1 00:00:01,550 --> 00:00:04,600 So let us look at ... 2 00:00:07,300 --> 00:00:09,600 ... geometry of a surface. 3 00:00:11,000 --> 00:00:18,400 Because algebraically Taylor series suggests that any function can be linearized ... 4 00:00:18,900 --> 00:00:25,000 ... by dropping all the higher terms of Taylor series. And then you can study that function algebraically. 5 00:00:25,300 --> 00:00:27,100 That linear function. 6 00:00:28,100 --> 00:00:30,700 What about geometry of that? 7 00:00:31,650 --> 00:00:33,400 What if you take ... 8 00:00:35,800 --> 00:00:38,800 ... a graph of a function in three-space? 9 00:00:39,650 --> 00:00:41,900 So this surface ... 10 00:00:42,350 --> 00:00:45,200 ... that looks like a saddle, right? So ... 11 00:00:45,300 --> 00:00:47,300 It's curved one way ... 12 00:00:47,600 --> 00:00:49,350 ... and the other way. 13 00:00:49,450 --> 00:00:50,450 Er ... 14 00:00:51,600 --> 00:00:53,800 So why is it that ... 15 00:00:53,900 --> 00:00:57,850 ... localization -- zooming in at a point gives you a plane. 16 00:00:57,950 --> 00:01:03,000 Well you just have to zoom in. And what I am doing is ... 17 00:01:03,100 --> 00:01:08,300 ... is honest recalculating of the same graph, of the same function ... 18 00:01:08,400 --> 00:01:12,600 ... just changing the scale and zooming in to that point. 19 00:01:12,700 --> 00:01:16,800 And from faraway that point looks curved. 20 00:01:17,700 --> 00:01:19,800 But if you zoom in ... 21 00:01:24,500 --> 00:01:28,600 ... then you see how this surface, this pink surface ... 22 00:01:28,950 --> 00:01:32,600 ... becomes flatter and flatter. 23 00:01:34,500 --> 00:01:38,750 And now those curves -- blue and green curves ... 24 00:01:38,850 --> 00:01:40,600 ... they are almost straight. 25 00:01:40,700 --> 00:01:44,400 And the pink surface containing those curves ... 26 00:01:49,450 --> 00:01:51,350 We don't even distinguish ... 27 00:01:51,600 --> 00:01:54,800 Well you don't even see the difference between the plane and ... 28 00:01:55,150 --> 00:01:58,300 ... that [...] curve. So ... 29 00:02:03,000 --> 00:02:05,000 So that's an example of ... 30 00:02:07,000 --> 00:02:08,600 ... geometric process of zooming in. 31 00:02:08,700 --> 00:02:15,600 And no matter where we zoom in we expect the surface to look like a plane locally. 32 00:02:20,500 --> 00:02:22,400 So that's the ... 33 00:02:22,700 --> 00:02:25,900 ... simple justification of why we look at ... 34 00:02:27,000 --> 00:02:30,300 ... planes as linearizations of surfaces. 35 00:02:30,400 --> 00:02:33,050 And of course not all the surfaces can be linearized. 36 00:02:33,150 --> 00:02:40,000 If you look at special examples like a cone ... 37 00:02:40,650 --> 00:02:44,500 ... then looking at the vertex of the cone and zooming in ... 38 00:02:44,600 --> 00:02:49,300 ... no matter how you zoom in you'll see exactly the same cone. 39 00:02:50,200 --> 00:02:53,250 Right? The angle of the cone will never change. 40 00:02:53,350 --> 00:02:54,450 It'll stay the same. 41 00:02:54,550 --> 00:02:57,800 And you zoom in, and zoom in and you will always see this cone. 42 00:02:57,900 --> 00:03:00,900 And this cone at that point will never become a plane. 43 00:03:01,000 --> 00:03:03,000 So that is ... 44 00:03:03,350 --> 00:03:06,700 ... the simplest example of a surface that cannot be linearized. 45 00:03:06,800 --> 00:03:09,650 Well of course that surface defines a function. 46 00:03:09,750 --> 00:03:14,000 Right? If you put that surface in that x, y, z ... 47 00:03:15,450 --> 00:03:19,000 ... space in this way ... 48 00:03:23,800 --> 00:03:28,300 ... [...] of course for every x, y on x, y plane ... 49 00:03:28,400 --> 00:03:31,400 ... you will have some point on the cone above it. 50 00:03:31,500 --> 00:03:32,950 And that defines a function. 51 00:03:33,050 --> 00:03:35,600 And the typical example would be ... 52 00:03:36,150 --> 00:03:37,150 Er ... 53 00:03:39,200 --> 00:03:40,850 ... the polar distance actually. 54 00:03:40,950 --> 00:03:43,200 The distance from the origin. 55 00:03:43,300 --> 00:03:45,400 So z equals ... 56 00:03:46,000 --> 00:03:50,300 ... the magnitude of the vector x, y. 57 00:03:50,400 --> 00:03:55,300 Or in other words the square root of x squared plus y squared 58 00:03:57,100 --> 00:03:59,250 So that function ... 59 00:03:59,350 --> 00:04:01,400 ... that algebraic function ... 60 00:04:04,800 --> 00:04:10,200 ... has a graph that looks like a cone. And it's not linearizable at zero. 61 00:04:12,900 --> 00:04:19,900 So we'll try to stay away from polar coordinates for as long as possible. 62 00:04:20,000 --> 00:04:21,800 And that's exactly the reason. 63 00:04:21,900 --> 00:04:24,000 And the ... 64 00:04:24,700 --> 00:04:26,700 ... even in er ... 65 00:04:27,400 --> 00:04:29,800 ... the variables x and y ... 66 00:04:30,200 --> 00:04:34,750 ... the function y equals absolute value of x is not good. 67 00:04:34,850 --> 00:04:38,700 Right? Because the graph of that function on the plane has that corner. 68 00:04:38,800 --> 00:04:42,850 And we have to stay away from that studying calculus. 69 00:04:42,950 --> 00:04:47,400 Keeping in mind that yes you can study that function using calculus but there are some ... 70 00:04:47,500 --> 00:04:50,800 ... exceptional things you can do ... well you have to do. |

Video Lectures > Calculus III (2011 Summer) > Lecture 2011.07.13 Functions of Two Variables. Differentials > Part 05 Geometric Linearization of the Graph of a Function >