1 00:00:00,700 --> 00:00:06,000 Er ... Geometry gives meaning to these concepts. 2 00:00:06,600 --> 00:00:10,600 So what we can think about geometrically ... 3 00:00:11,550 --> 00:00:18,900 ... is a function from a plane to a plane ... 4 00:00:19,700 --> 00:00:23,200 ... as a way to transform a plane. 5 00:00:23,300 --> 00:00:26,200 So we can think of transformation ... 6 00:00:31,700 --> 00:00:37,700 ... that takes a point of the plane to another point of the plane. 7 00:00:38,050 --> 00:00:40,700 So what are the examples? 8 00:00:42,000 --> 00:00:44,650 Examples of transformations. 9 00:00:44,750 --> 00:00:47,400 Well let's take a piece of paper. Imagine that ... 10 00:00:47,500 --> 00:00:50,450 ... it's not just a piece of paper, it's a whole plane. 11 00:00:50,550 --> 00:00:53,900 And what is it that we can do with that plane? 12 00:00:54,000 --> 00:00:57,100 Well the simplest thing we can do is we can shift it. 13 00:00:57,200 --> 00:01:01,000 Right? So we shift it for example to the right. 14 00:01:01,100 --> 00:01:03,400 And we imagine the whole plane being shifted. 15 00:01:03,500 --> 00:01:08,600 And each point is going to the right by the same amount. 16 00:01:09,800 --> 00:01:11,200 So that's ... 17 00:01:11,300 --> 00:01:13,100 ... the first example. 18 00:01:13,300 --> 00:01:15,700 Let's call it translation. 19 00:01:19,600 --> 00:01:23,300 Geometrically translation is a motion of a plane that ... 20 00:01:23,400 --> 00:01:27,950 ... moves every point in the same direction by the same amount. 21 00:01:28,050 --> 00:01:31,050 Now what else? Well you can think about rotation. 22 00:01:31,150 --> 00:01:35,300 Fix a point. And rotate by a certain angle. 23 00:01:35,600 --> 00:01:40,400 Right? So what happens is that one point in the center stays the same. 24 00:01:40,500 --> 00:01:47,500 And any other point is going to rotate about the center by a certain degree. 25 00:01:47,700 --> 00:01:51,300 By certain angle. And that angle of course is the same for all the points. 26 00:01:51,400 --> 00:01:53,950 So some points move just by a little bit. 27 00:01:54,050 --> 00:01:59,100 And some other points move a lot. And the further you go from the center ... 28 00:01:59,200 --> 00:02:02,500 ... the more you have of that motion. 29 00:02:03,000 --> 00:02:05,600 So that's another example. 30 00:02:12,500 --> 00:02:14,600 And well, what else can you do? 31 00:02:14,700 --> 00:02:16,600 Well you can also ... 32 00:02:16,750 --> 00:02:19,350 Let's give this more exotic example. 33 00:02:19,450 --> 00:02:24,100 You can also for example take a line in the plane. 34 00:02:26,150 --> 00:02:27,400 This line. 35 00:02:27,500 --> 00:02:29,200 And reflect. 36 00:02:29,300 --> 00:02:33,900 Reflect the bottom pert of the plane to the top. 37 00:02:34,000 --> 00:02:39,650 So that the upper part of the plane stays the same. All the points here don't move anywhere. 38 00:02:39,750 --> 00:02:42,600 But all the points from below ... 39 00:02:43,400 --> 00:02:48,200 ... flip and find their new home in the upper part of the plane. 40 00:02:48,300 --> 00:02:54,100 So this is the ... another example of transformation. Bum, reflection. 41 00:02:54,700 --> 00:02:56,100 Now, what else? 42 00:02:56,200 --> 00:02:58,100 [...] more complicated examples. 43 00:02:58,200 --> 00:03:02,500 Well, you can take the plane -- the whole plane. 44 00:03:02,600 --> 00:03:05,200 And can do something to it. 45 00:03:05,300 --> 00:03:07,000 And then ... 