1

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So let's go ahead and talk about the Nabila operator and the previous lecture, we have already introduced



2

00:00:06,490 --> 00:00:10,410

the operator as being a vector of the partial derivatives.



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00:00:11,430 --> 00:00:15,380

And alternatively, you could also write it down like this to save some space.



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00:00:15,420 --> 00:00:20,640

So this t just means you transpose this line vector in such a column vector.



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00:00:21,730 --> 00:00:27,730

And of course, if you just take this KNOBLER operator by itself, it doesn't mean anything, you always



6

00:00:27,730 --> 00:00:30,720

had to let this operator act on the function.



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00:00:30,730 --> 00:00:33,730

So that makes sense then.



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00:00:34,270 --> 00:00:39,970

Other text books or on websites, you may also find other notations.



9

00:00:40,390 --> 00:00:47,430

For example, you could write down does not an operator as partial derivative with respect to a vector,



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00:00:48,010 --> 00:00:54,610

which I don't really like because it would somehow indicate that you are dividing by a vector, which



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00:00:54,610 --> 00:00:55,640

doesn't make any sense.



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So I prefer this KNOBLER operator.



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But the downside of this Knapp's operator, is that you don't really know with respect to which variables



14

00:01:04,480 --> 00:01:05,530

you differentiate.



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So typically it's just X, Y and Z.



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00:01:09,280 --> 00:01:15,340

But if you have some more difficult problem where you have multiple vectors included in your function



17

00:01:15,580 --> 00:01:19,630

and you maybe also want to derive with respect to different variables.



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For example, when you have a function that depends on the vector, which is X, Y, Z, but also on



19

00:01:27,160 --> 00:01:34,060

the vector M, which is M, X and Y and Z, and you want to differentiate with respect to M, then you



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00:01:34,060 --> 00:01:35,320

could write it down like this.



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00:01:35,770 --> 00:01:39,720

You write down the block operator and you write down M on the bottom.



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00:01:39,730 --> 00:01:45,850

And this indicates that you don't want to derive with respect to X, Y, Z, but with respect to different



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00:01:45,850 --> 00:01:46,510

variables.



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And if you write down nothing here, then it's typically assumed that you derive with respect to X,



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00:01:53,320 --> 00:01:54,070

Y and Z.



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00:01:56,100 --> 00:02:02,100

Now, we have also learned already about the gradient, which is when you take your Anabella operator



27

00:02:02,460 --> 00:02:05,430

and you let this operator act on the scalar function.



28

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So this function ideally depends on all of these variables X, Y and Z.



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00:02:11,460 --> 00:02:15,940

And it could also depend on other variables that are not included in this KNOBLER operator.



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And then this is what we get.



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We get a vector with three components.



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So now let's see what this looks like, for our example, so let's take again our mountain.



33

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And this time I want to show you the top down view of our mountain.



34

00:02:35,060 --> 00:02:37,400

And what we get is something like this.



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00:02:37,400 --> 00:02:44,000

So we look from the top and you can see here the color in this 3D plot corresponds to this color here.



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00:02:44,970 --> 00:02:51,660

And so we have several lines to go around in such oval shapes, so it goes around the top of the mountain



37

00:02:51,660 --> 00:02:56,960

here, for example, and on these lines, the height of a mountain is equal.



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00:02:57,120 --> 00:03:00,920

So on every point of this line and then we go to a different line.



39

00:03:00,930 --> 00:03:02,190

So the other line is different.



40

00:03:02,220 --> 00:03:06,240

But on this line, the height is again constant.



41

00:03:08,170 --> 00:03:14,620

So this is something you would, for example, see on maps where you have a mountain and then this is



42

00:03:14,620 --> 00:03:21,760

typically indicated with such a plot because it's not really sensible to plot a map in 3D.



43

00:03:22,990 --> 00:03:26,040

So now what we can do is we can calculate our gradient.



44

00:03:26,710 --> 00:03:30,450

So we have already calculated the partial derivatives in the previous lecture.



45

00:03:30,850 --> 00:03:32,350

So I will just show you the results.



46

00:03:32,950 --> 00:03:38,350

Of course, here we would write partial derivative with respect to Z of F, but since our first two



47

00:03:38,350 --> 00:03:41,890

dimensional and does not depend on Z, this just gives us zero.



