examples like the interior of a circle and so on."},{"Start":"05:22.505 ","End":"05:27.200","Text":"With certain restrictions, the reverse is also true."},{"Start":"05:27.200 ","End":"05:29.885","Text":"But even if we know that this is true,"},{"Start":"05:29.885 ","End":"05:32.840","Text":"how do we find the potential function,"},{"Start":"05:32.840 ","End":"05:34.565","Text":"little f for big F?"},{"Start":"05:34.565 ","End":"05:37.500","Text":"Let\u0027s do this with an example."}],"ID":8752},{"Watched":false,"Name":"Conservative Vector Fields - 2D Cont.","Duration":"12m 53s","ChapterTopicVideoID":8596,"CourseChapterTopicPlaylistID":4965,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.760","Text":"Well, just remember that the grad of f is just the derivative of f with respect to x,"},{"Start":"00:05.760 ","End":"00:09.030","Text":"derivative of f with respect to y."},{"Start":"00:09.030 ","End":"00:12.735","Text":"What we have is that this has to be our P,"},{"Start":"00:12.735 ","End":"00:16.050","Text":"the first one, and this has to be our Q, the second one."},{"Start":"00:16.050 ","End":"00:17.670","Text":"We get 2 equations."},{"Start":"00:17.670 ","End":"00:21.060","Text":"Derivative of f with respect to x is the first bit,"},{"Start":"00:21.060 ","End":"00:25.860","Text":"2x cubed y^4 plus x."},{"Start":"00:25.860 ","End":"00:36.490","Text":"The second equation is derivative of f with respect to y is equal to 2x^4 y cubed plus y."},{"Start":"00:36.490 ","End":"00:38.690","Text":"Let\u0027s begin with the first one."},{"Start":"00:38.690 ","End":"00:41.540","Text":"If this is the derivative with respect to x,"},{"Start":"00:41.540 ","End":"00:48.500","Text":"then f, let me write it as f of x and y,"},{"Start":"00:48.500 ","End":"00:50.885","Text":"just to be clearer,"},{"Start":"00:50.885 ","End":"00:53.085","Text":"has got to be equal to the integral of this."},{"Start":"00:53.085 ","End":"01:03.525","Text":"It\u0027s the integral of 2x cubed y^4 plus x with respect to x."},{"Start":"01:03.525 ","End":"01:06.140","Text":"Now, this may look like a simple integral,"},{"Start":"01:06.140 ","End":"01:09.535","Text":"but there\u0027s a catch, and you\u0027ll see."},{"Start":"01:09.535 ","End":"01:13.430","Text":"The integral of this with respect to x,"},{"Start":"01:13.430 ","End":"01:15.230","Text":"I raise the power by 1,"},{"Start":"01:15.230 ","End":"01:18.590","Text":"it\u0027s x^4, and divide it by 2."},{"Start":"01:18.590 ","End":"01:24.935","Text":"So I get 1/2 x^4,2 over 4 is a 1/2,"},{"Start":"01:24.935 ","End":"01:29.100","Text":"and y^4, that\u0027s a constant."},{"Start":"01:29.100 ","End":"01:39.955","Text":"The integral of x is a 1/2 x squared plus a constant."},{"Start":"01:39.955 ","End":"01:44.840","Text":"This constant is not really a constant because we did an integral with respect to x."},{"Start":"01:44.840 ","End":"01:51.860","Text":"Actually, the constant has to be any function of y. I don\u0027t want to use the letter C,"},{"Start":"01:51.860 ","End":"01:56.165","Text":"so I\u0027ll use a different letter, say h,"},{"Start":"01:56.165 ","End":"01:58.910","Text":"and it\u0027s a function of y because whenever I"},{"Start":"01:58.910 ","End":"02:03.525","Text":"differentiate this with respect to x, I\u0027ll get 0."},{"Start":"02:03.525 ","End":"02:09.140","Text":"Now, I want to get an equation and I\u0027m heading towards derivative of f with respect to y."},{"Start":"02:09.140 ","End":"02:17.660","Text":"Let\u0027s now figure out what is f with respect to y. I\u0027ll write the x, y again."},{"Start":"02:17.660 ","End":"02:21.380","Text":"This is equal to differentiating we get,"},{"Start":"02:21.380 ","End":"02:25.390","Text":"with respect to y^4 with the 1/2 gives me 2,"},{"Start":"02:25.390 ","End":"02:27.795","Text":"and its y cubed,"},{"Start":"02:27.795 ","End":"02:30.325","Text":"but the x^4 stays."},{"Start":"02:30.325 ","End":"02:34.430","Text":"The derivative of a 1/2 x squared with respect to y is nothing,"},{"Start":"02:34.430 ","End":"02:40.920","Text":"and I\u0027m left with just h prime of y."},{"Start":"02:40.920 ","End":"02:47.085","Text":"Now, I\u0027ve got 2 expressions for f_y, here and here."},{"Start":"02:47.085 ","End":"02:49.320","Text":"I\u0027m going to compare these 2."},{"Start":"02:49.320 ","End":"02:52.860","Text":"These 2 have got to be equal."},{"Start":"02:52.860 ","End":"02:55.140","Text":"The first bit is equal."},{"Start":"02:55.140 ","End":"03:01.370","Text":"What this gives us is that h prime of y is equal to y,"},{"Start":"03:01.370 ","End":"03:04.145","Text":"and this is a regular derivative of a function of y."},{"Start":"03:04.145 ","End":"03:05.510","Text":"If we integrate this,"},{"Start":"03:05.510 ","End":"03:07.490","Text":"h of y is the integral of this,"},{"Start":"03:07.490 ","End":"03:12.090","Text":"the antiderivative, which is a 1/2y squared."},{"Start":"03:12.090 ","End":"03:14.040","Text":"But this times plus a constant,"},{"Start":"03:14.040 ","End":"03:16.600","Text":"and it\u0027s really a constant, a number."},{"Start":"03:16.600 ","End":"03:24.155","Text":"All that remains now is to substitute this h into here."},{"Start":"03:24.155 ","End":"03:31.380","Text":"We get, finally, that f of x,"},{"Start":"03:31.380 ","End":"03:32.970","Text":"y is equal to,"},{"Start":"03:32.970 ","End":"03:34.680","Text":"and I\u0027m coping from here first,"},{"Start":"03:34.680 ","End":"03:44.280","Text":"1/2x^4 y^4, plus 1/2x squared,"},{"Start":"03:44.280 ","End":"03:50.040","Text":"plus h of y, which is a 1/2y squared"},{"Start":"03:50.040 ","End":"03:55.440","Text":"plus C. This is our answer."},{"Start":"03:55.440 ","End":"04:01.250","Text":"This is the potential function for the vector field f,"},{"Start":"04:01.250 ","End":"04:02.930","Text":"but it still involves a constant."},{"Start":"04:02.930 ","End":"04:05.920","Text":"No, we can\u0027t get it exactly."},{"Start":"04:05.920 ","End":"04:12.815","Text":"We can always check at the end that the derivative of this with respect to x is this."},{"Start":"04:12.815 ","End":"04:15.580","Text":"If you differentiate it, you\u0027ll see that it\u0027s this."},{"Start":"04:15.580 ","End":"04:20.570","Text":"That the derivative of all of this with respect to y is equal to this."},{"Start":"04:20.570 ","End":"04:24.030","Text":"That is the answer."},{"Start":"04:24.030 ","End":"04:27.910","Text":"Now, let\u0027s take a third example."},{"Start":"04:28.060 ","End":"04:36.690","Text":"My third example will be F of x,"},{"Start":"04:36.690 ","End":"04:45.245","Text":"y is equal to minus y over x squared plus y squared,"},{"Start":"04:45.245 ","End":"04:52.330","Text":"x over x squared plus y squared."},{"Start":"04:52.700 ","End":"04:56.460","Text":"Let\u0027s call this one P and this one Q,"},{"Start":"04:56.460 ","End":"04:59.265","Text":"and then I have to compute what Px,"},{"Start":"04:59.265 ","End":"05:01.430","Text":"derivative of P with respect to x,"},{"Start":"05:01.430 ","End":"05:03.890","Text":"partial derivative, what is that equal?"},{"Start":"05:03.890 ","End":"05:08.225","Text":"Then we\u0027ll do a derivative of Q with respect."},{"Start":"05:08.225 ","End":"05:11.550","Text":"Again, I\u0027ve got it backwards, sorry."},{"Start":"05:11.990 ","End":"05:14.570","Text":"Learn from my mistake."},{"Start":"05:14.570 ","End":"05:16.175","Text":"The first one with respect to y,"},{"Start":"05:16.175 ","End":"05:17.510","Text":"second one with respect to"},{"Start":"05:17.510 ","End":"05:21.800","Text":"x. I better give you the quotient rule in case you\u0027ve forgotten it."},{"Start":"05:21.800 ","End":"05:23.929","Text":"The derivative of a quotient,"},{"Start":"05:23.929 ","End":"05:26.435","Text":"so u over v in general,"},{"Start":"05:26.435 ","End":"05:31.310","Text":"is the derivative of u times v"},{"Start":"05:31.310 ","End":"05:37.720","Text":"minus u times the derivative of v over v squared."},{"Start":"05:37.720 ","End":"05:43.165","Text":"What we get here is the derivative of the numerator is minus 1,"},{"Start":"05:43.165 ","End":"05:47.449","Text":"times the denominator, x squared plus y squared,"},{"Start":"05:47.449 ","End":"05:52.445","Text":"minus the numerator as is,"},{"Start":"05:52.445 ","End":"05:57.725","Text":"times derivative of the denominator with respect to y is 2y."},{"Start":"05:57.725 ","End":"06:05.245","Text":"All of this over x squared plus y squared squared."},{"Start":"06:05.245 ","End":"06:07.515","Text":"If I simplify this,"},{"Start":"06:07.515 ","End":"06:09.515","Text":"let\u0027s see what we get."},{"Start":"06:09.515 ","End":"06:17.375","Text":"We get minus x squared minus y squared plus 2y squared."},{"Start":"06:17.375 ","End":"06:19.310","Text":"If you think about it just a second,"},{"Start":"06:19.310 ","End":"06:22.960","Text":"it\u0027s y squared minus x squared"},{"Start":"06:22.960 ","End":"06:29.640","Text":"over x squared plus y squared squared."},{"Start":"06:29.640 ","End":"06:31.780","Text":"Now, what about the other one?"},{"Start":"06:31.780 ","End":"06:34.160","Text":"Well, we get the derivative of x,"},{"Start":"06:34.160 ","End":"06:42.500","Text":"which is 1 times x squared plus y squared minus x as is."},{"Start":"06:42.500 ","End":"06:47.795","Text":"The derivative of this with respect to x,"},{"Start":"06:47.795 ","End":"06:57.920","Text":"it would be 2x all over x squared plus y squared squared."},{"Start":"06:57.920 ","End":"06:59.120","Text":"If we simplify this,"},{"Start":"06:59.120 ","End":"07:03.845","Text":"we get x squared plus y squared minus 2x squared."},{"Start":"07:03.845 ","End":"07:11.070","Text":"We also get y squared minus x squared over same thing here."},{"Start":"07:11.590 ","End":"07:16.520","Text":"Now, these 2 certainly are equal."},{"Start":"07:16.520 ","End":"07:19.655","Text":"But does that mean that F is conservative?"},{"Start":"07:19.655 ","End":"07:21.815","Text":"Well, we\u0027ve forgotten about the domain,"},{"Start":"07:21.815 ","End":"07:23.060","Text":"or we haven\u0027t forgotten,"},{"Start":"07:23.060 ","End":"07:24.815","Text":"we\u0027ll discuss it now."},{"Start":"07:24.815 ","End":"07:30.810","Text":"The domain, which is the part of the plane where this is defined,"},{"Start":"07:31.030 ","End":"07:36.125","Text":"is everywhere except where the denominator is 0."},{"Start":"07:36.125 ","End":"07:39.820","Text":"Now, when is x squared plus y squared equal 0?"},{"Start":"07:39.820 ","End":"07:43.325","Text":"If x squared plus y squared equal 0,"},{"Start":"07:43.325 ","End":"07:46.360","Text":"and each of these terms is non-negative,"},{"Start":"07:46.360 ","End":"07:48.180","Text":"meaning 0 or positive."},{"Start":"07:48.180 ","End":"07:50.855","Text":"The only way this can be 0 is if they\u0027re both 0."},{"Start":"07:50.855 ","End":"07:54.200","Text":"In other words, x, y is the origin,"},{"Start":"07:54.200 ","End":"07:58.500","Text":"is the point 0, 0, but that\u0027s excluded."},{"Start":"07:59.210 ","End":"08:03.350","Text":"In other words, we have a plane with a hole in it,"},{"Start":"08:03.350 ","End":"08:04.985","Text":"the origin is missing."},{"Start":"08:04.985 ","End":"08:06.740","Text":"Because of the hole,"},{"Start":"08:06.740 ","End":"08:09.200","Text":"it is not simply connected."},{"Start":"08:09.200 ","End":"08:10.820","Text":"It may be open,"},{"Start":"08:10.820 ","End":"08:12.260","Text":"but I need both."},{"Start":"08:12.260 ","End":"08:14.810","Text":"I can say already that it\u0027s not simply connected."},{"Start":"08:14.810 ","End":"08:22.030","Text":"I can\u0027t automatically say whether this is conservative or not."},{"Start":"08:22.030 ","End":"08:26.180","Text":"At this stage, we got to say, don\u0027t know."},{"Start":"08:26.180 ","End":"08:28.430","Text":"It could be either."},{"Start":"08:28.430 ","End":"08:32.675","Text":"But I\u0027m going to show you that the answer is actually no."},{"Start":"08:32.675 ","End":"08:35.180","Text":"This is not conservative."},{"Start":"08:35.180 ","End":"08:39.270","Text":"Now, I\u0027ll show you why it\u0027s not conservative."},{"Start":"08:39.580 ","End":"08:42.965","Text":"Let me just scroll a bit here."},{"Start":"08:42.965 ","End":"08:47.490","Text":"If it\u0027s conservative, I\u0027ll write the word again,"},{"Start":"08:47.490 ","End":"08:51.925","Text":"then we know it has the property that it\u0027s line independent."},{"Start":"08:51.925 ","End":"08:53.765","Text":"If it\u0027s line independent,"},{"Start":"08:53.765 ","End":"08:58.650","Text":"then the integral around a closed curve of"},{"Start":"08:58.650 ","End":"09:05.205","Text":"F dot dr is going to equal 0 for any closed curve."},{"Start":"09:05.205 ","End":"09:09.230","Text":"I\u0027m going to show you a closed curve that is not 0."},{"Start":"09:09.230 ","End":"09:10.580","Text":"But to have any chance,"},{"Start":"09:10.580 ","End":"09:13.400","Text":"it has to go around the hole, the origin."},{"Start":"09:13.400 ","End":"09:15.924","Text":"Let\u0027s just take the unit circle."},{"Start":"09:15.924 ","End":"09:24.890","Text":"If I take the curve C or the path to be x equals cosine t,"},{"Start":"09:24.890 ","End":"09:33.285","Text":"y equals sine t and t goes from 0 to 2Pi,"},{"Start":"09:33.285 ","End":"09:37.525","Text":"that\u0027s the unit circle in the usual direction."},{"Start":"09:37.525 ","End":"09:43.410","Text":"Let\u0027s compute the integral of this vector field over this curve."},{"Start":"09:45.100 ","End":"09:49.115","Text":"Well, this is really r of t equals,"},{"Start":"09:49.115 ","End":"09:50.690","Text":"which is x of t,"},{"Start":"09:50.690 ","End":"09:59.195","Text":"y of t. We know that r prime of t will be equal to,"},{"Start":"09:59.195 ","End":"10:02.660","Text":"and I\u0027ll write it in not parametric but in vector form,"},{"Start":"10:02.660 ","End":"10:05.540","Text":"would be the derivative of this,"},{"Start":"10:05.540 ","End":"10:08.245","Text":"is minus sine t,"},{"Start":"10:08.245 ","End":"10:15.130","Text":"and here cosine t. Let me just arrange this."},{"Start":"10:15.130 ","End":"10:20.295","Text":"I just wrote r of t properly to the curve,"},{"Start":"10:20.295 ","End":"10:24.300","Text":"and of course copy that also between 0 and 2Pi."},{"Start":"10:24.300 ","End":"10:31.770","Text":"This this integral over the closed curve would"},{"Start":"10:31.770 ","End":"10:40.350","Text":"be the integral from 0 to 2 Pi of,"},{"Start":"10:40.350 ","End":"10:43.845","Text":"first of all, F, which is this,"},{"Start":"10:43.845 ","End":"10:46.520","Text":"but I have to put y and x in terms of this."},{"Start":"10:46.520 ","End":"10:49.145","Text":"So we get from here,"},{"Start":"10:49.145 ","End":"10:53.300","Text":"minus y is minus sine t,"},{"Start":"10:53.300 ","End":"11:02.550","Text":"and over here we get sine squared t plus cosine squared t,"},{"Start":"11:03.100 ","End":"11:12.835","Text":"and then x is cosine t over same thing."},{"Start":"11:12.835 ","End":"11:15.915","Text":"That\u0027s a dot product here."},{"Start":"11:15.915 ","End":"11:19.365","Text":"dr is r prime of t, dt."},{"Start":"11:19.365 ","End":"11:22.455","Text":"I\u0027ll just put a dt here."},{"Start":"11:22.455 ","End":"11:30.195","Text":"It\u0027s minus sine t, cosine t, dt."},{"Start":"11:30.195 ","End":"11:36.400","Text":"But of course you know that sine squared plus cosine squared is 1."},{"Start":"11:36.400 ","End":"11:40.600","Text":"This whole denominator can just disappear because it\u0027s 1,"},{"Start":"11:40.600 ","End":"11:44.390","Text":"and this denominator can disappear because it\u0027s 1."},{"Start":"11:44.390 ","End":"11:50.370","Text":"What we get is the integral from"},{"Start":"11:50.370 ","End":"11:56.690","Text":"0 to 2Pi minus sine times minus sine is sine squared t,"},{"Start":"11:56.690 ","End":"11:59.955","Text":"cosine times cosine, and I am adding,"},{"Start":"11:59.955 ","End":"12:03.215","Text":"plus cosine squared t, dt."},{"Start":"12:03.215 ","End":"12:07.805","Text":"Once again, this thing is equal to 1."},{"Start":"12:07.805 ","End":"12:16.410","Text":"This integral of 1 is t taken between 0 and 2Pi."},{"Start":"12:16.410 ","End":"12:19.005","Text":"When t is 2Pi, then t is 2Pi,"},{"Start":"12:19.005 ","End":"12:21.585","Text":"and when t is 0, t is 0."},{"Start":"12:21.585 ","End":"12:23.790","Text":"It\u0027s 2Pi minus 0,"},{"Start":"12:23.790 ","End":"12:25.755","Text":"which is just 2Pi."},{"Start":"12:25.755 ","End":"12:29.370","Text":"The point is not that it\u0027s 2Pi,"},{"Start":"12:29.370 ","End":"12:33.120","Text":"but that it\u0027s not equal to 0."},{"Start":"12:33.120 ","End":"12:35.510","Text":"If it was conservative,"},{"Start":"12:35.510 ","End":"12:39.840","Text":"then integral over a closed curve would be 0,"},{"Start":"12:39.840 ","End":"12:41.780","Text":"and because this doesn\u0027t happen,"},{"Start":"12:41.780 ","End":"12:43.880","Text":"then it\u0027s not conservative."},{"Start":"12:43.880 ","End":"12:47.390","Text":"This open and simply connected was there for a reason."},{"Start":"12:47.390 ","End":"12:54.000","Text":"After the break, we\u0027ll talk about conservative vector fields in 3D."}],"ID":8753},{"Watched":false,"Name":"Conservative Vector Fields - 3D","Duration":"8m 18s","ChapterTopicVideoID":8597,"CourseChapterTopicPlaylistID":4965,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.080","Text":"Up to now we\u0027ve only discussed conservative vector fields in"},{"Start":"00:04.080 ","End":"00:09.045","Text":"2D and now I want to talk a bit about 3D."},{"Start":"00:09.045 ","End":"00:15.150","Text":"In 3D, let\u0027s say we have a vector field F of x, y, and z,"},{"Start":"00:15.150 ","End":"00:18.450","Text":"and it\u0027s given by components P,"},{"Start":"00:18.450 ","End":"00:22.409","Text":"Q and R. I\u0027ll continue using the angular bracket notation,"},{"Start":"00:22.409 ","End":"00:26.860","Text":"although you could write it as PI plus QJ plus RK."},{"Start":"00:26.960 ","End":"00:33.435","Text":"Let\u0027s suppose that we know that it\u0027s conservative."},{"Start":"00:33.435 ","End":"00:38.340","Text":"I\u0027m going to show an example of how we find its potential function."},{"Start":"00:38.340 ","End":"00:40.875","Text":"Remember that if F is conservative,"},{"Start":"00:40.875 ","End":"00:44.880","Text":"then F is grad of f for"},{"Start":"00:44.880 ","End":"00:50.130","Text":"some scalar function of F. In case you\u0027ve forgotten what grad f is,"},{"Start":"00:50.130 ","End":"00:53.025","Text":"it\u0027s just the partial derivatives."},{"Start":"00:53.025 ","End":"00:56.025","Text":"I\u0027ll use the other notation this time for a change."},{"Start":"00:56.025 ","End":"00:58.500","Text":"The derivative with respect to x,"},{"Start":"00:58.500 ","End":"01:02.145","Text":"the partial derivative with respect to y,"},{"Start":"01:02.145 ","End":"01:06.195","Text":"and the partial derivative with respect to z,"},{"Start":"01:06.195 ","End":"01:07.980","Text":"which we all saw as F,"},{"Start":"01:07.980 ","End":"01:10.420","Text":"x and so on."},{"Start":"01:10.670 ","End":"01:18.240","Text":"We haven\u0027t yet got a way to tell if a vector field in 3D is conservative or not."},{"Start":"01:18.240 ","End":"01:20.880","Text":"We had a formula with 2D."},{"Start":"01:20.880 ","End":"01:23.370","Text":"We don\u0027t yet have it in 3D,"},{"Start":"01:23.370 ","End":"01:26.070","Text":"but let\u0027s suppose that I know that it is."},{"Start":"01:26.070 ","End":"01:29.730","Text":"How would I find from F, the vector field,"},{"Start":"01:29.730 ","End":"01:33.510","Text":"how would I find its potential, the scalar f?"},{"Start":"01:33.510 ","End":"01:39.750","Text":"An example, I\u0027ll take my vector field F to be,"},{"Start":"01:39.750 ","End":"01:48.540","Text":"P is going to be 2xy cubed, z^4."},{"Start":"01:48.540 ","End":"01:56.430","Text":"Q is going to be 3x squared y squared z^4,"},{"Start":"01:56.430 ","End":"02:02.010","Text":"and R will be 4x squared,"},{"Start":"02:02.010 ","End":"02:07.740","Text":"y cubed, z cubed."},{"Start":"02:07.740 ","End":"02:10.995","Text":"Our task is to find f,"},{"Start":"02:10.995 ","End":"02:13.215","Text":"at least up to a constant."},{"Start":"02:13.215 ","End":"02:21.180","Text":"We\u0027ll use the same technique as in 2D and it\u0027ll just be a little bit more involved in 3D,"},{"Start":"02:21.180 ","End":"02:23.565","Text":"but essentially the same idea."},{"Start":"02:23.565 ","End":"02:26.430","Text":"Basically we have 3 equations."