1 00:00:00,070 --> 00:00:06,810 Let us now talk about derivatives for functions with multiple arguments at this lecture is about the 2 00:00:06,810 --> 00:00:08,980 so-called partial derivatives. 3 00:00:09,810 --> 00:00:16,790 And before we can define what partial derivatives are in multi argument functions, that was first considered 4 00:00:16,790 --> 00:00:18,120 the one dimensional case. 5 00:00:18,820 --> 00:00:24,780 So here we have the function F of X, which could, for example, be this blue function that you can 6 00:00:24,780 --> 00:00:26,520 see here, which is a Kubic function. 7 00:00:27,540 --> 00:00:35,010 Now, if you want to to find the derivative, we have to look at the Points X, which is, for example, 8 00:00:35,010 --> 00:00:36,300 there's Redpoint here. 9 00:00:37,300 --> 00:00:44,640 So we take the value of this function F of X, and then we also consider different point X plus H. 10 00:00:44,760 --> 00:00:49,870 So this could, for example, be this orange point here where the distance in the X direction is H. 11 00:00:50,400 --> 00:00:55,770 And so we take the value of the function at this point, which is does one here take the difference? 12 00:00:56,010 --> 00:01:04,320 And we divide by the distance in X direction, which is H, and the result is the slope of this orange 13 00:01:04,320 --> 00:01:07,080 function here, which goes through both of these points. 14 00:01:08,340 --> 00:01:10,680 Now this is not yet the derivative. 15 00:01:11,220 --> 00:01:14,250 We first have to make h smaller and smaller. 16 00:01:14,260 --> 00:01:21,960 We have to take the limit of H to zero so that the Orange Point moves along the blue curve and approaches 17 00:01:21,960 --> 00:01:22,940 the Red Point here. 18 00:01:23,910 --> 00:01:30,930 And if we do this, we get a red function here, which has to slope, which is the derivative of the 19 00:01:30,930 --> 00:01:32,510 function at this point. 20 00:01:34,050 --> 00:01:39,950 And this is exactly the same definition as the conventional derivative in one dimension. 21 00:01:40,620 --> 00:01:46,800 And alternatively, you could also use this definition where you go into the negative direction with 22 00:01:46,800 --> 00:01:47,400 the slope. 23 00:01:47,970 --> 00:01:52,350 And you could also take this one here in the limit of zero. 24 00:01:52,360 --> 00:01:55,830 All of these expressions would get this would give you the same value. 25 00:01:56,130 --> 00:01:58,050 You can just take whatever you prefer. 26 00:01:59,280 --> 00:02:06,070 And as you may notice, this is exactly the same as the conventional derivative in one dimension. 27 00:02:06,300 --> 00:02:13,200 So this partial derivative, which is denoted by this this expression here is exactly the same as the 28 00:02:13,200 --> 00:02:14,430 conventional derivative. 29 00:02:15,450 --> 00:02:22,170 And to make it a bit easier in the notation, you could also write it down like this, and sometimes 30 00:02:22,170 --> 00:02:24,090 it's also written down like this. 31 00:02:24,570 --> 00:02:30,790 So all of this are just different notations for the same thing in one dimension in 2D. 32 00:02:31,050 --> 00:02:37,500 This is a bit different because here we have a function that depends on multiple arguments, for example, 33 00:02:37,500 --> 00:02:38,880 here, X and Y. 34 00:02:39,620 --> 00:02:46,230 So we cannot simply write F and then this dash here, because then it's not really clear what we mean 35 00:02:46,230 --> 00:02:46,710 by that. 36 00:02:47,670 --> 00:02:51,440 So in two dimensions, we really need to consider these partial derivatives. 37 00:02:51,990 --> 00:02:58,680 For example, here, if we calculate the partial derivative of F of X and Y with respect to X, then 38 00:02:58,680 --> 00:03:05,080 this means we only derive with respect to this coordinate and not with respect to Y. 39 00:03:05,970 --> 00:03:13,800 So this means in this limited definition here, we just add a value H to the X variable and not to Y, 40 00:03:14,430 --> 00:03:16,710 and the rest is then the same as previously. 