1 00:00:05,400 --> 00:00:09,000 OK. So let me look at ... 2 00:00:09,300 --> 00:00:11,100 [...] 3 00:00:12,100 --> 00:00:14,700 Let me now look at this question. 4 00:00:24,700 --> 00:00:26,600 The question I mentioned er ... 5 00:00:26,700 --> 00:00:32,850 If you start at some point x zero, y zero on the map ... 6 00:00:32,950 --> 00:00:36,300 ... and you want to go and make a step ... 7 00:00:37,400 --> 00:00:38,700 ... in a certain direction ... 8 00:00:38,800 --> 00:00:44,400 ... and your direction is not in direction of x or y -- some direction, right? 9 00:00:44,500 --> 00:00:49,500 So then this step can be denoted as a vector. 10 00:00:50,100 --> 00:00:52,700 Right? And what kind of vector? 11 00:00:52,800 --> 00:00:56,400 Well the vector in the coordinates x and y ... 12 00:00:57,200 --> 00:01:00,300 ... is going to be ... is going to mean ... 13 00:01:00,600 --> 00:01:03,700 The x-coordinate will be change of x. 14 00:01:03,800 --> 00:01:06,050 The y-coordinate will be change of y. 15 00:01:06,150 --> 00:01:12,100 We can safely call those dx and dy or delta x and delta y. 16 00:01:12,200 --> 00:01:13,850 Those are the same things. 17 00:01:13,950 --> 00:01:19,200 So you apply that vector to the point x zero, y zero. 18 00:01:19,300 --> 00:01:21,700 And you look at what happens there. 19 00:01:21,800 --> 00:01:23,800 So what does it mean? 20 00:01:24,000 --> 00:01:29,600 You look at what happens. You look at the change of the values f of ... 21 00:01:30,100 --> 00:01:37,200 ... x plus dx, y plus dy minus f of ... 22 00:01:38,700 --> 00:01:40,900 ... x zero, y zero ... 23 00:01:41,000 --> 00:01:43,900 [...] x zero and y zero there. 24 00:01:44,000 --> 00:01:48,800 So the difference of values at these two points ... 25 00:01:49,400 --> 00:01:52,500 ... related to the ... 26 00:01:53,500 --> 00:01:54,500 What? 27 00:01:55,400 --> 00:01:57,100 ... to the magnitude of that vector. 28 00:01:57,200 --> 00:02:00,400 To the length of your step as you measure it on a map. 29 00:02:02,750 --> 00:02:06,600 Magnitude of dx, dy. 30 00:02:09,500 --> 00:02:14,300 That's going to be the approximately the rate of change in that direction. 31 00:02:15,500 --> 00:02:16,500 And er ... 32 00:02:18,000 --> 00:02:23,700 The good thing about this quantity is that it can be ... 33 00:02:24,100 --> 00:02:27,800 ... easily expressed using that ... 34 00:02:31,900 --> 00:02:34,000 ... formula for differentials. 35 00:02:34,150 --> 00:02:40,500 So as we replace the difference of values of f with df. 36 00:02:48,350 --> 00:02:49,800 What is it that we see. 37 00:02:49,900 --> 00:02:53,000 You see that df is equal to ... 38 00:02:53,200 --> 00:02:58,800 ... f_x dx plus f_y dy ... 39 00:02:59,900 --> 00:03:04,200 ... all that is related to the magnitude of dx, dy. 40 00:03:09,800 --> 00:03:17,500 And then [...] momentarily recognize algebraically the numerator as ... 41 00:03:20,000 --> 00:03:22,000 ... dot product, yes! 42 00:03:22,900 --> 00:03:24,900 Isn't it a dot product? 43 00:03:26,150 --> 00:03:28,300 So how do you put these together? 44 00:03:28,400 --> 00:03:30,200 How do you like them to be together? 45 00:03:30,300 --> 00:03:32,200 Making two vectors. 46 00:03:34,850 --> 00:03:37,000 And dot product within them. 47 00:03:38,500 --> 00:03:41,300 Student: [...]. 48 00:03:41,400 --> 00:03:43,900 You put partials together of course. 49 00:03:44,050 --> 00:03:46,400 And you put dx, dy together. 50 00:03:48,050 --> 00:03:52,000 And then dividing by the magnitude of dx, dy ... 51 00:03:52,800 --> 00:03:54,800 ... makes perfect sense. 52 00:03:57,800 --> 00:04:04,300 Because then the dot product with that vector divided by the magnitude of that vector ... 53 00:04:04,400 --> 00:04:05,400 ... is ... 54 00:04:06,650 --> 00:04:07,650 ... component ... 55 00:04:09,500 --> 00:04:16,350 ... of the vector f_x, f_y of the vector made by partial derivatives ... 56 00:04:16,450 --> 00:04:19,800 ... in direction of dx, dy. 57 00:04:21,200 --> 00:04:23,500 Is that any special direction? 58 00:04:23,600 --> 00:04:28,000 Well that's exactly the direction that we follow, right? 59 00:04:28,100 --> 00:04:33,750 So if you want to go from that point in that direction ... 60 00:04:33,850 --> 00:04:36,000 ... then this component ... 61 00:04:36,800 --> 00:04:39,900 ... tells you the rate of change in that direction. 62 00:04:42,950 --> 00:04:43,950 And ... 63 00:04:45,150 --> 00:04:48,600 That not only gives you a computable value. 64 00:04:50,900 --> 00:04:54,200 It tells you, hints you ... 65 00:04:55,100 --> 00:04:59,100 ... that this vector has some importance. 66 00:05:00,300 --> 00:05:04,000 If you take ... well you never did this, you never put ... 67 00:05:04,100 --> 00:05:08,300 ... partial derivatives of a function together as a vector. 68 00:05:09,000 --> 00:05:10,600 And now we do. 69 00:05:11,100 --> 00:05:14,600 So for given function f of x, y ... 70 00:05:14,900 --> 00:05:18,500 ... if you take partial derivatives ... 71 00:05:18,600 --> 00:05:21,850 ... and you put them together and make a vector ... 72 00:05:21,950 --> 00:05:23,800 ... and the function of ... 73 00:05:23,900 --> 00:05:25,400 The function is of two variables. 74 00:05:25,500 --> 00:05:27,200 You have two partial derivatives. 75 00:05:27,300 --> 00:05:29,800 It is going to be a two dimensional vector. 76 00:05:29,900 --> 00:05:32,900 Then that's called the gradient vector. 77 00:05:39,500 --> 00:05:45,200 And that is actually considered as the derivative ... 78 00:05:51,450 --> 00:05:52,900 ... of f. 79 00:06:01,400 --> 00:06:03,100 Why do you think it's ... 80 00:06:03,200 --> 00:06:06,650 ... it should be considered as derivative? What should be a hint? 81 00:06:09,000 --> 00:06:11,700 Why is that the derivative? 82 00:06:18,050 --> 00:06:20,100 [...] as a question. 83 00:06:20,800 --> 00:06:21,800 I'll tell you tomorrow. |

Video Lectures > Calculus III (2011 Summer) > Lecture 2011.07.13 Functions of Two Variables. Differentials > Part 10 Rate of Change of a Function in a Given Direction >