},{"Start":"02:26.430 ","End":"02:31.995","Text":"We know that the partial derivative with respect to x of our function f,"},{"Start":"02:31.995 ","End":"02:38.730","Text":"which we\u0027re looking for, is this 2xy cubed z^4."},{"Start":"02:38.730 ","End":"02:43.350","Text":"We also know that the partial derivative of f with respect to"},{"Start":"02:43.350 ","End":"02:49.575","Text":"y is this 3x squared, y squared z^4."},{"Start":"02:49.575 ","End":"02:55.080","Text":"We know that the partial derivative of f with respect to z will"},{"Start":"02:55.080 ","End":"03:00.915","Text":"be 4x squared y cubed z cubed."},{"Start":"03:00.915 ","End":"03:03.780","Text":"Let\u0027s start with them one at a time."},{"Start":"03:03.780 ","End":"03:05.880","Text":"Let\u0027s start with this one."},{"Start":"03:05.880 ","End":"03:10.950","Text":"To find F, we\u0027d have to take an integral with respect to x. I would say"},{"Start":"03:10.950 ","End":"03:14.730","Text":"that F is equal to the integral of"},{"Start":"03:14.730 ","End":"03:22.320","Text":"2x y cubed z^4 dx."},{"Start":"03:22.320 ","End":"03:25.169","Text":"Now remember when we\u0027re talking about dx,"},{"Start":"03:25.169 ","End":"03:28.960","Text":"y and z are constants."},{"Start":"03:29.000 ","End":"03:38.320","Text":"I\u0027m looking at x and I know that the integral of 2x is x squared so I get x squared,"},{"Start":"03:38.540 ","End":"03:44.385","Text":"y cubed, z^4 plus constant."},{"Start":"03:44.385 ","End":"03:46.575","Text":"But what kind of a constant?"},{"Start":"03:46.575 ","End":"03:49.605","Text":"Remember this is a function of 3 variables."},{"Start":"03:49.605 ","End":"03:53.085","Text":"We\u0027re doing dx, which means that y and z are constants."},{"Start":"03:53.085 ","End":"03:54.420","Text":"Instead of a constant,"},{"Start":"03:54.420 ","End":"04:01.290","Text":"we can have a general function in the variables y and z,"},{"Start":"04:01.290 ","End":"04:03.930","Text":"because anything like this,"},{"Start":"04:03.930 ","End":"04:07.620","Text":"when we differentiate, it gives 0."},{"Start":"04:07.620 ","End":"04:12.825","Text":"Now what we can do is differentiate this,"},{"Start":"04:12.825 ","End":"04:17.565","Text":"this is again F. Now I can take the next one."},{"Start":"04:17.565 ","End":"04:19.650","Text":"I have the df by dy is this."},{"Start":"04:19.650 ","End":"04:21.105","Text":"Let\u0027s take it also from here,"},{"Start":"04:21.105 ","End":"04:25.380","Text":"I can compute df by dy from this expression."},{"Start":"04:25.380 ","End":"04:30.080","Text":"This will be y is the variable."},{"Start":"04:30.080 ","End":"04:31.850","Text":"Now x and z are constants,"},{"Start":"04:31.850 ","End":"04:33.440","Text":"so this will be 3y squared."},{"Start":"04:33.440 ","End":"04:37.310","Text":"I\u0027ll put the 3 in front and this becomes y squared,"},{"Start":"04:37.310 ","End":"04:39.110","Text":"then everything else is the same."},{"Start":"04:39.110 ","End":"04:47.930","Text":"Plus the derivative of this with respect to y. I\u0027ll just call it dh by dy,"},{"Start":"04:47.930 ","End":"04:50.780","Text":"where h is a function of y and z."},{"Start":"04:50.780 ","End":"04:52.525","Text":"Now if I look here,"},{"Start":"04:52.525 ","End":"04:54.090","Text":"and I look here,"},{"Start":"04:54.090 ","End":"04:56.820","Text":"I see that these 2 are the same,"},{"Start":"04:56.820 ","End":"05:00.730","Text":"except for the dh by dy."},{"Start":"05:02.210 ","End":"05:10.230","Text":"I get that dh by dy equals 0,"},{"Start":"05:10.230 ","End":"05:18.240","Text":"which means that h is the integral of 0 is a constant."},{"Start":"05:18.240 ","End":"05:22.185","Text":"But now the constant is going to be a function of z,"},{"Start":"05:22.185 ","End":"05:24.195","Text":"because h is a function,"},{"Start":"05:24.195 ","End":"05:31.570","Text":"write it again, of y and z."},{"Start":"05:32.480 ","End":"05:36.000","Text":"If its partial derivative is 0,"},{"Start":"05:36.000 ","End":"05:38.010","Text":"it could still be a function of z."},{"Start":"05:38.010 ","End":"05:39.525","Text":"Let\u0027s use another letter,"},{"Start":"05:39.525 ","End":"05:43.360","Text":"say g of z."},{"Start":"05:44.270 ","End":"05:50.140","Text":"Now I have that my function f,"},{"Start":"05:51.920 ","End":"05:54.030","Text":"if I substitute it here,"},{"Start":"05:54.030 ","End":"06:02.445","Text":"I\u0027ll get now that F is equal to x squared y cubed z^4,"},{"Start":"06:02.445 ","End":"06:04.890","Text":"instead of h of y and z,"},{"Start":"06:04.890 ","End":"06:09.375","Text":"I\u0027ve got some function just of z."},{"Start":"06:09.375 ","End":"06:16.920","Text":"Now I can again differentiate with respect to z and I would"},{"Start":"06:16.920 ","End":"06:24.270","Text":"get df by dz is equal to derivative with respect to z."},{"Start":"06:24.270 ","End":"06:27.375","Text":"This first part, x and y is constant."},{"Start":"06:27.375 ","End":"06:29.925","Text":"z^4, gives me 4z cubed."},{"Start":"06:29.925 ","End":"06:39.225","Text":"I\u0027ve got 4, this part is the same and z cubed plus g prime of z."},{"Start":"06:39.225 ","End":"06:41.835","Text":"Now I compare this and this,"},{"Start":"06:41.835 ","End":"06:43.440","Text":"and these 2 are equal."},{"Start":"06:43.440 ","End":"06:48.735","Text":"I get that g prime of z is equal to 0."},{"Start":"06:48.735 ","End":"06:53.010","Text":"If this equals this, you can see it."},{"Start":"06:53.010 ","End":"06:59.250","Text":"Then that means now that g of z is just really a constant."},{"Start":"06:59.250 ","End":"07:01.080","Text":"Because if I take the integral of 0 dz,"},{"Start":"07:01.080 ","End":"07:02.414","Text":"there\u0027s no other variables,"},{"Start":"07:02.414 ","End":"07:07.210","Text":"so g of z is going to be equal to a constant."},{"Start":"07:07.370 ","End":"07:12.735","Text":"What I finally get is that,"},{"Start":"07:12.735 ","End":"07:16.305","Text":"let me just write the answer in a different color."},{"Start":"07:16.305 ","End":"07:21.390","Text":"I\u0027ll just take this as here except I replace g and z,"},{"Start":"07:21.390 ","End":"07:23.385","Text":"so f, not put it on x,"},{"Start":"07:23.385 ","End":"07:27.734","Text":"y and z is equal to x squared"},{"Start":"07:27.734 ","End":"07:34.815","Text":"y cubed z^4 plus a constant."},{"Start":"07:34.815 ","End":"07:38.040","Text":"Of course at the end, we can verify that this"},{"Start":"07:38.040 ","End":"07:40.665","Text":"is a potential function for the vector field"},{"Start":"07:40.665 ","End":"07:46.170","Text":"f. If I differentiate this with respect to any of them,"},{"Start":"07:46.170 ","End":"07:47.835","Text":"the constant is going to disappear."},{"Start":"07:47.835 ","End":"07:49.410","Text":"In this, with respect to x,"},{"Start":"07:49.410 ","End":"07:52.695","Text":"I get 2x y cubed z^4,"},{"Start":"07:52.695 ","End":"07:53.910","Text":"which is written here."},{"Start":"07:53.910 ","End":"07:55.560","Text":"With respect to y, I get this."},{"Start":"07:55.560 ","End":"08:00.910","Text":"With respect to z, I get this and that\u0027s the process."},{"Start":"08:01.130 ","End":"08:07.680","Text":"This was a fairly easy example because in each case we kept getting 0s."},{"Start":"08:07.680 ","End":"08:10.050","Text":"We got a 0 here and we got a 0 here."},{"Start":"08:10.050 ","End":"08:12.195","Text":"It could get a bit more complicated."},{"Start":"08:12.195 ","End":"08:16.545","Text":"But I\u0027m going to leave it at that and we\u0027ll settle for this example."},{"Start":"08:16.545 ","End":"08:19.360","Text":"We\u0027re done for this."}],"ID":8754},{"Watched":false,"Name":"Exercise 1 part a","Duration":"4m 48s","ChapterTopicVideoID":8604,"CourseChapterTopicPlaylistID":4965,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"In this exercise, we\u0027re given F as below."},{"Start":"00:02.970 ","End":"00:08.520","Text":"F is the vector function of 2 variables in 2 variables."},{"Start":"00:08.520 ","End":"00:11.550","Text":"We have to decide if it\u0027s conservative."},{"Start":"00:11.550 ","End":"00:13.590","Text":"Assuming the answer is yes,"},{"Start":"00:13.590 ","End":"00:17.250","Text":"then we have to find a function phi,"},{"Start":"00:17.250 ","End":"00:19.020","Text":"this is called the potential function,"},{"Start":"00:19.020 ","End":"00:22.125","Text":"such that the gradient of phi is the given"},{"Start":"00:22.125 ","End":"00:26.970","Text":"F. Now there\u0027s a standard test to determine this."},{"Start":"00:26.970 ","End":"00:30.000","Text":"If we let the first component of F be P,"},{"Start":"00:30.000 ","End":"00:32.790","Text":"and the second 1 Q, well,"},{"Start":"00:32.790 ","End":"00:34.460","Text":"to be precise, P of x,"},{"Start":"00:34.460 ","End":"00:37.070","Text":"y and Q of x, y,"},{"Start":"00:37.070 ","End":"00:38.315","Text":"don\u0027t always do that,"},{"Start":"00:38.315 ","End":"00:39.905","Text":"then the test is,"},{"Start":"00:39.905 ","End":"00:43.220","Text":"if the derivative of P partial with respect to"},{"Start":"00:43.220 ","End":"00:47.975","Text":"y is equal to the derivative of Q with respect to x."},{"Start":"00:47.975 ","End":"00:50.300","Text":"We don\u0027t know this. This is what we\u0027re going to check,"},{"Start":"00:50.300 ","End":"00:51.650","Text":"but if the answer is yes,"},{"Start":"00:51.650 ","End":"00:54.890","Text":"then F is conservative. Let\u0027s see."},{"Start":"00:54.890 ","End":"00:59.300","Text":"What is P with respect to y."},{"Start":"00:59.300 ","End":"01:02.270","Text":"This is equal to 6x plus 5y,"},{"Start":"01:02.270 ","End":"01:04.880","Text":"the derivative with respect to 5."},{"Start":"01:04.880 ","End":"01:11.075","Text":"If we look at this and differentiate Q with respect to x, then we also get 5."},{"Start":"01:11.075 ","End":"01:14.210","Text":"Indeed, this equality holds,"},{"Start":"01:14.210 ","End":"01:17.650","Text":"and so this is conservative?"},{"Start":"01:17.650 ","End":"01:19.740","Text":"Yes, it is."},{"Start":"01:19.740 ","End":"01:21.980","Text":"Now that we know that it\u0027s conservative,"},{"Start":"01:21.980 ","End":"01:25.265","Text":"let\u0027s go about finding the function phi."},{"Start":"01:25.265 ","End":"01:29.495","Text":"Let\u0027s remember what the gradient operator of phi is."},{"Start":"01:29.495 ","End":"01:37.040","Text":"The gradient of phi is just the vector whose first component is phi with respect to x,"},{"Start":"01:37.040 ","End":"01:41.090","Text":"and the second component is phi with respect to y."},{"Start":"01:41.090 ","End":"01:45.475","Text":"But this has to be our original F. In other words,"},{"Start":"01:45.475 ","End":"01:49.625","Text":"what we get, where is it?"},{"Start":"01:49.625 ","End":"02:01.079","Text":"If this is equal to F and F is equal to P comma Q,"},{"Start":"02:01.280 ","End":"02:04.520","Text":"then what we need is 2 equations."},{"Start":"02:04.520 ","End":"02:09.340","Text":"We need that phi with respect to x is equal to P,"},{"Start":"02:09.340 ","End":"02:14.090","Text":"and phi with respect to y is equal to Q."},{"Start":"02:14.090 ","End":"02:16.580","Text":"Let\u0027s start with the first 1."},{"Start":"02:16.580 ","End":"02:20.610","Text":"Phi with respect to x equals P,"},{"Start":"02:20.910 ","End":"02:28.280","Text":"and P is 6x plus 5y."},{"Start":"02:28.340 ","End":"02:33.980","Text":"Phi must be the integral of this with respect to x."},{"Start":"02:33.980 ","End":"02:42.030","Text":"The integral of 6x plus 5y with respect to x."},{"Start":"02:42.030 ","End":"02:47.550","Text":"6x will give us 6x squared over 2 or,"},{"Start":"02:47.550 ","End":"02:49.395","Text":"in short, 3x squared,"},{"Start":"02:49.395 ","End":"02:51.780","Text":"5y gives us 5yx."},{"Start":"02:51.780 ","End":"02:53.850","Text":"I\u0027ll write this as 5xy,"},{"Start":"02:53.850 ","End":"02:55.970","Text":"and then there\u0027s a constant,"},{"Start":"02:55.970 ","End":"02:58.110","Text":"but it\u0027s not an ordinary constant."},{"Start":"02:58.110 ","End":"03:02.704","Text":"Because y is also considered a constant as far as x is concerned,"},{"Start":"03:02.704 ","End":"03:05.675","Text":"we take some arbitrary function of y."},{"Start":"03:05.675 ","End":"03:07.490","Text":"It\u0027s a constant as far as x goes,"},{"Start":"03:07.490 ","End":"03:09.545","Text":"but any function of y we\u0027ll do here,"},{"Start":"03:09.545 ","End":"03:12.790","Text":"because it\u0027s derivative with respect to x, would be 0."},{"Start":"03:12.790 ","End":"03:16.250","Text":"Now, we just took care of the first equation."},{"Start":"03:16.250 ","End":"03:17.315","Text":"Let\u0027s do the second,"},{"Start":"03:17.315 ","End":"03:24.720","Text":"phi with respect to y is Q. Phi with respect to y which is,"},{"Start":"03:24.720 ","End":"03:29.180","Text":"lets see, this will give me nothing because that doesn\u0027t contain y,"},{"Start":"03:29.180 ","End":"03:32.795","Text":"here I get 5x with respect to y,"},{"Start":"03:32.795 ","End":"03:35.919","Text":"and here I get C prime of y."},{"Start":"03:35.919 ","End":"03:40.590","Text":"This has to equal Q which is, where is it?"},{"Start":"03:40.590 ","End":"03:44.140","Text":"Here it is, 5x plus 4y."},{"Start":"03:45.980 ","End":"03:52.940","Text":"5x cancels with 5x so we just have derivative of something in y."},{"Start":"03:52.940 ","End":"04:00.230","Text":"We can integrate this and get C of y equals the anti-derivative of this,"},{"Start":"04:00.230 ","End":"04:05.880","Text":"but I\u0027ll write it as integral of 4ydy."},{"Start":"04:05.980 ","End":"04:14.780","Text":"This is equal to 4y squared over 2 is 2y squared plus a constant,"},{"Start":"04:14.780 ","End":"04:16.280","Text":"but this time a real constant."},{"Start":"04:16.280 ","End":"04:18.620","Text":"I don\u0027t want to use the same letter C,"},{"Start":"04:18.620 ","End":"04:22.795","Text":"I\u0027ll use the letter K. Now that I have C,"},{"Start":"04:22.795 ","End":"04:29.010","Text":"I can substitute this in here and now I can get that phi,"},{"Start":"04:29.010 ","End":"04:30.690","Text":"or when I write it in full,"},{"Start":"04:30.690 ","End":"04:37.460","Text":"phi of x and y is equal to 3x squared plus"},{"Start":"04:37.460 ","End":"04:42.190","Text":"5xy plus 2y squared"},{"Start":"04:42.190 ","End":"04:49.120","Text":"plus K. That\u0027s the answer and we\u0027re done."}],"ID":8755},{"Watched":false,"Name":"Exercise 1 part b","Duration":"6m 19s","ChapterTopicVideoID":8605,"CourseChapterTopicPlaylistID":4965,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.040","Text":"In this exercise, we\u0027re given a vector function F of 2 variables,"},{"Start":"00:08.040 ","End":"00:10.800","Text":"x and y as follows."},{"Start":"00:10.800 ","End":"00:17.130","Text":"We have to decide if F is a conservative field, vector field."},{"Start":"00:17.130 ","End":"00:21.030","Text":"If so, we have to find a function Phi,"},{"Start":"00:21.030 ","End":"00:24.285","Text":"also called the potential function,"},{"Start":"00:24.285 ","End":"00:28.680","Text":"such that the gradient of Phi is our function F."},{"Start":"00:28.680 ","End":"00:31.875","Text":"But the standard way of doing this,"},{"Start":"00:31.875 ","End":"00:34.950","Text":"we just give names to each of the 2 component functions,"},{"Start":"00:34.950 ","End":"00:46.410","Text":"say P and Q, or more precisely, P of x, y is 2x cosine y minus y, cosine x."},{"Start":"00:46.410 ","End":"00:57.350","Text":"Q of x and y will be the other 1 minus x squared sine y minus sine x."},{"Start":"00:57.350 ","End":"01:02.940","Text":"What the theory says is that F will be a conservative,"},{"Start":"01:02.940 ","End":"01:06.590","Text":"if we can show that the derivative of P,"},{"Start":"01:06.590 ","End":"01:09.500","Text":"partial derivative with respect to y is"},{"Start":"01:09.500 ","End":"01:12.845","Text":"equal to the partial derivative of Q with respect to x."},{"Start":"01:12.845 ","End":"01:14.510","Text":"At the moment this is the question mark,"},{"Start":"01:14.510 ","End":"01:15.800","Text":"that\u0027s what I\u0027m going to check."},{"Start":"01:15.800 ","End":"01:16.880","Text":"So let\u0027s see."},{"Start":"01:16.880 ","End":"01:21.305","Text":"Let\u0027s go for P with respect to y."},{"Start":"01:21.305 ","End":"01:25.025","Text":"So we look at this and remember that x is a constant."},{"Start":"01:25.025 ","End":"01:30.470","Text":"The derivative of cosine y is minus sine y and the constant sticks,"},{"Start":"01:30.470 ","End":"01:35.580","Text":"so it\u0027s minus 2x sine y."},{"Start":"01:35.580 ","End":"01:39.165","Text":"As for this, again,"},{"Start":"01:39.165 ","End":"01:46.010","Text":"the x is constant so the y just drops and we have cosine x."},{"Start":"01:46.010 ","End":"01:48.220","Text":"It\u0027s like the coefficient of y."},{"Start":"01:48.220 ","End":"01:49.350","Text":"Let\u0027s see."},{"Start":"01:49.350 ","End":"01:53.790","Text":"What is Q with respect to x?"},{"Start":"01:53.790 ","End":"01:57.180","Text":"Well, sine y is a constant,"},{"Start":"01:57.180 ","End":"02:07.235","Text":"and so we get minus 2x sine y and the derivative of sine x with respect to x is cosine x."},{"Start":"02:07.235 ","End":"02:10.220","Text":"Yes, these 2 look the same,"},{"Start":"02:10.220 ","End":"02:14.885","Text":"they are equal, so conservative field, yes."},{"Start":"02:14.885 ","End":"02:18.880","Text":"That means that we have to now look for the function Phi."},{"Start":"02:18.880 ","End":"02:22.610","Text":"Now, what does it mean that the gradient of Phi is F?"},{"Start":"02:22.610 ","End":"02:34.380","Text":"It means the gradient of Phi means Phi with respect to x, Phi with respect to y,"},{"Start":"02:34.380 ","End":"02:36.480","Text":"if we\u0027re using the brackets notation,"},{"Start":"02:36.480 ","End":"02:38.690","Text":"not the Ij notation."},{"Start":"02:38.690 ","End":"02:50.250","Text":"If this is going to equal F, well, F is just saying it\u0027s P comma Q."},{"Start":"02:50.250 ","End":"02:52.790","Text":"So if 2 things are equal vector functions,"},{"Start":"02:52.790 ","End":"02:54.040","Text":"we have to have each of them equal,"},{"Start":"02:54.040 ","End":"02:55.550","Text":"so this has to equal this,"},{"Start":"02:55.550 ","End":"02:57.155","Text":"and this has to equal this,"},{"Start":"02:57.155 ","End":"02:59.500","Text":"so we get 2 equations."},{"Start":"02:59.500 ","End":"03:05.520","Text":"Phi with respect to x is going to equal P,"},{"Start":"03:05.520 ","End":"03:07.925","Text":"which is what\u0027s written here,"},{"Start":"03:07.925 ","End":"03:14.745","Text":"2x cosine y minus y cosine x."},{"Start":"03:14.745 ","End":"03:20.720","Text":"If we know the derivative with respect to x then we can get Phi itself,"},{"Start":"03:20.720 ","End":"03:23.990","Text":"I should write in brackets of x and y,"},{"Start":"03:23.990 ","End":"03:29.745","Text":"it\u0027s going to be the integral of this thing, dx."},{"Start":"03:29.745 ","End":"03:31.680","Text":"Let me just copy it,"},{"Start":"03:31.680 ","End":"03:37.490","Text":"2x cosine y minus y cosine x."},{"Start":"03:37.490 ","End":"03:42.805","Text":"Yeah, that was dx."},{"Start":"03:42.805 ","End":"03:44.600","Text":"So what is this?"},{"Start":"03:44.600 ","End":"03:49.560","Text":"Now y is a constant, the integral of 2x is x squared,"},{"Start":"03:49.560 ","End":"03:53.935","Text":"so this part becomes x squared cosine y."},{"Start":"03:53.935 ","End":"03:57.140","Text":"Now the integral of cosine x is sine x."},{"Start":"03:57.140 ","End":"04:02.060","Text":"Y is just a constant, so we have minus y sine x."},{"Start":"04:02.060 ","End":"04:05.090","Text":"Normally, if it was just 1 variable,"},{"Start":"04:05.090 ","End":"04:06.920","Text":"we would put plus C,"},{"Start":"04:06.920 ","End":"04:10.450","Text":"but because we\u0027re doing integral dx and y is a constant,"},{"Start":"04:10.450 ","End":"04:14.995","Text":"we put a sum general function of just y."},{"Start":"04:14.995 ","End":"04:18.440","Text":"The derivative of this with respect to x will be 0."},{"Start":"04:18.440 ","End":"04:20.930","Text":"Okay, So we\u0027ve almost got Phi,"},{"Start":"04:20.930 ","End":"04:23.120","Text":"what we haven\u0027t gotten c of y,"},{"Start":"04:23.120 ","End":"04:25.475","Text":"and for this, we\u0027ll use the other equation."},{"Start":"04:25.475 ","End":"04:29.440","Text":"So Phi with respect to y equals Q,"},{"Start":"04:29.440 ","End":"04:36.079","Text":"so we have that Phi with respect to y,"},{"Start":"04:36.079 ","End":"04:38.924","Text":"which is this here,"},{"Start":"04:38.924 ","End":"04:40.865","Text":"let\u0027s see what that is."