Directional Derivatives
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[{"Name":"Directional Derivatives","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Directional derivatives 1","Duration":"7m 56s","ChapterTopicVideoID":8638,"CourseChapterTopicPlaylistID":4960,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8638.jpeg","UploadDate":"2020-02-26T11:53:18.0500000","DurationForVideoObject":"PT7M56S","Description":null,"MetaTitle":"Directional derivatives 1: Video + Workbook | Proprep","MetaDescription":"Directional Drivatives - Directional Derivatives. 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/directional-drivatives/directional-derivatives/vid8984","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.870","Text":"In this clip, we\u0027ll learn about directional derivatives,"},{"Start":"00:03.870 ","End":"00:08.760","Text":"which are an extension of the concept of partial derivatives."},{"Start":"00:08.760 ","End":"00:10.065","Text":"That\u0027s 1 way of looking at it."},{"Start":"00:10.065 ","End":"00:14.790","Text":"Anyway, we\u0027re going to mostly be concentrating on 2D and in 3D,"},{"Start":"00:14.790 ","End":"00:18.585","Text":"although this exists in any number of dimensions."},{"Start":"00:18.585 ","End":"00:26.355","Text":"Let me first start by a quick refresher on the concept of a gradient of a function."},{"Start":"00:26.355 ","End":"00:31.395","Text":"There\u0027s a gradient, and this is the symbol that we\u0027re going to use."},{"Start":"00:31.395 ","End":"00:34.785","Text":"That\u0027s called Grad or Del or Nabla."},{"Start":"00:34.785 ","End":"00:36.030","Text":"There\u0027s different names for it."},{"Start":"00:36.030 ","End":"00:37.605","Text":"We\u0027ll just say Grad."},{"Start":"00:37.605 ","End":"00:46.865","Text":"I want to remind you in 2D that if I have a function f of x and y, a scalar function,"},{"Start":"00:46.865 ","End":"00:49.280","Text":"each x and y gives us a number,"},{"Start":"00:49.280 ","End":"00:56.160","Text":"then we get from this a vector function called Grad f,"},{"Start":"00:56.160 ","End":"00:58.890","Text":"and what it does to x and y,"},{"Start":"00:58.890 ","End":"01:01.445","Text":"is it gives us a 2D vector,"},{"Start":"01:01.445 ","End":"01:09.430","Text":"which is the derivative with respect to x. I\u0027ll write at x and y,"},{"Start":"01:09.430 ","End":"01:11.035","Text":"although this is often omitted,"},{"Start":"01:11.035 ","End":"01:16.730","Text":"and this will be in the i direction plus derivative of f with"},{"Start":"01:16.730 ","End":"01:22.850","Text":"respect to y at that point in the j direction."},{"Start":"01:22.850 ","End":"01:26.950","Text":"Of course, we can write it with the other notation with brackets."},{"Start":"01:26.950 ","End":"01:28.915","Text":"I\u0027ll leave it like this."},{"Start":"01:28.915 ","End":"01:31.910","Text":"I\u0027ll mention that in 3D,"},{"Start":"01:31.910 ","End":"01:37.150","Text":"if we have f of x, y, and z,"},{"Start":"01:37.150 ","End":"01:41.430","Text":"then we also get a vector function,"},{"Start":"01:41.430 ","End":"01:45.625","Text":"this time in 3D then Grad f of x,"},{"Start":"01:45.625 ","End":"01:51.265","Text":"y and z will equal f with respect to x."},{"Start":"01:51.265 ","End":"01:54.925","Text":"You know what? I\u0027m going to skip the xyz,"},{"Start":"01:54.925 ","End":"01:59.530","Text":"although it is of xyz in the i direction,"},{"Start":"01:59.530 ","End":"02:05.430","Text":"plus f_y of xyz in the j direction,"},{"Start":"02:05.430 ","End":"02:10.100","Text":"plus the last component,"},{"Start":"02:10.100 ","End":"02:15.000","Text":"f with respect to z in the k direction,"},{"Start":"02:15.000 ","End":"02:18.645","Text":"and it works in any number of variables."},{"Start":"02:18.645 ","End":"02:22.480","Text":"We\u0027ll just be focusing on 2D and 3D."},{"Start":"02:23.660 ","End":"02:27.195","Text":"I\u0027m going to start out mostly in 2D."},{"Start":"02:27.195 ","End":"02:30.095","Text":"There\u0027s another concept I want to talk about,"},{"Start":"02:30.095 ","End":"02:38.355","Text":"that is the concept of a direction vector."},{"Start":"02:38.355 ","End":"02:44.080","Text":"But not only that, I also want to talk about a unit direction vector."},{"Start":"02:44.080 ","End":"02:45.680","Text":"First of all, direction vector,"},{"Start":"02:45.680 ","End":"02:47.855","Text":"then we\u0027ll introduce the word unit."},{"Start":"02:47.855 ","End":"02:51.019","Text":"Well, direction vector is just a vector in 2D,"},{"Start":"02:51.019 ","End":"02:52.340","Text":"and I\u0027ll give an example."},{"Start":"02:52.340 ","End":"02:54.320","Text":"Let\u0027s suppose I have,"},{"Start":"02:54.320 ","End":"02:59.600","Text":"say, vector v is equal to,"},{"Start":"02:59.600 ","End":"03:10.290","Text":"let\u0027s say, I\u0027ll use the i-j notation, 2i plus 3j."},{"Start":"03:10.290 ","End":"03:11.820","Text":"It\u0027s a certain vector in the plane,"},{"Start":"03:11.820 ","End":"03:15.065","Text":"and it has a directional vectors have magnitude and direction."},{"Start":"03:15.065 ","End":"03:19.715","Text":"The thing is that the same direction is shared by many vectors,"},{"Start":"03:19.715 ","End":"03:24.940","Text":"because if I take constant k times v,"},{"Start":"03:24.940 ","End":"03:28.710","Text":"but it has to be a positive constant,"},{"Start":"03:29.170 ","End":"03:32.810","Text":"then we also get a vector in the same direction."},{"Start":"03:32.810 ","End":"03:35.105","Text":"It could be 3v,"},{"Start":"03:35.105 ","End":"03:39.665","Text":"100v, it would just be longer but still have the same direction."},{"Start":"03:39.665 ","End":"03:44.150","Text":"The reason k has to be positive is if I take a negative k, for example,"},{"Start":"03:44.150 ","End":"03:47.300","Text":"minus v, it\u0027s on the same line,"},{"Start":"03:47.300 ","End":"03:49.715","Text":"but it\u0027s in the opposite direction."},{"Start":"03:49.715 ","End":"03:53.450","Text":"I\u0027m just giving a brief review of vectors."},{"Start":"03:53.450 ","End":"03:56.555","Text":"So when we take a direction vector,"},{"Start":"03:56.555 ","End":"03:59.765","Text":"an equivalent 1 would be a constant times it."},{"Start":"03:59.765 ","End":"04:03.360","Text":"Since there so many, we want to choose 1 specific 1,"},{"Start":"04:03.360 ","End":"04:06.800","Text":"and it\u0027s customary to choose the unit direction vector,"},{"Start":"04:06.800 ","End":"04:08.920","Text":"the 1 with length 1."},{"Start":"04:08.920 ","End":"04:10.970","Text":"The way we get that,"},{"Start":"04:10.970 ","End":"04:19.440","Text":"we sometimes use the notation a little hat over the letter,"},{"Start":"04:19.440 ","End":"04:22.260","Text":"it\u0027s sometimes called a caret, C-A-R-E-T,"},{"Start":"04:22.260 ","End":"04:23.280","Text":"and on the keyboard,"},{"Start":"04:23.280 ","End":"04:25.275","Text":"it\u0027s usually above the 6."},{"Start":"04:25.275 ","End":"04:28.855","Text":"Anyway, I\u0027ll say v-hat sometimes or v-caret."},{"Start":"04:28.855 ","End":"04:33.710","Text":"This is equal to the vector v divided"},{"Start":"04:33.710 ","End":"04:39.410","Text":"by the magnitude of v. We take any vector and divide it by its magnitude,"},{"Start":"04:39.410 ","End":"04:42.730","Text":"then we get a unit vector,"},{"Start":"04:42.730 ","End":"04:46.130","Text":"and it\u0027s in the same direction because the magnitude is positive,"},{"Start":"04:46.130 ","End":"04:48.050","Text":"so we\u0027re dividing by a positive number,"},{"Start":"04:48.050 ","End":"04:51.020","Text":"so k is like 1 over a positive number."},{"Start":"04:51.020 ","End":"04:52.805","Text":"That\u0027s a unit vector."},{"Start":"04:52.805 ","End":"04:58.535","Text":"Of course, the magnitude of this unit vector would equal 1."},{"Start":"04:58.535 ","End":"05:05.365","Text":"Now I\u0027m going to define a directional derivative in the direction of a unit vector."},{"Start":"05:05.365 ","End":"05:07.670","Text":"Now, before I give you the recipe,"},{"Start":"05:07.670 ","End":"05:09.170","Text":"let me give you the ingredients."},{"Start":"05:09.170 ","End":"05:11.240","Text":"What do we need for a directional derivative?"},{"Start":"05:11.240 ","End":"05:18.360","Text":"We need a function of 2 variables in the case of 2D, later 3D."},{"Start":"05:18.360 ","End":"05:25.540","Text":"I need a unit vector."},{"Start":"05:27.820 ","End":"05:30.950","Text":"This is enough for doing it in general,"},{"Start":"05:30.950 ","End":"05:33.410","Text":"but often we want the directional derivative of"},{"Start":"05:33.410 ","End":"05:37.055","Text":"a function in a certain direction at a given point."},{"Start":"05:37.055 ","End":"05:41.170","Text":"Sometimes we\u0027re given also a specific point,"},{"Start":"05:41.170 ","End":"05:45.870","Text":"and then, let me do it in specific and then in general."},{"Start":"05:46.100 ","End":"05:52.360","Text":"We define the directional derivative,"},{"Start":"05:52.360 ","End":"06:00.505","Text":"capital D in the direction u of the function f,"},{"Start":"06:00.505 ","End":"06:04.415","Text":"at the point x Naught, y Naught,"},{"Start":"06:04.415 ","End":"06:13.015","Text":"is going to be defined as the gradient of f at the point x Naught,"},{"Start":"06:13.015 ","End":"06:17.400","Text":"y Naught, dot product,"},{"Start":"06:17.400 ","End":"06:19.845","Text":"and I hope you remember dot product,"},{"Start":"06:19.845 ","End":"06:25.330","Text":"with the unit vector u."},{"Start":"06:25.330 ","End":"06:28.130","Text":"This is a unit vector."},{"Start":"06:28.820 ","End":"06:32.230","Text":"Well, I\u0027d say, if we wear it a little hat over it,"},{"Start":"06:32.230 ","End":"06:33.850","Text":"it means automatically unit."},{"Start":"06:33.850 ","End":"06:37.450","Text":"In general, we don\u0027t write it like this,"},{"Start":"06:37.450 ","End":"06:39.280","Text":"we don\u0027t often want it as a specific point,"},{"Start":"06:39.280 ","End":"06:40.600","Text":"but more in general."},{"Start":"06:40.600 ","End":"06:45.350","Text":"So I would just write that derivative in"},{"Start":"06:45.350 ","End":"06:52.620","Text":"the direction u of f is equal to Grad f.u,"},{"Start":"06:52.850 ","End":"06:56.650","Text":"the more condensed formula."},{"Start":"06:56.650 ","End":"06:59.720","Text":"I\u0027ll highlight the more condensed form."},{"Start":"06:59.720 ","End":"07:01.985","Text":"Although this works at any given point,"},{"Start":"07:01.985 ","End":"07:03.140","Text":"we plug in the point here,"},{"Start":"07:03.140 ","End":"07:05.555","Text":"you just plug in the point here."},{"Start":"07:05.555 ","End":"07:09.710","Text":"This actually works in any number of dimensions."},{"Start":"07:09.710 ","End":"07:14.825","Text":"Here I did a 2D version. That\u0027s the 2D."},{"Start":"07:14.825 ","End":"07:18.080","Text":"But I could also write 1 in 3D,"},{"Start":"07:18.080 ","End":"07:22.610","Text":"that the directional derivative in the direction of"},{"Start":"07:22.610 ","End":"07:28.310","Text":"a 3-dimensional unit vector of a function of 3 variables at a given point,"},{"Start":"07:28.310 ","End":"07:29.570","Text":"x Naught, y Naught,"},{"Start":"07:29.570 ","End":"07:35.190","Text":"z Naught is equal to the Grad of f at the point x Naught,"},{"Start":"07:35.190 ","End":"07:37.635","Text":"y Naught, z Naught,"},{"Start":"07:37.635 ","End":"07:41.295","Text":"dot product with u."},{"Start":"07:41.295 ","End":"07:43.855","Text":"That\u0027s the 3D version."},{"Start":"07:43.855 ","End":"07:46.340","Text":"We\u0027ll just remember it in general."},{"Start":"07:46.340 ","End":"07:50.089","Text":"Directional derivative give a direction and a function,"},{"Start":"07:50.089 ","End":"07:54.170","Text":"it\u0027s the gradient of the function dot product with the unit vector."},{"Start":"07:54.170 ","End":"07:57.269","Text":"We\u0027ll go straight to an example."}],"ID":8984},{"Watched":false,"Name":"Directional derivatives 2","Duration":"5m 19s","ChapterTopicVideoID":8639,"CourseChapterTopicPlaylistID":4960,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.770 ","End":"00:06.250","Text":"Let\u0027s see, time for the example, I\u0027ll just scroll down a bit."},{"Start":"00:07.040 ","End":"00:10.560","Text":"This is going to be a typical example of a question."},{"Start":"00:10.560 ","End":"00:16.635","Text":"I\u0027ll say, find the directional derivative of"},{"Start":"00:16.635 ","End":"00:23.280","Text":"the function f in 2 variables,"},{"Start":"00:23.280 ","End":"00:26.400","Text":"f of x, y equals x squared y."},{"Start":"00:26.400 ","End":"00:31.905","Text":"I\u0027m going to give you 2 more pieces of information at which point?"},{"Start":"00:31.905 ","End":"00:40.840","Text":"At the point, 1,2 and then I need to give you the direction."