48

00:03:42,550 --> 00:03:45,360

So we get this one here as our gradient.



49

00:03:46,360 --> 00:03:49,630

And now you see this is a field of vectors.



50

00:03:50,320 --> 00:03:52,670

So at every position, X and Y.



51

00:03:52,870 --> 00:03:59,350

So for every position on this map, we get a vector which can be indicated by an arrow.



52

00:04:00,370 --> 00:04:04,750

And now if we plot this vector field here, this gradient, it looks like this.



53

00:04:05,920 --> 00:04:08,440

And here we see a special feature of the gradient.



54

00:04:09,220 --> 00:04:17,220

All of these arrows are always pointing perpendicular to such and such a line of equal height.



55

00:04:18,070 --> 00:04:20,320

And that's something that's generally true.



56

00:04:20,470 --> 00:04:24,060

The gradient is always perpendicular to these lines.



57

00:04:24,880 --> 00:04:30,650

And what this means is the gradient always tells us the direction of the steepest slope.



58

00:04:31,480 --> 00:04:36,760

For example, imagine you're standing here on the mountain at this point and you want to know which



59

00:04:36,760 --> 00:04:38,230

direction is the steepest.



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Then you just calculate the gradient and it gives you the vector along which you have to go so that



61

00:04:44,890 --> 00:04:46,030

you have the steepest slope.



62

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So the gradient points always along the direction of the steepest slope.



63

00:04:52,940 --> 00:04:58,500

Now, besides the gradient, we can also define different operations with the naval operator.



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00:04:59,210 --> 00:05:00,800

One of them is the divergence.



65

00:05:01,960 --> 00:05:09,290

So far, we have considered scalar functions, which are no vectors themselves, but just Skylar's.



66

00:05:09,640 --> 00:05:14,160

But now let's consider a different function, let's call it H and H is a vector.



67

00:05:14,260 --> 00:05:16,750

So it's it has multiple components.



68

00:05:17,110 --> 00:05:19,270

And here we want to go to three dimensions.



69

00:05:19,270 --> 00:05:23,110

So it has H, X, X, Y and Z.



70

00:05:24,880 --> 00:05:31,120

Now, if we calculate the divergence of H, we have to calculate the dot product of our Nabala vector



71

00:05:31,480 --> 00:05:33,340

with this function H.



72

00:05:34,060 --> 00:05:39,460

So this means we have to calculate this DOT product, which of course means we have to calculate the



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00:05:39,460 --> 00:05:44,370

partial derivative of H X with respect to X plus these other two terms.



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00:05:44,410 --> 00:05:49,600

And you can see here and now we go ahead and do an exercise.



75

00:05:50,350 --> 00:05:55,000

So let's go ahead and calculate, calculate these two divergences.



76

00:05:55,900 --> 00:06:03,850

So we start with a very simple one, the divergence of the position vector R, where our vector R is



77

00:06:03,850 --> 00:06:07,610

just to vector X, Y and Z.



78

00:06:08,950 --> 00:06:18,280

So of course then the divergence of R is the sum of partial derivative with respect to X of X plus the



79

00:06:18,280 --> 00:06:19,330

other two terms.



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00:06:19,840 --> 00:06:24,340

Here we have Y and here we have C.



81

00:06:26,110 --> 00:06:31,270

And of course you notice the derivative of X with respect to X is just one.



82

00:06:32,140 --> 00:06:40,720

So we have one plus here we have also one plus one which is three.



83

00:06:41,350 --> 00:06:44,660

So the divergence of the position Vector R is three.



84

00:06:45,910 --> 00:06:50,950

Now we can use this result to calculate divergence of a more difficult function we have here.



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00:06:50,950 --> 00:06:59,920

The position vector are divided by R, so R is the absolute value of the position vector, which is



86

00:07:00,130 --> 00:07:04,450

the square root of X square, y square, the square.



87

00:07:05,890 --> 00:07:13,300

And we could also write this down as this brackets here, which is on the and this square root to the



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00:07:13,300 --> 00:07:15,940

power of one half.