},{"Start":"04:40.865 ","End":"04:43.860","Text":"I mean, this here derived with respect to y."},{"Start":"04:43.860 ","End":"04:46.050","Text":"We get what?"},{"Start":"04:46.050 ","End":"04:49.890","Text":"Minus x squared sine y,"},{"Start":"04:49.890 ","End":"04:51.840","Text":"we\u0027re differentiating with respect to y,"},{"Start":"04:51.840 ","End":"04:53.715","Text":"cosine is minus sine."},{"Start":"04:53.715 ","End":"04:55.970","Text":"Here, with respect to y,"},{"Start":"04:55.970 ","End":"04:58.745","Text":"we\u0027re just left with sine x,"},{"Start":"04:58.745 ","End":"05:03.545","Text":"and here we have the derivative of C of y."},{"Start":"05:03.545 ","End":"05:09.000","Text":"On the other hand, this is equal to Q, which is here."},{"Start":"05:12.400 ","End":"05:24.940","Text":"This Q is minus x squared sine y minus sine x."},{"Start":"05:24.940 ","End":"05:28.350","Text":"Well, this cancels with this,"},{"Start":"05:28.350 ","End":"05:30.910","Text":"this cancels with this."},{"Start":"05:30.910 ","End":"05:39.290","Text":"This just gives us C prime of y equals 0, so C, the integral of 0."},{"Start":"05:39.290 ","End":"05:42.950","Text":"In other words, c of y is just some constant."},{"Start":"05:42.950 ","End":"05:44.090","Text":"I don\u0027t want to use letter C,"},{"Start":"05:44.090 ","End":"05:45.590","Text":"I\u0027ll use the letter K."},{"Start":"05:45.590 ","End":"05:49.830","Text":"So C of y is k and that\u0027s all we need now."},{"Start":"05:49.830 ","End":"05:54.830","Text":"Plug that in here and I\u0027ll just write it at the side."},{"Start":"05:54.830 ","End":"05:57.130","Text":"So now we have that Phi,"},{"Start":"05:57.130 ","End":"06:06.050","Text":"I\u0027ll write it fully, of x and y is equal to x squared cosine of y"},{"Start":"06:06.050 ","End":"06:12.960","Text":"minus y sine x plus some constant number K."},{"Start":"06:12.960 ","End":"06:19.110","Text":"That is the answer and so we\u0027re done."}],"ID":8756},{"Watched":false,"Name":"Exercise 1 part c","Duration":"10m 9s","ChapterTopicVideoID":8606,"CourseChapterTopicPlaylistID":4965,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.354","Text":"In the previous exercises we had the 2 dimensional version."},{"Start":"00:05.354 ","End":"00:11.925","Text":"Here we have vector field in 3 dimensions of 3 variables,"},{"Start":"00:11.925 ","End":"00:14.205","Text":"F of x, y, z."},{"Start":"00:14.205 ","End":"00:16.455","Text":"We have the 3 components,"},{"Start":"00:16.455 ","End":"00:19.725","Text":"z squared, e to the minus y and 2x z."},{"Start":"00:19.725 ","End":"00:25.245","Text":"We have to find if F is conservative as a vector field and if so,"},{"Start":"00:25.245 ","End":"00:28.980","Text":"to find the function Phi also called the potential function,"},{"Start":"00:28.980 ","End":"00:32.610","Text":"so that the gradient of Phi is our original"},{"Start":"00:32.610 ","End":"00:37.910","Text":"F. Now what we do is we give these 3 component functions names."},{"Start":"00:37.910 ","End":"00:40.170","Text":"Let\u0027s say, this one is P,"},{"Start":"00:40.170 ","End":"00:47.870","Text":"this one is Q and this one R. There\u0027s more than one version of the test."},{"Start":"00:47.870 ","End":"00:53.675","Text":"One of them involves saying that the curl of"},{"Start":"00:53.675 ","End":"00:55.970","Text":"the vector field F is"},{"Start":"00:55.970 ","End":"01:00.170","Text":"0 but in case you\u0027ve forgotten what curl is, it doesn\u0027t really matter."},{"Start":"01:00.170 ","End":"01:05.970","Text":"This actually is equivalent to 3 conditions and"},{"Start":"01:05.970 ","End":"01:09.440","Text":"the 3 conditions are that the partial derivative of P with respect to"},{"Start":"01:09.440 ","End":"01:13.430","Text":"y equals partial derivative of Q with respect to x,"},{"Start":"01:13.430 ","End":"01:20.240","Text":"that\u0027s 1, and the partial derivative of P with respect"},{"Start":"01:20.240 ","End":"01:27.905","Text":"to z is equal to the partial derivative of R with respect to x,"},{"Start":"01:27.905 ","End":"01:33.110","Text":"and the partial derivative of Q with respect to z equals,"},{"Start":"01:33.110 ","End":"01:35.370","Text":"the partial derivative of"},{"Start":"01:39.070 ","End":"01:45.310","Text":"R with respect to y."},{"Start":"01:45.310 ","End":"01:50.600","Text":"This is the same thing so forget about curl if you forgotten already."},{"Start":"01:50.600 ","End":"01:52.940","Text":"If we just check that these 3 hold,"},{"Start":"01:52.940 ","End":"01:58.580","Text":"then it will be a conservative field and then we\u0027ll go about finding the function Phi."},{"Start":"01:58.580 ","End":"02:02.865","Text":"Let\u0027s check each one."},{"Start":"02:02.865 ","End":"02:06.870","Text":"Now, P maybe I\u0027ll write them out."},{"Start":"02:06.870 ","End":"02:13.545","Text":"P of x, y, z is Z squared."},{"Start":"02:13.545 ","End":"02:18.110","Text":"Q of x,"},{"Start":"02:18.110 ","End":"02:25.975","Text":"y z is e to the minus y. I think it\u0027s better to write it out longhand,"},{"Start":"02:25.975 ","End":"02:28.180","Text":"and R of x,"},{"Start":"02:28.180 ","End":"02:32.510","Text":"y, z is 2xz."},{"Start":"02:33.140 ","End":"02:35.955","Text":"Let\u0027s check the first one,"},{"Start":"02:35.955 ","End":"02:38.925","Text":"P with respect to y."},{"Start":"02:38.925 ","End":"02:41.670","Text":"Mean while we put question marks on these."},{"Start":"02:41.670 ","End":"02:45.005","Text":"I don\u0027t know that they are, but I\u0027m checking if all these 3 things hold,"},{"Start":"02:45.005 ","End":"02:46.655","Text":"then it\u0027ll be conservative."},{"Start":"02:46.655 ","End":"02:52.080","Text":"P with respect to y is 0."},{"Start":"02:55.190 ","End":"03:03.960","Text":"Q with respect to x is also equal to 0 and this is true."},{"Start":"03:03.960 ","End":"03:07.150","Text":"The answer here is yes."},{"Start":"03:07.150 ","End":"03:09.095","Text":"Now the next one,"},{"Start":"03:09.095 ","End":"03:12.110","Text":"P with respect to z,"},{"Start":"03:12.110 ","End":"03:20.100","Text":"is 2z and R with respect to x,"},{"Start":"03:20.100 ","End":"03:24.945","Text":"also x is the variable 2z is a constant."},{"Start":"03:24.945 ","End":"03:30.825","Text":"It\u0027s also 2 z and these are equal so yes,"},{"Start":"03:30.825 ","End":"03:32.565","Text":"for the second one."},{"Start":"03:32.565 ","End":"03:36.485","Text":"Let\u0027s go for the third Q with respect to z."},{"Start":"03:36.485 ","End":"03:38.660","Text":"Well, this is a function just of y,"},{"Start":"03:38.660 ","End":"03:45.140","Text":"so that\u0027s 0 and is it equal to R with respect to y?"},{"Start":"03:45.140 ","End":"03:46.640","Text":"Well, there\u0027s no y here,"},{"Start":"03:46.640 ","End":"03:47.720","Text":"so this is a constant,"},{"Start":"03:47.720 ","End":"03:49.310","Text":"so this is equal to 0."},{"Start":"03:49.310 ","End":"03:52.190","Text":"Yes here also."},{"Start":"03:52.190 ","End":"03:55.010","Text":"We can check off, yes,"},{"Start":"03:55.010 ","End":"03:58.930","Text":"this is conservative as a field."},{"Start":"03:58.930 ","End":"04:03.214","Text":"Now let\u0027s go about looking for the function Phi."},{"Start":"04:03.214 ","End":"04:08.300","Text":"Now what does it mean the grad operator."},{"Start":"04:08.300 ","End":"04:14.660","Text":"Let me remind you, the grad of Phi is just the vector of partial derivatives."},{"Start":"04:14.660 ","End":"04:17.915","Text":"It\u0027s Phi with respect to x."},{"Start":"04:17.915 ","End":"04:24.950","Text":"We\u0027re using the i j k notation so I plus Phi with respect to y in"},{"Start":"04:24.950 ","End":"04:33.545","Text":"the j direction and the j component plus Phi with respect to z k. Now,"},{"Start":"04:33.545 ","End":"04:36.995","Text":"we want this equation,"},{"Start":"04:36.995 ","End":"04:38.660","Text":"grad of phi equals F."},{"Start":"04:38.660 ","End":"04:41.255","Text":"This has to equal F,"},{"Start":"04:41.255 ","End":"04:45.110","Text":"which is P. Well,"},{"Start":"04:45.110 ","End":"04:47.000","Text":"I could write it out in full."},{"Start":"04:47.000 ","End":"04:49.085","Text":"Just, let me just copy that here."},{"Start":"04:49.085 ","End":"04:54.630","Text":"This is going to equal Z squared i plus"},{"Start":"04:54.630 ","End":"05:01.780","Text":"e to the minus y j plus 2xzk."},{"Start":"05:02.570 ","End":"05:09.090","Text":"These are vectors either write them in bold or you put an arrow over both."},{"Start":"05:09.090 ","End":"05:11.195","Text":"If these are going to be equal,"},{"Start":"05:11.195 ","End":"05:13.250","Text":"I want 3 qualities."},{"Start":"05:13.250 ","End":"05:18.735","Text":"I want this to be equal and this, and this."},{"Start":"05:18.735 ","End":"05:20.865","Text":"Let\u0027s start with this one."},{"Start":"05:20.865 ","End":"05:25.535","Text":"The Phi with respect to x is going to equal z squared."},{"Start":"05:25.535 ","End":"05:35.490","Text":"Well, if Phi with respect to x is equal to Z squared,"},{"Start":"05:35.490 ","End":"05:41.540","Text":"then we can get what Phi is by integration is the integral of Z"},{"Start":"05:41.540 ","End":"05:47.970","Text":"squared dx and this is equal to Z squared is just a constant as far as x goes."},{"Start":"05:47.970 ","End":"05:52.625","Text":"It\u0027s Z squared x plus and our constant,"},{"Start":"05:52.625 ","End":"05:55.310","Text":"but a constant as far as x goes,"},{"Start":"05:55.310 ","End":"06:00.800","Text":"which means any function of y and z,"},{"Start":"06:00.800 ","End":"06:04.955","Text":"that\u0027s a constant as far as x goes on it\u0027s derivative would be 0."},{"Start":"06:04.955 ","End":"06:09.980","Text":"Now we\u0027re going to see what we can find out about this function of y and z."},{"Start":"06:09.980 ","End":"06:13.729","Text":"We\u0027re going to use up more information,"},{"Start":"06:13.729 ","End":"06:16.880","Text":"which is this equation here."},{"Start":"06:16.880 ","End":"06:21.395","Text":"What we get if we compare these is that I\u0027ll start with this side,"},{"Start":"06:21.395 ","End":"06:28.155","Text":"e to the minus y is equal to Phi with respect to y."},{"Start":"06:28.155 ","End":"06:32.115","Text":"Well, this is Phi and with respect to y,"},{"Start":"06:32.115 ","End":"06:41.460","Text":"Z squared x is just nothing maybe I write that as 0 plus."},{"Start":"06:41.460 ","End":"06:46.940","Text":"Then this with respect to y is just the partial derivative of C with respect to y,"},{"Start":"06:46.940 ","End":"06:50.220","Text":"of y and z."},{"Start":"06:50.220 ","End":"06:53.780","Text":"This will give us another equation."},{"Start":"06:53.780 ","End":"06:59.825","Text":"That function C of y and z is the"},{"Start":"06:59.825 ","End":"07:09.130","Text":"integral of e to the minus y with respect to y and this is equal to,"},{"Start":"07:09.130 ","End":"07:13.210","Text":"this integral is minus e to the minus y."},{"Start":"07:13.210 ","End":"07:16.825","Text":"Again, instead of just a constant number,"},{"Start":"07:16.825 ","End":"07:20.515","Text":"it\u0027s going to be a constant as far as y and z go,"},{"Start":"07:20.515 ","End":"07:26.620","Text":"which means some unknown function of Z. I\u0027ll use the letter C again"},{"Start":"07:26.620 ","End":"07:29.170","Text":"because it won\u0027t be any confusion maybe I\u0027ll make it a bit"},{"Start":"07:29.170 ","End":"07:34.010","Text":"bolder because this C will be a function of just Z."},{"Start":"07:34.010 ","End":"07:37.320","Text":"Now we want to see what we can find out about this."},{"Start":"07:37.320 ","End":"07:39.560","Text":"We still have another piece of information,"},{"Start":"07:39.560 ","End":"07:40.850","Text":"the equation we haven\u0027t used,"},{"Start":"07:40.850 ","End":"07:42.995","Text":"and that\u0027s this equation here."},{"Start":"07:42.995 ","End":"07:49.660","Text":"I\u0027ll start with this side at 2xz is equal to."},{"Start":"07:49.660 ","End":"07:54.275","Text":"Now, let\u0027s meanwhile gather what we have about Phi."},{"Start":"07:54.275 ","End":"07:58.410","Text":"I can write it more specifically since I have this,"},{"Start":"07:58.880 ","End":"08:01.980","Text":"I know what C of y, z is."},{"Start":"08:01.980 ","End":"08:06.110","Text":"We get C squared x plus, well,"},{"Start":"08:06.110 ","End":"08:15.525","Text":"not plus because it\u0027s a minus e to the minus y plus C of Z."},{"Start":"08:15.525 ","End":"08:19.025","Text":"Now if I take this 2xz from here,"},{"Start":"08:19.025 ","End":"08:26.535","Text":"it\u0027s equal to the derivative of this with respect to Z."},{"Start":"08:26.535 ","End":"08:32.230","Text":"I get derivative of Z squared x"},{"Start":"08:32.230 ","End":"08:39.610","Text":"is 2zx or write it as 2xz because it would look like this."},{"Start":"08:39.610 ","End":"08:46.045","Text":"C the derivative of e to the minus y with respect to Z is just"},{"Start":"08:46.045 ","End":"08:53.900","Text":"0 and then we have C prime of z."},{"Start":"08:54.290 ","End":"09:06.070","Text":"This cancels out with this and so what we get is that C prime of Z is just 0."},{"Start":"09:06.070 ","End":"09:09.630","Text":"This thing comes out to be just 0 and"},{"Start":"09:09.630 ","End":"09:17.855","Text":"so this is my solid C. The one variable C. This gives us,"},{"Start":"09:17.855 ","End":"09:21.275","Text":"of course that C of z is some constant."},{"Start":"09:21.275 ","End":"09:28.810","Text":"I\u0027ll use the letter K. Is just K and now I can plug that in here."},{"Start":"09:30.500 ","End":"09:33.380","Text":"In fact not in here, I mean,"},{"Start":"09:33.380 ","End":"09:36.590","Text":"but all the way up to here."},{"Start":"09:36.590 ","End":"09:41.465","Text":"Now I\u0027ll get this organized and write Phi properly,"},{"Start":"09:41.465 ","End":"09:51.830","Text":"we have that Phi is equal to Z squared x minus e"},{"Start":"09:51.830 ","End":"09:59.210","Text":"to the minus y plus K. This is"},{"Start":"09:59.210 ","End":"10:03.920","Text":"the answer for the potential function Phi after we showed that it\u0027s"},{"Start":"10:03.920 ","End":"10:10.160","Text":"a conservative field and that concludes this question."}],"ID":8757},{"Watched":false,"Name":"Exercise 1 part d","Duration":"2m 9s","ChapterTopicVideoID":8607,"CourseChapterTopicPlaylistID":4965,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"In this exercise, we\u0027re given a vector field F"},{"Start":"00:03.720 ","End":"00:08.010","Text":"in 3 variables and 3 dimensions, and here it is."},{"Start":"00:08.010 ","End":"00:12.000","Text":"We\u0027re using the angular bracket notation for vectors,"},{"Start":"00:12.000 ","End":"00:15.810","Text":"and the first thing we want to know is if F is conservative,"},{"Start":"00:15.810 ","End":"00:20.050","Text":"let\u0027s answer that and then we\u0027ll look at the next part of the question."},{"Start":"00:21.860 ","End":"00:28.700","Text":"One of the conditions or formulas for telling if F is conservative,"},{"Start":"00:28.700 ","End":"00:30.040","Text":"I\u0027ll write it in a moment,"},{"Start":"00:30.040 ","End":"00:32.300","Text":"let\u0027s just introduce some notation."},{"Start":"00:32.300 ","End":"00:35.610","Text":"Let\u0027s call the first function P,"},{"Start":"00:35.610 ","End":"00:37.500","Text":"the second function Q,"},{"Start":"00:37.500 ","End":"00:38.640","Text":"each of them is of x, y,"},{"Start":"00:38.640 ","End":"00:41.744","Text":"z, the third function R,"},{"Start":"00:41.744 ","End":"00:48.350","Text":"and I copied the condition from a previous question,"},{"Start":"00:48.350 ","End":"00:53.690","Text":"that F is conservative if the following 3 equalities all hold at the moment,"},{"Start":"00:53.690 ","End":"00:55.415","Text":"I don\u0027t know that they do."},{"Start":"00:55.415 ","End":"00:57.050","Text":"If these 3 hold,"},{"Start":"00:57.050 ","End":"00:59.720","Text":"then the field is conservative."},{"Start":"00:59.720 ","End":"01:02.690","Text":"These are partial derivatives."},{"Start":"01:02.690 ","End":"01:08.635","Text":"So let\u0027s try them 1 by 1. Let\u0027s see."},{"Start":"01:08.635 ","End":"01:13.035","Text":"P with respect to y is 0,"},{"Start":"01:13.035 ","End":"01:17.990","Text":"Q with respect to x is also 0."},{"Start":"01:17.990 ","End":"01:20.090","Text":"So far so good."},{"Start":"01:20.090 ","End":"01:24.905","Text":"P with respect to z is"},{"Start":"01:24.905 ","End":"01:33.745","Text":"1 and R with respect to x is 0."},{"Start":"01:33.745 ","End":"01:36.375","Text":"This is not equal,"},{"Start":"01:36.375 ","End":"01:39.105","Text":"as soon as 1 of them fails,"},{"Start":"01:39.105 ","End":"01:41.570","Text":"then the answer is no,"},{"Start":"01:41.570 ","End":"01:44.635","Text":"it is not conservative,"},{"Start":"01:44.635 ","End":"01:49.820","Text":"and so we don\u0027t even have to continue because it says if so, well,"},{"Start":"01:49.820 ","End":"01:52.625","Text":"I don\u0027t care what it says because it\u0027s not so,"},{"Start":"01:52.625 ","End":"01:55.225","Text":"because F is not conservative,"},{"Start":"01:55.225 ","End":"01:56.570","Text":"and so there\u0027s no point,"},{"Start":"01:56.570 ","End":"02:03.680","Text":"they wanted us to find a potential function phi whose gradient is F, but no,"},{"Start":"02:03.680 ","End":"02:05.810","Text":"as soon as it\u0027s not conservative,"},{"Start":"02:05.810 ","End":"02:08.700","Text":"we\u0027re done. That\u0027s it."}],"ID":8758},{"Watched":false,"Name":"Exercise 2 part a","Duration":"4m 56s","ChapterTopicVideoID":8608,"CourseChapterTopicPlaylistID":4965,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"In this exercise, we have a type 2 line integral."},{"Start":"00:04.320 ","End":"00:06.720","Text":"It\u0027s from this point to this point."},{"Start":"00:06.720 ","End":"00:09.073","Text":"Normally, this wouldn\u0027t make sense because"},{"Start":"00:09.073 ","End":"00:12.600","Text":"it could depend on which path I take from here to here."},{"Start":"00:12.600 ","End":"00:14.580","Text":"That\u0027s where part a comes in,"},{"Start":"00:14.580 ","End":"00:19.515","Text":"that we first have to show that this integral is path independent,"},{"Start":"00:19.515 ","End":"00:22.920","Text":"independent of the path that joins these 2 points."},{"Start":"00:22.920 ","End":"00:24.480","Text":"Then it\u0027ll make sense."},{"Start":"00:24.480 ","End":"00:34.020","Text":"Now, notice that I could rewrite this integral in the form vector field F.dr."},{"Start":"00:34.020 ","End":"00:35.790","Text":"I\u0027ll show you what I mean."},{"Start":"00:35.790 ","End":"00:43.799","Text":"Let\u0027s say that we call this function here P or P of x, y,"},{"Start":"00:43.799 ","End":"00:49.980","Text":"and this one we\u0027ll call Q for short or Q of x, y for long,"},{"Start":"00:49.980 ","End":"00:59.085","Text":"and then this thing just becomes the integral."},{"Start":"00:59.085 ","End":"01:06.160","Text":"First of all of P dx plus Q dy."},{"Start":"01:06.830 ","End":"01:10.230","Text":"I\u0027m less concerned with the path at the moment."},{"Start":"01:10.230 ","End":"01:12.855","Text":"This is equal to,"},{"Start":"01:12.855 ","End":"01:16.440","Text":"let\u0027s say I use the ij notation,"},{"Start":"01:16.440 ","End":"01:39.780","Text":"P i plus Q j. product with i dx or dx i plus j dy."},{"Start":"01:39.780 ","End":"01:42.410","Text":"I should have maybe written it the other way around."},{"Start":"01:42.410 ","End":"01:44.360","Text":"There, because dot product means,"},{"Start":"01:44.360 ","End":"01:47.900","Text":"we take the first the i component with the i component"},{"Start":"01:47.900 ","End":"01:50.090","Text":"plus the j component with the j component,"},{"Start":"01:50.090 ","End":"01:52.180","Text":"just what\u0027s written here."},{"Start":"01:52.180 ","End":"01:55.430","Text":"This is a vector field."},{"Start":"01:55.430 ","End":"01:58.025","Text":"We could call this vector field F,"},{"Start":"01:58.025 ","End":"01:59.930","Text":"which is made up of 2 parts,"},{"Start":"01:59.930 ","End":"02:01.970","Text":"the P and the Q components,"},{"Start":"02:01.970 ","End":"02:06.001","Text":"and this is what is dr."},{"Start":"02:06.001 ","End":"02:10.770","Text":"Dr is dx, dy, If we\u0027re using angular brackets,"},{"Start":"02:10.770 ","End":"02:13.010","Text":"it would be dx,di and so on."