},{"Start":"00:40.970 ","End":"00:46.980","Text":"In the direction of,"},{"Start":"00:46.980 ","End":"00:52.365","Text":"let\u0027s say 3i plus 4j,"},{"Start":"00:52.365 ","End":"00:57.840","Text":"let\u0027s put some arrows and"},{"Start":"00:57.840 ","End":"01:04.620","Text":"all I really need is this formula or maybe better the longer formula if they were in 2D."},{"Start":"01:04.620 ","End":"01:08.970","Text":"So this formula here."},{"Start":"01:08.970 ","End":"01:13.050","Text":"Let\u0027s see what I have and what I don\u0027t have, x naught,"},{"Start":"01:13.050 ","End":"01:17.160","Text":"y naught we have is 1,2 we have f,"},{"Start":"01:17.160 ","End":"01:26.175","Text":"but we need to compute grad f. Also, we don\u0027t have unit vector u,"},{"Start":"01:26.175 ","End":"01:30.000","Text":"we have this direction vector we have"},{"Start":"01:30.000 ","End":"01:33.965","Text":"the convert it to a unit vector using this formula here."},{"Start":"01:33.965 ","End":"01:36.310","Text":"That\u0027s to the easy stuff first."},{"Start":"01:36.310 ","End":"01:42.245","Text":"My vector u will be this vector here,"},{"Start":"01:42.245 ","End":"01:51.435","Text":"3i plus 4j and I need to divide it by the magnitude of this,"},{"Start":"01:51.435 ","End":"01:54.110","Text":"I\u0027ll just go straight away to the definition of"},{"Start":"01:54.110 ","End":"01:57.080","Text":"the magnitude of this thing the computation is going to be"},{"Start":"01:57.080 ","End":"02:03.270","Text":"the square root of 3 squared plus 4 squared and this will come out,"},{"Start":"02:03.270 ","End":"02:04.440","Text":"we do this mentally look,"},{"Start":"02:04.440 ","End":"02:06.090","Text":"9 plus 16 is 25,"},{"Start":"02:06.090 ","End":"02:07.710","Text":"square root is 5."},{"Start":"02:07.710 ","End":"02:18.300","Text":"3 over 5, 4 over 5 so I get 3/5i plus 4/5j, that\u0027s the u."},{"Start":"02:18.300 ","End":"02:22.470","Text":"Now, I need grad f,"},{"Start":"02:22.470 ","End":"02:24.015","Text":"so let\u0027s do that."},{"Start":"02:24.015 ","End":"02:28.170","Text":"Grad f is equal to."},{"Start":"02:28.170 ","End":"02:34.065","Text":"The definition of somewhere above, but remember, take the derivative with respect to x."},{"Start":"02:34.065 ","End":"02:40.090","Text":"Well, I\u0027ll write that. It\u0027s f with respect to xi plus f with respect"},{"Start":"02:40.090 ","End":"02:46.015","Text":"to y j and this is equal to where are we?"},{"Start":"02:46.015 ","End":"02:50.300","Text":"Here\u0027s f, with respect to x is 2xy,"},{"Start":"02:50.670 ","End":"02:55.180","Text":"y is the constant, and with respect to y,"},{"Start":"02:55.180 ","End":"02:57.025","Text":"it\u0027s just x squared,"},{"Start":"02:57.025 ","End":"02:59.305","Text":"so it\u0027s x squared j."},{"Start":"02:59.305 ","End":"03:03.610","Text":"Now, what I want is not the general grad f,"},{"Start":"03:03.610 ","End":"03:07.670","Text":"I need grad f at the point."},{"Start":"03:08.330 ","End":"03:12.690","Text":"Grad f at the point, where are we?"},{"Start":"03:12.690 ","End":"03:17.820","Text":"1,2 is going to equal x is 1,"},{"Start":"03:17.820 ","End":"03:21.974","Text":"that\u0027s the substitute for x_1 and for y,"},{"Start":"03:21.974 ","End":"03:26.415","Text":"substitute 2, so we get 2xy,"},{"Start":"03:26.415 ","End":"03:31.650","Text":"2 times 1 times 2 is 4,"},{"Start":"03:31.650 ","End":"03:37.860","Text":"i and x squared is 1 squared is 1 plus j."},{"Start":"03:37.860 ","End":"03:48.915","Text":"I have this bit grad f at the point that\u0027s this one, and I have"},{"Start":"03:48.915 ","End":"03:52.350","Text":"u the unit vector which is this"},{"Start":"03:52.350 ","End":"03:56.525","Text":"and all I have to do now is dot product these two together."},{"Start":"03:56.525 ","End":"03:59.340","Text":"Our answer will be"},{"Start":"04:01.750 ","End":"04:07.110","Text":"3/5i plus 4/5j."},{"Start":"04:07.110 ","End":"04:10.110","Text":"Well, I did it backwards, it doesn\u0027t matter the dot product"},{"Start":"04:10.110 ","End":"04:15.930","Text":"is or maybe here there\u0027s a slit it over"},{"Start":"04:15.930 ","End":"04:24.315","Text":"dots, I\u0027m working backwards, and here, it\u0027s this one here for i plus j"},{"Start":"04:24.315 ","End":"04:33.900","Text":"and that\u0027s the dot product that I need from here,"},{"Start":"04:33.900 ","End":"04:37.770","Text":"this dot product or this dot product."},{"Start":"04:37.770 ","End":"04:42.105","Text":"We just multiply 4 times 3/5 is,"},{"Start":"04:42.105 ","End":"04:46.650","Text":"I\u0027ll just write it 4 times 3/5 plus this is 1 j,"},{"Start":"04:46.650 ","End":"04:49.790","Text":"so it\u0027s 1 times 4/5."},{"Start":"04:49.790 ","End":"04:51.530","Text":"Let\u0027s see if I put it all over 5,"},{"Start":"04:51.530 ","End":"04:59.210","Text":"I have 4 times 3 is 12 plus 4 is 16 over 5 and I could leave it like that,"},{"Start":"04:59.210 ","End":"05:02.700","Text":"or I could do it in decimal,"},{"Start":"05:02.700 ","End":"05:05.040","Text":"5 into 16 goes 3,"},{"Start":"05:05.040 ","End":"05:13.560","Text":"3/5, 3.2 or 3/5."},{"Start":"05:13.560 ","End":"05:16.970","Text":"I\u0027ll highlight this from the improper fraction,"},{"Start":"05:16.970 ","End":"05:20.280","Text":"and that\u0027s the answer."}],"ID":8985},{"Watched":false,"Name":"Directional derivatives 3","Duration":"6m 40s","ChapterTopicVideoID":8640,"CourseChapterTopicPlaylistID":4960,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.220","Text":"Now I\u0027m basically done,"},{"Start":"00:05.220 ","End":"00:08.070","Text":"except I\u0027m going to continue with an optional part,"},{"Start":"00:08.070 ","End":"00:10.470","Text":"which is to give you some intuition with"},{"Start":"00:10.470 ","End":"00:15.660","Text":"diagrams of what the directional derivative might mean."},{"Start":"00:15.660 ","End":"00:19.845","Text":"If you\u0027re comfortable with just formulas and without any visual aid,"},{"Start":"00:19.845 ","End":"00:24.395","Text":"then you can skip the rest of this."},{"Start":"00:24.395 ","End":"00:27.305","Text":"But for those who would like to get some intuition,"},{"Start":"00:27.305 ","End":"00:29.225","Text":"I\u0027m going to continue."},{"Start":"00:29.225 ","End":"00:31.845","Text":"I\u0027ll just clear some space here,"},{"Start":"00:31.845 ","End":"00:36.080","Text":"maybe I\u0027ll try and keep this definition meanwhile."},{"Start":"00:36.080 ","End":"00:39.035","Text":"Before bringing the picture,"},{"Start":"00:39.035 ","End":"00:43.470","Text":"let\u0027s look again at this definition here,"},{"Start":"00:44.270 ","End":"00:47.315","Text":"the what grad f is."},{"Start":"00:47.315 ","End":"00:56.765","Text":"Notice that if I take grad f and dot product it,"},{"Start":"00:56.765 ","End":"01:00.710","Text":"instead of with the unit vector u,"},{"Start":"01:00.710 ","End":"01:04.390","Text":"with the vector i,"},{"Start":"01:04.390 ","End":"01:10.290","Text":"what I get is just f with respect to x,"},{"Start":"01:10.290 ","End":"01:11.565","Text":"but I\u0027ll spell it out."},{"Start":"01:11.565 ","End":"01:16.770","Text":"It\u0027s f with respect to xi plus f with"},{"Start":"01:16.770 ","End":"01:22.940","Text":"respect to yj dot product with i,"},{"Start":"01:22.940 ","End":"01:29.315","Text":"but i is just to spell it out 1i plus 0j."},{"Start":"01:29.315 ","End":"01:32.180","Text":"If I multiply this with this plus this with this,"},{"Start":"01:32.180 ","End":"01:36.565","Text":"basically I just get f with respect to x."},{"Start":"01:36.565 ","End":"01:42.365","Text":"Similarly, if I take grad f dot product with j,"},{"Start":"01:42.365 ","End":"01:46.295","Text":"I just get the derivative of f with respect to y."},{"Start":"01:46.295 ","End":"01:53.700","Text":"Now, i and j are unit vectors."},{"Start":"01:57.110 ","End":"02:02.195","Text":"I could have written them with a hat over them like I did with the u."},{"Start":"02:02.195 ","End":"02:04.040","Text":"What I\u0027m saying is,"},{"Start":"02:04.040 ","End":"02:09.155","Text":"is that we have 2 directional vectors that we already know."},{"Start":"02:09.155 ","End":"02:18.340","Text":"The directional vector in the direction of i is the partial derivative with respect to x,"},{"Start":"02:18.340 ","End":"02:22.400","Text":"and the directional derivative in the direction of j, or if you like,"},{"Start":"02:22.400 ","End":"02:24.545","Text":"in the direction of the positive y-axis,"},{"Start":"02:24.545 ","End":"02:28.430","Text":"is the partial derivative with respect to y."},{"Start":"02:28.430 ","End":"02:31.610","Text":"Now, i and j are not the only unit vectors,"},{"Start":"02:31.610 ","End":"02:33.620","Text":"so this generalizes it instead of in"},{"Start":"02:33.620 ","End":"02:37.825","Text":"the x direction and in the y direction to any direction."},{"Start":"02:37.825 ","End":"02:44.860","Text":"I\u0027m going to bring in a picture to illustrate the partial derivative in the x direction."},{"Start":"02:44.860 ","End":"02:47.285","Text":"Here is the picture."},{"Start":"02:47.285 ","End":"02:49.220","Text":"Just scroll a bit."},{"Start":"02:49.220 ","End":"02:54.380","Text":"What we see here is a certain function."},{"Start":"02:54.380 ","End":"02:55.610","Text":"It doesn\u0027t matter what it is."},{"Start":"02:55.610 ","End":"02:58.115","Text":"Sum of f of xy is the surface,"},{"Start":"02:58.115 ","End":"03:01.175","Text":"the x-axis, y-axis, and z-axis."},{"Start":"03:01.175 ","End":"03:05.870","Text":"What we do is we have a certain point,"},{"Start":"03:05.870 ","End":"03:10.095","Text":"that would be the point that\u0027s directly below,"},{"Start":"03:10.095 ","End":"03:16.135","Text":"I imagine that this would be the point and somewhere directly below it."},{"Start":"03:16.135 ","End":"03:19.055","Text":"You just go down vertically."},{"Start":"03:19.055 ","End":"03:24.140","Text":"This is probably somewhere here is the point x,"},{"Start":"03:24.140 ","End":"03:28.610","Text":"y and we take"},{"Start":"03:28.610 ","End":"03:35.209","Text":"a cross-section in the direction of the x-axis."},{"Start":"03:35.209 ","End":"03:38.419","Text":"In other words, it\u0027s parallel to"},{"Start":"03:38.419 ","End":"03:45.320","Text":"the zx-axis and it"},{"Start":"03:45.320 ","End":"03:50.760","Text":"cuts the surface in the curve."},{"Start":"03:53.690 ","End":"03:59.875","Text":"Well, it\u0027s just the slope of the tangent and the x direction,"},{"Start":"03:59.875 ","End":"04:02.215","Text":"this is in the way,"},{"Start":"04:02.215 ","End":"04:11.610","Text":"is in this direction 1 unit that would be a vector i. I put a hat on,"},{"Start":"04:11.610 ","End":"04:12.940","Text":"it, I didn\u0027t mean to,"},{"Start":"04:12.940 ","End":"04:16.495","Text":"but actually it\u0027s quite right because i is a unit vector."},{"Start":"04:16.495 ","End":"04:24.355","Text":"Similarly, if we did a cross-section in the direction of y parallel to the yz,"},{"Start":"04:24.355 ","End":"04:28.395","Text":"then we\u0027d get also a curve and"},{"Start":"04:28.395 ","End":"04:33.250","Text":"a tangent and the slope would be the partial derivative with respect to y."},{"Start":"04:33.250 ","End":"04:40.055","Text":"But the thing is that we don\u0027t have to just take directions i and j,"},{"Start":"04:40.055 ","End":"04:43.370","Text":"we could take any unit vector."},{"Start":"04:43.370 ","End":"04:45.560","Text":"That\u0027s the generalization."},{"Start":"04:45.560 ","End":"04:50.600","Text":"Let\u0027s see if I can find another picture from a different source."},{"Start":"04:50.600 ","End":"04:51.980","Text":"I just searched the Internet."},{"Start":"04:51.980 ","End":"04:54.410","Text":"You can find lots of stuff there."},{"Start":"04:54.410 ","End":"04:59.225","Text":"Well, we had, here\u0027s the x, y plane."},{"Start":"04:59.225 ","End":"05:01.790","Text":"I called it x nought, y nought,"},{"Start":"05:01.790 ","End":"05:03.290","Text":"here it\u0027s called p, q."},{"Start":"05:03.290 ","End":"05:04.430","Text":"It doesn\u0027t matter."},{"Start":"05:04.430 ","End":"05:06.155","Text":"We have the surface,"},{"Start":"05:06.155 ","End":"05:08.435","Text":"z is a function of x and y."},{"Start":"05:08.435 ","End":"05:09.920","Text":"We have a certain point,"},{"Start":"05:09.920 ","End":"05:12.555","Text":"let me just emphasize it."},{"Start":"05:12.555 ","End":"05:16.010","Text":"Then we have a point above it."},{"Start":"05:16.010 ","End":"05:22.390","Text":"I\u0027m highlighting, yeah, this and the point above it."},{"Start":"05:22.880 ","End":"05:27.184","Text":"This is the vector u, the unit vector."