1 00:00:00,420 --> 00:00:06,420 So now we will consider a curl into divergence for this example, function vector are divided by Ah, 2 00:00:06,990 --> 00:00:14,340 and I've programmed this already here and we can have a look at the figure once again. 3 00:00:14,370 --> 00:00:17,010 So you see, it's along the radial direction, always. 4 00:00:17,430 --> 00:00:21,360 And now let's go ahead and calculate the divergence and the curl. 5 00:00:22,650 --> 00:00:32,330 So the definition of the divergence of G at the position R and H, so this will be the step size again. 6 00:00:32,340 --> 00:00:36,060 And these are three arguments that we always must provide. 7 00:00:36,870 --> 00:00:45,000 And as previously, I will just label these components of art as X, Y and Z, and then I can just go 8 00:00:45,000 --> 00:00:53,370 ahead and calculate the G X D X. So this will be the partial derivative of the X component of G. 9 00:00:53,640 --> 00:00:54,630 With respect to X. 10 00:00:55,530 --> 00:00:58,560 And so this will once again be a central difference as methods. 11 00:00:59,670 --> 00:01:00,720 This will be G 12 00:01:03,660 --> 00:01:06,450 four and P Dot Parade. 13 00:01:06,990 --> 00:01:12,540 So now we must provide the vector of the position, which will be initially X, Y and Z. 14 00:01:13,800 --> 00:01:21,360 And since we want to have here the X component of G, we must now use something like this. 15 00:01:21,370 --> 00:01:28,270 So we accessed the first component and now we do central differences with respect to X. 16 00:01:28,290 --> 00:01:36,570 So we write plus h for the X component, and then we copy the same thing minus and minus h. 17 00:01:36,960 --> 00:01:40,210 And we divide it by two times h. 18 00:01:41,640 --> 00:01:50,670 And now we do the other two terms, which will be g y DUI and GC DC. 19 00:01:52,050 --> 00:01:57,330 So now we must just be careful to shift here to h for the central differences. 20 00:01:58,680 --> 00:02:06,390 But not only that, we must also change here the index from here to the Y component for this one and 21 00:02:06,390 --> 00:02:09,120 two, the Z component and here as well. 22 00:02:10,259 --> 00:02:11,940 So now we have the individual terms. 23 00:02:12,120 --> 00:02:20,810 Let me show you again these ones here g x the X G Y de Y GC DC. 24 00:02:20,850 --> 00:02:23,910 And so for the divergence, we have to add these three up. 25 00:02:25,740 --> 00:02:28,560 So I return 26 00:02:31,710 --> 00:02:45,720 on d g x d x plus DG y d y plus DG, C D C, and then we can check what is the value for our specific 27 00:02:45,720 --> 00:02:46,260 position. 28 00:02:46,800 --> 00:02:51,780 Then we have to find to be 0.5, minus 1.2 and minus eight. 29 00:02:51,870 --> 00:02:53,040 Just some random numbers. 30 00:02:54,600 --> 00:02:55,650 This would be that. 31 00:02:56,730 --> 00:03:01,830 And when we compare it with the analytical solution, which is actually pretty difficult to get when 32 00:03:01,830 --> 00:03:05,730 you have never done it before, the analytical solution gives us 33 00:03:10,920 --> 00:03:11,490 this number. 34 00:03:12,720 --> 00:03:15,710 So once again, we have a pretty high accuracy. 35 00:03:15,720 --> 00:03:22,440 You see, at some points after the dots, there is some difference, but we can of course, improve 36 00:03:22,440 --> 00:03:28,290 this by just increasing or sorry, by decreasing the step size and then we increase the accuracy. 37 00:03:29,040 --> 00:03:31,140 But I think this is really sufficient. 38 00:03:31,180 --> 00:03:32,860 And yeah, it works. 39 00:03:32,880 --> 00:03:34,590 We have implemented the divergence. 40 00:03:36,020 --> 00:03:44,040 And now for the curl, the curl is actually pretty similar to the divergence because we must just calculate 41 00:03:44,040 --> 00:03:45,510 the other partial derivatives. 