},{"Start":"02:13.010 ","End":"02:17.705","Text":"Now, the reason I\u0027m writing it in this form is that there\u0027s a theorem about"},{"Start":"02:17.705 ","End":"02:23.480","Text":"conservative vector fields and path independence that basically says that"},{"Start":"02:23.480 ","End":"02:34.650","Text":"if F, the vector field is conservative then we have path independence."},{"Start":"02:34.650 ","End":"02:37.685","Text":"I\u0027ll just write the word path independence,"},{"Start":"02:37.685 ","End":"02:42.590","Text":"which means basically what it says here"},{"Start":"02:42.590 ","End":"02:50.710","Text":"that the integral and the integral from 2 points will give the same result."},{"Start":"02:51.440 ","End":"02:54.830","Text":"That we can use the test we\u0027ve been using"},{"Start":"02:54.830 ","End":"03:01.320","Text":"in all previous exercises about F being conservative,"},{"Start":"03:01.320 ","End":"03:08.630","Text":"at least the 2-dimensional test when F is a function to a 2D vector field,"},{"Start":"03:08.630 ","End":"03:12.020","Text":"that this thing will be conservative"},{"Start":"03:12.020 ","End":"03:17.150","Text":"if and only if the partial derivative of P with respect to y"},{"Start":"03:17.150 ","End":"03:20.710","Text":"is equal to the partial derivative of Q with respect to x."},{"Start":"03:20.710 ","End":"03:23.150","Text":"I\u0027ll just write this with a question mark at the moment"},{"Start":"03:23.150 ","End":"03:24.815","Text":"because this is what we\u0027re going to show."},{"Start":"03:24.815 ","End":"03:26.210","Text":"If this is true,"},{"Start":"03:26.210 ","End":"03:29.989","Text":"then we have that vector field F is conservative,"},{"Start":"03:29.989 ","End":"03:32.764","Text":"and then we have the path independence of integrals."},{"Start":"03:32.764 ","End":"03:34.925","Text":"So this is just a simple check."},{"Start":"03:34.925 ","End":"03:44.330","Text":"P with respect to y is equal to x is a constant."},{"Start":"03:44.330 ","End":"03:51.400","Text":"So from here, we get just 6x times 2y,"},{"Start":"03:51.400 ","End":"03:55.815","Text":"or you can write it as 6x then we have the 2y from here."},{"Start":"03:55.815 ","End":"04:00.365","Text":"With respect to y here we have minus 3y squared."},{"Start":"04:00.365 ","End":"04:01.865","Text":"On the other hand,"},{"Start":"04:01.865 ","End":"04:04.730","Text":"Qx is equal to,"},{"Start":"04:04.730 ","End":"04:07.039","Text":"let\u0027s see this time y is a constant,"},{"Start":"04:07.039 ","End":"04:10.350","Text":"so we just get the 6x squared."},{"Start":"04:14.760 ","End":"04:16.750","Text":"Sorry, my apologies."},{"Start":"04:16.750 ","End":"04:19.900","Text":"Differentiating with respect to x, y is the constant,"},{"Start":"04:19.900 ","End":"04:26.460","Text":"so y stays, and from here we get 12 x times the y."},{"Start":"04:26.460 ","End":"04:30.435","Text":"From the other 1, let\u0027s see, y is the constant,"},{"Start":"04:30.435 ","End":"04:38.015","Text":"so derivative of 3x is 3 and the constant y squared stays."},{"Start":"04:38.015 ","End":"04:43.550","Text":"These 2 are equal to me because 6 times 2 is 12,"},{"Start":"04:43.550 ","End":"04:45.560","Text":"so these are equal."},{"Start":"04:45.560 ","End":"04:49.155","Text":"The answer to this is yes,"},{"Start":"04:49.155 ","End":"04:51.870","Text":"Py equals Qx, super conservative,"},{"Start":"04:51.870 ","End":"04:53.410","Text":"so it\u0027s path independence."},{"Start":"04:53.410 ","End":"04:56.970","Text":"That concludes part a."}],"ID":8759},{"Watched":false,"Name":"Exercise 2 part b","Duration":"15m 45s","ChapterTopicVideoID":8609,"CourseChapterTopicPlaylistID":4965,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.310","Text":"We just did part a,"},{"Start":"00:02.310 ","End":"00:07.680","Text":"now we\u0027re on to part b. I kept some things I needed."},{"Start":"00:07.680 ","End":"00:09.795","Text":"I guess I don\u0027t need this either."},{"Start":"00:09.795 ","End":"00:12.900","Text":"The important thing is that we showed that"},{"Start":"00:12.900 ","End":"00:16.740","Text":"we had a vector field F, which was conservative."},{"Start":"00:16.740 ","End":"00:19.875","Text":"F was just here."},{"Start":"00:19.875 ","End":"00:24.790","Text":"P times i plus Q times j."},{"Start":"00:24.790 ","End":"00:29.025","Text":"The thing is that when F is conservative,"},{"Start":"00:29.025 ","End":"00:36.910","Text":"then what we know is that it\u0027s the potential of some function Phi ie"},{"Start":"00:36.910 ","End":"00:40.535","Text":"that the vector field F is the gradient"},{"Start":"00:40.535 ","End":"00:45.930","Text":"of a potential function Phi over Phi is Phi of xy."},{"Start":"00:46.600 ","End":"00:49.790","Text":"This integral in general,"},{"Start":"00:49.790 ","End":"00:52.880","Text":"let\u0027s say this was a point A just to be a bit more general,"},{"Start":"00:52.880 ","End":"00:54.875","Text":"and this was the point B."},{"Start":"00:54.875 ","End":"01:07.410","Text":"Then this integral of F dot dr,"},{"Start":"01:07.410 ","End":"01:13.190","Text":"say from A to B, can be computed by taking the function Phi"},{"Start":"01:13.190 ","End":"01:20.135","Text":"at B minus the potential function at A."},{"Start":"01:20.135 ","End":"01:26.865","Text":"That\u0027s going to be 1 way of computing the line integral."},{"Start":"01:26.865 ","End":"01:31.545","Text":"Let\u0027s call it method a with the potential function."},{"Start":"01:31.545 ","End":"01:36.145","Text":"Our first job is to find this potential function Phi."},{"Start":"01:36.145 ","End":"01:40.805","Text":"Remember that grad, or sometimes even written with a vector sign,"},{"Start":"01:40.805 ","End":"01:49.650","Text":"the gradient of Phi is equal to, first of all,"},{"Start":"01:49.650 ","End":"01:53.370","Text":"Phi with respect to x times i,"},{"Start":"01:53.370 ","End":"01:57.329","Text":"and then Phi with respect to y times j,"},{"Start":"01:57.329 ","End":"01:59.820","Text":"and using the i, j notation."},{"Start":"01:59.820 ","End":"02:05.195","Text":"If we have this equality and F is here,"},{"Start":"02:05.195 ","End":"02:08.125","Text":"then this equals this."},{"Start":"02:08.125 ","End":"02:09.800","Text":"When 2 vectors are equal,"},{"Start":"02:09.800 ","End":"02:11.090","Text":"both components are equal,"},{"Start":"02:11.090 ","End":"02:13.475","Text":"so we get 2 equations."},{"Start":"02:13.475 ","End":"02:19.105","Text":"This gives us the Phi with respect to x is p,"},{"Start":"02:19.105 ","End":"02:25.015","Text":"and Phi with respect to y is equal to q."},{"Start":"02:25.015 ","End":"02:27.135","Text":"This will help us to find Phi."},{"Start":"02:27.135 ","End":"02:28.425","Text":"Let\u0027s start with 1 of them,"},{"Start":"02:28.425 ","End":"02:31.040","Text":"Phi with respect to x is p,"},{"Start":"02:31.040 ","End":"02:32.545","Text":"and where is p?"},{"Start":"02:32.545 ","End":"02:34.579","Text":"It is here."},{"Start":"02:34.579 ","End":"02:41.385","Text":"This is equal to 6xy squared minus y cubed."},{"Start":"02:41.385 ","End":"02:49.230","Text":"From here, we can conclude that Phi is just the integral of this dx."},{"Start":"02:49.230 ","End":"02:58.475","Text":"We have the integral of 6xy squared minus y cubed,"},{"Start":"02:58.475 ","End":"03:01.360","Text":"and this is dx."},{"Start":"03:01.360 ","End":"03:05.380","Text":"This gives us y is a constant."},{"Start":"03:05.380 ","End":"03:09.530","Text":"The integral of just x is x squared over 2,"},{"Start":"03:09.530 ","End":"03:16.110","Text":"so altogether we get from here 3x squared."},{"Start":"03:16.640 ","End":"03:19.260","Text":"Integral of x is x squared over 2,"},{"Start":"03:19.260 ","End":"03:20.850","Text":"2 with the 6x here,"},{"Start":"03:20.850 ","End":"03:24.150","Text":"3 times x squared, y squared."},{"Start":"03:24.150 ","End":"03:29.825","Text":"From here, just y cubed times xy cubed is a constant."},{"Start":"03:29.825 ","End":"03:35.885","Text":"But the constant that we add here is a constant with respect to x,"},{"Start":"03:35.885 ","End":"03:39.190","Text":"which means it could be any function of y."},{"Start":"03:39.190 ","End":"03:44.045","Text":"So it\u0027s c of y because the derivative of this with respect to x is 0."},{"Start":"03:44.045 ","End":"03:47.570","Text":"We still have to find out more about c,"},{"Start":"03:47.570 ","End":"03:49.755","Text":"otherwise, we haven\u0027t found Phi."},{"Start":"03:49.755 ","End":"03:52.000","Text":"So we\u0027ll use the 2nd equation,"},{"Start":"03:52.000 ","End":"03:55.925","Text":"Phi with respect to y is q. I want to start with the right-hand side,"},{"Start":"03:55.925 ","End":"03:58.445","Text":"Q. Q is written here,"},{"Start":"03:58.445 ","End":"04:07.470","Text":"is 6x squared y minus 3xy squared."},{"Start":"04:07.480 ","End":"04:12.795","Text":"This has got to equal Phi with respect to y."},{"Start":"04:12.795 ","End":"04:14.280","Text":"Now, I have Phi,"},{"Start":"04:14.280 ","End":"04:16.095","Text":"it\u0027s what\u0027s written here."},{"Start":"04:16.095 ","End":"04:23.525","Text":"If I take this and differentiate this with respect to y, what do I get?"},{"Start":"04:23.525 ","End":"04:26.905","Text":"From the first bit, I get,"},{"Start":"04:26.905 ","End":"04:34.260","Text":"let\u0027s see, for y squared gives me 2y, 2y times 3."},{"Start":"04:34.260 ","End":"04:36.675","Text":"I\u0027ll just put the 2 up front."},{"Start":"04:36.675 ","End":"04:38.750","Text":"The 2 upfront will give me 6."},{"Start":"04:38.750 ","End":"04:44.270","Text":"I\u0027ll leave the y here and the x squared in the middle just so it\u0027ll look like this."},{"Start":"04:44.270 ","End":"04:47.030","Text":"Then from y cubed dx,"},{"Start":"04:47.030 ","End":"04:51.500","Text":"I get 3y squared x."},{"Start":"04:51.500 ","End":"04:53.810","Text":"But again, to make it look like this,"},{"Start":"04:53.810 ","End":"04:58.865","Text":"I\u0027ll just re-change the order and get 3xy squared."},{"Start":"04:58.865 ","End":"05:03.620","Text":"But I also have c-prime of y."},{"Start":"05:03.620 ","End":"05:08.195","Text":"Now, left-hand side and right-hand side are practically the same."},{"Start":"05:08.195 ","End":"05:10.040","Text":"This is equal to this,"},{"Start":"05:10.040 ","End":"05:11.975","Text":"and this is equal to this."},{"Start":"05:11.975 ","End":"05:18.280","Text":"It follows from this that c-prime of y is equal to 0."},{"Start":"05:18.280 ","End":"05:23.420","Text":"If the derivative of the function c is 0,"},{"Start":"05:23.420 ","End":"05:30.229","Text":"then that gives us that c of y is equal to an actual constant."},{"Start":"05:30.229 ","End":"05:32.644","Text":"I\u0027ll use letter k for that constant."},{"Start":"05:32.644 ","End":"05:33.950","Text":"Now that I have this,"},{"Start":"05:33.950 ","End":"05:35.855","Text":"I can plug that in here."},{"Start":"05:35.855 ","End":"05:40.860","Text":"Now we have found the function Phi."},{"Start":"05:41.600 ","End":"05:44.055","Text":"I\u0027ll rewrite it here,"},{"Start":"05:44.055 ","End":"05:49.014","Text":"Phi, it\u0027s a function of x and y,"},{"Start":"05:49.014 ","End":"05:54.685","Text":"is equal to 3x squared y squared"},{"Start":"05:54.685 ","End":"06:03.585","Text":"minus y cubed x plus the k. What I want now,"},{"Start":"06:03.585 ","End":"06:15.610","Text":"method a is the integral from 1,2 to"},{"Start":"06:15.610 ","End":"06:19.930","Text":"3,4 of F dot dr"},{"Start":"06:21.200 ","End":"06:28.320","Text":"is just equal to Phi at the 1 endpoint,"},{"Start":"06:28.320 ","End":"06:34.995","Text":"the upper limit of 3,4 minus Phi of 1, 2."},{"Start":"06:34.995 ","End":"06:37.790","Text":"This is equal to just have to substitute here."},{"Start":"06:37.790 ","End":"06:40.405","Text":"Let\u0027s do the 3,4."},{"Start":"06:40.405 ","End":"06:46.049","Text":"I\u0027ll put the brackets for this part 3 times 3 squared,"},{"Start":"06:46.049 ","End":"06:56.220","Text":"4 squared minus y cubed is 4 cubed times x, which is 3."},{"Start":"06:56.220 ","End":"07:00.325","Text":"We are not bothering with the plus k because we\u0027ll get a k in both of them."},{"Start":"07:00.325 ","End":"07:02.920","Text":"When we subtract, it\u0027ll cancel out."},{"Start":"07:02.920 ","End":"07:10.630","Text":"I don\u0027t really need this k. Now minus the Phi of 1,2 is"},{"Start":"07:10.630 ","End":"07:18.840","Text":"3 times 1 squared times 2 squared minus 2 cubed times 1."},{"Start":"07:18.840 ","End":"07:22.195","Text":"Just have to do a computation here."},{"Start":"07:22.195 ","End":"07:31.080","Text":"This part is equal to 3 times 3 squared is 27."},{"Start":"07:31.080 ","End":"07:34.680","Text":"27 times 16."},{"Start":"07:34.680 ","End":"07:41.210","Text":"Here, I\u0027ll get 4 cubed is 64 times 3."},{"Start":"07:41.210 ","End":"07:47.915","Text":"I will just do the whole thing on the calculator and give you the final answer 236."},{"Start":"07:47.915 ","End":"07:52.230","Text":"That was 1 method and we got the answer 236, and now,"},{"Start":"07:52.230 ","End":"07:55.650","Text":"let\u0027s do the second method on a clean page,"},{"Start":"07:55.650 ","End":"07:59.070","Text":"but we\u0027ll remember the answer 236."},{"Start":"07:59.070 ","End":"08:02.970","Text":"I clean the board and back up we go."},{"Start":"08:02.970 ","End":"08:10.230","Text":"The second method, we\u0027ll use the path independence that we found and we\u0027ll take a path"},{"Start":"08:10.230 ","End":"08:17.685","Text":"from 1,2 to 3,4 and actually compute the integral directly along that path."},{"Start":"08:17.685 ","End":"08:21.105","Text":"I\u0027ll just make a note that we\u0027re on method 2 of part"},{"Start":"08:21.105 ","End":"08:25.905","Text":"B. I\u0027m going to draw a sketch now that we want to"},{"Start":"08:25.905 ","End":"08:28.530","Text":"take a path from the point 1,2"},{"Start":"08:28.530 ","End":"08:38.220","Text":"to the point 3,4."},{"Start":"08:38.220 ","End":"08:41.550","Text":"1 of the easiest things to do is to take a path that"},{"Start":"08:41.550 ","End":"08:45.630","Text":"goes in pieces with horizontal and vertical only."},{"Start":"08:45.630 ","End":"08:50.490","Text":"For example, we could go first across from here to here,"},{"Start":"08:50.490 ","End":"08:57.680","Text":"and then up from here to here and then because this has the constant y,"},{"Start":"08:57.680 ","End":"09:02.570","Text":"this is going to be something ,2 and because vertical keeps the x,"},{"Start":"09:02.570 ","End":"09:05.960","Text":"so this will be the point 3,2."},{"Start":"09:05.960 ","End":"09:07.265","Text":"We\u0027ll go from here to here,"},{"Start":"09:07.265 ","End":"09:09.770","Text":"and then from here to here and we\u0027ll get a lot of"},{"Start":"09:09.770 ","End":"09:13.525","Text":"0s when we do vertical and horizontal paths."},{"Start":"09:13.525 ","End":"09:20.130","Text":"We\u0027ll call this path C_1 and this path C_2 and together it\u0027ll"},{"Start":"09:20.130 ","End":"09:26.505","Text":"give us the path C from this point to this point."},{"Start":"09:26.505 ","End":"09:29.895","Text":"We want to take the integral along"},{"Start":"09:29.895 ","End":"09:35.220","Text":"C_1 plus the integral of C_2 of whatever it is that\u0027s written here,"},{"Start":"09:35.220 ","End":"09:38.055","Text":"I\u0027m just making a note that that\u0027s what we\u0027re going to do."},{"Start":"09:38.055 ","End":"09:40.380","Text":"Let\u0027s start with C_1 first,"},{"Start":"09:40.380 ","End":"09:43.600","Text":"and then we\u0027ll parameterize that."},{"Start":"09:43.910 ","End":"09:49.380","Text":"The easiest way to parameterize C_1 is,"},{"Start":"09:49.380 ","End":"09:53.160","Text":"I don\u0027t need to use the formula for the path between 2 points."},{"Start":"09:53.160 ","End":"09:54.675","Text":"When I\u0027m going horizontally,"},{"Start":"09:54.675 ","End":"09:57.060","Text":"I know that my y is staying constant,"},{"Start":"09:57.060 ","End":"10:01.020","Text":"it\u0027s constantly 2, and x is moving from 1-3."},{"Start":"10:01.020 ","End":"10:11.070","Text":"So I can say x equals t and t moves from 1-3 and I also will need"},{"Start":"10:11.070 ","End":"10:16.980","Text":"when I substitute here dx so I\u0027ll have that dx is equal to"},{"Start":"10:16.980 ","End":"10:27.615","Text":"dt and dy will equal just 0 or 0 dt, whatever."},{"Start":"10:27.615 ","End":"10:32.205","Text":"Now when we substitute the integral into C_1,"},{"Start":"10:32.205 ","End":"10:36.165","Text":"we get the integral according to the parameter t here"},{"Start":"10:36.165 ","End":"10:42.600","Text":"from 1-3 and we just substitute everything here."},{"Start":"10:42.600 ","End":"10:46.770","Text":"We get 6 and then looking here,"},{"Start":"10:46.770 ","End":"10:49.380","Text":"x is t, y is 2."},{"Start":"10:49.380 ","End":"10:57.840","Text":"6t2 squared minus 2 cubed and then dx,"},{"Start":"10:57.840 ","End":"11:01.485","Text":"which is dt plus,"},{"Start":"11:01.485 ","End":"11:03.990","Text":"since dy is 0,"},{"Start":"11:03.990 ","End":"11:08.880","Text":"here, it doesn\u0027t add anything, plus 0."},{"Start":"11:08.880 ","End":"11:13.170","Text":"I\u0027ll just make a note that l haven\u0027t forgotten the second piece."},{"Start":"11:13.170 ","End":"11:16.515","Text":"Now this is a straightforward integral."},{"Start":"11:16.515 ","End":"11:20.535","Text":"This is equal to the integral from 1-3."},{"Start":"11:20.535 ","End":"11:31.590","Text":"All we have here is 6 times 2 squared is 6 times 4 is 24t minus 8 dt"},{"Start":"11:31.590 ","End":"11:36.370","Text":"and this is equal to"},{"Start":"11:36.830 ","End":"11:46.200","Text":"24t gives me 12t squared minus 8t from 1-3."},{"Start":"11:46.200 ","End":"11:50.010","Text":"If I plug in 3,"},{"Start":"11:50.010 ","End":"11:58.380","Text":"I get 12 times 9 is 108 minus 24 is 84, minus 1."},{"Start":"11:58.380 ","End":"12:04.215","Text":"I plug in 1, I get 12 minus 8 is 4."},{"Start":"12:04.215 ","End":"12:07.185","Text":"This is 80."},{"Start":"12:07.185 ","End":"12:09.780","Text":"I\u0027ll just highlight this 80,"},{"Start":"12:09.780 ","End":"12:11.670","Text":"and now we\u0027ll go to the other path,"},{"Start":"12:11.670 ","End":"12:20.625","Text":"C_2 and for C_2 I can easily parameterize it without using the formula,"},{"Start":"12:20.625 ","End":"12:28.995","Text":"again, because here x stays constant at 3 and y goes from 2-4."},{"Start":"12:28.995 ","End":"12:38.120","Text":"I can say that y equals t and t goes from 2-4 and then I also have that dx here is"},{"Start":"12:38.120 ","End":"12:48.045","Text":"equal to 0 and dy is just equal to dt and now I need to substitute."},{"Start":"12:48.045 ","End":"12:50.580","Text":"I\u0027ve lost the original exercise."},{"Start":"12:50.580 ","End":"12:57.120","Text":"Let\u0027s see if I can scroll up and see it. Let\u0027s just see."},{"Start":"12:57.120 ","End":"13:05.295","Text":"The C_2 part gives me the integral from,"},{"Start":"13:05.295 ","End":"13:12.110","Text":"and this time it\u0027s going to be just by the parameter from 2-4"},{"Start":"13:12.110 ","End":"13:21.105","Text":"of 6xy squared is 6,"},{"Start":"13:21.105 ","End":"13:25.920","Text":"x is 3, 6 3,"},{"Start":"13:25.920 ","End":"13:45.930","Text":"y is t, t squared minus t cubed plus"},{"Start":"13:45.930 ","End":"13:51.615","Text":"6x, 6 times 3 squared times"},{"Start":"13:51.615 ","End":"13:57.750","Text":"t minus 3 times 3 times t squared,"},{"Start":"13:57.750 ","End":"14:00.550","Text":"dy, which is dt."},{"Start":"14:01.280 ","End":"14:03.975","Text":"Now we have that."},{"Start":"14:03.975 ","End":"14:07.410","Text":"This is equal to,"},{"Start":"14:07.410 ","End":"14:09.705","Text":"this part is just nothing."},{"Start":"14:09.705 ","End":"14:12.930","Text":"We have the integral from 2-4."},{"Start":"14:12.930 ","End":"14:26.850","Text":"3 squared times 6 is 54t minus 9t squared dt,"},{"Start":"14:26.850 ","End":"14:29.610","Text":"which is equal to,"},{"Start":"14:29.610 ","End":"14:35.040","Text":"let\u0027s see, this thing, t squared over 2,"},{"Start":"14:35.040 ","End":"14:40.920","Text":"so it\u0027s 27t squared minus,"},{"Start":"14:40.920 ","End":"14:50.670","Text":"and then t cubed over 3 that will leave me 3t cubed from 2-4 and this is equal to,"},{"Start":"14:50.670 ","End":"14:55.965","Text":"when I put in 4, 4 squared is 16."},{"Start":"14:55.965 ","End":"14:58.710","Text":"I won\u0027t waste time with the calculations."},{"Start":"14:58.710 ","End":"15:03.045","Text":"When you plug in 4, we\u0027ll get 240."},{"Start":"15:03.045 ","End":"15:04.830","Text":"When we plug in 2,"},{"Start":"15:04.830 ","End":"15:14.445","Text":"we\u0027ll get 84 and this will give us a 156 and I\u0027ll highlight that."