},{"Start":"05:27.184 ","End":"05:30.365","Text":"We take the plane that goes through"},{"Start":"05:30.365 ","End":"05:34.400","Text":"this line determined by the point and the vector and vertical,"},{"Start":"05:34.400 ","End":"05:37.440","Text":"meaning parallel to the z-axis also."},{"Start":"05:37.790 ","End":"05:43.679","Text":"Where this plane cuts the surface,"},{"Start":"05:43.679 ","End":"05:47.130","Text":"we get a curve and a point."},{"Start":"05:47.130 ","End":"05:51.000","Text":"Also the slope of the tangent here,"},{"Start":"05:51.000 ","End":"05:54.555","Text":"there\u0027s some tangent line here."},{"Start":"05:54.555 ","End":"05:57.380","Text":"It\u0027s not the greatest picture,"},{"Start":"05:57.380 ","End":"06:02.375","Text":"but the slope of this tangent line would be the directional derivative."},{"Start":"06:02.375 ","End":"06:08.570","Text":"In other words, instead of a vertical plane parallel to 1 of the 2 axes,"},{"Start":"06:08.570 ","End":"06:12.380","Text":"we take a vertical plane in any direction and we get"},{"Start":"06:12.380 ","End":"06:17.510","Text":"a curve and its slope of the tangent at that point is the directional derivative."},{"Start":"06:17.510 ","End":"06:22.905","Text":"You could see it as a generalization of partial derivatives,"},{"Start":"06:22.905 ","End":"06:24.800","Text":"so instead of just i and j,"},{"Start":"06:24.800 ","End":"06:27.325","Text":"we have any unit vector."},{"Start":"06:27.325 ","End":"06:29.564","Text":"I hope this helps,"},{"Start":"06:29.564 ","End":"06:34.070","Text":"but you can certainly get by without the intuition and the diagrams."},{"Start":"06:34.070 ","End":"06:36.620","Text":"It\u0027s just an extra teaching aid,"},{"Start":"06:36.620 ","End":"06:37.730","Text":"if you like it,"},{"Start":"06:37.730 ","End":"06:40.680","Text":"good. Okay, I\u0027m done."}],"ID":8986},{"Watched":false,"Name":"Exercise 1 part a","Duration":"4m 10s","ChapterTopicVideoID":8653,"CourseChapterTopicPlaylistID":4960,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.025","Text":"In this exercise, we\u0027re given a function of 2 variables here."},{"Start":"00:05.025 ","End":"00:07.500","Text":"We actually asked 3 things:"},{"Start":"00:07.500 ","End":"00:11.280","Text":"First of all, to compute the gradient of the function,"},{"Start":"00:11.280 ","End":"00:16.200","Text":"then to figure out its length at the point 3,"},{"Start":"00:16.200 ","End":"00:21.365","Text":"4 and to explain the results; what is its meaning?"},{"Start":"00:21.365 ","End":"00:23.685","Text":"Let\u0027s start with the gradient,"},{"Start":"00:23.685 ","End":"00:25.380","Text":"but I\u0027ll start even further back."},{"Start":"00:25.380 ","End":"00:29.040","Text":"Let\u0027s remember what it means in general."},{"Start":"00:29.040 ","End":"00:33.660","Text":"If we have a function f of x and y,"},{"Start":"00:33.660 ","End":"00:37.770","Text":"then the gradient of f sometimes is"},{"Start":"00:37.770 ","End":"00:43.895","Text":"written optionally with an arrow on top of it to remind us that it\u0027s actually a vector."},{"Start":"00:43.895 ","End":"00:45.650","Text":"F is a scalar function,"},{"Start":"00:45.650 ","End":"00:49.460","Text":"but this is a vector function or a vector operator"},{"Start":"00:49.460 ","End":"00:53.780","Text":"which takes a scalar function and gives us a vector function if you want to be precise."},{"Start":"00:53.780 ","End":"01:01.405","Text":"This thing is equal to the vector f with respect to x,"},{"Start":"01:01.405 ","End":"01:04.085","Text":"f with respect to y."},{"Start":"01:04.085 ","End":"01:06.155","Text":"Partial derivatives."},{"Start":"01:06.155 ","End":"01:09.230","Text":"There\u0027s other ways of writing the partial derivative, of course,"},{"Start":"01:09.230 ","End":"01:11.435","Text":"but I\u0027ll write it like this."},{"Start":"01:11.435 ","End":"01:21.105","Text":"In our case, we have f of x, I\u0027ll just repeat that is x squared plus y squared."},{"Start":"01:21.105 ","End":"01:25.360","Text":"Grad f, or Dell"},{"Start":"01:26.510 ","End":"01:33.360","Text":"at the point x, y in general is equal to, let\u0027s see."},{"Start":"01:33.360 ","End":"01:43.440","Text":"Df by dx here is 2x and df by dy is 2y."},{"Start":"01:43.440 ","End":"01:48.750","Text":"That\u0027s in general but at that specific point 3,"},{"Start":"01:48.750 ","End":"01:53.220","Text":"4, I just substitute grad f at the point 3,"},{"Start":"01:53.220 ","End":"01:55.440","Text":"4 is equal to,"},{"Start":"01:55.440 ","End":"01:57.220","Text":"I just substitute Instead of x,"},{"Start":"01:57.220 ","End":"02:00.340","Text":"I\u0027ll put in 3 and instead of y I\u0027ll put in 4,"},{"Start":"02:00.340 ","End":"02:04.940","Text":"so we get 6, 8."},{"Start":"02:04.940 ","End":"02:09.100","Text":"That\u0027s the gradient of f in general."},{"Start":"02:11.150 ","End":"02:15.235","Text":"The gradient of f at the point here is length."},{"Start":"02:15.235 ","End":"02:23.615","Text":"The length is sometimes denoted by this sign, magnitude."},{"Start":"02:23.615 ","End":"02:28.275","Text":"Magnitude or length is the same thing of 6,"},{"Start":"02:28.275 ","End":"02:33.025","Text":"8 is equal to, there\u0027s a formula,"},{"Start":"02:33.025 ","End":"02:34.810","Text":"just write it at the side in general,"},{"Start":"02:34.810 ","End":"02:38.365","Text":"the magnitude at least in 2 dimensions of the vector a,"},{"Start":"02:38.365 ","End":"02:43.450","Text":"b is just the square root of a squared plus b squared."},{"Start":"02:43.450 ","End":"02:45.355","Text":"In 3D, if it\u0027s a, b, c,"},{"Start":"02:45.355 ","End":"02:47.470","Text":"it\u0027s a squared plus b squared plus c squared and so on."},{"Start":"02:47.470 ","End":"02:52.940","Text":"Here we have the square root of 6 squared plus 8 squared."},{"Start":"02:52.940 ","End":"02:54.980","Text":"We can do it in our head. 6 squared is 36,"},{"Start":"02:54.980 ","End":"03:00.470","Text":"this is 64, together it\u0027s 100, square root of a 100 is 10."},{"Start":"03:00.470 ","End":"03:03.950","Text":"Let me just highlight these results."},{"Start":"03:03.950 ","End":"03:07.460","Text":"That was the answer to the gradient at 3, 4,"},{"Start":"03:07.460 ","End":"03:11.180","Text":"this is its length."},{"Start":"03:11.180 ","End":"03:13.330","Text":"The meaning, I could say a lot,"},{"Start":"03:13.330 ","End":"03:19.730","Text":"but basically, what it is is that if we look at all possible directional derivatives,"},{"Start":"03:19.730 ","End":"03:26.665","Text":"which we write d_u of the function f at the point 3, 4,"},{"Start":"03:26.665 ","End":"03:36.515","Text":"then the greatest value of a directional derivative here is 10."},{"Start":"03:36.515 ","End":"03:44.010","Text":"This is achieved in the direction of the gradient vector,"},{"Start":"03:44.010 ","End":"03:47.400","Text":"which is 6, 8."},{"Start":"03:47.400 ","End":"03:50.190","Text":"If we go in this direction,"},{"Start":"03:50.190 ","End":"03:52.580","Text":"we would take a unit vector,"},{"Start":"03:52.580 ","End":"03:54.560","Text":"we would divide this by its size if we wanted"},{"Start":"03:54.560 ","End":"03:57.635","Text":"a unit vector, but in this general direction,"},{"Start":"03:57.635 ","End":"04:04.750","Text":"parallel to this, then we have the greatest value,"},{"Start":"04:04.750 ","End":"04:06.810","Text":"and it is 10."},{"Start":"04:06.810 ","End":"04:09.730","Text":"That\u0027s about as much as I\u0027m going to say."}],"ID":8987},{"Watched":false,"Name":"Exercise 1 part b","Duration":"5m 59s","ChapterTopicVideoID":8654,"CourseChapterTopicPlaylistID":4960,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.835","Text":"This exercise is part 2,"},{"Start":"00:02.835 ","End":"00:07.080","Text":"and it\u0027s continuation of the previous exercise where we already computed"},{"Start":"00:07.080 ","End":"00:13.080","Text":"the gradient to this function at this point 3, 4."},{"Start":"00:13.080 ","End":"00:15.450","Text":"Now we\u0027re asked, in addition,"},{"Start":"00:15.450 ","End":"00:21.120","Text":"to show that this gradient is normal to the level curve passing through that point."},{"Start":"00:21.120 ","End":"00:25.290","Text":"Let me just remind you what we had in the previous exercise."},{"Start":"00:25.290 ","End":"00:29.010","Text":"We had that the gradient, grad f,"},{"Start":"00:29.010 ","End":"00:32.160","Text":"came out in general to be 2x,"},{"Start":"00:32.160 ","End":"00:35.430","Text":"2y, the general point x, y."},{"Start":"00:35.430 ","End":"00:37.155","Text":"But in our specific case,"},{"Start":"00:37.155 ","End":"00:39.110","Text":"the grad f at the point 3,"},{"Start":"00:39.110 ","End":"00:42.600","Text":"4 came out to be 6,"},{"Start":"00:42.600 ","End":"00:46.095","Text":"8, which we got by substituting 3, 4 in here."},{"Start":"00:46.095 ","End":"00:48.620","Text":"Now let\u0027s address the matter of the level curve,"},{"Start":"00:48.620 ","End":"00:52.610","Text":"sometimes called a contour through 3, 4."},{"Start":"00:52.610 ","End":"00:56.270","Text":"First thing to do is to find the value of the function at 3, 4."},{"Start":"00:56.270 ","End":"00:59.210","Text":"So I need f of 3, 4."},{"Start":"00:59.210 ","End":"01:04.490","Text":"This is equal to 3 squared plus 4 squared, which is 25."},{"Start":"01:04.490 ","End":"01:05.600","Text":"I\u0027m not even going to write that."},{"Start":"01:05.600 ","End":"01:07.730","Text":"We can do it in our heads."},{"Start":"01:07.760 ","End":"01:14.960","Text":"The level curve is now going to be defined by f of x,"},{"Start":"01:14.960 ","End":"01:19.690","Text":"y equals 25, which,"},{"Start":"01:19.690 ","End":"01:28.210","Text":"in our case, is going to be x squared plus y squared equals 25."},{"Start":"01:28.210 ","End":"01:32.265","Text":"Now, 25 is 5 squared."},{"Start":"01:32.265 ","End":"01:37.645","Text":"This is an equation of a circle centered at the origin with radius 5."},{"Start":"01:37.645 ","End":"01:40.950","Text":"I\u0027ll bring in a quick sketch and some axes,"},{"Start":"01:40.950 ","End":"01:45.620","Text":"that\u0027s going to be the y-direction, the x-direction."},{"Start":"01:45.620 ","End":"01:52.140","Text":"Let\u0027s say this is 5 here and 5 here."},{"Start":"01:52.640 ","End":"01:56.430","Text":"Here\u0027s a circle and here\u0027s the point,"},{"Start":"01:56.430 ","End":"02:03.220","Text":"let\u0027s say, 3, 4 is here."},{"Start":"02:05.240 ","End":"02:13.520","Text":"This is actually the level curve of the function passing through this point."},{"Start":"02:13.520 ","End":"02:23.195","Text":"The gradient 6, 8 is a vector somewhere in this direction."},{"Start":"02:23.195 ","End":"02:27.545","Text":"That\u0027s the vector 6, 8."},{"Start":"02:27.545 ","End":"02:32.090","Text":"Now, what does it mean for this to be normal to this curve?"},{"Start":"02:32.090 ","End":"02:36.860","Text":"It means normal to the tangent line at this point."},{"Start":"02:36.860 ","End":"02:39.320","Text":"We have to find the tangent or at least its slope."},{"Start":"02:39.320 ","End":"02:42.050","Text":"We actually just need the slope of the tangent."},{"Start":"02:42.050 ","End":"02:45.590","Text":"The slope of the tangent is given by y prime."},{"Start":"02:45.590 ","End":"02:48.630","Text":"Let me just write. This is the tangent."},{"Start":"02:48.980 ","End":"03:00.950","Text":"The slope of the tangent at that point is given by the derivative y prime."},{"Start":"03:03.420 ","End":"03:09.790","Text":"Now, we have an implicit function here so we need to do an implicit differentiation."},{"Start":"03:09.790 ","End":"03:12.610","Text":"I suppose you could also extract y in terms of x,"},{"Start":"03:12.610 ","End":"03:17.185","Text":"but the most obvious thing to do is the implicit differentiation."},{"Start":"03:17.185 ","End":"03:20.470","Text":"From here, I\u0027ll get with respect to x,"},{"Start":"03:20.470 ","End":"03:22.705","Text":"derivative of x squared is 2_x."},{"Start":"03:22.705 ","End":"03:26.350","Text":"The derivative of y-squared is not just 2_y,"},{"Start":"03:26.350 ","End":"03:29.680","Text":"it\u0027s times y-prime because it\u0027s an implicit differentiation,"},{"Start":"03:29.680 ","End":"03:32.120","Text":"y is the function of x supposedly,"},{"Start":"03:32.120 ","End":"03:35.980","Text":"and then this will equal 0 because it\u0027s a constant."},{"Start":"03:35.980 ","End":"03:39.855","Text":"If we now extract y prime,"},{"Start":"03:39.855 ","End":"03:42.765","Text":"canceled by 2, of course."},{"Start":"03:42.765 ","End":"03:50.780","Text":"Then we get that y prime equals minus x and then divided by y, which is this."},{"Start":"03:50.780 ","End":"03:53.930","Text":"So y prime at the point 3,"},{"Start":"03:53.930 ","End":"03:59.410","Text":"4 is equal to minus 3 over 4."