41 00:03:17,730 --> 00:03:25,560 So what this means is that when we have such a function here where we have X and Y at these X's and 42 00:03:25,560 --> 00:03:29,210 the height of this plot is the value of F. 43 00:03:29,220 --> 00:03:31,350 So this is something like. 44 00:03:31,800 --> 00:03:32,640 Yeah, like Mount. 45 00:03:32,640 --> 00:03:37,860 And you could imagine where you have two coordinates and the third coordinate the height, then the 46 00:03:37,860 --> 00:03:38,820 value of the function. 47 00:03:39,330 --> 00:03:46,530 So if we want to calculate here the slope along the X direction at some point, for example here, then 48 00:03:46,530 --> 00:03:51,000 we have to calculate the partial derivative of F with respect to X. 49 00:03:52,530 --> 00:03:58,160 And alternatively, we could, of course, also calculate the partial derivative with respect to why 50 00:03:58,860 --> 00:04:05,280 and in this case, we have to add this small value of age, which we make small and smaller to the Y 51 00:04:05,280 --> 00:04:05,910 argument. 52 00:04:06,600 --> 00:04:14,040 And then it gives us the slope of this line here, which measures the slope of this mountain along the 53 00:04:14,040 --> 00:04:14,970 Y direction. 54 00:04:15,960 --> 00:04:20,940 Now, here we had only shown what the derivative means for one specific point. 55 00:04:21,300 --> 00:04:27,240 But of course, like in one dimension, we can write down the derivative in a general way where we are 56 00:04:27,240 --> 00:04:29,070 not restricted to a single point. 57 00:04:29,700 --> 00:04:32,860 So this derivative then becomes a function itself. 58 00:04:33,570 --> 00:04:36,300 So let's consider this as an example here. 59 00:04:36,780 --> 00:04:39,660 So let's take this mountain that we also had on the previous slide. 60 00:04:39,960 --> 00:04:47,520 And here the function is given by the expression here we have basically a caution function in the X 61 00:04:47,520 --> 00:04:52,410 direction, which is an exponential function with the argument minus X square. 62 00:04:53,070 --> 00:04:57,090 So this gives us such a caution shape here and then in the right direction. 63 00:04:57,210 --> 00:04:59,370 I have chosen a bit of a different profile. 64 00:04:59,370 --> 00:05:05,850 I have chosen the exponential function with the argument minus Y to the power of four, which gives 65 00:05:05,850 --> 00:05:08,220 us more of a peculiar shape here. 66 00:05:09,000 --> 00:05:14,940 So just I wanted to make it a bit more interesting and not just a boring Gorshin function in every direction. 67 00:05:16,200 --> 00:05:19,950 So now we can calculate the partial derivative with respect to X. 68 00:05:20,670 --> 00:05:27,030 So what we have to do is we have to derive this expression here with respect to X and we have to leave 69 00:05:27,300 --> 00:05:29,650 Y like a number. 70 00:05:30,120 --> 00:05:35,160 So if we would derive this with respect to X, it would give zero. 71 00:05:36,330 --> 00:05:43,800 So what we have here, we could also imagine something like minus X squared, plus two, for example. 72 00:05:44,790 --> 00:05:47,050 Now, how would we derive such an expression? 73 00:05:47,790 --> 00:05:55,530 First of all, we must apply the chain rule because we have an exponential function of some argument. 74 00:05:55,950 --> 00:05:59,130 And then the argument itself is, again, a function. 75 00:05:59,700 --> 00:06:05,160 This is this a function would be minus X squared, minus Y to the power of four. 76 00:06:06,120 --> 00:06:11,850 So first we have to calculate the outer derivative and the outer derivative of an exponential function 77 00:06:11,850 --> 00:06:16,830 is the exponential function because the exponential function does not change and the derivative. 78 00:06:17,850 --> 00:06:23,220 But then we also have to multiply with the derivative of the inner function because of the chain rule 79 00:06:24,000 --> 00:06:25,980 and the derivative of this expression. 