89

00:07:17,380 --> 00:07:21,820

OK, so here we have a quotient or a product.



90

00:07:21,820 --> 00:07:25,890

We could also say and we can apply the product rule in this case.



91

00:07:26,650 --> 00:07:34,870

So we write down the product run one of our times, the derivative of of.



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00:07:34,870 --> 00:07:35,290

Ah.



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00:07:35,350 --> 00:07:41,380

So this is because our first function is our and our other function is one of our.



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00:07:42,400 --> 00:07:47,020

So this is our first term and then we get the other term, which is the vector R, so we leave that



95

00:07:47,020 --> 00:07:50,740

invariant and we derive with respect to the other function.



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00:07:51,130 --> 00:07:59,050

So now we write down gradient of one over R, so this one here, we know already this is all three,



97

00:07:59,170 --> 00:08:00,400

so let's write it down.



98

00:08:00,760 --> 00:08:12,010

We get three of our class, our times, the gradient of one over R so let's calculate the gradient of



99

00:08:12,010 --> 00:08:13,670

one of our artists a bit more difficult.



100

00:08:14,170 --> 00:08:26,110

So gradient of one of our is the gradient of X squared plus Y squared plus Z square to the power of



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00:08:26,110 --> 00:08:30,780

one half and now we get of course a vector.



102

00:08:31,540 --> 00:08:36,230

And I want to start with calculating just the X component.



103

00:08:37,090 --> 00:08:43,360

So we have to calculate the partial derivative with respect to X of this whole expression here.



104

00:08:44,320 --> 00:08:51,010

So we first have to calculate the outer derivative, which is one 1/2 times.



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00:08:53,090 --> 00:08:54,470

Sorry I made a mistake here.



106

00:08:54,590 --> 00:08:59,420

This is, of course, one of ours, so this is not one half year, it's minus one half.



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00:09:00,470 --> 00:09:07,940

So we get the outer derivative, which is minus one half times this bracket.



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00:09:11,030 --> 00:09:19,880

And then we have to decrease the exponential by one, so we get minus three over two, and then we also



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00:09:19,880 --> 00:09:25,580

have to consider the inner derivative and the inner derivative of X squared plus Y square, blowsy square



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00:09:25,700 --> 00:09:34,760

is two X, so we get to X and here for Y, we get the same thing for the outer derivative and we get



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00:09:34,760 --> 00:09:40,220

to Y for the inner derivative T and the same thing times to Z.



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00:09:42,190 --> 00:09:49,450

So what this is, is actually we can take here just one 1/2 and this too, so they cancel out.



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00:09:49,450 --> 00:09:56,390

So we get minus, um, in all cases we have this expression here.



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00:09:56,800 --> 00:09:59,140

So this is ah.



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00:09:59,140 --> 00:10:00,880

To the power of minus.



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Three, because our is this expression in the brackets to the power of one half, and then we have the



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00:10:10,270 --> 00:10:13,360

vector X, Y, Z.



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00:10:15,060 --> 00:10:21,020

So we can also write this down as being, um.



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Minors are to the past three, and then here we have our Victor R..



120

00:10:34,000 --> 00:10:41,110

OK, yeah, so here, of course, make sure that this is also the victor, so now we can go ahead and



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continue calculating this divergence of this whole term where we have to make sure that this one that



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00:10:46,900 --> 00:10:48,140

we calculate this one here.



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00:10:48,910 --> 00:10:49,810

So let's go back.



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Let's calculate the divergence of our divided by hour.



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So we already had our first term, three of our plus, and now we have our vector dot products with



126

00:11:05,380 --> 00:11:09,970

minus factor are divided by our to the power of three.



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00:11:10,900 --> 00:11:17,830

And so this one here, our times are is because these two are parallel because they are the same vectors,



128

00:11:17,830 --> 00:11:18,910

just our square.



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So we get our square divided by to the power of three which is one over.



130

00:11:24,700 --> 00:11:29,920

So we get three of our minus one over our.



131

00:11:30,400 --> 00:11:31,510

So this is.



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00:11:32,760 --> 00:11:36,240

Two of our and this is our result.



133

00:11:36,860 --> 00:11:38,880

OK, so let's go ahead and plot this.