},{"Start":"03:59.410 ","End":"04:01.745","Text":"That\u0027s the slope of the tangent."},{"Start":"04:01.745 ","End":"04:03.260","Text":"That\u0027s my first result."},{"Start":"04:03.260 ","End":"04:05.460","Text":"I\u0027m going to highlight this."},{"Start":"04:06.100 ","End":"04:15.250","Text":"The second thing I want to compare with is this direction."},{"Start":"04:15.250 ","End":"04:18.590","Text":"What I need is the slope of the vector."},{"Start":"04:18.590 ","End":"04:20.080","Text":"What do I mean by slope of vector?"},{"Start":"04:20.080 ","End":"04:24.395","Text":"In 2D, the slope is just rise over run."},{"Start":"04:24.395 ","End":"04:27.120","Text":"It\u0027s this over this."},{"Start":"04:27.170 ","End":"04:31.635","Text":"If this is the vector 6, 8,"},{"Start":"04:31.635 ","End":"04:36.705","Text":"like this is 8 and this is 6,"},{"Start":"04:36.705 ","End":"04:38.920","Text":"it\u0027s not to scale."},{"Start":"04:40.700 ","End":"04:48.540","Text":"The slope of this is equal to 8 over 6,"},{"Start":"04:48.540 ","End":"04:50.340","Text":"and I can cancel,"},{"Start":"04:50.340 ","End":"04:52.755","Text":"make it 4 over 3."},{"Start":"04:52.755 ","End":"04:56.790","Text":"That\u0027s my second result. I\u0027m going to highlight it."},{"Start":"04:56.790 ","End":"04:57.890","Text":"Now I have 2 slopes,"},{"Start":"04:57.890 ","End":"05:02.105","Text":"the slope of the tangent and the slope of the vector."},{"Start":"05:02.105 ","End":"05:05.710","Text":"How do I know if these 2 are normal?"},{"Start":"05:05.710 ","End":"05:13.200","Text":"If this and this are perpendicular or this is normal to this."},{"Start":"05:13.200 ","End":"05:16.780","Text":"Are these 2 perpendicular?"},{"Start":"05:17.900 ","End":"05:24.119","Text":"There\u0027s a simple test for when 2 slopes are perpendicular,"},{"Start":"05:24.119 ","End":"05:26.730","Text":"the product has to be minus 1."},{"Start":"05:26.730 ","End":"05:31.410","Text":"Meaning, if we have m_1 for 1 slope and m_2 is the other slope,"},{"Start":"05:31.410 ","End":"05:34.560","Text":"and if m_1, m_2 is minus 1, then they\u0027re perpendicular."},{"Start":"05:34.560 ","End":"05:37.530","Text":"Let\u0027s try what it happens in our case."},{"Start":"05:37.530 ","End":"05:43.049","Text":"We get minus 3-quarters times 4 over 3,"},{"Start":"05:43.049 ","End":"05:46.110","Text":"and we get minus 12 over 12."},{"Start":"05:46.110 ","End":"05:48.420","Text":"This is equal to minus 1."},{"Start":"05:48.420 ","End":"05:55.820","Text":"Yes, this gradient really is normal to the contour i.e."},{"Start":"05:55.820 ","End":"05:59.640","Text":"to its tangent at that point. We\u0027re done."}],"ID":8988},{"Watched":false,"Name":"Exercise 2","Duration":"4m 18s","ChapterTopicVideoID":8655,"CourseChapterTopicPlaylistID":4960,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.365","Text":"In this exercise, we\u0027re given a function of 2 variables here."},{"Start":"00:04.365 ","End":"00:09.360","Text":"We have to compute the directional derivative of f at the point 1,"},{"Start":"00:09.360 ","End":"00:12.465","Text":"2, and then the direction of the vector,"},{"Start":"00:12.465 ","End":"00:16.065","Text":"3, 4 or 3i plus 4j."},{"Start":"00:16.065 ","End":"00:18.870","Text":"The thing is that directional derivative,"},{"Start":"00:18.870 ","End":"00:21.540","Text":"the formula is defined for a unit vector."},{"Start":"00:21.540 ","End":"00:25.520","Text":"This is not a unit vector but if it was a unit vector,"},{"Start":"00:25.520 ","End":"00:32.730","Text":"the directional derivative of a unit vector at the point 1,"},{"Start":"00:32.730 ","End":"00:37.220","Text":"2 would be given by the gradient of"},{"Start":"00:37.220 ","End":"00:43.580","Text":"the function f dot product with the unit vector."},{"Start":"00:43.580 ","End":"00:45.470","Text":"We need to compute 2 things."},{"Start":"00:45.470 ","End":"00:49.400","Text":"First of all, the unit vector that corresponds to this vector that\u0027s parallel to it,"},{"Start":"00:49.400 ","End":"00:51.819","Text":"the same direction and parallel."},{"Start":"00:51.819 ","End":"00:54.680","Text":"Then we going to get the gradient."},{"Start":"00:54.680 ","End":"00:56.780","Text":"Let\u0027s do the easy 1 first,"},{"Start":"00:56.780 ","End":"00:59.074","Text":"let\u0027s figure out the unit vector."},{"Start":"00:59.074 ","End":"01:02.374","Text":"To make a vector into a unit vector,"},{"Start":"01:02.374 ","End":"01:10.470","Text":"we just take the vector and we divide it by its length or magnitude."},{"Start":"01:10.540 ","End":"01:13.175","Text":"Now, the magnitude or view,"},{"Start":"01:13.175 ","End":"01:20.760","Text":"this part here is just the square root of 3 squared plus 4 squared,"},{"Start":"01:20.760 ","End":"01:23.550","Text":"9 plus 16 is 25,"},{"Start":"01:23.550 ","End":"01:27.420","Text":"so that comes out to be 5."},{"Start":"01:27.420 ","End":"01:34.300","Text":"What we get is that this is equal to 1/5 of,"},{"Start":"01:34.300 ","End":"01:39.250","Text":"where is it, yeah, 3i plus 4j."},{"Start":"01:40.250 ","End":"01:48.720","Text":"You know what? I actually like the angular brackets notation 3,4."},{"Start":"01:49.690 ","End":"01:59.160","Text":"This is equal to 3/5, 4/5."},{"Start":"01:59.160 ","End":"02:01.185","Text":"I think I\u0027ll move it down here."},{"Start":"02:01.185 ","End":"02:03.225","Text":"We\u0027ve got 1 part,"},{"Start":"02:03.225 ","End":"02:06.195","Text":"we\u0027ve got the u caret,"},{"Start":"02:06.195 ","End":"02:09.240","Text":"it\u0027s called, u with a hat on it."},{"Start":"02:09.240 ","End":"02:12.760","Text":"That\u0027s the unit vector in the same direction as this."},{"Start":"02:12.760 ","End":"02:14.715","Text":"Now, the other bit."},{"Start":"02:14.715 ","End":"02:16.870","Text":"Let\u0027s see, I\u0027ll do it over here."},{"Start":"02:16.870 ","End":"02:19.120","Text":"Grad f in general,"},{"Start":"02:19.120 ","End":"02:25.525","Text":"the point xy is going to equal the derivative with respect to x,"},{"Start":"02:25.525 ","End":"02:34.274","Text":"which is 6x and then y is a constant, so it sticks."},{"Start":"02:34.274 ","End":"02:37.429","Text":"Then the derivative with respect to y,"},{"Start":"02:37.429 ","End":"02:39.530","Text":"though the 3x squared is a constant,"},{"Start":"02:39.530 ","End":"02:41.750","Text":"constant times y is just the constant,"},{"Start":"02:41.750 ","End":"02:44.239","Text":"so it\u0027s 3x squared."},{"Start":"02:44.270 ","End":"02:51.180","Text":"I should\u0027ve written here at the point 1, 2."},{"Start":"02:51.180 ","End":"02:56.595","Text":"We have that grad f at the point 1,"},{"Start":"02:56.595 ","End":"02:59.010","Text":"2 means we let x equals 1,"},{"Start":"02:59.010 ","End":"03:06.980","Text":"y equals 2, so we have 6 times 1 times 2 is 12,"},{"Start":"03:06.980 ","End":"03:11.030","Text":"and 3 times 1 squared is 3."},{"Start":"03:11.030 ","End":"03:13.910","Text":"We now have the second piece of what we wanted."},{"Start":"03:13.910 ","End":"03:17.960","Text":"This is this and now,"},{"Start":"03:17.960 ","End":"03:20.994","Text":"all I have to do is take the dot product."},{"Start":"03:20.994 ","End":"03:27.380","Text":"This is 12,3 dot product"},{"Start":"03:27.380 ","End":"03:34.465","Text":"with the unit vector, 3/5, 4/5."},{"Start":"03:34.465 ","End":"03:37.970","Text":"Remember that the dot product this times this plus this times this."},{"Start":"03:37.970 ","End":"03:42.935","Text":"We get, 12 times 3 over 5 is 36 over 5,"},{"Start":"03:42.935 ","End":"03:52.320","Text":"plus 3 times 4 is 12 over 5 so we get, let\u0027s see,"},{"Start":"03:52.320 ","End":"04:02.040","Text":"48 over 5, and I could leave it like that or I could say 5 goes into 45,"},{"Start":"04:02.040 ","End":"04:06.435","Text":"9 times, 9 and 3 leftover,"},{"Start":"04:06.435 ","End":"04:12.620","Text":"9 and 3/5 or if you like decimals, 9.6."},{"Start":"04:12.620 ","End":"04:15.005","Text":"I\u0027ll go with 9 and 3/5."},{"Start":"04:15.005 ","End":"04:18.870","Text":"I\u0027ll just highlight that and we are done."}],"ID":8989},{"Watched":false,"Name":"Exercise 3","Duration":"5m 45s","ChapterTopicVideoID":8656,"CourseChapterTopicPlaylistID":4960,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.244","Text":"In this exercise, we\u0027re given a function of 2 variables as follows,"},{"Start":"00:04.244 ","End":"00:08.430","Text":"and we want the directional derivative of this function at this point"},{"Start":"00:08.430 ","End":"00:11.549","Text":"in the direction of this vector."},{"Start":"00:11.549 ","End":"00:14.039","Text":"Now when we do directional derivatives,"},{"Start":"00:14.039 ","End":"00:15.929","Text":"we work with unit vectors."},{"Start":"00:15.929 ","End":"00:19.835","Text":"But I happen to note this already is a unit vector, and I\u0027ll show you."},{"Start":"00:19.835 ","End":"00:25.940","Text":"If we take the magnitude of u it\u0027s equal to 1/2"},{"Start":"00:25.940 ","End":"00:32.130","Text":"squared plus root 3 over 2 squared,"},{"Start":"00:32.130 ","End":"00:34.415","Text":"and then the square root of all that."},{"Start":"00:34.415 ","End":"00:42.600","Text":"This is just equal to the square root of 1/4 plus 3/4,"},{"Start":"00:42.600 ","End":"00:44.730","Text":"which is the square root of 1,"},{"Start":"00:44.730 ","End":"00:47.160","Text":"which is just 1."},{"Start":"00:47.160 ","End":"00:54.495","Text":"We can actually label this as u with a hat on it,"},{"Start":"00:54.495 ","End":"00:58.120","Text":"u carrot, which is the unit vector."},{"Start":"00:58.760 ","End":"01:05.060","Text":"Now all we need to do is remember what directional derivative of f at a point is."},{"Start":"01:05.060 ","End":"01:15.754","Text":"The directional derivative in the direction of u of f at the point,"},{"Start":"01:15.754 ","End":"01:20.510","Text":"1, Pi over 2 is given by the formula"},{"Start":"01:20.510 ","End":"01:30.450","Text":"the grad of f at the point 1, Pi over 2 dot with this unit vector."},{"Start":"01:30.490 ","End":"01:33.575","Text":"Now this we have already,"},{"Start":"01:33.575 ","End":"01:38.890","Text":"because our original u is the unit vector so all I need is this."},{"Start":"01:39.710 ","End":"01:44.415","Text":"The gradient of the function f,"},{"Start":"01:44.415 ","End":"01:47.130","Text":"I\u0027ll use the ij notation, well, let me just write it,"},{"Start":"01:47.130 ","End":"01:56.145","Text":"is the derivative of f with respect to x times i."},{"Start":"01:56.145 ","End":"02:01.670","Text":"Actually even i can be written with a hat on it because i and j are unit vectors,"},{"Start":"02:01.670 ","End":"02:05.850","Text":"plus f with respect to y j."},{"Start":"02:07.600 ","End":"02:12.680","Text":"We\u0027re more used to using the bar over it."},{"Start":"02:12.680 ","End":"02:16.685","Text":"But bold face is an alternative way of doing it."},{"Start":"02:16.685 ","End":"02:20.300","Text":"This is equal to, derivative of f with respect to x."},{"Start":"02:20.300 ","End":"02:24.960","Text":"Let\u0027s see, that\u0027s 1 minus,"},{"Start":"02:24.960 ","End":"02:32.430","Text":"now the sine of xy would normally be the cosine of xy."},{"Start":"02:32.430 ","End":"02:33.780","Text":"But it\u0027s not xy,"},{"Start":"02:33.780 ","End":"02:35.220","Text":"it\u0027s x times a constant,"},{"Start":"02:35.220 ","End":"02:37.490","Text":"so we need to multiply by that constant,"},{"Start":"02:37.490 ","End":"02:41.450","Text":"which I\u0027ll put in front instead of at the end."},{"Start":"02:41.450 ","End":"02:48.440","Text":"That\u0027s times i plus derivative with respect to y."},{"Start":"02:48.440 ","End":"02:52.280","Text":"The x disappears and it\u0027s not even a plus,"},{"Start":"02:52.280 ","End":"02:57.375","Text":"well it becomes a minus there."},{"Start":"02:57.375 ","End":"03:02.245","Text":"Once again, we have the cosine of xy."},{"Start":"03:02.245 ","End":"03:04.790","Text":"But this time y is the variable,"},{"Start":"03:04.790 ","End":"03:07.855","Text":"so I have to multiply it by x,"},{"Start":"03:07.855 ","End":"03:11.315","Text":"and this is times j."},{"Start":"03:11.315 ","End":"03:13.925","Text":"Did I want brackets?"},{"Start":"03:13.925 ","End":"03:16.400","Text":"Yeah, why not? Extra"},{"Start":"03:16.400 ","End":"03:18.230","Text":"brackets won\u0027t hurt."},{"Start":"03:18.230 ","End":"03:24.595","Text":"Now, that\u0027s the general gradient at the point xy."},{"Start":"03:24.595 ","End":"03:31.130","Text":"We want specifically the gradient of f at the point 1, Pi over 2."},{"Start":"03:31.130 ","End":"03:35.235","Text":"We need to replace x by this,"},{"Start":"03:35.235 ","End":"03:37.755","Text":"and y by that in this formula here."