},{"Start":"15:14.445 ","End":"15:20.910","Text":"That\u0027s the 2 separate paths and altogether our path from here to here,"},{"Start":"15:20.910 ","End":"15:23.295","Text":"we add the 2 bits together,"},{"Start":"15:23.295 ","End":"15:29.070","Text":"we\u0027ll get 80 plus 156,"},{"Start":"15:29.070 ","End":"15:35.240","Text":"which equals 236 and that"},{"Start":"15:35.240 ","End":"15:38.210","Text":"is exactly what we got in the first part when"},{"Start":"15:38.210 ","End":"15:41.465","Text":"we did it in the first method using the potential functions."},{"Start":"15:41.465 ","End":"15:45.930","Text":"So we\u0027re all right and we are done."}],"ID":8760},{"Watched":false,"Name":"Exercise 3","Duration":"6m 7s","ChapterTopicVideoID":8598,"CourseChapterTopicPlaylistID":4965,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.465","Text":"In this exercise, we have to compute the following type 2 line integral,"},{"Start":"00:06.465 ","End":"00:09.705","Text":"but it doesn\u0027t quite make sense."},{"Start":"00:09.705 ","End":"00:13.440","Text":"We\u0027re asked to compute this from this point to this point,"},{"Start":"00:13.440 ","End":"00:18.075","Text":"but it doesn\u0027t give us the path of how to get from this point to this point,"},{"Start":"00:18.075 ","End":"00:20.760","Text":"and it could depend on the path."},{"Start":"00:20.760 ","End":"00:25.185","Text":"The idea is to first show that this is path independent,"},{"Start":"00:25.185 ","End":"00:26.955","Text":"and so it makes sense."},{"Start":"00:26.955 ","End":"00:29.100","Text":"Let\u0027s do that first."},{"Start":"00:29.100 ","End":"00:34.455","Text":"If we let this part be P of xy,"},{"Start":"00:34.455 ","End":"00:41.420","Text":"this bit here, and this bit here we\u0027ll call it Q or Q of xy."},{"Start":"00:41.420 ","End":"00:45.500","Text":"If you recall, there\u0027s a condition that says that if the derivative of P with"},{"Start":"00:45.500 ","End":"00:50.495","Text":"respect to y equals the derivative of Q with respect to x,"},{"Start":"00:50.495 ","End":"00:56.970","Text":"then this integral is path independent."},{"Start":"00:56.970 ","End":"01:00.360","Text":"Let\u0027s show this first. Let\u0027s see."},{"Start":"01:00.360 ","End":"01:04.065","Text":"P with respect to y. X is a constant,"},{"Start":"01:04.065 ","End":"01:08.340","Text":"so it\u0027s 2x times 3y squared,"},{"Start":"01:08.340 ","End":"01:15.180","Text":"altogether 6xy squared, and Q with respect to x,"},{"Start":"01:15.180 ","End":"01:16.995","Text":"1 gives me nothing,"},{"Start":"01:16.995 ","End":"01:19.200","Text":"y squared is a constant,"},{"Start":"01:19.200 ","End":"01:23.320","Text":"so I get 6xy squared."},{"Start":"01:23.320 ","End":"01:25.425","Text":"These 2 are indeed equal."},{"Start":"01:25.425 ","End":"01:26.860","Text":"Up to now, I was checking,"},{"Start":"01:26.860 ","End":"01:29.900","Text":"is this equal? Yes, it is."},{"Start":"01:30.230 ","End":"01:32.670","Text":"We have path independence,"},{"Start":"01:32.670 ","End":"01:33.925","Text":"but more than that,"},{"Start":"01:33.925 ","End":"01:35.800","Text":"when we have this path independence,"},{"Start":"01:35.800 ","End":"01:43.195","Text":"we also know that there\u0027s a potential function Phi of x and y,"},{"Start":"01:43.195 ","End":"01:48.570","Text":"such that the derivative of Phi with respect to x is"},{"Start":"01:48.570 ","End":"01:56.160","Text":"P and the derivative of Phi with respect to y is Q."},{"Start":"01:56.160 ","End":"02:02.360","Text":"More than that, that we can evaluate the line integral by taking this Phi,"},{"Start":"02:02.360 ","End":"02:06.915","Text":"the potential function, let\u0027s write the word it\u0027s called a potential function,"},{"Start":"02:06.915 ","End":"02:09.350","Text":"and instead of taking the line integral,"},{"Start":"02:09.350 ","End":"02:16.990","Text":"we can say that the line integral of Pdx plus Qdy,"},{"Start":"02:16.990 ","End":"02:20.655","Text":"just call this point A and B in general,"},{"Start":"02:20.655 ","End":"02:23.370","Text":"this is my A, this is my B,"},{"Start":"02:23.370 ","End":"02:30.855","Text":"will equal Phi at the point B minus Phi at the point A."},{"Start":"02:30.855 ","End":"02:33.380","Text":"This is the method I\u0027m going to use."},{"Start":"02:33.380 ","End":"02:36.470","Text":"It is possible to also just choose some path or"},{"Start":"02:36.470 ","End":"02:40.210","Text":"combinations of paths to go from here to here."},{"Start":"02:40.210 ","End":"02:43.190","Text":"There was the previous exercise where we compute it both ways,"},{"Start":"02:43.190 ","End":"02:45.650","Text":"using the potential function and using the path,"},{"Start":"02:45.650 ","End":"02:47.125","Text":"we got the same answer."},{"Start":"02:47.125 ","End":"02:51.320","Text":"I prefer to use the method of the potential function Phi,"},{"Start":"02:51.320 ","End":"02:53.615","Text":"and that\u0027s our first task is to find it."},{"Start":"02:53.615 ","End":"02:55.880","Text":"Well, we have 2 equations here."},{"Start":"02:55.880 ","End":"03:04.050","Text":"Phi_x or partial derivative of Phi with respect to x is P. I\u0027ll just write that,"},{"Start":"03:04.050 ","End":"03:08.850","Text":"that Phi with respect to x is 2xy cubed,"},{"Start":"03:08.850 ","End":"03:18.280","Text":"and that will give us the Phi is just the integral of 2xy cubed with respect to x."},{"Start":"03:18.620 ","End":"03:21.265","Text":"This is equal to,"},{"Start":"03:21.265 ","End":"03:24.390","Text":"remember that y is a constant, so with respect to x,"},{"Start":"03:24.390 ","End":"03:26.400","Text":"2x gives us x squared,"},{"Start":"03:26.400 ","End":"03:29.950","Text":"so we\u0027ve got x squared, y cubed."},{"Start":"03:29.950 ","End":"03:31.510","Text":"But that\u0027s not all,"},{"Start":"03:31.510 ","End":"03:34.255","Text":"normally we would just write plus a constant."},{"Start":"03:34.255 ","End":"03:36.700","Text":"But because this is a partial derivative,"},{"Start":"03:36.700 ","End":"03:38.290","Text":"this with respect to x,"},{"Start":"03:38.290 ","End":"03:43.185","Text":"a constant is any function of just y,"},{"Start":"03:43.185 ","End":"03:46.580","Text":"then that\u0027s like a constant as far as x goes."},{"Start":"03:46.580 ","End":"03:49.190","Text":"But we need to know what this function is,"},{"Start":"03:49.190 ","End":"03:54.035","Text":"and that\u0027s where we\u0027ll use the second equation that Phi with respect to y is Q."},{"Start":"03:54.035 ","End":"04:00.030","Text":"Let\u0027s see, Q is 1 plus 3x squared y squared,"},{"Start":"04:00.030 ","End":"04:02.460","Text":"I started with this side,"},{"Start":"04:02.460 ","End":"04:04.795","Text":"and Phi with respect to y,"},{"Start":"04:04.795 ","End":"04:07.790","Text":"I just differentiate this with respect to y."},{"Start":"04:07.790 ","End":"04:15.255","Text":"The first part gives us 3x squared y squared plus,"},{"Start":"04:15.255 ","End":"04:18.245","Text":"and then C prime of y."},{"Start":"04:18.245 ","End":"04:21.995","Text":"Now this bit\u0027s the same on both sides,"},{"Start":"04:21.995 ","End":"04:25.655","Text":"and so I just get C prime of y is 1,"},{"Start":"04:25.655 ","End":"04:30.600","Text":"which means that C of y is the integral of 1dy,"},{"Start":"04:30.610 ","End":"04:33.845","Text":"which is just equal to y,"},{"Start":"04:33.845 ","End":"04:36.110","Text":"and this time plus an actual constant,"},{"Start":"04:36.110 ","End":"04:40.834","Text":"I\u0027ll call it K. Now if I put this in here,"},{"Start":"04:40.834 ","End":"04:43.520","Text":"then I have my potential function,"},{"Start":"04:43.520 ","End":"04:50.465","Text":"Phi of xy is equal to x squared y cubed"},{"Start":"04:50.465 ","End":"04:58.765","Text":"plus y plus K. I don\u0027t really need the K because we\u0027re going to be subtracting,"},{"Start":"04:58.765 ","End":"05:01.260","Text":"so let\u0027s just evaluate this."},{"Start":"05:01.260 ","End":"05:07.640","Text":"Our line integral, what we need is Phi of the point B, which is 3,"},{"Start":"05:07.640 ","End":"05:13.515","Text":"1 minus Phi at the lower limit,"},{"Start":"05:13.515 ","End":"05:15.570","Text":"which is 1, 4,"},{"Start":"05:15.570 ","End":"05:16.890","Text":"and this will be our answer."},{"Start":"05:16.890 ","End":"05:21.435","Text":"Let\u0027s see. Substituting here 3,"},{"Start":"05:21.435 ","End":"05:23.940","Text":"1 we have 3 squared,"},{"Start":"05:23.940 ","End":"05:27.645","Text":"1 cubed plus 1,"},{"Start":"05:27.645 ","End":"05:30.075","Text":"we don\u0027t need the K because it would cancel,"},{"Start":"05:30.075 ","End":"05:32.610","Text":"minus and then I\u0027ll put in 1,"},{"Start":"05:32.610 ","End":"05:35.375","Text":"4, so I\u0027ve got 1 squared,"},{"Start":"05:35.375 ","End":"05:39.550","Text":"4 cubed plus 4."},{"Start":"05:39.550 ","End":"05:41.970","Text":"Let\u0027s see, this equals."},{"Start":"05:41.970 ","End":"05:47.090","Text":"The first bit is just 3 squared is 9 times 1 is 9 plus"},{"Start":"05:47.090 ","End":"05:53.000","Text":"1 is 10 minus here I have 4 cubed is 64,"},{"Start":"05:53.000 ","End":"05:56.820","Text":"plus 4 is 68."},{"Start":"05:57.380 ","End":"06:03.210","Text":"Altogether that would be minus 58,"},{"Start":"06:03.210 ","End":"06:07.060","Text":"and that is our answer, we\u0027re done."}],"ID":8761},{"Watched":false,"Name":"Exercise 4","Duration":"7m 39s","ChapterTopicVideoID":8599,"CourseChapterTopicPlaylistID":4965,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.470","Text":"In this exercise, we\u0027re given this line integral to compute."},{"Start":"00:04.470 ","End":"00:08.880","Text":"We\u0027re given a start point,1,0 and an end point, 2,1."},{"Start":"00:08.880 ","End":"00:12.615","Text":"But we\u0027re not given the path of how to get from here to here."},{"Start":"00:12.615 ","End":"00:18.135","Text":"So that leads us to suspect that maybe this is path independent."},{"Start":"00:18.135 ","End":"00:21.585","Text":"Then it makes sense and we can choose any path."},{"Start":"00:21.585 ","End":"00:28.800","Text":"There is an easy condition when we have these continuous differentiable functions."},{"Start":"00:28.940 ","End":"00:34.320","Text":"If we call this function of x and y,"},{"Start":"00:34.320 ","End":"00:39.195","Text":"P, and we call this function of x and y, Q,"},{"Start":"00:39.195 ","End":"00:41.900","Text":"then we know that a necessary and sufficient condition for"},{"Start":"00:41.900 ","End":"00:44.540","Text":"path independence is that P with"},{"Start":"00:44.540 ","End":"00:46.985","Text":"respect to y partial derivative"},{"Start":"00:46.985 ","End":"00:50.585","Text":"is equal to the partial derivative of Q with respect to x."},{"Start":"00:50.585 ","End":"00:53.590","Text":"Let\u0027s check if this is the case."},{"Start":"00:53.590 ","End":"00:57.795","Text":"P with respect to y is equal to,"},{"Start":"00:57.795 ","End":"00:59.295","Text":"x is a constant,"},{"Start":"00:59.295 ","End":"01:04.270","Text":"from here we get just the 2x"},{"Start":"01:04.270 ","End":"01:11.060","Text":"and with respect to y here we get 4y cubed, and here nothing."},{"Start":"01:11.060 ","End":"01:15.915","Text":"On the other hand, Q with respect to x is equal to,"},{"Start":"01:15.915 ","End":"01:20.775","Text":"from here we get 2x and y is a constant."},{"Start":"01:20.775 ","End":"01:27.480","Text":"We just get the 4 with the y cubed, minus 4y cubed."},{"Start":"01:27.480 ","End":"01:31.350","Text":"These are indeed equal, you can see."},{"Start":"01:31.350 ","End":"01:33.150","Text":"The answer to this is yes,"},{"Start":"01:33.150 ","End":"01:36.630","Text":"and so we have path independence."},{"Start":"01:36.630 ","End":"01:40.205","Text":"One way of computing this line integral,"},{"Start":"01:40.205 ","End":"01:43.400","Text":"there is another we\u0027ve done previously using the potential function,"},{"Start":"01:43.400 ","End":"01:47.645","Text":"but here I want to actually find a path from here to here."},{"Start":"01:47.645 ","End":"01:49.945","Text":"Let\u0027s make a quick sketch."},{"Start":"01:49.945 ","End":"01:54.550","Text":"Let\u0027s say that this is the point 1,0"},{"Start":"01:55.160 ","End":"02:02.095","Text":"and that this is the point 2,1."},{"Start":"02:02.095 ","End":"02:07.140","Text":"Then we could choose a straight line path from here to here."},{"Start":"02:07.540 ","End":"02:12.845","Text":"It comes out nicer if you do horizontal and vertical,"},{"Start":"02:12.845 ","End":"02:14.225","Text":"either this way and this way,"},{"Start":"02:14.225 ","End":"02:16.355","Text":"or you could have gone up and then across."},{"Start":"02:16.355 ","End":"02:21.740","Text":"Let\u0027s do it this way and find this middle point and go from here to here."},{"Start":"02:21.740 ","End":"02:23.540","Text":"Now this point here,"},{"Start":"02:23.540 ","End":"02:26.600","Text":"since we went to cross, is going to have the same y as this."},{"Start":"02:26.600 ","End":"02:29.000","Text":"So the y will be 0."},{"Start":"02:29.000 ","End":"02:30.620","Text":"Since we\u0027re here, we\u0027re going up,"},{"Start":"02:30.620 ","End":"02:31.790","Text":"it has the same axis."},{"Start":"02:31.790 ","End":"02:34.985","Text":"This is going to be the point 2,0."},{"Start":"02:34.985 ","End":"02:38.330","Text":"If I call this whole path C,"},{"Start":"02:38.330 ","End":"02:42.830","Text":"then C is going to mainly made up of 2 bits, C_1 plus C_2."},{"Start":"02:42.830 ","End":"02:44.450","Text":"If this bit is C_1,"},{"Start":"02:44.450 ","End":"02:48.350","Text":"this is C_2, the horizontal, the vertical."},{"Start":"02:48.350 ","End":"02:54.860","Text":"We want to just add the integral along this path plus the integral along this path."},{"Start":"02:54.860 ","End":"02:57.695","Text":"Let\u0027s start by parametrizing each one,"},{"Start":"02:57.695 ","End":"03:01.940","Text":"C_1, let\u0027s see x equals y equals."},{"Start":"03:01.940 ","End":"03:08.115","Text":"There is a standard formula for path between 2 points."},{"Start":"03:08.115 ","End":"03:10.430","Text":"When it\u0027s horizontal or vertical,"},{"Start":"03:10.430 ","End":"03:13.205","Text":"it\u0027s usually simpler to do without the formula."},{"Start":"03:13.205 ","End":"03:17.310","Text":"Here, x goes from 1-2,"},{"Start":"03:17.310 ","End":"03:19.830","Text":"y stays 0 I\u0027ll start with that,"},{"Start":"03:19.830 ","End":"03:21.720","Text":"and x just goes from 1-2."},{"Start":"03:21.720 ","End":"03:25.590","Text":"I\u0027ll say x is t and t goes from 1-2."},{"Start":"03:25.590 ","End":"03:28.680","Text":"That\u0027s the parametrization for C_1."},{"Start":"03:28.680 ","End":"03:34.865","Text":"For C_2, in this case the x remains constant and the y changes."},{"Start":"03:34.865 ","End":"03:38.365","Text":"Here we have that x is equal to 2,"},{"Start":"03:38.365 ","End":"03:40.995","Text":"but y goes from 0-2."},{"Start":"03:40.995 ","End":"03:47.400","Text":"Instead of saying that, we say y equals t and t goes from 0-1."},{"Start":"03:47.400 ","End":"03:49.875","Text":"That\u0027s the parametrizations."},{"Start":"03:49.875 ","End":"03:54.615","Text":"I know we\u0027ll need dx and dy for each of them."},{"Start":"03:54.615 ","End":"04:02.910","Text":"Here, dx equals dt and dy equals 0 dt, or just 0."},{"Start":"04:02.910 ","End":"04:05.070","Text":"But here x is the constant,"},{"Start":"04:05.070 ","End":"04:10.080","Text":"so dx here is equal to 0 and y equals t."},{"Start":"04:10.080 ","End":"04:15.210","Text":"Here dy equals dt."},{"Start":"04:15.210 ","End":"04:18.795","Text":"Getting a bit crowded and maybe I\u0027ll just put a separator here."},{"Start":"04:18.795 ","End":"04:26.360","Text":"Just to remind you, the integral along the path C from here to here is just"},{"Start":"04:26.360 ","End":"04:33.745","Text":"the integral along C_1 plus the integral along C_2 of whatever it is; Pdx plus Qdy."},{"Start":"04:33.745 ","End":"04:35.530","Text":"Let\u0027s do this one first,"},{"Start":"04:35.530 ","End":"04:39.395","Text":"the C_1 part will give us the integral."},{"Start":"04:39.395 ","End":"04:42.095","Text":"Now we just use the parameter,"},{"Start":"04:42.095 ","End":"04:45.275","Text":"the parameter t goes from 1-2."},{"Start":"04:45.275 ","End":"04:55.060","Text":"From 1-2 and we replace x by t and y by 0 in here."},{"Start":"04:55.730 ","End":"05:00.740","Text":"If we replace x by t and y by 0,"},{"Start":"05:00.740 ","End":"05:04.685","Text":"we get, I\u0027ll write it fully first of all,"},{"Start":"05:04.685 ","End":"05:13.400","Text":"2 times t times 0 minus 0^4 plus 3,"},{"Start":"05:13.400 ","End":"05:18.180","Text":"dx is equal to dt,"},{"Start":"05:18.860 ","End":"05:22.035","Text":"and here dy is 0."},{"Start":"05:22.035 ","End":"05:26.780","Text":"It\u0027s just 0 I won\u0027t even do it at all,"},{"Start":"05:26.780 ","End":"05:29.790","Text":"because dy here is 0."},{"Start":"05:32.410 ","End":"05:36.215","Text":"Altogether we only have 3 here."},{"Start":"05:36.215 ","End":"05:42.000","Text":"We have the integral from 1-2 of 3dt,"},{"Start":"05:42.890 ","End":"05:47.175","Text":"and this comes out to be 3t."},{"Start":"05:47.175 ","End":"05:49.080","Text":"Plug in 2 is 6,"},{"Start":"05:49.080 ","End":"05:51.240","Text":"plug in 1 is 3, all together,"},{"Start":"05:51.240 ","End":"05:54.555","Text":"we\u0027ve got 6 minus 3, it\u0027s just 3."},{"Start":"05:54.555 ","End":"05:57.630","Text":"Now let\u0027s get on to C_2."},{"Start":"05:57.630 ","End":"06:00.810","Text":"In C_2, we have the integral."},{"Start":"06:00.810 ","End":"06:06.450","Text":"This time it\u0027s from 0-1 and we"},{"Start":"06:06.450 ","End":"06:12.690","Text":"replace x equals 2 and y equals t. But dx is 0,"},{"Start":"06:12.690 ","End":"06:17.505","Text":"so the first bit is 0 for the dx bit,"},{"Start":"06:17.505 ","End":"06:23.340","Text":"and then here x squared is that 2 squared"},{"Start":"06:23.340 ","End":"06:31.260","Text":"minus 4"},{"Start":"06:31.260 ","End":"06:33.975","Text":"times 2 times t cubed"},{"Start":"06:33.975 ","End":"06:35.790","Text":"and this is dy,"},{"Start":"06:35.790 ","End":"06:44.365","Text":"which is dt, and altogether this is equal to, let\u0027s see,"},{"Start":"06:44.365 ","End":"06:48.500","Text":"what we have is the integral from 0-1 of"},{"Start":"06:48.500 ","End":"06:56.580","Text":"4 minus 8t cubed dt."},{"Start":"06:57.620 ","End":"07:01.905","Text":"This is equal to 40."},{"Start":"07:01.905 ","End":"07:04.560","Text":"Now here, t^4/4."},{"Start":"07:04.560 ","End":"07:12.630","Text":"I\u0027m just left with 8/4 is 2t^4 from 0-1."},{"Start":"07:12.630 ","End":"07:14.570","Text":"At 0, I get nothing."},{"Start":"07:14.570 ","End":"07:22.435","Text":"At 1, I get 4 minus 2, which is 2."},{"Start":"07:22.435 ","End":"07:25.415","Text":"We have the integral over C_1 here,"},{"Start":"07:25.415 ","End":"07:27.905","Text":"we have the integral over C_2 here."},{"Start":"07:27.905 ","End":"07:29.989","Text":"I want to add these two together,"},{"Start":"07:29.989 ","End":"07:33.905","Text":"so I get 3 plus 2,"},{"Start":"07:33.905 ","End":"07:39.689","Text":"which equals 5, and that is our answer."}],"ID":8762},{"Watched":false,"Name":"Exercise 5","Duration":"6m 20s","ChapterTopicVideoID":8600,"CourseChapterTopicPlaylistID":4965,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.885","Text":"In this exercise, it\u0027s one of those from physics involving work and force fields."},{"Start":"00:06.885 ","End":"00:10.635","Text":"We are given the force field F,"},{"Start":"00:10.635 ","End":"00:13.995","Text":"we\u0027re given it in i j terms."},{"Start":"00:13.995 ","End":"00:18.380","Text":"E^y in the first component, xe^y in the second component."},{"Start":"00:18.380 ","End":"00:24.210","Text":"We have a particle which moves from this point to this point along this curve."},{"Start":"00:24.210 ","End":"00:26.040","Text":"We have to, as I say,"},{"Start":"00:26.040 ","End":"00:28.290","Text":"compute the work done by,"},{"Start":"00:28.290 ","End":"00:30.810","Text":"first of all, a sketch."},{"Start":"00:30.810 ","End":"00:34.260","Text":"Here\u0027s the point 1,0,"},{"Start":"00:34.260 ","End":"00:37.630","Text":"here is the point minus 1,0."},{"Start":"00:38.050 ","End":"00:42.710","Text":"This if we rewrite it as part of x-squared plus y-squared"},{"Start":"00:42.710 ","End":"00:46.910","Text":"equals 1 and it\u0027s the upper semicircle,"},{"Start":"00:46.910 ","End":"00:50.405","Text":"y equals square root of 1 minus x squared."