80 00:06:26,190 --> 00:06:32,980 If we derive the respect to X is minus two times X and this one is zero. 81 00:06:34,050 --> 00:06:40,920 So we get as a whole partial derivative minus two times x2 in the derivative times, the outer derivative, 82 00:06:40,920 --> 00:06:42,370 which is this function itself. 83 00:06:43,680 --> 00:06:50,070 And if we plot this and calculate the partial derivative along the X direction, then we get something 84 00:06:50,070 --> 00:06:50,590 like this. 85 00:06:51,060 --> 00:06:58,340 So this HitIer tells us the value of this partial derivative and every position, X and Y. 86 00:06:59,280 --> 00:07:05,040 So you could imagine like this, you, your you as a human stand here at the bottom of the mountain 87 00:07:05,040 --> 00:07:10,640 and you want to go up and now you want to plot how steep is the slope of this mountain. 88 00:07:11,100 --> 00:07:14,370 And of course, here in this area, it's especially steep. 89 00:07:14,370 --> 00:07:18,750 So you get large value and at the top of the mountain you have a plateau. 90 00:07:18,930 --> 00:07:20,790 So there is no slope at all. 91 00:07:20,790 --> 00:07:22,800 Does it see this wide region in between? 92 00:07:23,430 --> 00:07:27,390 And then it goes down and here it's very steep and negative. 93 00:07:27,660 --> 00:07:29,490 So you get this blue area here. 94 00:07:30,570 --> 00:07:37,380 And when you are, for example, here and you go just along the X direction, there is almost no slope, 95 00:07:37,380 --> 00:07:38,490 just maybe a bit here. 96 00:07:38,490 --> 00:07:41,310 But or if you go here and it's really zero. 97 00:07:41,580 --> 00:07:44,740 So this is why this is here zero in all cases. 98 00:07:46,260 --> 00:07:48,750 Now, imagine you want to take a different trip. 99 00:07:48,960 --> 00:07:51,540 You don't want to stand here and go along the X direction. 100 00:07:51,960 --> 00:07:54,330 You want to approach the mountain from a different side. 101 00:07:54,340 --> 00:07:57,360 You stand here and you go along the Y direction. 102 00:07:58,440 --> 00:08:04,140 So if you want to calculate the slope, you have to calculate the partial derivative with respect to 103 00:08:04,140 --> 00:08:05,340 Y in this case. 104 00:08:06,120 --> 00:08:08,550 So this is a very similar procedure. 105 00:08:08,790 --> 00:08:13,980 But this time we have to consider that this one here as a constant and this one is our argument. 106 00:08:14,800 --> 00:08:21,240 So our partial derivative would be again the outer derivative, which is just exponential function times, 107 00:08:21,300 --> 00:08:26,460 you know, derivative, which is minus four times Y to the power of three. 108 00:08:27,690 --> 00:08:29,250 So this is a partial derivative. 109 00:08:29,640 --> 00:08:31,810 And if we plotted, we get something like this. 110 00:08:32,730 --> 00:08:40,920 So here you can see we first approach this very, very steep slope here where this derivative has a 111 00:08:40,920 --> 00:08:41,610 large peak. 112 00:08:42,630 --> 00:08:49,020 And then when you approach the top of the mountain slope gets less steep. 113 00:08:49,020 --> 00:08:51,330 And then at the plateau, the slope. 114 00:08:51,440 --> 00:08:58,520 Zero, this is why you get here a very large region where the river turf is basically zero and then 115 00:08:58,520 --> 00:08:59,390 it goes back down. 116 00:08:59,390 --> 00:09:00,830 So you get this negative value. 117 00:09:01,430 --> 00:09:07,640 So you can see in principle, it's the same function as here, but it's rotated and the slopes are much 118 00:09:07,640 --> 00:09:12,950 more steeper because we have here this way to the poor for dependents and here we have to x to the power 119 00:09:12,950 --> 00:09:14,090 of two dependents. 120 00:09:14,540 --> 00:09:21,410 OK, so we have learned what partial derivatives are and what they mean in terms of approaching a mountain 121 00:09:21,410 --> 00:09:23,210 along the X or the Y direction. 122 00:09:24,050 --> 00:09:26,630 Now, of course, you could also go a different path. 123 00:09:27,050 --> 00:09:30,960 You could stand here and go along this diagonal direction. 