},{"Start":"03:37.755 ","End":"03:38.640","Text":"Let\u0027s see."},{"Start":"03:38.640 ","End":"03:41.220","Text":"What is x times y?"},{"Start":"03:41.220 ","End":"03:43.530","Text":"x times y is Pi over 2,"},{"Start":"03:43.530 ","End":"03:48.610","Text":"cosine of Pi over 2 is 0,"},{"Start":"03:49.870 ","End":"03:53.660","Text":"one of those well-known angles."},{"Start":"03:53.660 ","End":"03:57.500","Text":"This part is 0, so altogether,"},{"Start":"03:57.500 ","End":"03:59.840","Text":"I\u0027m just left with 1 times i."},{"Start":"03:59.840 ","End":"04:02.735","Text":"I\u0027ll write the 1, it\u0027s 1i."},{"Start":"04:02.735 ","End":"04:07.640","Text":"Sometimes I like the arrows on them."},{"Start":"04:07.640 ","End":"04:10.710","Text":"I\u0027ll go with arrows."},{"Start":"04:12.700 ","End":"04:17.689","Text":"1 times i plus,"},{"Start":"04:17.689 ","End":"04:19.730","Text":"and let\u0027s see what we get in the other one."},{"Start":"04:19.730 ","End":"04:24.140","Text":"The same thing, cosine of xy is again cosine 90, that\u0027s 0,"},{"Start":"04:24.140 ","End":"04:25.490","Text":"but here there is nothing else,"},{"Start":"04:25.490 ","End":"04:27.650","Text":"so it\u0027s just 0j."},{"Start":"04:29.620 ","End":"04:33.260","Text":"If we want to write this in the other notation,"},{"Start":"04:33.260 ","End":"04:35.685","Text":"it would be 1, 0."},{"Start":"04:35.685 ","End":"04:39.610","Text":"In fact, you know what, I\u0027ll write the other one in this notation also"},{"Start":"04:39.610 ","End":"04:53.200","Text":"that u carrot is actually equal to 1/2, root 3 over 2."},{"Start":"04:53.200 ","End":"04:56.755","Text":"Now that I have the 2 pieces I need,"},{"Start":"04:56.755 ","End":"05:00.729","Text":"I\u0027ve got u here,"},{"Start":"05:00.729 ","End":"05:07.810","Text":"and I have this bit here in the angular bracket notation."},{"Start":"05:07.810 ","End":"05:13.165","Text":"All that remains is to do this dot product."},{"Start":"05:13.165 ","End":"05:30.440","Text":"What I get is from here, I\u0027ve got 1, 0 dot 1/2, root 3 over 2."},{"Start":"05:30.440 ","End":"05:33.140","Text":"It\u0027s this times this plus this times this,"},{"Start":"05:33.140 ","End":"05:34.640","Text":"well 0 times something,"},{"Start":"05:34.640 ","End":"05:36.980","Text":"it doesn\u0027t matter, so 1 times a 1/2."},{"Start":"05:36.980 ","End":"05:41.990","Text":"This is equal to 1/2,"},{"Start":"05:41.990 ","End":"05:43.280","Text":"and that\u0027s the answer."},{"Start":"05:43.280 ","End":"05:45.960","Text":"We\u0027re done."}],"ID":8990},{"Watched":false,"Name":"Exercise 4","Duration":"4m 19s","ChapterTopicVideoID":8657,"CourseChapterTopicPlaylistID":4960,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.725","Text":"In this exercise, we\u0027re given the following function of x and y,"},{"Start":"00:04.725 ","End":"00:10.680","Text":"and we have to compute the directional derivative of this function at this point."},{"Start":"00:10.680 ","End":"00:13.410","Text":"We\u0027re not given the unit vector,"},{"Start":"00:13.410 ","End":"00:20.190","Text":"we\u0027re just given that the unit vector forms a 45 degree angle with the positive x-axis."},{"Start":"00:20.190 ","End":"00:22.410","Text":"I\u0027ll just illustrate this."},{"Start":"00:22.410 ","End":"00:27.585","Text":"Remember that angles are taken from the x-axis counterclockwise."},{"Start":"00:27.585 ","End":"00:29.865","Text":"This is the positive x-axis."},{"Start":"00:29.865 ","End":"00:36.495","Text":"If I go 45 degrees in this direction,"},{"Start":"00:36.495 ","End":"00:44.975","Text":"then this might be the unit vector u, provided that its length is 1."},{"Start":"00:44.975 ","End":"00:48.125","Text":"Now, there\u0027s a simple formula that computes this."},{"Start":"00:48.125 ","End":"00:53.075","Text":"All we need, if we\u0027re given the angle, let me do this over here,"},{"Start":"00:53.075 ","End":"00:59.900","Text":"unit vector u is always the cosine of the angle,"},{"Start":"00:59.900 ","End":"01:02.180","Text":"in this case 45 degrees,"},{"Start":"01:02.180 ","End":"01:05.480","Text":"sine of 45 degrees."},{"Start":"01:05.480 ","End":"01:07.130","Text":"It\u0027s always a unit vector,"},{"Start":"01:07.130 ","End":"01:10.670","Text":"whatever angle it is here, because cosine squared plus sine squared is 1,"},{"Start":"01:10.670 ","End":"01:12.770","Text":"that\u0027s just the way it is."},{"Start":"01:12.770 ","End":"01:17.270","Text":"Evaluating this, we can work in degrees,"},{"Start":"01:17.270 ","End":"01:18.830","Text":"you can set your calculator for degrees,"},{"Start":"01:18.830 ","End":"01:22.790","Text":"or you can just remember that this is equal to 1 over the square root of 2,"},{"Start":"01:22.790 ","End":"01:27.005","Text":"and so is this, they\u0027re both equal to 1 over the square root of 2."},{"Start":"01:27.005 ","End":"01:29.365","Text":"That\u0027s the unit vector."},{"Start":"01:29.365 ","End":"01:33.815","Text":"Now, we remember the definition of the directional derivative."},{"Start":"01:33.815 ","End":"01:39.560","Text":"Directional derivative d in the direction of u at the point 1,"},{"Start":"01:39.560 ","End":"01:47.385","Text":"2 is the gradient of f at the point 1,"},{"Start":"01:47.385 ","End":"01:53.940","Text":"2 dot product with the directional unit vector u."},{"Start":"01:53.940 ","End":"01:56.445","Text":"This we have already."},{"Start":"01:56.445 ","End":"01:59.729","Text":"This bit is just this."},{"Start":"01:59.729 ","End":"02:02.610","Text":"All I need now is this bit."},{"Start":"02:02.610 ","End":"02:08.095","Text":"The gradient of f in general at a point x,"},{"Start":"02:08.095 ","End":"02:16.040","Text":"y is equal to the partial derivative of f with respect to x at xy,"},{"Start":"02:16.040 ","End":"02:19.280","Text":"partial derivative of f with respect to y."},{"Start":"02:19.280 ","End":"02:23.955","Text":"In other words, f with respect to x, let\u0027s see."},{"Start":"02:23.955 ","End":"02:32.025","Text":"With respect to x, I\u0027ve got 4x minus 3y and that\u0027s it."},{"Start":"02:32.025 ","End":"02:34.230","Text":"With respect to y, that\u0027s a constant."},{"Start":"02:34.230 ","End":"02:36.675","Text":"I\u0027ve got minus 3x,"},{"Start":"02:36.675 ","End":"02:39.880","Text":"and then plus 10y."},{"Start":"02:40.520 ","End":"02:48.990","Text":"Now, I have to find the gradient of f at the point 1, 2."},{"Start":"02:48.990 ","End":"02:52.460","Text":"I just have to let x equal 1,"},{"Start":"02:52.460 ","End":"02:54.965","Text":"y equals 2 in this formula."},{"Start":"02:54.965 ","End":"02:57.950","Text":"Let\u0027s see, x is 1, y is 2,"},{"Start":"02:57.950 ","End":"03:01.620","Text":"that\u0027s 4 minus 6."},{"Start":"03:02.000 ","End":"03:11.200","Text":"Let\u0027s see here, minus 3 and then plus 10 is plus 20."},{"Start":"03:12.890 ","End":"03:18.495","Text":"This, of course, is minus 2, 17."},{"Start":"03:18.495 ","End":"03:22.640","Text":"This is the piece I\u0027m missing for this."},{"Start":"03:22.640 ","End":"03:27.320","Text":"Now, all that remains is to do this dot product here."},{"Start":"03:27.320 ","End":"03:31.770","Text":"So it\u0027s this one, minus 2,"},{"Start":"03:31.770 ","End":"03:36.165","Text":"17 dot product with this one,"},{"Start":"03:36.165 ","End":"03:39.360","Text":"which is 1 over root 2,"},{"Start":"03:39.360 ","End":"03:42.730","Text":"1 over root 2."},{"Start":"03:43.000 ","End":"03:47.000","Text":"Perhaps I\u0027ll use an alternative form, which is more convenient,"},{"Start":"03:47.000 ","End":"03:50.975","Text":"this thing is exactly the same as root 2 over 2,"},{"Start":"03:50.975 ","End":"03:54.020","Text":"as you can see by multiplying top and bottom by root 2,"},{"Start":"03:54.020 ","End":"03:56.705","Text":"and so is this root 2 over 2."},{"Start":"03:56.705 ","End":"03:58.820","Text":"So let\u0027s use this form instead,"},{"Start":"03:58.820 ","End":"04:02.850","Text":"I\u0027d rather have square roots on the numerator."},{"Start":"04:04.400 ","End":"04:06.780","Text":"If you figure this out,"},{"Start":"04:06.780 ","End":"04:09.480","Text":"we\u0027re multiplying each of these by root 2 over 2,"},{"Start":"04:09.480 ","End":"04:12.199","Text":"so I\u0027ve got minus 2 plus 17 is 15."},{"Start":"04:12.199 ","End":"04:16.685","Text":"Basically, we get 15 root 2 over 2,"},{"Start":"04:16.685 ","End":"04:20.640","Text":"and that is the answer."}],"ID":8991},{"Watched":false,"Name":"Exercise 5","Duration":"6m 15s","ChapterTopicVideoID":8658,"CourseChapterTopicPlaylistID":4960,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.115","Text":"In this exercise we\u0027re given a function of x and y as follows."},{"Start":"00:05.115 ","End":"00:10.410","Text":"We want to compute the directional derivative of the function f at the point 1,"},{"Start":"00:10.410 ","End":"00:13.725","Text":"3, and in the direction of this point."},{"Start":"00:13.725 ","End":"00:16.890","Text":"What does it mean from this point and in the direction of this point?"},{"Start":"00:16.890 ","End":"00:20.310","Text":"It means if I have one point and I have another point,"},{"Start":"00:20.310 ","End":"00:23.310","Text":"so this is the point A and this is the point B."},{"Start":"00:23.310 ","End":"00:26.130","Text":"Then in the direction means parallel to"},{"Start":"00:26.130 ","End":"00:32.200","Text":"the displacement vector that takes me from A to B."},{"Start":"00:32.720 ","End":"00:37.380","Text":"When the plane that\u0027s A, so we have A,"},{"Start":"00:37.380 ","End":"00:43.190","Text":"let\u0027s call it a_ x and a_y suppose these are the coordinates and"},{"Start":"00:43.190 ","End":"00:49.070","Text":"B has the coordinates the x of b and the y of b."},{"Start":"00:49.070 ","End":"00:51.800","Text":"Maybe I could have used better letters, this will do."},{"Start":"00:51.800 ","End":"00:56.375","Text":"This displacement vector actually that takes me from A to B,"},{"Start":"00:56.375 ","End":"01:02.410","Text":"I subtract the head minus the tail of the x components,"},{"Start":"01:02.410 ","End":"01:05.120","Text":"so it\u0027s b_x minus a_x,"},{"Start":"01:05.120 ","End":"01:10.800","Text":"and then the head minus the tail of the y component."},{"Start":"01:12.850 ","End":"01:15.920","Text":"In general, I don\u0027t remember the actual letters."},{"Start":"01:15.920 ","End":"01:17.840","Text":"I Just remember that you take the coordinates of"},{"Start":"01:17.840 ","End":"01:20.765","Text":"the head and subtract the coordinates of the tail."},{"Start":"01:20.765 ","End":"01:23.870","Text":"In our case, the position vector,"},{"Start":"01:23.870 ","End":"01:27.210","Text":"let\u0027s call it u. I keep saying position,"},{"Start":"01:27.210 ","End":"01:31.640","Text":"I mean displacement that\u0027s what it\u0027s called when it takes you from one point to another,"},{"Start":"01:31.640 ","End":"01:36.875","Text":"is going to equal the vector."},{"Start":"01:36.875 ","End":"01:38.450","Text":"Now, first of all,"},{"Start":"01:38.450 ","End":"01:39.995","Text":"the x minus the x,"},{"Start":"01:39.995 ","End":"01:41.990","Text":"this is the head, this is the tail,"},{"Start":"01:41.990 ","End":"01:47.960","Text":"It\u0027s 4 minus 1 and then 5 minus 3."},{"Start":"01:47.960 ","End":"01:53.760","Text":"Our position vector is in fact 3, 2."},{"Start":"01:53.760 ","End":"01:56.855","Text":"Now whenever you\u0027re doing directional derivatives,"},{"Start":"01:56.855 ","End":"01:58.835","Text":"we need unit vectors."},{"Start":"01:58.835 ","End":"02:04.940","Text":"We want a unit vector u in the same direction as u,"},{"Start":"02:04.940 ","End":"02:06.635","Text":"with the arrow on top,"},{"Start":"02:06.635 ","End":"02:09.710","Text":"but with magnitude 1."},{"Start":"02:09.710 ","End":"02:13.835","Text":"I take the vector divided by its magnitude,"},{"Start":"02:13.835 ","End":"02:16.775","Text":"then it brings me to a unit vector."},{"Start":"02:16.775 ","End":"02:25.665","Text":"In our case we\u0027ll get 3, 2 over,"},{"Start":"02:25.665 ","End":"02:30.165","Text":"and let\u0027s see, magnitude will be the square root of this squared plus this squared,"},{"Start":"02:30.165 ","End":"02:33.355","Text":"3 squared plus 2 squared,"},{"Start":"02:33.355 ","End":"02:35.920","Text":"that\u0027s 9 plus 4 is 13."},{"Start":"02:35.920 ","End":"02:43.110","Text":"In other words, we get 3 over square root of 13,"},{"Start":"02:43.110 ","End":"02:46.415","Text":"2 over square root of 13."},{"Start":"02:46.415 ","End":"02:48.340","Text":"Now we\u0027ve got a unit vector."},{"Start":"02:48.340 ","End":"02:50.300","Text":"Now, what about the directional derivative?"