},{"Start":"00:50.405 ","End":"00:53.540","Text":"The path that we want is along here,"},{"Start":"00:53.540 ","End":"00:56.255","Text":"and let\u0027s call this path c."},{"Start":"00:56.255 ","End":"01:03.260","Text":"The formula from physics that the work done is just the type 2 line integral,"},{"Start":"01:03.260 ","End":"01:09.420","Text":"the integral along c of the force field F.dr."},{"Start":"01:10.630 ","End":"01:19.055","Text":"Now dr is just equal to dxdy,"},{"Start":"01:19.055 ","End":"01:24.020","Text":"or dxi plus dyj,"},{"Start":"01:24.020 ","End":"01:30.360","Text":"where r is just a parametrization of the curve."},{"Start":"01:30.550 ","End":"01:35.240","Text":"Now the thing is that this comes out to be quite a difficult"},{"Start":"01:35.240 ","End":"01:39.500","Text":"integral to compute and so here\u0027s the plan."},{"Start":"01:39.500 ","End":"01:44.570","Text":"If we show that this force field is conservative,"},{"Start":"01:44.570 ","End":"01:49.190","Text":"then the line integral is going to be path independent."},{"Start":"01:49.190 ","End":"01:52.640","Text":"Then we can choose a nice a path from here to here, for example,"},{"Start":"01:52.640 ","End":"01:55.955","Text":"the straight line path and that will be easier to compute."},{"Start":"01:55.955 ","End":"01:57.710","Text":"That\u0027s the strategy."},{"Start":"01:57.710 ","End":"02:02.065","Text":"Now let\u0027s show that F is conservative."},{"Start":"02:02.065 ","End":"02:06.740","Text":"This would just mean to show that P with"},{"Start":"02:06.740 ","End":"02:11.840","Text":"respect to y partial derivative is equal to q with respect to x."},{"Start":"02:11.840 ","End":"02:13.295","Text":"If we show this,"},{"Start":"02:13.295 ","End":"02:15.830","Text":"then F is conservative and like I said,"},{"Start":"02:15.830 ","End":"02:18.665","Text":"that the integral becomes path independent."},{"Start":"02:18.665 ","End":"02:21.590","Text":"Let\u0027s check P with respect to y,"},{"Start":"02:21.590 ","End":"02:26.630","Text":"it\u0027s just e^y and q with respect"},{"Start":"02:26.630 ","End":"02:31.940","Text":"to x. X is the variable,"},{"Start":"02:31.940 ","End":"02:33.320","Text":"so e^y is a constant,"},{"Start":"02:33.320 ","End":"02:35.810","Text":"so it\u0027s just also equal to e^y."},{"Start":"02:35.810 ","End":"02:37.835","Text":"These are equal, yes,"},{"Start":"02:37.835 ","End":"02:41.390","Text":"conservative vector field path independence."},{"Start":"02:41.390 ","End":"02:43.910","Text":"Instead of this path c,"},{"Start":"02:43.910 ","End":"02:45.905","Text":"I\u0027ll choose a variation,"},{"Start":"02:45.905 ","End":"02:49.740","Text":"will choose the path along here."},{"Start":"02:50.270 ","End":"02:58.050","Text":"That this be the C. I\u0027ll call"},{"Start":"02:58.050 ","End":"03:05.920","Text":"this one C old and this now I\u0027ll take as the path C is a straight line from here to here."},{"Start":"03:05.920 ","End":"03:11.710","Text":"There are several formulas to parameterize a line segment."},{"Start":"03:11.710 ","End":"03:18.595","Text":"1 of them is to say that for each x and y,"},{"Start":"03:18.595 ","End":"03:20.860","Text":"we take the start point,"},{"Start":"03:20.860 ","End":"03:22.600","Text":"in this case, 1."},{"Start":"03:22.600 ","End":"03:30.384","Text":"In this case, 0 plus t times the end minus the start."},{"Start":"03:30.384 ","End":"03:35.485","Text":"In this case, for x we have minus 1 minus 1,"},{"Start":"03:35.485 ","End":"03:44.870","Text":"and in this case it\u0027s 0 minus 0 and t always goes from 0 to 1 if I use this formula."},{"Start":"03:45.050 ","End":"03:48.015","Text":"I\u0027ll just rewrite this."},{"Start":"03:48.015 ","End":"03:53.215","Text":"We get that x equals 1 minus 2t."},{"Start":"03:53.215 ","End":"03:55.880","Text":"Here I get y equals 0,"},{"Start":"03:55.880 ","End":"04:03.890","Text":"which is pretty clear because we\u0027re going along the x-axis and t goes from 0-1."},{"Start":"04:03.890 ","End":"04:12.130","Text":"We\u0027ll also need dx and dy."},{"Start":"04:13.010 ","End":"04:15.900","Text":"Let\u0027s go back here a moment."},{"Start":"04:15.900 ","End":"04:20.110","Text":"This w, which is F.dr,"},{"Start":"04:20.840 ","End":"04:24.060","Text":"is the PQ components."},{"Start":"04:24.060 ","End":"04:27.620","Text":"Here we have the dx,dy components."},{"Start":"04:27.620 ","End":"04:35.120","Text":"It just comes out to be Pdx"},{"Start":"04:35.120 ","End":"04:44.275","Text":"plus Qdy along the path c and this is equal to,"},{"Start":"04:44.275 ","End":"04:46.995","Text":"I\u0027ll just write what P and Q are."},{"Start":"04:46.995 ","End":"04:56.400","Text":"It\u0027s e^ydx plus xe^ydy."},{"Start":"04:56.400 ","End":"04:59.220","Text":"Now remember, the c we\u0027re taking is this one,"},{"Start":"04:59.220 ","End":"05:03.830","Text":"the straight line and it\u0027s parametrized here."},{"Start":"05:03.830 ","End":"05:11.135","Text":"This is the parametrization for the new C. Like I said, we need dx and dy."},{"Start":"05:11.135 ","End":"05:17.520","Text":"From here we get that dx is minus 2 dt."},{"Start":"05:18.070 ","End":"05:22.710","Text":"But dy is equal to 0 dt,"},{"Start":"05:22.710 ","End":"05:25.110","Text":"which is just 0."},{"Start":"05:25.110 ","End":"05:29.165","Text":"Actually this part will drop out."},{"Start":"05:29.165 ","End":"05:33.235","Text":"Just put a line through it because dy is 0."},{"Start":"05:33.235 ","End":"05:41.380","Text":"What we\u0027re left with is the integral and the parameters from 0 to 1 at t,"},{"Start":"05:41.380 ","End":"05:43.975","Text":"and then we have e^y."},{"Start":"05:43.975 ","End":"05:47.380","Text":"Well, y is 0, e^0,"},{"Start":"05:47.380 ","End":"05:59.140","Text":"and dx is minus 2dt."},{"Start":"05:59.140 ","End":"06:01.475","Text":"Minus 2 I can pull out in front."},{"Start":"06:01.475 ","End":"06:06.350","Text":"The integral from 0 to 1, e^0 is 1dt."},{"Start":"06:06.350 ","End":"06:10.610","Text":"Well, the integral of 1 is just the upper minus the lower."},{"Start":"06:10.610 ","End":"06:12.635","Text":"It\u0027s 1 minus 0 is 1."},{"Start":"06:12.635 ","End":"06:16.800","Text":"Altogether we get minus 2."},{"Start":"06:17.140 ","End":"06:20.250","Text":"That\u0027s the answer."}],"ID":8763},{"Watched":false,"Name":"Exercise 6","Duration":"13m 46s","ChapterTopicVideoID":8601,"CourseChapterTopicPlaylistID":4965,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.240","Text":"In this exercise we have to compute the following type 2 line integral that\u0027s in 3D."},{"Start":"00:07.520 ","End":"00:10.620","Text":"There appears at first to be something missing."},{"Start":"00:10.620 ","End":"00:14.830","Text":"We\u0027re told from where to start and where to end,"},{"Start":"00:14.830 ","End":"00:18.690","Text":"but we\u0027re not given the actual path or curve to get from here to here."},{"Start":"00:18.690 ","End":"00:21.720","Text":"This leads us to suspect that perhaps this could be 1 of"},{"Start":"00:21.720 ","End":"00:24.030","Text":"those path independent ones and then it"},{"Start":"00:24.030 ","End":"00:26.610","Text":"doesn\u0027t matter how we get from this point to this point."},{"Start":"00:26.610 ","End":"00:29.320","Text":"Can we just introduce some notation,"},{"Start":"00:29.320 ","End":"00:31.455","Text":"what we have is the integral."},{"Start":"00:31.455 ","End":"00:35.250","Text":"You know what, I\u0027ll call the top point B and"},{"Start":"00:35.250 ","End":"00:39.675","Text":"the lower point A. I\u0027ll mark them also here, A and B."},{"Start":"00:39.675 ","End":"00:44.195","Text":"Now give these names or let this 1 be P,"},{"Start":"00:44.195 ","End":"00:49.410","Text":"Q and R. It\u0027s actually P of x, y, z,"},{"Start":"00:49.410 ","End":"00:58.270","Text":"but for short I\u0027ll just call it P dx plus Q dy, plus R dz."},{"Start":"01:00.140 ","End":"01:09.820","Text":"There\u0027s a way of telling that this is line independent if the following 3 equations hold,"},{"Start":"01:09.820 ","End":"01:12.770","Text":"the partial derivative of P with respect to"},{"Start":"01:12.770 ","End":"01:16.970","Text":"y equals partial derivative of Q with respect to x."},{"Start":"01:16.970 ","End":"01:19.070","Text":"We don\u0027t know this, we\u0027re going to check."},{"Start":"01:19.070 ","End":"01:21.215","Text":"In 2D that\u0027s all there is."},{"Start":"01:21.215 ","End":"01:23.480","Text":"But in 3D there\u0027s 3 checks."},{"Start":"01:23.480 ","End":"01:25.130","Text":"It\u0027s not even hard to remember,"},{"Start":"01:25.130 ","End":"01:26.330","Text":"you just take 2 at a time."},{"Start":"01:26.330 ","End":"01:32.050","Text":"Let\u0027s say, I take next P and R. So I write P and I write R. I"},{"Start":"01:32.050 ","End":"01:39.105","Text":"take P in the opposition it\u0027s with respect to z and R gets the x."},{"Start":"01:39.105 ","End":"01:42.270","Text":"The third 1 we take Q and R,"},{"Start":"01:42.270 ","End":"01:50.144","Text":"and the Q is with respect to z and the R I take with respect to y."},{"Start":"01:50.144 ","End":"01:53.575","Text":"If all these 3 turn out to be,"},{"Start":"01:53.575 ","End":"02:00.865","Text":"then we have path independence of the integral and it tells us other things too."},{"Start":"02:00.865 ","End":"02:04.860","Text":"But we\u0027ll get to that. Let\u0027s first of all do the check."},{"Start":"02:05.170 ","End":"02:11.660","Text":"P with respect to y is, let\u0027s see,"},{"Start":"02:11.660 ","End":"02:18.215","Text":"this is 0 because there\u0027s no y here and this is just 6."},{"Start":"02:18.215 ","End":"02:22.100","Text":"On the other hand, Q with respect to x,"},{"Start":"02:22.100 ","End":"02:23.820","Text":"there\u0027s no x here so that\u0027s 0,"},{"Start":"02:23.820 ","End":"02:27.335","Text":"it\u0027s also just 6. That\u0027s okay."},{"Start":"02:27.335 ","End":"02:31.370","Text":"Next, P with respect to z. I don\u0027t have anything"},{"Start":"02:31.370 ","End":"02:35.310","Text":"from the second term but from the first I get 2"},{"Start":"02:35.310 ","End":"02:40.025","Text":"times 3 is 6xz"},{"Start":"02:40.025 ","End":"02:46.440","Text":"squared and R with respect to x, here nothing."},{"Start":"02:46.440 ","End":"02:53.745","Text":"Here I get also 2 times 3 is 6xz squared, same thing."},{"Start":"02:53.745 ","End":"02:56.984","Text":"Q with respect to z."},{"Start":"02:56.984 ","End":"03:02.175","Text":"Here with respect to z I get minus 2y,"},{"Start":"03:02.175 ","End":"03:04.380","Text":"and R with respect to y,"},{"Start":"03:04.380 ","End":"03:06.780","Text":"this is nothing., form here minus 2y."},{"Start":"03:06.780 ","End":"03:09.730","Text":"All 3 of them check out."},{"Start":"03:10.340 ","End":"03:15.290","Text":"We have path independence and we could choose a path from here to here,"},{"Start":"03:15.290 ","End":"03:17.705","Text":"but there\u0027s another method that we learned."},{"Start":"03:17.705 ","End":"03:22.550","Text":"When we have the path independence we also have a vector field that\u0027s conservative."},{"Start":"03:22.550 ","End":"03:29.155","Text":"If I take P times I plus Q times J plus R times K,"},{"Start":"03:29.155 ","End":"03:31.760","Text":"it\u0027s a conservative vector field."},{"Start":"03:31.760 ","End":"03:35.330","Text":"To cut the long story short what it tells us is that there is"},{"Start":"03:35.330 ","End":"03:42.360","Text":"a potential function Phi of x, y and z."},{"Start":"03:42.360 ","End":"03:46.340","Text":"Path independence is pretty much the same as being conservative"},{"Start":"03:46.340 ","End":"03:50.510","Text":"and there being a potential function, almost the same."},{"Start":"03:50.510 ","End":"03:52.970","Text":"The conditions here do hold."},{"Start":"03:52.970 ","End":"04:00.065","Text":"We know that there\u0027s such a function such that the derivative with respect to x is P,"},{"Start":"04:00.065 ","End":"04:04.475","Text":"the derivative of it with respect to y is Q,"},{"Start":"04:04.475 ","End":"04:07.850","Text":"and the derivative of Phi with respect to z is"},{"Start":"04:07.850 ","End":"04:14.340","Text":"R. Using these 3 equations we can find Phi and then the"},{"Start":"04:14.340 ","End":"04:20.850","Text":"integral from A to B of whatever it"},{"Start":"04:20.850 ","End":"04:30.465","Text":"is will just equal Phi at the point B minus Phi at the point A."},{"Start":"04:30.465 ","End":"04:34.085","Text":"All we have to do now is figure out what this function is."},{"Start":"04:34.085 ","End":"04:37.025","Text":"We\u0027ll take each of these 3 equations 1 at a time."},{"Start":"04:37.025 ","End":"04:45.620","Text":"The first 1, Phi with respect to x equals P. Let\u0027s see, where is P?"},{"Start":"04:45.620 ","End":"04:52.640","Text":"I\u0027ll just write it another way."},{"Start":"04:52.640 ","End":"05:00.815","Text":"Phi is going to equal the integral with respect to x of Pdx,"},{"Start":"05:00.815 ","End":"05:03.590","Text":"and I\u0027ll just replace P by what it\u0027s equal to."},{"Start":"05:03.590 ","End":"05:05.585","Text":"Well, I\u0027ll just wright first, Pdx,"},{"Start":"05:05.585 ","End":"05:15.290","Text":"and now P is 2xz cubed plus 6y dx and this equals,"},{"Start":"05:15.290 ","End":"05:18.560","Text":"remember y and z are constants as far as x goes,"},{"Start":"05:18.560 ","End":"05:26.455","Text":"so from here I just get x squared and z cubed remains."},{"Start":"05:26.455 ","End":"05:29.590","Text":"From the 6y I get 6yx,"},{"Start":"05:30.770 ","End":"05:36.590","Text":"and then I also get a constant as far as x goes,"},{"Start":"05:36.590 ","End":"05:41.870","Text":"which means some function that only involves y and z."},{"Start":"05:41.870 ","End":"05:45.875","Text":"Not a constant, it\u0027s only constant as far as x goes."},{"Start":"05:45.875 ","End":"05:47.930","Text":"That\u0027s what we got from this."},{"Start":"05:47.930 ","End":"05:50.360","Text":"Now, let\u0027s use the second equation."},{"Start":"05:50.360 ","End":"05:52.900","Text":"Phi with respect to y."},{"Start":"05:52.900 ","End":"06:00.855","Text":"Here\u0027s Phi, and with respect to y this will give me nothing,"},{"Start":"06:00.855 ","End":"06:07.650","Text":"0 plus this with respect to y is 6x,"},{"Start":"06:07.650 ","End":"06:14.285","Text":"and this with respect to y is partial derivative of z with respect to y of y, z."},{"Start":"06:14.285 ","End":"06:18.380","Text":"All this is equal to Q. Q is here,"},{"Start":"06:18.380 ","End":"06:22.010","Text":"which is 6x minus 2yz."},{"Start":"06:22.010 ","End":"06:28.845","Text":"Now the 6x, and we don\u0027t need the 0,"},{"Start":"06:28.845 ","End":"06:38.110","Text":"cancels out, so we have the partial derivative of C with respect to y is this."},{"Start":"06:38.110 ","End":"06:46.785","Text":"We can get what C is by taking the integral with respect to y,"},{"Start":"06:46.785 ","End":"06:50.620","Text":"so it\u0027s the integral of minus 2yzdy."},{"Start":"06:54.200 ","End":"07:00.000","Text":"Now, the 2y gives us y squared so we just get"},{"Start":"07:00.000 ","End":"07:06.245","Text":"minus y squared z and also plus a constant."},{"Start":"07:06.245 ","End":"07:10.850","Text":"But now we\u0027re only dealing in y and z so this constant is a"},{"Start":"07:10.850 ","End":"07:16.655","Text":"constant involving only z. I\u0027ll use the same letter C,"},{"Start":"07:16.655 ","End":"07:18.890","Text":"even though we shouldn\u0027t really use the same letter twice,"},{"Start":"07:18.890 ","End":"07:23.940","Text":"I\u0027ll make it bold to show it\u0027s different of z."},{"Start":"07:24.730 ","End":"07:33.079","Text":"Now, I can put this in here."},{"Start":"07:33.079 ","End":"07:37.280","Text":"I should have written here C of yz, fixed."},{"Start":"07:37.280 ","End":"07:39.080","Text":"If we\u0027re going to use it in 2 different ways,"},{"Start":"07:39.080 ","End":"07:41.885","Text":"at least let me be clear which C we\u0027re talking about here."},{"Start":"07:41.885 ","End":"07:45.090","Text":"I can put that in here and now that will give us"},{"Start":"07:45.090 ","End":"07:51.380","Text":"a clearer form for Phi which will equal, let\u0027s see,"},{"Start":"07:51.380 ","End":"07:57.085","Text":"we have x squared z cubed plus 6yx,"},{"Start":"07:57.085 ","End":"08:03.420","Text":"I\u0027ll write it as 6xy and C yz from here,"},{"Start":"08:03.420 ","End":"08:07.185","Text":"so that\u0027s minus y squared z,"},{"Start":"08:07.185 ","End":"08:12.450","Text":"plus some function of z."},{"Start":"08:12.450 ","End":"08:15.860","Text":"It\u0027s only constant as far as the other variables go."},{"Start":"08:15.860 ","End":"08:19.850","Text":"We still haven\u0027t used this last equation and I\u0027ll apply"},{"Start":"08:19.850 ","End":"08:25.035","Text":"this last equation to this Phi to its present form."},{"Start":"08:25.035 ","End":"08:30.225","Text":"Differentiating with respect to z,"},{"Start":"08:30.225 ","End":"08:32.255","Text":"this Phi with respect to z,"},{"Start":"08:32.255 ","End":"08:36.350","Text":"the z cubed gives me 3z squared and the x squared is a"},{"Start":"08:36.350 ","End":"08:41.980","Text":"constant so it\u0027s 3x squared, z squared."},{"Start":"08:41.980 ","End":"08:46.800","Text":"6xy just gives me nothing with respect to z,"},{"Start":"08:46.800 ","End":"08:50.845","Text":"and here I\u0027ll get minus y squared."},{"Start":"08:50.845 ","End":"08:57.785","Text":"That\u0027s Phi with respect to z equals R. R I can copy from here, for example,"},{"Start":"08:57.785 ","End":"09:05.195","Text":"and I\u0027ll get 3x squared z squared minus y squared."},{"Start":"09:05.195 ","End":"09:15.200","Text":"Sorry, I forgot here to write plus C prime of z in here, wait a minute."},{"Start":"09:19.370 ","End":"09:26.200","Text":"Now, this side is the same as this side except for the C prime of z."},{"Start":"09:26.200 ","End":"09:30.755","Text":"What we get is that C prime of z equals 0,"},{"Start":"09:30.755 ","End":"09:37.090","Text":"which gives us that C of z is just an actual constant,"},{"Start":"09:37.090 ","End":"09:45.800","Text":"and let\u0027s call that K. I can put that K here."},{"Start":"09:45.800 ","End":"09:49.480","Text":"I\u0027ll just once again write what Phi is."},{"Start":"09:49.480 ","End":"09:55.340","Text":"Phi is x squared z"},{"Start":"09:55.340 ","End":"10:00.840","Text":"cubed plus 6xy minus y squared"},{"Start":"10:00.840 ","End":"10:06.990","Text":"z plus K. What I want is this expression here,"},{"Start":"10:06.990 ","End":"10:18.150","Text":"our line integral is Phi of B minus Phi of A,"},{"Start":"10:18.190 ","End":"10:21.530","Text":"which is, we know what B and A are."},{"Start":"10:21.530 ","End":"10:28.380","Text":"B is the point 2, 1 minus 1,"},{"Start":"10:29.210 ","End":"10:33.405","Text":"and the point A,"},{"Start":"10:33.405 ","End":"10:34.740","Text":"that\u0027s the lower limit,"},{"Start":"10:34.740 ","End":"10:41.335","Text":"is 1 minus 1, 1."},{"Start":"10:41.335 ","End":"10:44.440","Text":"Let\u0027s do the computation now."},{"Start":"10:44.440 ","End":"10:50.400","Text":"Let\u0027s see. 2, 1 minus 1. What do I get?"},{"Start":"10:50.400 ","End":"11:00.630","Text":"2 squared minus 1 cubed"},{"Start":"11:00.630 ","End":"11:08.620","Text":"plus 6 times, xy is 1."},{"Start":"11:08.990 ","End":"11:13.420","Text":"I\u0027m substituting in the wrong 1. Wait a minute."},{"Start":"11:15.260 ","End":"11:18.570","Text":"Xy is just 2 times 1,"},{"Start":"11:18.570 ","End":"11:29.040","Text":"and then minus y squared z. Y squared z is 1 squared times minus 1."},{"Start":"11:29.040 ","End":"11:34.965","Text":"That\u0027s just the top 1,"},{"Start":"11:34.965 ","End":"11:40.305","Text":"and I have to subtract the other 1 which is from here,"},{"Start":"11:40.305 ","End":"11:42.120","Text":"1 minus 1, 1."},{"Start":"11:42.120 ","End":"11:52.860","Text":"It\u0027s 1 squared 1 cubed plus 6xy is"},{"Start":"11:52.860 ","End":"12:00.630","Text":"minus 6 times 1 times 1 and minus y"},{"Start":"12:00.630 ","End":"12:10.085","Text":"squared z is minus minus 1 squared times 1."},{"Start":"12:10.085 ","End":"12:13.790","Text":"Notice I didn\u0027t need to use the K because the K"},{"Start":"12:13.790 ","End":"12:18.300","Text":"would appear here and here it would cancel itself out."},{"Start":"12:19.070 ","End":"12:23.810","Text":"Anyway, I\u0027ll give you the final result of this computation,"},{"Start":"12:23.810 ","End":"12:31.020","Text":"I make it 15 and that is what the answer is."