42 00:03:45,540 --> 00:03:51,210 So for example, GC, D, Y, G, Y, DC and so on and so on. 43 00:03:51,510 --> 00:03:56,460 And then we must construct three components and calculate these differences here. 44 00:03:57,750 --> 00:04:00,740 So this is really nothing special. 45 00:04:00,750 --> 00:04:07,020 It's very similar actually to this one, but it's a bit more work to write everything down. 46 00:04:08,100 --> 00:04:13,860 So for this reason, I will now just stop the video and write it down. 47 00:04:13,860 --> 00:04:17,820 And then if you want, you can write it in your own notebook as well. 48 00:04:18,029 --> 00:04:19,140 So see you in a second. 49 00:04:19,470 --> 00:04:23,340 So you see pretty similar to the divergence I have now created to curl. 50 00:04:23,670 --> 00:04:30,300 Once again, I've labeled X, Y and Z, and I have created the six remaining partial derivatives. 51 00:04:30,420 --> 00:04:36,420 So for example, G X DUI, G X DC and so on and so on. 52 00:04:36,420 --> 00:04:37,860 So all these mixed terms. 53 00:04:38,940 --> 00:04:47,080 And for example, to calculate the derivative of G X with respect to why we must take the X components. 54 00:04:47,080 --> 00:04:49,470 So here the component index will be zero. 55 00:04:49,890 --> 00:04:58,170 And since we derive with respect to why we must add the H to Y and subtract the H from Y, and then 56 00:04:58,170 --> 00:04:59,760 you do the same thing for the other. 57 00:04:59,880 --> 00:05:04,980 Five components, and then you must have a look at how the gradient. 58 00:05:05,440 --> 00:05:09,510 I'm sorry, sorry, how the curl is constructed. 59 00:05:09,930 --> 00:05:15,390 This is just by taking these three components here and you always pairwise subtract them. 60 00:05:16,110 --> 00:05:20,280 And for the X component, you calculate GC. 61 00:05:21,090 --> 00:05:24,990 So DGC the Y minus the GeoEye DC. 62 00:05:25,820 --> 00:05:33,330 And this is why I let the GC the y minus DG y these c, so exactly what I just said. 63 00:05:34,080 --> 00:05:40,320 And then, of course, for the other components, you do the same thing and you see already the analytical 64 00:05:40,320 --> 00:05:43,860 solution tells us that this is zero for every point in space. 65 00:05:43,860 --> 00:05:45,780 So let's see if this really is true. 66 00:05:46,980 --> 00:05:51,060 G ah, and h, yes, indeed. 67 00:05:51,240 --> 00:05:58,260 So you see, there's a some value times 10 to the power of minus 12, 12, 13, so this is essentially 68 00:05:58,260 --> 00:05:58,830 zero. 69 00:06:00,060 --> 00:06:05,370 So I hope you're not disappointed that this example didn't give us such a cool result for the curl. 70 00:06:05,730 --> 00:06:12,640 But actually in the next section where we will discuss integration, I will in the end consider a really 71 00:06:12,640 --> 00:06:19,200 a physical example where we will calculate the magnetic fields of a charged wire. 72 00:06:19,800 --> 00:06:25,020 And there we will first calculate a so-called vector potential using the integral method that we will 73 00:06:25,020 --> 00:06:26,370 establish in the next section. 74 00:06:26,820 --> 00:06:33,000 And thereafter, we must calculate the magnetic field from this vector potential and this we do by calculating 75 00:06:33,000 --> 00:06:33,570 the curl. 76 00:06:34,260 --> 00:06:40,170 So in the next section, there will be a nice exercise for practicing how to calculate the curl where 77 00:06:40,170 --> 00:06:43,920 the result will something will be something different to zero. 78 00:06:44,790 --> 00:06:47,640 So are you still excited and I see you in the next lecture?