},{"Start":"02:50.300 ","End":"02:53.780","Text":"Now we can say directional derivative in the direction of"},{"Start":"02:53.780 ","End":"02:59.165","Text":"the unit vector u of the function f at the point 1,"},{"Start":"02:59.165 ","End":"03:02.210","Text":"3 is equal to,"},{"Start":"03:02.210 ","End":"03:08.075","Text":"it\u0027s always grad of the same function at the same point 1,"},{"Start":"03:08.075 ","End":"03:12.250","Text":"3 dot product with the unit vector."},{"Start":"03:12.250 ","End":"03:16.590","Text":"Now we need grad f. Let me do this at the side."},{"Start":"03:16.590 ","End":"03:22.910","Text":"Grad f, schematically is just the derivative with respect to x,"},{"Start":"03:22.910 ","End":"03:25.220","Text":"the derivative with respect to y."},{"Start":"03:25.220 ","End":"03:27.860","Text":"In our case, it\u0027s equal to,"},{"Start":"03:27.860 ","End":"03:31.235","Text":"the derivative with respect to x will be y squared,"},{"Start":"03:31.235 ","End":"03:34.500","Text":"the derivative with respect to y, 2xy."},{"Start":"03:35.090 ","End":"03:40.980","Text":"Grad f, the gradient at the point 1,"},{"Start":"03:40.980 ","End":"03:44.115","Text":"3, this is equal to,"},{"Start":"03:44.115 ","End":"03:47.655","Text":"put 1 instead of x and 3 instead of y,"},{"Start":"03:47.655 ","End":"03:48.930","Text":"so here we have,"},{"Start":"03:48.930 ","End":"03:50.910","Text":"let\u0027s see y squared is 9,"},{"Start":"03:50.910 ","End":"03:55.750","Text":"2xy, 2 times 1 times 3 is 6."},{"Start":"03:56.360 ","End":"04:00.035","Text":"To summarize, we want to compute this,"},{"Start":"04:00.035 ","End":"04:02.800","Text":"and we have the unit vector."},{"Start":"04:02.800 ","End":"04:04.690","Text":"Where is it now?"},{"Start":"04:04.690 ","End":"04:07.070","Text":"It is here."},{"Start":"04:08.010 ","End":"04:14.820","Text":"This gradient at the point that is here."},{"Start":"04:14.820 ","End":"04:17.130","Text":"Now all I need to do I have these 2,"},{"Start":"04:17.130 ","End":"04:18.730","Text":"I just need to take the dot product."},{"Start":"04:18.730 ","End":"04:22.795","Text":"The dot product will be the green bit is 9,"},{"Start":"04:22.795 ","End":"04:26.420","Text":"6 dot-product with this."},{"Start":"04:26.420 ","End":"04:31.640","Text":"Let\u0027s see."},{"Start":"04:31.640 ","End":"04:33.310","Text":"This is this here,"},{"Start":"04:33.310 ","End":"04:41.795","Text":"and this is what we computed over here."},{"Start":"04:41.795 ","End":"04:46.220","Text":"The only thing remaining to compute the directional derivative is the dot product."},{"Start":"04:46.220 ","End":"04:47.765","Text":"I\u0027ll compute that here."},{"Start":"04:47.765 ","End":"04:50.395","Text":"We want the green,"},{"Start":"04:50.395 ","End":"04:53.130","Text":"well this bit, 9,"},{"Start":"04:53.130 ","End":"04:56.760","Text":"6 dot-product with this,"},{"Start":"04:56.760 ","End":"05:03.140","Text":"you know what, I\u0027ll take the 1 over square root of 13 outside"},{"Start":"05:03.140 ","End":"05:09.900","Text":"the product and then we\u0027ll just get the 3, 2."},{"Start":"05:10.210 ","End":"05:14.045","Text":"I took the one 1 over square root of 13 outside."},{"Start":"05:14.045 ","End":"05:21.380","Text":"Now let\u0027s see, 9 times 3 is 27,"},{"Start":"05:21.380 ","End":"05:23.660","Text":"2 times 6 is 12."},{"Start":"05:23.660 ","End":"05:27.725","Text":"Let me just see, 27 plus 12,"},{"Start":"05:27.725 ","End":"05:31.090","Text":"that comes out to be 39,"},{"Start":"05:31.090 ","End":"05:38.690","Text":"so we get that this is 39 over square root of 13."},{"Start":"05:38.690 ","End":"05:40.490","Text":"We could stop here, well,"},{"Start":"05:40.490 ","End":"05:45.350","Text":"I can multiply top and bottom by square root of 13,"},{"Start":"05:45.350 ","End":"05:51.725","Text":"and this is equal to root 13 times root 13,"},{"Start":"05:51.725 ","End":"05:55.800","Text":"this is just 13,"},{"Start":"05:56.780 ","End":"06:02.085","Text":"this goes in to 39, 3 times,"},{"Start":"06:02.085 ","End":"06:08.100","Text":"so 3 square root of 13."},{"Start":"06:08.100 ","End":"06:13.355","Text":"This is our answer for the directional derivative at that point,"},{"Start":"06:13.355 ","End":"06:16.050","Text":"so we are done."}],"ID":8992},{"Watched":false,"Name":"Exercise 6","Duration":"5m 20s","ChapterTopicVideoID":8659,"CourseChapterTopicPlaylistID":4960,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.965","Text":"In this exercise, we have a 3-dimensional example."},{"Start":"00:04.965 ","End":"00:06.720","Text":"We have a function f on x, y,"},{"Start":"00:06.720 ","End":"00:13.120","Text":"and z as follows and we need the directional derivative of f at the 3D point this,"},{"Start":"00:13.120 ","End":"00:15.540","Text":"and in the direction of vector, we have i,"},{"Start":"00:15.540 ","End":"00:18.225","Text":"j, and k here, so also 3D."},{"Start":"00:18.225 ","End":"00:20.310","Text":"You know that in directional derivatives,"},{"Start":"00:20.310 ","End":"00:21.900","Text":"we always need a unit vector."},{"Start":"00:21.900 ","End":"00:24.090","Text":"Why don\u0027t we even start with that?"},{"Start":"00:24.090 ","End":"00:29.310","Text":"I want the unit vector u corresponding to this given vector."},{"Start":"00:29.310 ","End":"00:34.350","Text":"What you do is you take the vector and you divide by its length or magnitude,"},{"Start":"00:34.350 ","End":"00:37.245","Text":"which is putting it in bars."},{"Start":"00:37.245 ","End":"00:41.165","Text":"What it actually means is to take the vector,"},{"Start":"00:41.165 ","End":"00:48.115","Text":"which in this case is 1i plus 2j plus 2k,"},{"Start":"00:48.115 ","End":"00:49.939","Text":"and divide by the magnitude."},{"Start":"00:49.939 ","End":"00:54.590","Text":"The magnitude is these all squared,"},{"Start":"00:54.590 ","End":"00:57.610","Text":"1 squared plus 2 squared plus 2 squared,"},{"Start":"00:57.610 ","End":"00:59.970","Text":"and in short what we get that,"},{"Start":"00:59.970 ","End":"01:02.280","Text":"this thing is 1 plus 4 plus 4 is 9,"},{"Start":"01:02.280 ","End":"01:04.125","Text":"square root of 9 is 3."},{"Start":"01:04.125 ","End":"01:16.950","Text":"I get 1/3i plus 2/3j plus 2/3 k."},{"Start":"01:16.950 ","End":"01:20.435","Text":"What we want,"},{"Start":"01:20.435 ","End":"01:22.610","Text":"directional derivative can be written as follows."},{"Start":"01:22.610 ","End":"01:25.025","Text":"D for the directional derivative,"},{"Start":"01:25.025 ","End":"01:28.265","Text":"and then we put the unit vector,"},{"Start":"01:28.265 ","End":"01:30.830","Text":"and then we put the name of the function f,"},{"Start":"01:30.830 ","End":"01:32.525","Text":"and at the point we want,"},{"Start":"01:32.525 ","End":"01:37.085","Text":"which 2, 1, 4."},{"Start":"01:37.085 ","End":"01:38.900","Text":"This is equal to,"},{"Start":"01:38.900 ","End":"01:41.395","Text":"by the formula, the gradient,"},{"Start":"01:41.395 ","End":"01:52.440","Text":"grad f at the point 2, 1, 4 dot product with the unit vector u."},{"Start":"01:52.440 ","End":"01:57.735","Text":"We have this is just this,"},{"Start":"01:57.735 ","End":"02:00.630","Text":"but we need also this,"},{"Start":"02:00.630 ","End":"02:02.790","Text":"so let\u0027s start computing that."},{"Start":"02:02.790 ","End":"02:07.980","Text":"Now in general, grad of f, the x, y,"},{"Start":"02:07.980 ","End":"02:13.220","Text":"z is equal to the derivative with respect to x in"},{"Start":"02:13.220 ","End":"02:18.510","Text":"the i direction plus f_y, j direction."},{"Start":"02:18.510 ","End":"02:19.520","Text":"Usually, we stop here,"},{"Start":"02:19.520 ","End":"02:22.040","Text":"but we\u0027re in 3D, so we have another one,"},{"Start":"02:22.040 ","End":"02:25.715","Text":"f_z in the k direction."},{"Start":"02:25.715 ","End":"02:27.305","Text":"That\u0027s in general."},{"Start":"02:27.305 ","End":"02:29.360","Text":"In our case, first of all,"},{"Start":"02:29.360 ","End":"02:32.030","Text":"our function f is x squared y squared z squared."},{"Start":"02:32.030 ","End":"02:36.780","Text":"This one comes out to be 2x y squared z,"},{"Start":"02:36.780 ","End":"02:38.790","Text":"those 2 are constants,"},{"Start":"02:38.790 ","End":"02:43.500","Text":"i plus with respect to y,"},{"Start":"02:43.500 ","End":"02:44.730","Text":"this is a 2y here,"},{"Start":"02:44.730 ","End":"02:46.125","Text":"bring the 2 in front,"},{"Start":"02:46.125 ","End":"02:51.645","Text":"2x squared yz, all this j direction."},{"Start":"02:51.645 ","End":"02:58.290","Text":"With respect to z, it\u0027s just x squared y squared in the k direction."},{"Start":"02:58.290 ","End":"03:00.600","Text":"That\u0027s in general for x, y, z."},{"Start":"03:00.600 ","End":"03:07.925","Text":"What we want is grad f specifically at the point 2, 1, 4."},{"Start":"03:07.925 ","End":"03:09.285","Text":"We just substitute."},{"Start":"03:09.285 ","End":"03:11.310","Text":"2 we substitute instead of x,"},{"Start":"03:11.310 ","End":"03:12.750","Text":"1 we substitute instead of y,"},{"Start":"03:12.750 ","End":"03:15.540","Text":"4 instead of z. Let\u0027s see."},{"Start":"03:15.540 ","End":"03:18.960","Text":"2 times 2 is 4,"},{"Start":"03:18.960 ","End":"03:20.805","Text":"times 1 is still 4,"},{"Start":"03:20.805 ","End":"03:23.440","Text":"times 4 is 16i."},{"Start":"03:23.570 ","End":"03:29.100","Text":"Now 2x squared, 2 times 4 is 8,"},{"Start":"03:29.100 ","End":"03:30.855","Text":"times y is still 8,"},{"Start":"03:30.855 ","End":"03:37.290","Text":"times 4 is 32j plus x squared y squared,"},{"Start":"03:37.290 ","End":"03:38.985","Text":"2 squared 1 squared,"},{"Start":"03:38.985 ","End":"03:50.610","Text":"that is 4 times k."},{"Start":"03:50.610 ","End":"03:55.915","Text":"That would be what we\u0027re looking for here."},{"Start":"03:55.915 ","End":"03:58.700","Text":"I just can highlight it now."},{"Start":"03:58.740 ","End":"04:03.505","Text":"All we\u0027re missing is this dot product here."},{"Start":"04:03.505 ","End":"04:05.400","Text":"I\u0027ll just compute it."},{"Start":"04:05.400 ","End":"04:14.600","Text":"Let\u0027s see. This bit is 16i plus 32j plus"},{"Start":"04:14.600 ","End":"04:22.680","Text":"4k dot with 1/3, this thing."},{"Start":"04:24.200 ","End":"04:28.065","Text":"I\u0027ll take the 1/3 out all the way to the front."},{"Start":"04:28.065 ","End":"04:30.325","Text":"Just erase that."},{"Start":"04:30.325 ","End":"04:32.885","Text":"Put the 1/3 in the front."},{"Start":"04:32.885 ","End":"04:36.305","Text":"Actually I should have kept this over 3, then it\u0027s easier."},{"Start":"04:36.305 ","End":"04:43.140","Text":"Then it\u0027s just 1i plus 2j plus 2k."},{"Start":"04:43.400 ","End":"04:46.040","Text":"This is easy to compute."},{"Start":"04:46.040 ","End":"04:48.020","Text":"We get 1/3, now let\u0027s see,"},{"Start":"04:48.020 ","End":"04:49.880","Text":"16 times 1 is 16,"},{"Start":"04:49.880 ","End":"04:54.120","Text":"32 times 2 is 64,"},{"Start":"04:54.120 ","End":"04:56.820","Text":"4 times 2 is 8."},{"Start":"04:56.820 ","End":"05:06.510","Text":"Let\u0027s see. 16 and 64 is 80 plus 8 it\u0027s 88, 88/3."},{"Start":"05:06.510 ","End":"05:08.315","Text":"This is the answer."},{"Start":"05:08.315 ","End":"05:10.390","Text":"For those of you who like mixed numbers,"},{"Start":"05:10.390 ","End":"05:12.730","Text":"3 goes into 87,"},{"Start":"05:12.730 ","End":"05:18.790","Text":"29 times and we\u0027re left with 1 over so it\u0027s 29 and 1/3."},{"Start":"05:18.790 ","End":"05:20.600","Text":"We are done."}],"ID":8993},{"Watched":false,"Name":"Exercise 7","Duration":"7m 28s","ChapterTopicVideoID":8660,"CourseChapterTopicPlaylistID":4960,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.700","Text":"Here we have a problem from physics,"},{"Start":"00:02.700 ","End":"00:06.300","Text":"but if you\u0027re not studying physics, don\u0027t be alarmed."},{"Start":"00:06.300 ","End":"00:09.555","Text":"You don\u0027t have to worry about what electric potential is."},{"Start":"00:09.555 ","End":"00:12.780","Text":"V is just some function of x and y as follows."},{"Start":"00:12.780 ","End":"00:13.950","Text":"That\u0027s one thing that\u0027s important."},{"Start":"00:13.950 ","End":"00:15.900","Text":"V is a function of x and y."},{"Start":"00:15.900 ","End":"00:25.830","Text":"Second thing is, when we say rate of change in the direction,"},{"Start":"00:25.830 ","End":"00:28.665","Text":"that\u0027s a cue for directional derivative."},{"Start":"00:28.665 ","End":"00:33.285","Text":"So we want the directional derivative at this point here,"},{"Start":"00:33.285 ","End":"00:37.065","Text":"and in the direction from this point to this point."},{"Start":"00:37.065 ","End":"00:42.060","Text":"In other words, we have here 3, 4,"},{"Start":"00:42.060 ","End":"00:48.405","Text":"and here we have a point, 2, 6."