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00:11:39,600 --> 00:11:46,320

So here's our function age, which we have just calculators, is the vector are divided by the absolute



135

00:11:46,320 --> 00:11:48,570

value of R, so that's a vector.



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00:11:48,600 --> 00:11:49,560

That's a scalar.



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00:11:50,510 --> 00:11:57,440

So this function depends on every every point in the three dimensional space, it's dependent on X,



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00:11:57,440 --> 00:12:02,700

Y and Z because we have learned what our vector and the absolute value of our are.



139

00:12:04,280 --> 00:12:11,150

So this means we get for every position in the three dimensional space, a value of our function.



140

00:12:11,780 --> 00:12:14,440

And our function itself is again a vector.



141

00:12:15,020 --> 00:12:16,790

So it has three components.



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00:12:17,240 --> 00:12:25,400

And a way to plot this is to position an arrow at every point in history, dimensional space and its



143

00:12:25,400 --> 00:12:28,950

direction and length corresponds to the value of this function.



144

00:12:29,990 --> 00:12:31,670

So you can see what this looks like.



145

00:12:32,060 --> 00:12:35,430

It is vector field that points along the radio direction.



146

00:12:35,810 --> 00:12:41,510

So here we have the center point and every arrow points outwards along the radial direction, which



147

00:12:41,510 --> 00:12:45,250

is of course, because we have here is dependent on the position Vector R.



148

00:12:46,490 --> 00:12:52,670

And also you can see, since we always divide by the absolute value of our all of these arrows have



149

00:12:52,670 --> 00:12:58,880

the same length and the only point where this field is undefined is the position of zero.



150

00:13:00,640 --> 00:13:07,240

OK, so now we can this is a bit looks a bit confusing, this is because that's a three dimensional



151

00:13:07,360 --> 00:13:10,090

plot with another three dimensional field.



152

00:13:11,410 --> 00:13:11,890

Yeah.



153

00:13:12,040 --> 00:13:13,320

Included in this plot.



154

00:13:13,660 --> 00:13:18,540

So it's too much information to really be recognizable easily.



155

00:13:19,180 --> 00:13:22,240

So what's more easy is to take a cut of this field.



156

00:13:22,540 --> 00:13:25,990

So that's a special case for setting Dessy Component zero.



157

00:13:26,710 --> 00:13:30,600

So in here, you can see already I have included this plane for Z equals zero.



158

00:13:31,090 --> 00:13:34,640

So this is what it looks like when we look from the top.



159

00:13:35,230 --> 00:13:38,620

This is what our field looks like for Z equals zero.



160

00:13:39,760 --> 00:13:43,600

So here you can see again this radial profile of our vector fields.



161

00:13:43,780 --> 00:13:49,180

All of these arrows point along the radial direction outwards and all of them have the same length.



162

00:13:51,070 --> 00:13:57,370

Now, if we calculate the divergence of this field, as we have done in the previous lecture, we get



163

00:13:57,580 --> 00:14:02,110

to Örvar and if we plot to over, we get this function here.



164

00:14:02,860 --> 00:14:09,760

So far, positions far away from the center, we get zero because then this R becomes large and this



165

00:14:09,760 --> 00:14:11,290

expression here becomes zero.



166

00:14:11,930 --> 00:14:19,510

But when we approach the zero point here, then this function becomes larger and larger and in fact



167

00:14:19,840 --> 00:14:21,270

it even goes to infinity.



168

00:14:21,760 --> 00:14:23,260

This is because this is why.



169

00:14:23,260 --> 00:14:28,900

This is why here I have just cut off the plot at some point because I cannot plot infinity.



170

00:14:30,260 --> 00:14:38,270

And what this means is that our vector fields are divided by our has a very strong divergence in the



171

00:14:38,270 --> 00:14:38,780

center.



172

00:14:39,610 --> 00:14:47,530

And this in terms of physics, for example, would be the electric field or the electric field of a



173

00:14:47,530 --> 00:14:48,130

charge.



174

00:14:48,250 --> 00:14:54,700

So if you have a charge and charge, it sits at some point, then the electric field will have a radio



175

00:14:54,790 --> 00:15:01,080

shape and it will have a divergence because the charge is the divergence of the electric field.