124 00:09:31,430 --> 00:09:34,490 So how would we calculate the slope along this direction? 125 00:09:35,240 --> 00:09:41,750 And here we have to use a directional derivative, which is, of course, related to these two individual 126 00:09:41,750 --> 00:09:42,500 derivatives. 127 00:09:43,490 --> 00:09:50,860 So let's consider this as a vector along which we go, which is yeah, it has one one component. 128 00:09:50,880 --> 00:09:52,970 So it's really along the diagonal. 129 00:09:53,240 --> 00:09:57,560 And then here I just normalized it so that the length of this vector is one. 130 00:09:58,970 --> 00:10:01,620 And the result will look like this. 131 00:10:02,150 --> 00:10:08,600 So when we go along this direction, we will get a slope profile that looks like this and how can we 132 00:10:08,600 --> 00:10:09,500 calculate this? 133 00:10:10,220 --> 00:10:17,690 So we have to take this directional vector Eve and we have to calculate the DOT product with this vector 134 00:10:17,690 --> 00:10:19,310 of the partial derivatives. 135 00:10:19,970 --> 00:10:26,240 So here we have the partial derivative, which is pegged to X, Y and Z of our function F. 136 00:10:28,710 --> 00:10:36,300 Now, what we can also do is we can take this to here and write it down like this, so this is an abbreviation 137 00:10:36,300 --> 00:10:41,220 of this whole expression and this triangle here is called Nabala Operator. 138 00:10:41,490 --> 00:10:44,610 And we will come to this operator in the next lecture in detail. 139 00:10:45,450 --> 00:10:52,200 But here, it's also important to realize that this directional vector eve is a unit vector, because 140 00:10:52,200 --> 00:10:57,490 if it would have a larger length, then we would artificially scale up our slope. 141 00:10:58,020 --> 00:11:05,400 So it's important to normalize this vector here to the length of one that I already told you this. 142 00:11:05,400 --> 00:11:07,270 This is called the Doppler operator. 143 00:11:07,650 --> 00:11:15,480 This is this vector after partial derivatives that you have to let's act on the function F so in general, 144 00:11:15,480 --> 00:11:21,600 our function F can depend on multiple arguments, for example, here on three three dimensions. 145 00:11:21,870 --> 00:11:28,390 And then it could of course also depend on another argument which is not included here in this Knobler 146 00:11:28,410 --> 00:11:28,910 vector. 147 00:11:30,000 --> 00:11:33,780 And then this knobler acting on F is also called the gradient. 148 00:11:34,020 --> 00:11:36,120 But once again we will come to this later. 149 00:11:37,340 --> 00:11:41,970 So let's go back to our case here and let's calculate this DOT product. 150 00:11:42,570 --> 00:11:51,570 So we calculate five times the gradient of F and the gradient of F is just a vector with this entry 151 00:11:51,570 --> 00:11:54,450 as the first entry and this year as the second entry. 152 00:11:55,110 --> 00:12:04,270 So we write one of the square root two times, one times this one, which is this plus one of a square 153 00:12:04,270 --> 00:12:07,230 root two times, one times this one here. 154 00:12:08,280 --> 00:12:12,120 And what this gives us, because in both cases we have this exponential function. 155 00:12:12,600 --> 00:12:14,100 We can round it down like this. 156 00:12:14,130 --> 00:12:18,200 We have in both cases the minus sign have in both cases the exponential function. 157 00:12:18,630 --> 00:12:23,580 The only difference is the two X and the four Y to the power of three. 158 00:12:24,510 --> 00:12:30,360 And if we plot this expression, we get exactly this one here, which is the directional derivative 159 00:12:30,600 --> 00:12:37,980 along this direction, which is today I and in the next lecture we will discuss the KNOBLER operator 160 00:12:37,980 --> 00:12:43,620 in detail and we will talk about the gradient, which you have already learned about a bit, but also 161 00:12:43,620 --> 00:12:44,400 divergence. 162 00:12:44,400 --> 00:12:44,970 And Karl.