},{"Start":"00:48.405 ","End":"00:54.170","Text":"What we need is the vector that takes me from this point to this point,"},{"Start":"00:54.170 ","End":"00:56.615","Text":"sometimes called the displacement vector."},{"Start":"00:56.615 ","End":"01:02.305","Text":"The way we get it is we subtract the head minus the tail."},{"Start":"01:02.305 ","End":"01:04.050","Text":"I\u0027m not going to repeat the formula."},{"Start":"01:04.050 ","End":"01:05.400","Text":"Just remember the head,"},{"Start":"01:05.400 ","End":"01:11.715","Text":"2 minus 3 is minus 1."},{"Start":"01:11.715 ","End":"01:13.590","Text":"Then again the head minus the tail,"},{"Start":"01:13.590 ","End":"01:16.230","Text":"6 minus 4 is 2."},{"Start":"01:16.230 ","End":"01:20.155","Text":"So this is the vector minus 1, 2."},{"Start":"01:20.155 ","End":"01:23.400","Text":"This is what takes me from this point to this point."},{"Start":"01:23.400 ","End":"01:27.165","Text":"The important things are: here\u0027s the function,"},{"Start":"01:27.165 ","End":"01:34.895","Text":"here\u0027s the point, and here is the direction."},{"Start":"01:34.895 ","End":"01:37.770","Text":"Maybe I\u0027ll just highlight those."},{"Start":"01:39.490 ","End":"01:48.140","Text":"The function, the point, and the direction."},{"Start":"01:48.140 ","End":"01:51.875","Text":"That\u0027s all you need for the directional derivative."},{"Start":"01:51.875 ","End":"01:54.560","Text":"Let\u0027s now be more formal."},{"Start":"01:54.560 ","End":"01:58.204","Text":"We have a direction vector u,"},{"Start":"01:58.204 ","End":"02:05.495","Text":"which is equal to minus 1, 2."},{"Start":"02:05.495 ","End":"02:08.510","Text":"You always know that with directional derivative,"},{"Start":"02:08.510 ","End":"02:09.800","Text":"you need a unit vector."},{"Start":"02:09.800 ","End":"02:15.565","Text":"So we\u0027re going to figure out u carrot, u hat, whatever,"},{"Start":"02:15.565 ","End":"02:21.640","Text":"is the same thing as u over the length minus 1,"},{"Start":"02:21.640 ","End":"02:26.580","Text":"2 divided by length or magnitude of the same thing,"},{"Start":"02:26.580 ","End":"02:29.240","Text":"and this is equal to, let\u0027s see,"},{"Start":"02:29.240 ","End":"02:34.610","Text":"1 squared plus 2 squared is 5 square root of 5."},{"Start":"02:34.610 ","End":"02:40.935","Text":"So it\u0027s minus 1 over square root of 5,"},{"Start":"02:40.935 ","End":"02:43.885","Text":"2 over square root of 5."},{"Start":"02:43.885 ","End":"02:47.600","Text":"In case you lost me in the magnitude,"},{"Start":"02:47.600 ","End":"02:53.725","Text":"I just did the square root of minus 1 squared plus 2 squared,"},{"Start":"02:53.725 ","End":"02:57.080","Text":"and then that\u0027s how I got to the square root of 5."},{"Start":"02:57.080 ","End":"02:59.285","Text":"Now, directional derivative."},{"Start":"02:59.285 ","End":"03:04.060","Text":"The directional derivative in the direction of a unit vector u of a function,"},{"Start":"03:04.060 ","End":"03:05.465","Text":"well, usually it\u0027s f,"},{"Start":"03:05.465 ","End":"03:10.505","Text":"but I\u0027m going to work with V today instead of f. I hope that\u0027s not too confusing."},{"Start":"03:10.505 ","End":"03:13.110","Text":"At the point 3,"},{"Start":"03:13.110 ","End":"03:15.150","Text":"4 is equal to,"},{"Start":"03:15.150 ","End":"03:17.280","Text":"by the formula, the gradient,"},{"Start":"03:17.280 ","End":"03:20.310","Text":"we usually have f but today we have V,"},{"Start":"03:20.310 ","End":"03:23.220","Text":"at the same point,"},{"Start":"03:23.220 ","End":"03:28.000","Text":"dot product with that unit vector."},{"Start":"03:28.720 ","End":"03:37.340","Text":"We have the unit vector because this is just what we wrote here."},{"Start":"03:37.520 ","End":"03:43.000","Text":"What we\u0027re missing is this bit here."},{"Start":"03:43.760 ","End":"03:46.065","Text":"I\u0027ll do that to the side,"},{"Start":"03:46.065 ","End":"03:49.110","Text":"grad of V, in general,"},{"Start":"03:49.110 ","End":"03:55.505","Text":"it\u0027s just V with respect to x and then V with respect to y."},{"Start":"03:55.505 ","End":"03:58.120","Text":"We need to do some differentiation."},{"Start":"03:58.120 ","End":"04:02.335","Text":"I can do this with a little bit of a shortcut."},{"Start":"04:02.335 ","End":"04:07.720","Text":"Notice that the natural log of square root of something, it doesn\u0027t matter,"},{"Start":"04:07.720 ","End":"04:10.834","Text":"say a for some positive number,"},{"Start":"04:10.834 ","End":"04:15.055","Text":"is just 1/2 the natural log of a."},{"Start":"04:15.055 ","End":"04:17.495","Text":"Why on earth did I say that?"},{"Start":"04:17.495 ","End":"04:21.015","Text":"That\u0027s because square root of a is a^1/2."},{"Start":"04:21.015 ","End":"04:23.210","Text":"When you take a logarithm of an exponent,"},{"Start":"04:23.210 ","End":"04:26.455","Text":"the exponent comes out in front, so to speak."},{"Start":"04:26.455 ","End":"04:31.240","Text":"We can actually write V as 1/2"},{"Start":"04:31.240 ","End":"04:36.380","Text":"natural log of x squared plus y squared without any square root."},{"Start":"04:36.380 ","End":"04:38.800","Text":"That already makes things easier."},{"Start":"04:38.800 ","End":"04:43.485","Text":"Grad of V, which we said is V_x, V_y."},{"Start":"04:43.485 ","End":"04:45.885","Text":"Well, the 1/2 is a constant."},{"Start":"04:45.885 ","End":"04:50.885","Text":"In fact, it can even be pulled right out of the vector."},{"Start":"04:50.885 ","End":"04:54.740","Text":"I have the derivative of this with respect to x."},{"Start":"04:54.740 ","End":"05:00.000","Text":"The derivative of the logarithm is 1 over this thing,"},{"Start":"05:00.000 ","End":"05:03.560","Text":"but then I have to multiply by the inner derivative,"},{"Start":"05:03.560 ","End":"05:06.770","Text":"which is 2x because it\u0027s with respect to x."},{"Start":"05:06.770 ","End":"05:14.630","Text":"Then very similarly, the other one is going to be just 2y over x squared plus y squared,"},{"Start":"05:14.630 ","End":"05:17.225","Text":"partial derivative with respect to y."},{"Start":"05:17.225 ","End":"05:21.155","Text":"Not only that, I can even cancel."},{"Start":"05:21.155 ","End":"05:26.590","Text":"I can get rid of this 1/2 along with this 2 and this 2."},{"Start":"05:26.590 ","End":"05:30.525","Text":"What we have left is fairly simple looking."},{"Start":"05:30.525 ","End":"05:34.970","Text":"At this point, I compute the gradient where I want,"},{"Start":"05:34.970 ","End":"05:36.905","Text":"which is at 3, 4,"},{"Start":"05:36.905 ","End":"05:40.850","Text":"which means that I put 3 instead of x and 4 instead of y here."},{"Start":"05:40.850 ","End":"05:42.690","Text":"So what we get is,"},{"Start":"05:42.690 ","End":"05:47.850","Text":"x is 3 and x squared plus y squared is 3 squared plus 4 square,"},{"Start":"05:47.850 ","End":"05:50.310","Text":"9 plus 16 is 25."},{"Start":"05:50.310 ","End":"05:52.625","Text":"The other one is y, which is 4,"},{"Start":"05:52.625 ","End":"05:56.700","Text":"same denominator, so also 25."},{"Start":"05:57.860 ","End":"06:00.380","Text":"This is now what we wanted here."},{"Start":"06:00.380 ","End":"06:02.315","Text":"This is the green bit."},{"Start":"06:02.315 ","End":"06:04.730","Text":"I have the yellow here."},{"Start":"06:04.730 ","End":"06:07.325","Text":"I just need a dot product of the 2."},{"Start":"06:07.325 ","End":"06:10.070","Text":"What do I get? Let\u0027s see."},{"Start":"06:10.070 ","End":"06:16.280","Text":"The green, that would be 3/25,"},{"Start":"06:16.280 ","End":"06:26.700","Text":"4/25 dot product, this kind of u is minus 1 over root 5,"},{"Start":"06:26.700 ","End":"06:30.310","Text":"2 over root 5."},{"Start":"06:30.530 ","End":"06:34.155","Text":"Let\u0027s see. This equals,"},{"Start":"06:34.155 ","End":"06:37.040","Text":"let me take stuff out the brackets."},{"Start":"06:37.040 ","End":"06:41.480","Text":"I can take 1 over 25 out from these,"},{"Start":"06:41.480 ","End":"06:45.710","Text":"and I can also take 1 over root 5 from here."},{"Start":"06:45.710 ","End":"06:53.300","Text":"I\u0027m just left with 3 times minus 1 is minus 3,"},{"Start":"06:53.300 ","End":"06:56.790","Text":"4 times 2 is 8."},{"Start":"06:56.950 ","End":"06:59.150","Text":"Minus 3 plus 8,"},{"Start":"06:59.150 ","End":"07:02.970","Text":"this bit is 5,"},{"Start":"07:02.970 ","End":"07:08.010","Text":"and then 5 into 25 just goes 5."},{"Start":"07:08.010 ","End":"07:10.770","Text":"We\u0027re left with 5 here."},{"Start":"07:10.770 ","End":"07:21.285","Text":"What I have is equal to 1 over 5 root 5,"},{"Start":"07:21.285 ","End":"07:23.820","Text":"and this is our answer."},{"Start":"07:23.820 ","End":"07:27.640","Text":"I\u0027ll leave it in this form. We\u0027re done."}],"ID":8994},{"Watched":false,"Name":"Exercise 8","Duration":"4m 7s","ChapterTopicVideoID":8661,"CourseChapterTopicPlaylistID":4960,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"In this exercise, we have to find the direction for"},{"Start":"00:03.450 ","End":"00:06.495","Text":"which the directional derivative of this function of"},{"Start":"00:06.495 ","End":"00:13.230","Text":"2 variables at the origin is maximal and to compute its value."},{"Start":"00:13.230 ","End":"00:21.075","Text":"I need to remind you of a theorem and here it is. Let\u0027s read it."},{"Start":"00:21.075 ","End":"00:27.910","Text":"The maximal value of the directional derivative of f in a direction of a unit vector u,"},{"Start":"00:27.910 ","End":"00:30.420","Text":"I need to explain what x is."},{"Start":"00:30.420 ","End":"00:35.970","Text":"X in 2D, it\u0027s the point x,"},{"Start":"00:35.970 ","End":"00:38.230","Text":"y, while in 3D,"},{"Start":"00:38.230 ","End":"00:41.390","Text":"it would be this, but we\u0027re taking the 2D version."},{"Start":"00:41.390 ","End":"00:44.885","Text":"Just instead of writing f of x and y,"},{"Start":"00:44.885 ","End":"00:47.000","Text":"we write f of x vector,"},{"Start":"00:47.000 ","End":"00:49.670","Text":"and then this thing is good for 2D and 3D."},{"Start":"00:49.670 ","End":"00:54.900","Text":"Similarly, in 3D, it\u0027s x, y, z."},{"Start":"00:56.710 ","End":"01:03.645","Text":"The maximum value and the maximum rate of change,"},{"Start":"01:03.645 ","End":"01:06.325","Text":"that\u0027s in brackets, is given by"},{"Start":"01:06.325 ","End":"01:11.325","Text":"the length or magnitude of the gradient of f at that point."},{"Start":"01:11.325 ","End":"01:15.415","Text":"The direction it occurs in is in the direction of the gradient."},{"Start":"01:15.415 ","End":"01:19.854","Text":"In other words, u would be the unit vector in the direction of the gradient."},{"Start":"01:19.854 ","End":"01:24.915","Text":"I\u0027m going to just filled in the details for 3D in case we need it in future."},{"Start":"01:24.915 ","End":"01:29.580","Text":"In our case, we have that f is this."},{"Start":"01:29.580 ","End":"01:32.820","Text":"So let\u0027s find the gradient of f. In 2D,"},{"Start":"01:32.820 ","End":"01:34.920","Text":"the gradient of f,"},{"Start":"01:34.920 ","End":"01:37.115","Text":"at a general point x, y,"},{"Start":"01:37.115 ","End":"01:42.530","Text":"is equal to the derivative of f with respect to x at the point x,"},{"Start":"01:42.530 ","End":"01:46.790","Text":"y, derivative of f with respect to y. I don\u0027t know which is right at the point x,"},{"Start":"01:46.790 ","End":"01:49.370","Text":"y, which saves time."},{"Start":"01:49.370 ","End":"01:52.175","Text":"This is equal to,"},{"Start":"01:52.175 ","End":"01:55.475","Text":"now let\u0027s see, we need the derivative with respect to x."},{"Start":"01:55.475 ","End":"02:00.810","Text":"Now, all this is a constant as far as x goes."},{"Start":"02:00.810 ","End":"02:04.170","Text":"The derivative of e^x is just e^x,"},{"Start":"02:04.170 ","End":"02:05.550","Text":"and the constant sticks,"},{"Start":"02:05.550 ","End":"02:07.410","Text":"so it\u0027s exactly what\u0027s written there,"},{"Start":"02:07.410 ","End":"02:11.085","Text":"cosine of y plus sine y."},{"Start":"02:11.085 ","End":"02:14.010","Text":"All this is with respect to x."},{"Start":"02:14.010 ","End":"02:16.170","Text":"Now with respect to y,"},{"Start":"02:16.170 ","End":"02:20.415","Text":"x is a constant so e^x is a constant,"},{"Start":"02:20.415 ","End":"02:23.070","Text":"and so e^x just stays,"},{"Start":"02:23.070 ","End":"02:25.220","Text":"and we just need to differentiate this."},{"Start":"02:25.220 ","End":"02:28.615","Text":"Cosine gives me minus sine,"},{"Start":"02:28.615 ","End":"02:32.350","Text":"and sine gives us cosine."},{"Start":"02:32.810 ","End":"02:36.150","Text":"So that\u0027s these 2."