176

00:15:02,070 --> 00:15:07,800

And there are other examples in physics and also in other fields of science where the divergence plays



177

00:15:07,800 --> 00:15:14,280

an important role and it's a very useful feature to characterize vector fields that look difficult.



178

00:15:14,640 --> 00:15:20,520

But you can really tell, OK, the zero position seems to be a special point by just looking at these



179

00:15:20,520 --> 00:15:21,090

vectors.



180

00:15:21,300 --> 00:15:26,570

And in fact, if we look at the divergence, then it's very large at this particular position.



181

00:15:28,420 --> 00:15:36,040

Now, the other thing that we can do with the operator and a vector function is to calculate the kernel



182

00:15:36,040 --> 00:15:37,390

instead of the divergence.



183

00:15:39,370 --> 00:15:45,100

So let us once again consider our function age, which is still a three dimensional function, and let's



184

00:15:45,100 --> 00:15:52,900

calculate the kernel so the kernel of our function age is just the vector product of the knuckler operator



185

00:15:53,290 --> 00:15:54,740

with the function age.



186

00:15:55,750 --> 00:15:57,370

So you can write it down like this.



187

00:15:57,520 --> 00:15:58,930

You have this vector product.



188

00:16:00,040 --> 00:16:03,060

Now you just have to know how to calculate a vector product.



189

00:16:03,730 --> 00:16:08,470

So you get as a result, a three component vector, which looks like this.



190

00:16:08,980 --> 00:16:14,590

For example, for the first component, you have to take the partial derivative with respect to Y and



191

00:16:14,590 --> 00:16:22,300

let it act on HHC and then subtract the partial derivative, which is big to see acting on H.



192

00:16:22,300 --> 00:16:27,590

Y and then the other terms you get in a very similar fashion.



193

00:16:28,810 --> 00:16:31,930

So let's go ahead and calculate an example.



194

00:16:32,980 --> 00:16:37,020

So here we have an example in cylindrical coordinates.



195

00:16:37,690 --> 00:16:40,360

So that's something that should not confuse you.



196

00:16:40,850 --> 00:16:47,780

Does this term role here is just the the length in the plane.



197

00:16:47,800 --> 00:16:51,850

So this is the square root of X squared plus Y square.



198

00:16:52,540 --> 00:16:58,540

So it's a bit different to R where we would have the same expression, but another plus C square here.



199

00:16:59,080 --> 00:17:02,140

So this is just the distance in the X Y plane.



200

00:17:03,450 --> 00:17:16,830

So now we can write this down as the rotation of our vector is minus Y divided by X squared plus Y square.



201

00:17:18,210 --> 00:17:23,490

And here we have X divided by X squared plus Y square.



202

00:17:25,470 --> 00:17:26,920

And here we have zero.



203

00:17:28,260 --> 00:17:36,450

All right, so if we calculate this, then we can look at our equation and here we have only X and Y



204

00:17:36,450 --> 00:17:37,300

components.



205

00:17:38,040 --> 00:17:44,580

So we do not have this one here and we do not have this one here.



206

00:17:46,430 --> 00:17:55,950

Furthermore, we have only a wider pendent X component and an X X dependent Y component.



207

00:17:56,630 --> 00:18:04,370

So this means the only terms that give results different from zeros are partial derivative of the X



208

00:18:04,370 --> 00:18:06,180

component with respect to Y.



209

00:18:07,070 --> 00:18:08,990

So let's search for this year.



210

00:18:08,990 --> 00:18:11,840

We have it X component which was packed.



211

00:18:11,850 --> 00:18:21,620

Why does one and the other term that gives non-zero is the X derivative of the Y component.



212

00:18:22,550 --> 00:18:26,060

So we have to search this is this one here.



213

00:18:27,560 --> 00:18:30,440

OK, and the other ones are also zero.



214

00:18:32,140 --> 00:18:40,150

So we have only a Z components, so instead of writing this down as a vector, I just write here we



215

00:18:40,150 --> 00:18:53,880

have our unit vector along Z, which is so easy is zero zero one and times this this expression here.