},{"Start":"12:31.020 ","End":"12:33.605","Text":"Remember there was another part of the question."},{"Start":"12:33.605 ","End":"12:37.925","Text":"Here it is, give a physical meaning to the result."},{"Start":"12:37.925 ","End":"12:41.860","Text":"We can give it a meaning in physics,"},{"Start":"12:41.860 ","End":"12:47.305","Text":"it\u0027s involving work of a particle."},{"Start":"12:47.305 ","End":"12:52.745","Text":"I just copy pasted the formula for work from another exercise."},{"Start":"12:52.745 ","End":"12:55.370","Text":"Basically, we need to rephrase this and say,"},{"Start":"12:55.370 ","End":"13:00.034","Text":"this is the work done by the force."},{"Start":"13:00.034 ","End":"13:02.060","Text":"Actually, we need to say which force,"},{"Start":"13:02.060 ","End":"13:10.020","Text":"the force is PI plus QJ plus RK."},{"Start":"13:13.370 ","End":"13:24.405","Text":"The path C is any path from A to B so as the particle moves from A to B,"},{"Start":"13:24.405 ","End":"13:29.120","Text":"this answer is the work done by this force on that particle."},{"Start":"13:29.120 ","End":"13:32.270","Text":"I just need to rephrase this."},{"Start":"13:32.270 ","End":"13:38.910","Text":"Work done by the force on a particle moving from A to B"},{"Start":"13:38.910 ","End":"13:46.870","Text":"where the force is this. We\u0027re done."}],"ID":8764},{"Watched":false,"Name":"Exercise 7","Duration":"19m 20s","ChapterTopicVideoID":8602,"CourseChapterTopicPlaylistID":4965,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.730","Text":"In this exercise, we\u0027re given a vector field in 2 dimensions f,"},{"Start":"00:05.730 ","End":"00:08.385","Text":"which is this, and I\u0027ve copied it over here."},{"Start":"00:08.385 ","End":"00:13.980","Text":"Also, I\u0027d like to abbreviate it to say that this is the first function,"},{"Start":"00:13.980 ","End":"00:19.860","Text":"I\u0027ll call it P. That\u0027s P. This is Q, P of x,"},{"Start":"00:19.860 ","End":"00:24.615","Text":"y is this times i plus Q of x,"},{"Start":"00:24.615 ","End":"00:28.720","Text":"y times j, and that\u0027s q."},{"Start":"00:28.850 ","End":"00:36.345","Text":"We want to compute 3 different line integrals."},{"Start":"00:36.345 ","End":"00:39.540","Text":"We\u0027re given 3 paths, L_1, L_2,"},{"Start":"00:39.540 ","End":"00:42.530","Text":"L_3, and we\u0027ll read the descriptions in a moment."},{"Start":"00:42.530 ","End":"00:47.435","Text":"They\u0027re all closed paths and we want to compute the integral."},{"Start":"00:47.435 ","End":"00:52.850","Text":"The circle means over a closed path for each of these L_1, L_2, L_3."},{"Start":"00:52.850 ","End":"00:56.315","Text":"I\u0027d better introduce a sketch here."},{"Start":"00:56.315 ","End":"00:59.730","Text":"Also, I\u0027ll need a bit more space."},{"Start":"00:59.950 ","End":"01:04.025","Text":"Okay, here\u0027s a pair of axes in the plane."},{"Start":"01:04.025 ","End":"01:06.305","Text":"Let\u0027s see what is L_1."},{"Start":"01:06.305 ","End":"01:08.780","Text":"It\u0027s a circle of radius 1,"},{"Start":"01:08.780 ","End":"01:11.345","Text":"x squared plus y squared equals 1 squared."},{"Start":"01:11.345 ","End":"01:14.545","Text":"It\u0027s something like this."},{"Start":"01:14.545 ","End":"01:19.745","Text":"The positive orientation goes round this way."},{"Start":"01:19.745 ","End":"01:22.230","Text":"That\u0027s L_1."},{"Start":"01:23.270 ","End":"01:25.770","Text":"What is L_2?"},{"Start":"01:25.770 ","End":"01:27.644","Text":"L_2 is an ellipse."},{"Start":"01:27.644 ","End":"01:31.635","Text":"This is 4 squared and this is 3 squared,"},{"Start":"01:31.635 ","End":"01:37.250","Text":"and so this ellipse goes from 4 to minus 4 in the x-direction,"},{"Start":"01:37.250 ","End":"01:45.560","Text":"from 3 to minus 3 in the y direction as it\u0027s just roughly say something like this."},{"Start":"01:45.560 ","End":"01:49.145","Text":"It really doesn\u0027t have to look great."},{"Start":"01:49.145 ","End":"01:51.695","Text":"We\u0027re just illustrating."},{"Start":"01:51.695 ","End":"01:58.440","Text":"This would be 1 and minus 1."},{"Start":"01:58.440 ","End":"02:01.280","Text":"Here\u0027s 4 and minus 4."},{"Start":"02:01.280 ","End":"02:10.460","Text":"Yeah, I know it looks more like an egg and this 1 is L_2 and it goes round this way,"},{"Start":"02:10.460 ","End":"02:15.030","Text":"the opposite direction is this and the third"},{"Start":"02:15.030 ","End":"02:20.165","Text":"1 is a circle with radius 1 and the center is that 10, 7."},{"Start":"02:20.165 ","End":"02:21.520","Text":"Well, it doesn\u0027t have to be to scale."},{"Start":"02:21.520 ","End":"02:24.995","Text":"Let\u0027s say 10, 7 is somewhere here."},{"Start":"02:24.995 ","End":"02:33.560","Text":"It\u0027s a circle of radius 1 around here and it\u0027s also the positive orientation."},{"Start":"02:33.560 ","End":"02:37.530","Text":"This would be our L_3."},{"Start":"02:37.910 ","End":"02:40.260","Text":"Notice, by the way,"},{"Start":"02:40.260 ","End":"02:42.000","Text":"that the domain for"},{"Start":"02:42.000 ","End":"02:51.070","Text":"this vector field is"},{"Start":"02:51.070 ","End":"02:56.600","Text":"everywhere except where the denominator is 0 and the denominator is 0 only at the origin."},{"Start":"02:56.600 ","End":"02:59.899","Text":"That\u0027s the only way x squared plus y squared can be 0."},{"Start":"02:59.899 ","End":"03:01.490","Text":"I\u0027ll make a note of that."},{"Start":"03:01.490 ","End":"03:03.620","Text":"I\u0027ll put the red dot here and say this is"},{"Start":"03:03.620 ","End":"03:08.285","Text":"the bad point and it\u0027s the only bad point and everywhere else,"},{"Start":"03:08.285 ","End":"03:13.310","Text":"the vector field is defined and it\u0027s continuous."},{"Start":"03:13.310 ","End":"03:17.160","Text":"I would like to check if it\u0027s conservative as a vector field,"},{"Start":"03:17.160 ","End":"03:21.260","Text":"and if so, it\u0027ll help us with solving the rest of the question."},{"Start":"03:21.260 ","End":"03:23.030","Text":"I remember that condition for"},{"Start":"03:23.030 ","End":"03:27.800","Text":"conservative is the condition that the partial derivative of P with"},{"Start":"03:27.800 ","End":"03:30.650","Text":"respect to y is equal to"},{"Start":"03:30.650 ","End":"03:34.970","Text":"the partial derivative of Q with respect to x. I want to check this."},{"Start":"03:34.970 ","End":"03:41.040","Text":"I\u0027m putting a question mark here but note that P can be rewritten,"},{"Start":"03:41.040 ","End":"03:48.540","Text":"P of x, y is equal to the x squared plus y squared over x squared plus y squared is 1."},{"Start":"03:48.540 ","End":"03:57.030","Text":"I can rewrite it as 1 minus y over x squared plus y squared."},{"Start":"03:57.030 ","End":"03:59.180","Text":"Before I do these derivatives,"},{"Start":"03:59.180 ","End":"04:02.825","Text":"let me remind you of the quotient rule because we\u0027ll be using that,"},{"Start":"04:02.825 ","End":"04:06.400","Text":"that the derivative of u over v,"},{"Start":"04:06.400 ","End":"04:08.160","Text":"in respect doesn\u0027t matter,"},{"Start":"04:08.160 ","End":"04:10.770","Text":"x or y in general,"},{"Start":"04:10.770 ","End":"04:14.085","Text":"is equal to on the denominator,"},{"Start":"04:14.085 ","End":"04:15.650","Text":"we have v squared, and here,"},{"Start":"04:15.650 ","End":"04:19.520","Text":"we have the derivative of u times v"},{"Start":"04:19.520 ","End":"04:26.940","Text":"minus u times the derivative of v. P with respect to y,"},{"Start":"04:26.940 ","End":"04:28.985","Text":"let\u0027s do the computation."},{"Start":"04:28.985 ","End":"04:32.660","Text":"A derivative of 1 is 0."},{"Start":"04:32.660 ","End":"04:35.690","Text":"I have minus from here,"},{"Start":"04:35.690 ","End":"04:37.910","Text":"and then using the quotient rule here,"},{"Start":"04:37.910 ","End":"04:43.780","Text":"I have the denominator squared x squared plus y squared, squared,"},{"Start":"04:43.780 ","End":"04:51.350","Text":"and then my u is y. I have u prime and it\u0027s with respect to y."},{"Start":"04:51.350 ","End":"04:54.455","Text":"It\u0027s 1 times v,"},{"Start":"04:54.455 ","End":"05:00.830","Text":"which is x squared plus y squared minus uv prime."},{"Start":"05:00.830 ","End":"05:09.990","Text":"U is y, and v prime is just 2y because x is a constant."},{"Start":"05:11.690 ","End":"05:14.940","Text":"Now look, from here,"},{"Start":"05:14.940 ","End":"05:17.740","Text":"we get x squared."},{"Start":"05:18.850 ","End":"05:22.180","Text":"Ignore this minus for the moment, just deal with that in a minute."},{"Start":"05:22.180 ","End":"05:25.405","Text":"Here, we get x squared plus y squared minus 2y squared."},{"Start":"05:25.405 ","End":"05:28.375","Text":"It\u0027s x squared minus y squared on the top."},{"Start":"05:28.375 ","End":"05:33.040","Text":"Because of the minus, I can switch directions and say it\u0027s y squared minus x"},{"Start":"05:33.040 ","End":"05:38.400","Text":"squared over this same thing squared."},{"Start":"05:38.400 ","End":"05:41.030","Text":"That\u0027s 1 of them. Now, let\u0027s check the other 1."},{"Start":"05:41.030 ","End":"05:45.050","Text":"What is the derivative of Q with respect to x?"},{"Start":"05:45.140 ","End":"05:49.840","Text":"This time we get from the quotient rule,"},{"Start":"05:49.840 ","End":"05:56.345","Text":"the same denominator, which is x squared plus y squared, squared."},{"Start":"05:56.345 ","End":"06:01.595","Text":"This time, u is x and v is still this thing."},{"Start":"06:01.595 ","End":"06:05.960","Text":"U prime is 1 times v,"},{"Start":"06:05.960 ","End":"06:12.020","Text":"which is x squared plus y squared minus u,"},{"Start":"06:12.020 ","End":"06:17.255","Text":"which is x, times derivative of v this time with respect to x,"},{"Start":"06:17.255 ","End":"06:25.615","Text":"which is 2x and this gives us x squared plus y squared minus 2x squared."},{"Start":"06:25.615 ","End":"06:35.625","Text":"It\u0027s again y squared minus x squared over x squared plus y squared squared."},{"Start":"06:35.625 ","End":"06:41.085","Text":"These are equal and so the answer to this is"},{"Start":"06:41.085 ","End":"06:46.700","Text":"yes and that means that the vector field F is conservative."},{"Start":"06:46.700 ","End":"06:51.900","Text":"I\u0027ll just write that F is conservative."},{"Start":"06:54.980 ","End":"07:03.960","Text":"Under certain conditions, conservative means path independent for the integrals"},{"Start":"07:03.960 ","End":"07:06.470","Text":"but that only applies to"},{"Start":"07:06.470 ","End":"07:17.075","Text":"a simply connected region."},{"Start":"07:17.075 ","End":"07:21.290","Text":"Simply connected means it has no holes."},{"Start":"07:21.290 ","End":"07:25.445","Text":"Now this region, certainly the plane has a hole in it,"},{"Start":"07:25.445 ","End":"07:28.120","Text":"so not simply connected."},{"Start":"07:28.120 ","End":"07:33.860","Text":"However, if I\u0027m looking for the integral L_3,"},{"Start":"07:33.860 ","End":"07:35.975","Text":"I can consider just part of the plane,"},{"Start":"07:35.975 ","End":"07:40.910","Text":"maybe even just the first quadrant without the axis or some region here,"},{"Start":"07:40.910 ","End":"07:43.680","Text":"it will be simply connected."},{"Start":"07:45.110 ","End":"07:47.210","Text":"I was going to scroll down,"},{"Start":"07:47.210 ","End":"07:48.710","Text":"but then I\u0027ll lose these equations."},{"Start":"07:48.710 ","End":"07:52.580","Text":"Hang on a minute. I\u0027ll just copy those equations here."},{"Start":"07:52.580 ","End":"08:01.670","Text":"This L_1 is x squared plus y squared equals 1."},{"Start":"08:01.670 ","End":"08:02.960","Text":"Here we had what?"},{"Start":"08:02.960 ","End":"08:08.375","Text":"X squared over 16 plus y squared over 9 equals 1,"},{"Start":"08:08.375 ","End":"08:11.860","Text":"and here x minus 10"},{"Start":"08:11.860 ","End":"08:19.920","Text":"squared plus y minus 7 squared equals 1,"},{"Start":"08:19.920 ","End":"08:23.260","Text":"and now I can scroll."},{"Start":"08:23.600 ","End":"08:32.450","Text":"What I was saying was that the easiest 1 to do is part C. For part C,"},{"Start":"08:32.450 ","End":"08:38.190","Text":"I can say that the integral over L_3 of f,"},{"Start":"08:38.440 ","End":"08:47.330","Text":"this actually should be dot."},{"Start":"08:47.330 ","End":"08:53.105","Text":"Same here, dot and dot and r is not in bold."},{"Start":"08:53.105 ","End":"08:55.270","Text":"I\u0027ll just emphasize that."},{"Start":"08:55.270 ","End":"08:57.990","Text":"I guess I should have made it bold."},{"Start":"08:57.990 ","End":"09:01.440","Text":"When it\u0027s bold it\u0027s also another way of saying it\u0027s a vector,"},{"Start":"09:01.440 ","End":"09:03.830","Text":"you don\u0027t have to put the arrow then."},{"Start":"09:03.830 ","End":"09:10.480","Text":"The integral of F dot dr and I\u0027ll indicate they\u0027re both vectors,"},{"Start":"09:10.480 ","End":"09:12.890","Text":"over L_3 equals 0,"},{"Start":"09:12.890 ","End":"09:18.440","Text":"because of the theorem that on a simply connected region,"},{"Start":"09:18.440 ","End":"09:27.360","Text":"conservative implies path independent."},{"Start":"09:28.140 ","End":"09:34.000","Text":"I wrote this very telegraphic, F is conservative."},{"Start":"09:34.000 ","End":"09:35.620","Text":"On a simply connected region,"},{"Start":"09:35.620 ","End":"09:38.200","Text":"then the integrals are path independent."},{"Start":"09:38.200 ","End":"09:41.245","Text":"Path independent means that on a closed circuit,"},{"Start":"09:41.245 ","End":"09:44.030","Text":"closed curve is 0."},{"Start":"09:44.730 ","End":"09:49.090","Text":"Let\u0027s do L_1 next and we can\u0027t say that this is 0."},{"Start":"09:49.090 ","End":"09:56.080","Text":"In fact, it\u0027s not going to come out 0 because no matter how I alter the region,"},{"Start":"09:56.080 ","End":"09:58.510","Text":"it\u0027s still going to go around the hole and it\u0027s"},{"Start":"09:58.510 ","End":"10:01.495","Text":"not going to be in a simply connected region."},{"Start":"10:01.495 ","End":"10:03.760","Text":"We have to compute it."},{"Start":"10:03.760 ","End":"10:07.795","Text":"Let\u0027s do that with polar coordinates."},{"Start":"10:07.795 ","End":"10:13.190","Text":"I can parametrize L_1 the usual way."},{"Start":"10:17.910 ","End":"10:21.865","Text":"I\u0027m just using polar as a parametric,"},{"Start":"10:21.865 ","End":"10:25.855","Text":"x equals r cosine Theta, but r is 1."},{"Start":"10:25.855 ","End":"10:29.890","Text":"It\u0027s just cosine and I\u0027ll use t as a parameter,"},{"Start":"10:29.890 ","End":"10:32.995","Text":"y equals sine t,"},{"Start":"10:32.995 ","End":"10:36.820","Text":"and t goes from 0 to 360 degrees."},{"Start":"10:36.820 ","End":"10:38.500","Text":"But we\u0027re using radians."},{"Start":"10:38.500 ","End":"10:45.320","Text":"As cosine and sine gives us a circle as we go around from 0 to 2 Pi."},{"Start":"10:46.650 ","End":"10:51.730","Text":"We\u0027ll also need dx and dy."},{"Start":"10:51.730 ","End":"10:57.175","Text":"Of course, dx is minus sine t,"},{"Start":"10:57.175 ","End":"11:03.110","Text":"dt and dy is cosine t, dt."},{"Start":"11:04.470 ","End":"11:06.550","Text":"What am I doing now?"},{"Start":"11:06.550 ","End":"11:10.660","Text":"Part A."},{"Start":"11:10.660 ","End":"11:15.250","Text":"In a, the integral over L_1 of F.dr,"},{"Start":"11:15.250 ","End":"11:18.310","Text":"which is just the same"},{"Start":"11:18.310 ","End":"11:27.350","Text":"as Pdx plus Qdy."},{"Start":"11:28.950 ","End":"11:36.655","Text":"We\u0027ll do this as the line integral type 2 parametrize."},{"Start":"11:36.655 ","End":"11:42.325","Text":"We go, t goes from 0 to 2 Pi."},{"Start":"11:42.325 ","End":"11:46.690","Text":"Where did I write that? Here it is."},{"Start":"11:46.690 ","End":"11:53.620","Text":"P is 1 minus,"},{"Start":"11:53.620 ","End":"12:00.085","Text":"y is sine t over,"},{"Start":"12:00.085 ","End":"12:09.170","Text":"now x squared plus y squared is just sine squared plus cosine squared is 1."},{"Start":"12:10.470 ","End":"12:13.704","Text":"I\u0027ll just make a note to the side."},{"Start":"12:13.704 ","End":"12:16.270","Text":"Made a note of it that explains this one here,"},{"Start":"12:16.270 ","End":"12:17.410","Text":"x squared plus y squared,"},{"Start":"12:17.410 ","End":"12:19.255","Text":"sine squared plus cosine squared,"},{"Start":"12:19.255 ","End":"12:21.565","Text":"or cosine squared plus sine squared."},{"Start":"12:21.565 ","End":"12:25.045","Text":"Then we have dx,"},{"Start":"12:25.045 ","End":"12:28.520","Text":"which is minus sine t, dt."},{"Start":"12:31.230 ","End":"12:39.340","Text":"Then we have the second bit, Qdy."},{"Start":"12:39.340 ","End":"12:40.929","Text":"I\u0027ll need more room."},{"Start":"12:40.929 ","End":"12:46.540","Text":"Let\u0027s see if I can squash it in."},{"Start":"12:46.540 ","End":"12:49.450","Text":"Well, Q is this here,"},{"Start":"12:49.450 ","End":"12:54.880","Text":"x is equal to cosine t, and the same thing,"},{"Start":"12:54.880 ","End":"12:59.750","Text":"x squared plus y squared is the same one we got before,"},{"Start":"13:00.870 ","End":"13:07.465","Text":"dy is cosine t, dt."},{"Start":"13:07.465 ","End":"13:10.130","Text":"Got it squeezed in here."},{"Start":"13:10.890 ","End":"13:13.375","Text":"Let\u0027s simplify this."},{"Start":"13:13.375 ","End":"13:18.500","Text":"We have the integral from 0 to 2 Pi."},{"Start":"13:20.250 ","End":"13:23.050","Text":"Let\u0027s see."},{"Start":"13:23.050 ","End":"13:26.155","Text":"We have to multiply minus sine t by each of these."},{"Start":"13:26.155 ","End":"13:35.425","Text":"I\u0027ve got minus sine t and minus sine times minus sine is plus sine squared t. The dt,"},{"Start":"13:35.425 ","End":"13:44.390","Text":"I\u0027ll take at the end plus cosine squared t and all this dt."},{"Start":"13:45.990 ","End":"13:54.130","Text":"Continuing, we have the integral cosine squared plus sine squared again is 1."},{"Start":"13:54.130 ","End":"14:04.420","Text":"We have 1 minus sine t dt from 0 to 2 Pi."},{"Start":"14:04.420 ","End":"14:09.160","Text":"Let\u0027s continue over here. Let\u0027s see."},{"Start":"14:09.160 ","End":"14:13.525","Text":"This equals t minus"},{"Start":"14:13.525 ","End":"14:21.040","Text":"cosine t from 0 to 2 Pi."},{"Start":"14:21.040 ","End":"14:26.335","Text":"Now, cosine 2 Pi and cosine 0 are the same."},{"Start":"14:26.335 ","End":"14:28.885","Text":"This will cancel each other out."},{"Start":"14:28.885 ","End":"14:34.630","Text":"T from 2 Pi minus 0,"},{"Start":"14:34.630 ","End":"14:38.090","Text":"it just comes out to be 2 pi."},{"Start":"14:39.120 ","End":"14:46.885","Text":"Now, I somehow want to use my result for part A to help me with part B,"},{"Start":"14:46.885 ","End":"14:49.225","Text":"which is the ellipse."},{"Start":"14:49.225 ","End":"14:54.504","Text":"Now, notice that if I put the arrow this way,"},{"Start":"14:54.504 ","End":"14:58.150","Text":"then this curve is not L_2."},{"Start":"14:58.150 ","End":"15:01.435","Text":"It\u0027s what we call minus L_2."},{"Start":"15:01.435 ","End":"15:12.580","Text":"L_1 and L_2 both go round the bad point just once and one complete circle around."},{"Start":"15:12.580 ","End":"15:14.620","Text":"I\u0027m going to use the theorem here,"},{"Start":"15:14.620 ","End":"15:19.480","Text":"which I think we may not have been covered in the tutorial."},{"Start":"15:19.480 ","End":"15:27.835","Text":"But basically, I\u0027ll just briefly mentioned it in picture form,"},{"Start":"15:27.835 ","End":"15:32.815","Text":"if I have a not quite a simply connected region,"},{"Start":"15:32.815 ","End":"15:34.600","Text":"one that\u0027s not simply connected,"},{"Start":"15:34.600 ","End":"15:37.405","Text":"but it just has a single hole in it."},{"Start":"15:37.405 ","End":"15:39.400","Text":"The hole doesn\u0027t have to be a point."},{"Start":"15:39.400 ","End":"15:41.230","Text":"It could be a larger hole,"},{"Start":"15:41.230 ","End":"15:44.470","Text":"but let\u0027s say we have the region and a hole in it,"},{"Start":"15:44.470 ","End":"15:49.315","Text":"and the region is including the interior,"},{"Start":"15:49.315 ","End":"15:55.945","Text":"but just not covering the hole."