},{"Start":"02:36.150 ","End":"02:38.610","Text":"Now we want at the origin,"},{"Start":"02:38.610 ","End":"02:40.380","Text":"at the point 0, 0."},{"Start":"02:40.380 ","End":"02:44.160","Text":"So grad f at 0,"},{"Start":"02:44.160 ","End":"02:45.480","Text":"0 is equal to."},{"Start":"02:45.480 ","End":"02:47.970","Text":"Just substitute for x, we substitute 0."},{"Start":"02:47.970 ","End":"02:50.770","Text":"For y, we substitute 0."},{"Start":"02:52.040 ","End":"02:54.975","Text":"E^0 is 1."},{"Start":"02:54.975 ","End":"03:03.044","Text":"Cosine of 0 is equal to 1,"},{"Start":"03:03.044 ","End":"03:07.320","Text":"and sine 0 is equal to 0,"},{"Start":"03:07.320 ","End":"03:10.725","Text":"and I see I\u0027d forgotten a bracket here."},{"Start":"03:10.725 ","End":"03:12.780","Text":"This is equal to."},{"Start":"03:12.780 ","End":"03:15.810","Text":"Sine is 0, cosine is 1,"},{"Start":"03:15.810 ","End":"03:20.024","Text":"e^x is 1, all this is 1,"},{"Start":"03:20.024 ","End":"03:26.640","Text":"and here I get also 1 from here minus 0,"},{"Start":"03:26.640 ","End":"03:30.855","Text":"and here 1, so it\u0027s 1, 1."},{"Start":"03:30.855 ","End":"03:34.140","Text":"This actually answers 1 part of the question."},{"Start":"03:34.140 ","End":"03:37.535","Text":"This 1, 1 is the direction"},{"Start":"03:37.535 ","End":"03:41.780","Text":"according to the theorem for which the directional derivative is maximal."},{"Start":"03:41.780 ","End":"03:48.360","Text":"Now the maximal value is given by this magnitude or length,"},{"Start":"03:48.360 ","End":"03:51.695","Text":"in our case, we just need the magnitude of 1,"},{"Start":"03:51.695 ","End":"03:53.810","Text":"1, and what is that?"},{"Start":"03:53.810 ","End":"03:57.290","Text":"That\u0027s just the square root of 1 squared plus 1 squared,"},{"Start":"03:57.290 ","End":"03:59.980","Text":"which is the square root of 2."},{"Start":"03:59.980 ","End":"04:04.400","Text":"This square root of 2 answers the second question of its value."},{"Start":"04:04.400 ","End":"04:08.190","Text":"We found both bits of information, and we\u0027re done."}],"ID":8995},{"Watched":false,"Name":"Exercise 9","Duration":"3m 28s","ChapterTopicVideoID":8662,"CourseChapterTopicPlaylistID":4960,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.585","Text":"This exercise is very similar to a couple of"},{"Start":"00:03.585 ","End":"00:08.655","Text":"previous exercises we did but this time it\u0027s in 3D."},{"Start":"00:08.655 ","End":"00:12.630","Text":"As before, we have to find the direction for which the directional derivative"},{"Start":"00:12.630 ","End":"00:16.570","Text":"of a function at a point is maximal and to compute its value."},{"Start":"00:16.570 ","End":"00:19.470","Text":"The difference is, as I said that here we have a function of x, y,"},{"Start":"00:19.470 ","End":"00:21.840","Text":"and z, and of course,"},{"Start":"00:21.840 ","End":"00:23.895","Text":"the point is also 3D."},{"Start":"00:23.895 ","End":"00:30.255","Text":"We\u0027re going to use the same theorem and I copy pasted from a previous exercise,"},{"Start":"00:30.255 ","End":"00:34.960","Text":"but of course here we\u0027ll need the 3D case."},{"Start":"00:36.980 ","End":"00:43.460","Text":"As before, we need to compute first of all the gradient and that will be"},{"Start":"00:43.460 ","End":"00:49.475","Text":"the direction for maximum change and its length will be the value of the maximal change."},{"Start":"00:49.475 ","End":"00:55.340","Text":"All we need now is the gradient of the function f at a general point x, y,"},{"Start":"00:55.340 ","End":"00:59.230","Text":"z is equal to, in general,"},{"Start":"00:59.230 ","End":"01:01.909","Text":"it\u0027s df by dx partial derivative,"},{"Start":"01:01.909 ","End":"01:06.415","Text":"df by dy, and df by dz."},{"Start":"01:06.415 ","End":"01:09.950","Text":"Now, the derivative of f with respect to x,"},{"Start":"01:09.950 ","End":"01:15.740","Text":"so z and y are constants, 6x squared y."},{"Start":"01:15.740 ","End":"01:19.180","Text":"With respect to y,"},{"Start":"01:19.180 ","End":"01:21.150","Text":"x and z are constants."},{"Start":"01:21.150 ","End":"01:27.930","Text":"Here I just get 2x cubed and from here 6y,"},{"Start":"01:27.930 ","End":"01:31.330","Text":"but the z sticks, 6yz."},{"Start":"01:32.420 ","End":"01:36.290","Text":"In the last component with respect to z,"},{"Start":"01:36.290 ","End":"01:37.310","Text":"x and y are constants,"},{"Start":"01:37.310 ","End":"01:39.170","Text":"so this is nothing."},{"Start":"01:39.170 ","End":"01:42.939","Text":"Constant times z minus 3y squared."},{"Start":"01:42.939 ","End":"01:47.945","Text":"I make it. Now let\u0027s substitute."},{"Start":"01:47.945 ","End":"01:51.275","Text":"We want the gradient of f at the point 1,"},{"Start":"01:51.275 ","End":"01:54.740","Text":"2, minus 1, which means we substitute x, y,"},{"Start":"01:54.740 ","End":"01:58.525","Text":"and z these values and we get,"},{"Start":"01:58.525 ","End":"02:02.115","Text":"if x is 1 and y is 2,"},{"Start":"02:02.115 ","End":"02:06.555","Text":"6,1 squared 2 is 12."},{"Start":"02:06.555 ","End":"02:10.155","Text":"Then 2x cubed is 2."},{"Start":"02:10.155 ","End":"02:13.950","Text":"Y times z is minus 2,"},{"Start":"02:13.950 ","End":"02:17.280","Text":"so it\u0027s 2 plus 6 times 2,"},{"Start":"02:17.280 ","End":"02:19.695","Text":"2 plus 12, 14."},{"Start":"02:19.695 ","End":"02:28.705","Text":"Minus 3y squared, y squared is 4 so minus 3 times 4 minus 12."},{"Start":"02:28.705 ","End":"02:31.640","Text":"This is the gradient."},{"Start":"02:32.040 ","End":"02:36.310","Text":"This answers the question about"},{"Start":"02:36.310 ","End":"02:40.030","Text":"the direction that we were asked here, that\u0027s the direction."},{"Start":"02:40.030 ","End":"02:42.385","Text":"Now we have to find the value."},{"Start":"02:42.385 ","End":"02:45.265","Text":"That\u0027s given by according to the theorem,"},{"Start":"02:45.265 ","End":"02:47.690","Text":"the magnitude of the same thing,"},{"Start":"02:47.690 ","End":"02:53.805","Text":"magnitude length 12, 14, minus 12."},{"Start":"02:53.805 ","End":"02:55.930","Text":"This is the computation,"},{"Start":"02:55.930 ","End":"02:57.050","Text":"I\u0027ll just do part of it."},{"Start":"02:57.050 ","End":"03:01.830","Text":"It\u0027s the square root of 12 squared is 144 plus"},{"Start":"03:01.830 ","End":"03:07.380","Text":"14 squared is 196 minus 12 squared is the same as 12 squared,"},{"Start":"03:07.380 ","End":"03:10.040","Text":"is 144 and that equals,"},{"Start":"03:10.040 ","End":"03:13.290","Text":"I\u0027ll leave it up to you to do it on the calculator."},{"Start":"03:14.170 ","End":"03:20.960","Text":"That answers the second question we were asked as to the value,"},{"Start":"03:20.960 ","End":"03:23.315","Text":"except that you need to compute it."},{"Start":"03:23.315 ","End":"03:28.440","Text":"Just leave it as the square root of something. We\u0027re done."}],"ID":8996},{"Watched":false,"Name":"Exercise 10","Duration":"4m 13s","ChapterTopicVideoID":8663,"CourseChapterTopicPlaylistID":4960,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.364","Text":"This exercise is a little bit different because it\u0027s a word problem."},{"Start":"00:04.364 ","End":"00:07.845","Text":"It\u0027s like physics, but you don\u0027t have to worry about that."},{"Start":"00:07.845 ","End":"00:09.510","Text":"Temperature could be anything."},{"Start":"00:09.510 ","End":"00:13.980","Text":"Basically, we have a function of 3 variables, x, y, and z."},{"Start":"00:13.980 ","End":"00:20.549","Text":"We have to find in which direction we should go to cool off as quickly as possible."},{"Start":"00:20.549 ","End":"00:24.000","Text":"Now, cool off means lower temperature."},{"Start":"00:24.000 ","End":"00:32.535","Text":"We want to find the direction where the function decreases most rapidly."},{"Start":"00:32.535 ","End":"00:36.675","Text":"Now, normally, we deal with increasing."},{"Start":"00:36.675 ","End":"00:41.060","Text":"Basically, what we want is the minimal rate of change,"},{"Start":"00:41.060 ","End":"00:43.820","Text":"whereas we usually look for the maximal rate of change."},{"Start":"00:43.820 ","End":"00:46.415","Text":"Let me bring in a theorem."},{"Start":"00:46.415 ","End":"00:48.950","Text":"This is the theorem for maximum."},{"Start":"00:48.950 ","End":"00:51.590","Text":"I don\u0027t remember if I gave the 1 for minimum."},{"Start":"00:51.590 ","End":"00:55.940","Text":"But we have this theorem about the maximum rate of change in"},{"Start":"00:55.940 ","End":"01:00.575","Text":"the direction of the gradient and the value given by the magnitude of the gradient."},{"Start":"01:00.575 ","End":"01:04.160","Text":"This was in 3D, where x is x, y, z,"},{"Start":"01:04.160 ","End":"01:08.870","Text":"so f of x vector is just x, y, z."},{"Start":"01:08.870 ","End":"01:13.400","Text":"Now, you don\u0027t have to remember 2 theorems"},{"Start":"01:13.400 ","End":"01:19.360","Text":"because if you just change the word maximum here and here to minimum,"},{"Start":"01:19.360 ","End":"01:24.935","Text":"so we\u0027re looking for the minimum value of the directional derivative,"},{"Start":"01:24.935 ","End":"01:28.660","Text":"which is the minimum rate of change,"},{"Start":"01:28.660 ","End":"01:32.300","Text":"then all you have to do to go from maximum to minimum"},{"Start":"01:32.300 ","End":"01:36.800","Text":"is to put a minus here and a minus here."},{"Start":"01:36.800 ","End":"01:41.360","Text":"That\u0027s all. If you go in the opposite direction of the maximum rate of change,"},{"Start":"01:41.360 ","End":"01:44.229","Text":"you get the minimum rate of change."},{"Start":"01:44.229 ","End":"01:48.450","Text":"That\u0027s all we need for now for this."},{"Start":"01:48.450 ","End":"01:51.770","Text":"We need the gradient."},{"Start":"01:51.770 ","End":"02:00.380","Text":"The gradient of f is equal to the vector of partial derivatives with respect to x,"},{"Start":"02:00.380 ","End":"02:03.895","Text":"with respect to y, and with respect to z."},{"Start":"02:03.895 ","End":"02:08.420","Text":"In this case, let\u0027s compute it with respect to x,"},{"Start":"02:08.420 ","End":"02:09.590","Text":"y and z are constants,"},{"Start":"02:09.590 ","End":"02:12.500","Text":"so it\u0027s just 6x."},{"Start":"02:12.500 ","End":"02:17.350","Text":"With respect to y, it\u0027s minus 10y,"},{"Start":"02:17.350 ","End":"02:23.805","Text":"and with respect to z, it\u0027s plus 4z."},{"Start":"02:23.805 ","End":"02:26.805","Text":"What we want is that our point,"},{"Start":"02:26.805 ","End":"02:31.770","Text":"which is 1/3, 1/5, 1/2,"},{"Start":"02:31.770 ","End":"02:37.340","Text":"and this will equal to just substituting for x, y, and z,"},{"Start":"02:37.340 ","End":"02:41.725","Text":"6 times 1/3 is 2,"},{"Start":"02:41.725 ","End":"02:45.810","Text":"1/5 times minus 10 is minus 2,"},{"Start":"02:45.810 ","End":"02:49.380","Text":"and 1/2 times 4 is 2."},{"Start":"02:49.380 ","End":"02:51.660","Text":"This is the vector."},{"Start":"02:51.660 ","End":"02:58.500","Text":"What we want is we want minus grad of f,"},{"Start":"02:58.630 ","End":"03:03.990","Text":"which is at that point, which is,"},{"Start":"03:04.070 ","End":"03:12.065","Text":"minus this is minus 2 plus 2 minus 2."},{"Start":"03:12.065 ","End":"03:21.890","Text":"The negative of this is the answer for the direction and in fact,"},{"Start":"03:21.890 ","End":"03:26.370","Text":"they didn\u0027t ask us for the value of this."},{"Start":"03:27.290 ","End":"03:29.930","Text":"This is the end of the question,"},{"Start":"03:29.930 ","End":"03:31.220","Text":"but I\u0027ll go ahead and say,"},{"Start":"03:31.220 ","End":"03:33.680","Text":"what\u0027s the rate of cooling off?"},{"Start":"03:33.680 ","End":"03:37.175","Text":"I\u0027ll do it in a different color just because it\u0027s a bonus."},{"Start":"03:37.175 ","End":"03:45.930","Text":"The value is minus the gradient of this of minus 2,"},{"Start":"03:45.930 ","End":"03:50.765","Text":"2, minus 2, actually it doesn\u0027t matter about the minuses in the computation."},{"Start":"03:50.765 ","End":"03:53.780","Text":"It\u0027s minus the square root of 2 squared,"},{"Start":"03:53.780 ","End":"03:57.425","Text":"which is 4 plus 4 plus 4."},{"Start":"03:57.425 ","End":"04:03.065","Text":"In other words, the answer is minus the square root of 12,"},{"Start":"04:03.065 ","End":"04:04.400","Text":"but we weren\u0027t asked,"},{"Start":"04:04.400 ","End":"04:06.155","Text":"so this is just extra."},{"Start":"04:06.155 ","End":"04:12.180","Text":"Of course, the maximum is plus the square root of 12. We\u0027re done."}],"ID":8997}],"Thumbnail":null,"ID":4960}]

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