46 00:03:07,700 --> 00:03:09,550 ... you'll smash it. 47 00:03:09,650 --> 00:03:12,500 Make it flat, totally flat. 48 00:03:12,600 --> 00:03:15,650 And then each point of the plane ... 49 00:03:15,750 --> 00:03:20,400 ... again finds its new home in that plane. 50 00:03:20,600 --> 00:03:24,000 So somehow any point of the plane gets ... 51 00:03:24,550 --> 00:03:30,800 ... transformed into another point of the plane by some rule. Right? So ... 52 00:03:30,900 --> 00:03:35,000 The reasonable geometric rule of how you translate the points ... 53 00:03:35,300 --> 00:03:40,600 ... transform, well. Far from being translation, or rotation, or reflection. 54 00:03:41,100 --> 00:03:43,350 But well, there is a rule. 55 00:03:44,351 --> 00:03:47,700 Now, how to ... 56 00:03:49,300 --> 00:03:52,300 ... relate this geometry to algebra? 57 00:03:52,400 --> 00:03:57,050 Well you can introduce coordinate system in the plane. 58 00:03:57,150 --> 00:04:02,000 And you can think about a point having coordinates x, y. 59 00:04:02,400 --> 00:04:06,300 And then applying ... 60 00:04:06,950 --> 00:04:11,700 ... the transformation to that point results in a point ... 61 00:04:12,650 --> 00:04:14,800 ... with coordinates u, v ... 62 00:04:14,900 --> 00:04:17,000 ... of the same plane. Right? 63 00:04:18,200 --> 00:04:21,000 So it's ... er ... 64 00:04:22,450 --> 00:04:25,500 It's about taking a point to a point. 65 00:04:26,500 --> 00:04:33,650 Now er ... it's important to remind here about this concept of transformation and about concept of a function. 66 00:04:33,750 --> 00:04:39,800 Is that you should not think of this function as a process ... 67 00:04:39,900 --> 00:04:43,450 ... because er ... a lot of examples here ... 68 00:04:43,550 --> 00:04:49,250 ... many examples seem to be a process like translation or rotation. 69 00:04:49,350 --> 00:04:52,900 Well function is not a process. Er ... 70 00:04:53,000 --> 00:04:56,000 Let's recall the concept of a function ... 71 00:04:56,350 --> 00:05:01,800 ... from Calc I when you have one variable x, one variable y and ... 72 00:05:02,200 --> 00:05:05,700 ... y equals f of x is a function. 73 00:05:05,800 --> 00:05:09,200 Well you never think about this function as ... 74 00:05:09,300 --> 00:05:13,000 ... a process of transforming this line ... 75 00:05:13,300 --> 00:05:17,350 ... bending it and then smashing to the y-line. 76 00:05:17,450 --> 00:05:22,600 You think of the function as a process of taking a point here -- x ... 77 00:05:22,700 --> 00:05:28,200 ... and then following some rule to result in a point there -- y. 78 00:05:29,600 --> 00:05:32,600 Right? You go straight -- here, there. 79 00:05:33,500 --> 00:05:35,900 Given x, result y. 80 00:05:36,000 --> 00:05:40,500 There is no process of transforming that you usually think about. 81 00:05:40,600 --> 00:05:44,300 And I'd like you not to think about any process. 82 00:05:45,900 --> 00:05:51,700 So translation, yes, it can be thought of as a process of moving point slowly to the right. 83 00:05:51,800 --> 00:05:57,950 But we will be interested in just going from the beginning straight to the end ... 84 00:05:58,050 --> 00:06:00,350 ... without thinking about the process of motion. 85 00:06:00,450 --> 00:06:03,500 Or without thinking about the process of rotating. 86 00:06:04,800 --> 00:06:08,800 Original point -- end point. That's it. No process. 87 00:06:09,300 --> 00:06:10,300 Er ... |

Video Lectures > Calculus III (2012 Spring) > Lecture 2012.01.27 Functions from a Plane to a Plane > Part 02 Geometric Approach >