},{"Start":"15:55.945 ","End":"16:03.010","Text":"Then if we have 2 line integrals, 2 different curves,"},{"Start":"16:03.010 ","End":"16:10.375","Text":"but each of them goes exactly one surround in the same direction,"},{"Start":"16:10.375 ","End":"16:13.600","Text":"say the counterclockwise positive direction,"},{"Start":"16:13.600 ","End":"16:17.500","Text":"I have a curve 1 and curve 2,"},{"Start":"16:17.500 ","End":"16:25.959","Text":"then the line integral over a curve 1 of whatever it was,"},{"Start":"16:25.959 ","End":"16:29.185","Text":"the same like Pdx plus Qdy or whatever,"},{"Start":"16:29.185 ","End":"16:34.255","Text":"will equal the line integral along the curve C_2."},{"Start":"16:34.255 ","End":"16:36.700","Text":"In other words, both of them have to go"},{"Start":"16:36.700 ","End":"16:42.745","Text":"exactly one surround in the same direction counterclockwise."},{"Start":"16:42.745 ","End":"16:45.505","Text":"I\u0027m not going to write it in words."},{"Start":"16:45.505 ","End":"16:49.220","Text":"You should remember this idea."},{"Start":"16:50.100 ","End":"16:53.605","Text":"You could use it to save time."},{"Start":"16:53.605 ","End":"16:55.825","Text":"In our case, what does it give us?"},{"Start":"16:55.825 ","End":"16:58.915","Text":"In our case, if I use this theorem,"},{"Start":"16:58.915 ","End":"17:09.010","Text":"I can conclude in part B that minus L_2 and L_1 are like C_1 and C_2 here."},{"Start":"17:09.010 ","End":"17:15.640","Text":"The integral of minus L_2 over minus L_2 of blah,"},{"Start":"17:15.640 ","End":"17:17.335","Text":"blah, blah, same thing,"},{"Start":"17:17.335 ","End":"17:21.985","Text":"equals the integral over L_1 of whatever it was."},{"Start":"17:21.985 ","End":"17:26.140","Text":"But when you take the integral of a minus a curve,"},{"Start":"17:26.140 ","End":"17:27.895","Text":"meaning in the opposite direction,"},{"Start":"17:27.895 ","End":"17:30.500","Text":"it just reverses it."},{"Start":"17:31.080 ","End":"17:33.490","Text":"Also, make a note of that at the side."},{"Start":"17:33.490 ","End":"17:39.310","Text":"In general, the integral of such type 2 line integral over a curve over a minus a curve,"},{"Start":"17:39.310 ","End":"17:42.385","Text":"is minus the integral over the curve."},{"Start":"17:42.385 ","End":"17:47.450","Text":"Here, this will give us that this is minus the integral over L_2."},{"Start":"17:47.520 ","End":"17:51.520","Text":"The integral over L_2 of whatever it is,"},{"Start":"17:51.520 ","End":"17:58.120","Text":"is minus the integral over L_1 and this we\u0027ve computed to be, what was it?"},{"Start":"17:58.120 ","End":"17:59.755","Text":"2 Pi."},{"Start":"17:59.755 ","End":"18:05.395","Text":"This integral will be minus 2 Pi."},{"Start":"18:05.395 ","End":"18:08.305","Text":"I don\u0027t know why I\u0027m squashing everything in."},{"Start":"18:08.305 ","End":"18:12.130","Text":"Now, we have all 3 results."},{"Start":"18:12.130 ","End":"18:15.460","Text":"For part C, we got the answer 0."},{"Start":"18:15.460 ","End":"18:25.210","Text":"For part A, we got the answer, where was it?"},{"Start":"18:26.120 ","End":"18:28.770","Text":"Here, 2 Pi."},{"Start":"18:28.770 ","End":"18:32.040","Text":"This separator confused me."},{"Start":"18:32.040 ","End":"18:35.325","Text":"This could have caused an awful of this part B here."},{"Start":"18:35.325 ","End":"18:36.705","Text":"This is continuing here."},{"Start":"18:36.705 ","End":"18:43.460","Text":"In part B, we got the answer minus 2 Pi."},{"Start":"18:44.460 ","End":"18:48.370","Text":"Basically, the important thing I could say in"},{"Start":"18:48.370 ","End":"18:51.685","Text":"general is that and let me go back up a bit,"},{"Start":"18:51.685 ","End":"18:54.865","Text":"if I have a loop that doesn\u0027t go round the bad bit,"},{"Start":"18:54.865 ","End":"18:57.485","Text":"the hole, it\u0027s going to be 0."},{"Start":"18:57.485 ","End":"19:02.960","Text":"If I could compute one of the integrals that goes once around the hole,"},{"Start":"19:02.960 ","End":"19:07.640","Text":"then I can do all the integrals that go once around the hole. It\u0027ll be the same."},{"Start":"19:07.640 ","End":"19:09.740","Text":"But if it goes around in the opposite direction,"},{"Start":"19:09.740 ","End":"19:15.775","Text":"it will just be minus that as we did here and we got minus 2 Pi."},{"Start":"19:15.775 ","End":"19:20.210","Text":"That\u0027s about it and we are done."}],"ID":8765},{"Watched":false,"Name":"Exercise 8","Duration":"13m 26s","ChapterTopicVideoID":8603,"CourseChapterTopicPlaylistID":4965,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.000","Text":"In this exercise, we\u0027re given a 2-dimensional vector field in the plane."},{"Start":"00:06.000 ","End":"00:10.905","Text":"We have 2 different paths from this point to this point."},{"Start":"00:10.905 ","End":"00:12.420","Text":"They are given as follows,"},{"Start":"00:12.420 ","End":"00:15.359","Text":"and I\u0027ll draw a sketch in a moment."},{"Start":"00:15.359 ","End":"00:22.785","Text":"In part A, we\u0027ll compute the integral of the same function F,"},{"Start":"00:22.785 ","End":"00:25.395","Text":"both going from here to here,"},{"Start":"00:25.395 ","End":"00:27.060","Text":"but along different paths,"},{"Start":"00:27.060 ","End":"00:31.185","Text":"and see if we get the same answer or different answers,"},{"Start":"00:31.185 ","End":"00:34.110","Text":"and part B, when we get to it."},{"Start":"00:34.110 ","End":"00:38.630","Text":"Note that both here and here we have the same equation,"},{"Start":"00:38.630 ","End":"00:40.100","Text":"x squared plus y squared equals 4,"},{"Start":"00:40.100 ","End":"00:43.340","Text":"which is a circle of radius 2 centered at the origin."},{"Start":"00:43.340 ","End":"00:47.540","Text":"This would be the point 2 and this would be the point minus 2."},{"Start":"00:47.540 ","End":"00:51.845","Text":"The thing is that L_1 is on the positive,"},{"Start":"00:51.845 ","End":"00:56.690","Text":"the upper semicircle and L_2 is the lower semicircle."},{"Start":"00:56.690 ","End":"00:58.520","Text":"I\u0027ll put arrows on them."},{"Start":"00:58.520 ","End":"01:02.030","Text":"They both go from 2,0 to minus 2,0."},{"Start":"01:02.030 ","End":"01:04.295","Text":"This 1 is L_1,"},{"Start":"01:04.295 ","End":"01:10.350","Text":"and this here would be L_2."},{"Start":"01:10.750 ","End":"01:14.930","Text":"I\u0027d just like to note that this vector field is"},{"Start":"01:14.930 ","End":"01:18.680","Text":"not defined whenever this denominator is 0,"},{"Start":"01:18.680 ","End":"01:21.230","Text":"which happens just at the origin."},{"Start":"01:21.230 ","End":"01:24.740","Text":"I\u0027m not going to relate to that right at the moment,"},{"Start":"01:24.740 ","End":"01:27.110","Text":"but I just wanted to mention it for the record that"},{"Start":"01:27.110 ","End":"01:30.485","Text":"the vector field is not defined at the origin."},{"Start":"01:30.485 ","End":"01:35.815","Text":"Now let\u0027s start with the first integral,"},{"Start":"01:35.815 ","End":"01:39.080","Text":"a technical point, I should have made dr bold or at least put"},{"Start":"01:39.080 ","End":"01:43.145","Text":"an arrow over it because dr is also a vector."},{"Start":"01:43.145 ","End":"01:46.220","Text":"F is bold, which is another way instead of"},{"Start":"01:46.220 ","End":"01:49.990","Text":"writing the arrow of saying something\u0027s a vector."},{"Start":"01:49.990 ","End":"01:58.685","Text":"Let\u0027s use parametrization, which will use like polar coordinates."},{"Start":"01:58.685 ","End":"02:05.100","Text":"L_1 could be described as, let\u0027s see,"},{"Start":"02:05.100 ","End":"02:14.054","Text":"I could take x equals 2 cosine Theta t and y"},{"Start":"02:14.054 ","End":"02:24.005","Text":"equals 2 sine t. The 2 sine would give me the whole circle as t goes from 0-2Pi."},{"Start":"02:24.005 ","End":"02:28.380","Text":"But here I\u0027m just going from t equals 0,"},{"Start":"02:28.380 ","End":"02:30.270","Text":"t is like Theta, the angle,"},{"Start":"02:30.270 ","End":"02:34.035","Text":"up to 180 degrees, which is Pi."},{"Start":"02:34.035 ","End":"02:38.310","Text":"There are the limits for this."},{"Start":"02:38.310 ","End":"02:42.590","Text":"Note that I\u0027ll also need dx."},{"Start":"02:42.590 ","End":"02:46.340","Text":"dx is equal to minus"},{"Start":"02:46.340 ","End":"02:56.490","Text":"2 sine t dt and dy is 2 cosine t dt."},{"Start":"02:58.280 ","End":"03:05.425","Text":"If I just call the first function P of xy,"},{"Start":"03:05.425 ","End":"03:07.175","Text":"and the second function,"},{"Start":"03:07.175 ","End":"03:09.815","Text":"I call it Q of xy,"},{"Start":"03:09.815 ","End":"03:15.890","Text":"then this type 2 line integral is just the integral"},{"Start":"03:15.890 ","End":"03:25.680","Text":"of Pdx plus Qdy over whichever 1 it is,"},{"Start":"03:25.680 ","End":"03:29.020","Text":"over L_1 or 2."},{"Start":"03:29.060 ","End":"03:32.625","Text":"Accordingly, I\u0027ll write anything,"},{"Start":"03:32.625 ","End":"03:35.440","Text":"L means 1 of these."},{"Start":"03:35.570 ","End":"03:43.280","Text":"What we get if we interpret this integral using"},{"Start":"03:43.280 ","End":"03:51.500","Text":"the parameter t is the range of the parameter is 0 to Pi,"},{"Start":"03:51.500 ","End":"03:55.910","Text":"and it\u0027s going counterclockwise, so we\u0027re okay."},{"Start":"03:55.910 ","End":"04:01.290","Text":"Then what we need is Pdx."},{"Start":"04:01.290 ","End":"04:07.160","Text":"Well, P from here is minus y over x squared plus"},{"Start":"04:07.160 ","End":"04:13.760","Text":"y squared minus 2 sine t over."},{"Start":"04:13.760 ","End":"04:17.150","Text":"Now, x squared plus y squared."},{"Start":"04:17.150 ","End":"04:20.380","Text":"We can see it here is just 4."},{"Start":"04:20.380 ","End":"04:24.425","Text":"You could compute 2 squared cosine squared,"},{"Start":"04:24.425 ","End":"04:26.630","Text":"sine squared, but we already have it here,"},{"Start":"04:26.630 ","End":"04:29.220","Text":"so it\u0027s over 4."},{"Start":"04:30.030 ","End":"04:39.750","Text":"Where are we here? Dx which is minus 2 sine t dt."},{"Start":"04:39.750 ","End":"04:41.845","Text":"Then the other bit,"},{"Start":"04:41.845 ","End":"04:48.179","Text":"the Qdy is Q is x over,"},{"Start":"04:48.179 ","End":"04:55.835","Text":"and x is just cosine t over the same, 4."},{"Start":"04:55.835 ","End":"05:05.420","Text":"Then I need the dy, which is 2 cosine t dt."},{"Start":"05:05.900 ","End":"05:13.920","Text":"Let\u0027s gather it all together as just something dt 0 to Pi."},{"Start":"05:14.780 ","End":"05:19.385","Text":"Now, from here, we have minus times minus is plus,"},{"Start":"05:19.385 ","End":"05:22.385","Text":"2 with 2 is 4."},{"Start":"05:22.385 ","End":"05:27.940","Text":"Basically what I\u0027m saying is minus 2 times minus 2 is 4 and that cancels."},{"Start":"05:27.940 ","End":"05:31.590","Text":"I have sine squared t,"},{"Start":"05:31.590 ","End":"05:33.555","Text":"and then from the other,"},{"Start":"05:33.555 ","End":"05:35.220","Text":"I forgot a 2 here,"},{"Start":"05:35.220 ","End":"05:38.025","Text":"sorry. That\u0027s a 2."},{"Start":"05:38.025 ","End":"05:42.150","Text":"This was x and I just forgot that too, forgive me."},{"Start":"05:42.150 ","End":"05:48.660","Text":"I get plus. Here I get the same cancellation, 2 times 2 is 4."},{"Start":"05:48.660 ","End":"05:53.190","Text":"Cosine squared t dt,"},{"Start":"05:53.190 ","End":"05:56.560","Text":"but this is 1,"},{"Start":"05:57.620 ","End":"06:03.965","Text":"and the integral of 1 is just the upper limit of integration minus the lower."},{"Start":"06:03.965 ","End":"06:07.650","Text":"So this is equal to Pi."},{"Start":"06:08.090 ","End":"06:13.625","Text":"That takes care of the integral over L_1."},{"Start":"06:13.625 ","End":"06:17.445","Text":"This 1 came out to be Pi."},{"Start":"06:17.445 ","End":"06:19.900","Text":"Now, what about the other?"},{"Start":"06:19.900 ","End":"06:24.150","Text":"I don\u0027t want to do all the work again."},{"Start":"06:24.220 ","End":"06:34.625","Text":"For L_2, all I would need to change are the limits of integration."},{"Start":"06:34.625 ","End":"06:39.155","Text":"Instead of t going from 0 to Pi,"},{"Start":"06:39.155 ","End":"06:45.230","Text":"it would be going the other way from 0 to minus Pi."},{"Start":"06:45.230 ","End":"06:47.210","Text":"Yes, it\u0027s going backwards."},{"Start":"06:47.210 ","End":"06:51.200","Text":"It\u0027s okay to have the upper limit smaller than the lower limit."},{"Start":"06:51.200 ","End":"06:57.830","Text":"It just means it\u0027s decreasing from 0 to minus Pi of exactly the same integral,"},{"Start":"06:57.830 ","End":"07:01.440","Text":"which is the 1 dt."},{"Start":"07:01.610 ","End":"07:06.395","Text":"This would come out to be minus Pi."},{"Start":"07:06.395 ","End":"07:10.535","Text":"Let me emphasize the minus here, in case you don\u0027t see it."},{"Start":"07:10.535 ","End":"07:14.130","Text":"So we get different answers."},{"Start":"07:15.020 ","End":"07:19.190","Text":"We\u0027re used to seeing situations where we\u0027re path-independent,"},{"Start":"07:19.190 ","End":"07:20.810","Text":"but that\u0027s not the rule,"},{"Start":"07:20.810 ","End":"07:22.415","Text":"that\u0027s the exception in here."},{"Start":"07:22.415 ","End":"07:27.305","Text":"Yet we got 2 different results going from the same point to the same point,"},{"Start":"07:27.305 ","End":"07:30.760","Text":"but different routes and that\u0027s fine."},{"Start":"07:30.950 ","End":"07:38.100","Text":"This 1 came out to be minus Pi and we\u0027ve answered part A."},{"Start":"07:38.100 ","End":"07:41.025","Text":"Now, what about part B?"},{"Start":"07:41.025 ","End":"07:45.185","Text":"In a moment I\u0027ll sketch this 1/2 annulus shape."},{"Start":"07:45.185 ","End":"07:49.760","Text":"But notice that we\u0027re not asked to prove that F is conservative in the whole plane."},{"Start":"07:49.760 ","End":"07:55.265","Text":"We couldn\u0027t possibly do that because there\u0027s a hole here."},{"Start":"07:55.265 ","End":"07:57.520","Text":"But even if there wasn\u0027t a hole,"},{"Start":"07:57.520 ","End":"07:59.720","Text":"it wouldn\u0027t expect it to be"},{"Start":"07:59.720 ","End":"08:02.690","Text":"conservative in the whole plane because if it was conservative,"},{"Start":"08:02.690 ","End":"08:04.460","Text":"then it would be path-independent,"},{"Start":"08:04.460 ","End":"08:06.290","Text":"and here we\u0027ve got 2 different results for"},{"Start":"08:06.290 ","End":"08:10.175","Text":"2 different paths from the same start and end points."},{"Start":"08:10.175 ","End":"08:16.350","Text":"Anyway, that\u0027s philosophy, let\u0027s just get to practicality and I\u0027ll"},{"Start":"08:16.350 ","End":"08:22.775","Text":"sketch D. Here\u0027s the sketch for part B."},{"Start":"08:22.775 ","End":"08:25.790","Text":"An annulus in general is a ring shape,"},{"Start":"08:25.790 ","End":"08:29.540","Text":"it\u0027s like 2 concentric circles and the bit"},{"Start":"08:29.540 ","End":"08:34.520","Text":"between them is what is called in mathematics an annulus,"},{"Start":"08:34.520 ","End":"08:36.530","Text":"just in case you were wondering."},{"Start":"08:36.530 ","End":"08:40.440","Text":"This is a 1/2 annulus because,"},{"Start":"08:40.820 ","End":"08:44.205","Text":"first of all, between 2 circles,"},{"Start":"08:44.205 ","End":"08:46.695","Text":"this is 3 squared and this is 1 squared."},{"Start":"08:46.695 ","End":"08:54.880","Text":"If I write here 3 and here 1 and here minus 1 and here minus 3,"},{"Start":"08:54.880 ","End":"08:58.940","Text":"where between the circle of radius"},{"Start":"08:58.940 ","End":"09:03.980","Text":"3 and the circle of radius 1 outside the 1 inside the other."},{"Start":"09:03.980 ","End":"09:09.440","Text":"But this bigger or equal to 0 is what makes it the upper 1/2."},{"Start":"09:09.440 ","End":"09:14.465","Text":"It includes the boundary and perhaps I\u0027ll shade the interior,"},{"Start":"09:14.465 ","End":"09:21.220","Text":"and we\u0027ll label it D. Now in some sense,"},{"Start":"09:21.220 ","End":"09:24.280","Text":"part B is a peculiar question because actually,"},{"Start":"09:24.280 ","End":"09:31.540","Text":"I\u0027m not going to use almost any of the properties or the equations that define the 1/2"},{"Start":"09:31.540 ","End":"09:39.770","Text":"annulus D. The only thing I really care about is that it doesn\u0027t contain this bad point."},{"Start":"09:39.770 ","End":"09:42.395","Text":"That the function F,"},{"Start":"09:42.395 ","End":"09:46.795","Text":"the vector field is defined and continuous everywhere in it."},{"Start":"09:46.795 ","End":"09:51.125","Text":"We\u0027ve lost F. Let\u0027s go back and see what F was."},{"Start":"09:51.125 ","End":"09:56.155","Text":"Here it is, perhaps I should copy it again."},{"Start":"09:56.155 ","End":"10:02.320","Text":"What we had up there was that F of x,y, the vector field,"},{"Start":"10:02.320 ","End":"10:10.920","Text":"was equal to P times i plus Q times j,"},{"Start":"10:10.920 ","End":"10:12.410","Text":"it wasn\u0027t written in that form,"},{"Start":"10:12.410 ","End":"10:14.390","Text":"but I wanted to just separate it,"},{"Start":"10:14.390 ","End":"10:24.545","Text":"where P was equal to minus y over"},{"Start":"10:24.545 ","End":"10:29.470","Text":"x squared plus y squared and Q of"},{"Start":"10:29.470 ","End":"10:36.180","Text":"x,y was equal to x over x squared plus y squared."},{"Start":"10:36.180 ","End":"10:42.630","Text":"It\u0027s defined everywhere on D. In order for it to be conservative,"},{"Start":"10:43.750 ","End":"10:50.750","Text":"the theorem that all I have to do is show that the partial derivative of"},{"Start":"10:50.750 ","End":"10:57.080","Text":"P with respect to y is equal to the partial derivative of Q with respect to x."},{"Start":"10:57.080 ","End":"10:58.985","Text":"This is what I want to check,"},{"Start":"10:58.985 ","End":"11:01.780","Text":"and then we\u0027ll be all set."},{"Start":"11:01.780 ","End":"11:06.885","Text":"First, P with respect to y is equal to,"},{"Start":"11:06.885 ","End":"11:10.770","Text":"using the quotient rule,"},{"Start":"11:10.770 ","End":"11:15.650","Text":"we have the denominator squared here,"},{"Start":"11:15.650 ","End":"11:22.805","Text":"and then we have the derivative of the numerator,"},{"Start":"11:22.805 ","End":"11:29.750","Text":"which is minus 1 times the denominator,"},{"Start":"11:29.750 ","End":"11:35.750","Text":"x squared plus y squared minus numerator,"},{"Start":"11:35.750 ","End":"11:41.209","Text":"which is minus y times"},{"Start":"11:41.209 ","End":"11:50.395","Text":"the derivative of the denominator, which is 2y."},{"Start":"11:50.395 ","End":"11:54.155","Text":"This comes out to be, well, let\u0027s see."},{"Start":"11:54.155 ","End":"12:00.785","Text":"We have minus x squared minus y squared plus 2y squared."},{"Start":"12:00.785 ","End":"12:07.595","Text":"Altogether, y squared minus x squared here and the same denominator."},{"Start":"12:07.595 ","End":"12:12.225","Text":"The other 1, Q with respect to x."},{"Start":"12:12.225 ","End":"12:13.830","Text":"Again, with the quotient rule,"},{"Start":"12:13.830 ","End":"12:16.190","Text":"but remember this time it\u0027s with respect to x."},{"Start":"12:16.190 ","End":"12:19.355","Text":"We get derivative of the numerator,"},{"Start":"12:19.355 ","End":"12:23.435","Text":"which is 1 times the denominator"},{"Start":"12:23.435 ","End":"12:30.350","Text":"minus numerator times derivative of the denominator,"},{"Start":"12:30.350 ","End":"12:33.835","Text":"which is this time 2x,"},{"Start":"12:33.835 ","End":"12:41.380","Text":"and all this over denominator x squared plus y squared squared,"},{"Start":"12:41.380 ","End":"12:46.970","Text":"and this is equal to x squared plus y squared minus 2x squared,"},{"Start":"12:46.970 ","End":"12:49.295","Text":"so it comes out to be the same thing,"},{"Start":"12:49.295 ","End":"12:56.080","Text":"y squared minus x squared over x squared plus y squared squared."},{"Start":"12:56.080 ","End":"13:06.990","Text":"Clearly, these 2 are equal and that means that F is conservative on"},{"Start":"13:06.990 ","End":"13:11.450","Text":"D. I\u0027ll write it again on D."},{"Start":"13:11.450 ","End":"13:15.860","Text":"The only thing I used about D is that it bypasses the bad points"},{"Start":"13:15.860 ","End":"13:19.670","Text":"and F is defined and continuous and differentiable"},{"Start":"13:19.670 ","End":"13:26.140","Text":"everywhere in D. That\u0027s it for part B and we\u0027re done."}],"ID":8766}],"Thumbnail":null,"ID":4965}]

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