216

00:18:53,890 --> 00:19:08,160

So we have partial derivative acts of the of the Y component, which is X divided by X square, y square



217

00:19:08,170 --> 00:19:19,100

square root minus um the Y derivative of the X component, which is this one here.



218

00:19:19,570 --> 00:19:28,180

So we have another minus sign, so we get a plus and then we have here Y divided by X squared plus Y



219

00:19:28,180 --> 00:19:28,600

square.



220

00:19:31,410 --> 00:19:36,640

OK, and then we make such a racket here and that's it.



221

00:19:37,200 --> 00:19:42,210

OK, now we just have to calculate these partial derivatives here so we get easy.



222

00:19:42,840 --> 00:19:49,950

And now as the first derivative we get, we have to apply the the product rule and then later also the



223

00:19:49,950 --> 00:19:50,630

chain rule.



224

00:19:51,090 --> 00:19:59,460

So we get a first term where we have to where we write one over square root, X squared plus Y square



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times.



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The derivative of this term here, which is one.



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So that's it.



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And then we have plus.



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And this time we leave the X here and we have to differentiate one over the square root of X squared



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plus Y square.



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So we get as the outer derivative in this case, minus one half and then one divided by X squared plus



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Y square to the power of three because he the exponent is minus one half.



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We have to decrease the exponent by one.



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So we get minus three 1/2, which is exactly this one here.



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And as a prefect we get minus one half and then we have to consider the inner derivative of X squared



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plus Y square, which is to X.



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So we have to multiply two X and then here we get very similar terms.



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That's just copy them.



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00:21:00,400 --> 00:21:02,920

So we get for the first time exactly the same.



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This is this one here.



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And for the other term we get, uh, y then minus one half.



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00:21:14,680 --> 00:21:22,450

This one is the same and here we get four in a derivative to Y and here we get X squared plus Y square



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00:21:22,840 --> 00:21:24,070

to the power of three.



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00:21:25,600 --> 00:21:30,250

So what we have is easy times.



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Then let's, let's uh, let's look at these individual terms.



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So we have twice this first term here.



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So this is two times and this one here is are two of our and now we only have the second and the fourth



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term left.



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So let's write them down.



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We have a minus sign in both cases.



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We have in both cases one half and then two later.



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So these cancel out and then we have here X Square and we we divide by, um, sorry, I made a mistake



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here.



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This is of course not r this is real.



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So this is what we introduced here.



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So this is not r this is real.



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And here we get to row again, but wrote to the power of three and here we have then the fourth term



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which looks exactly the same, but instead of X Square we get Y squared so we can write it down like



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00:22:29,020 --> 00:22:29,350

this.



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As you can see here, we have X squared plus Y square, which is row square, so we have zero square



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divided by row to the power of three, which is one overall.



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The result is two of a row, minus one overall is one of a row times easy.



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This is the rotation of this function here.



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Now let us look what this whole field looks like.



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So, first of all, let us plot our function H which was this vector here with minus Y and the X component



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and X and the Y component, and then we divide by the distance in the plane.



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And so our vector field looks like this.



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As you can see, we have no Z dependent's.



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So this means the field looks the same for every value of C.



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So we get something like a cube here.



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We can cut through this tube and the profile in every cut looks exactly the same.



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And if we look at such a cut here, you can see that it looks like such a rotating, rotating vector



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plot.



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00:23:39,030 --> 00:23:44,760

So once again, at the position zero, it's not defined because we then would divide by zero.



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But besides that, we we get such a plot here where we have these arrows, which all have the same length,



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and they go around the center in circles.



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So we have calculated the curl of this of this thing sometimes I also call it the rotation, because



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this is what we call it in my native language.



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And I think it also makes sense to call it the rotation, because when we have such a rotating vector



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00:24:14,870 --> 00:24:18,890

fields, you see that everything rotates around the center.



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00:24:19,520 --> 00:24:25,910

And if we plot the rotation, if we plot the Z component that we have just calculated, we get again



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00:24:26,060 --> 00:24:34,190

something that we had previously where we have a very large signal of this curl here of of this rotation



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00:24:34,490 --> 00:24:36,480

in the center for the position zero.



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00:24:37,010 --> 00:24:39,910

So this means this point here is special.



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00:24:39,920 --> 00:24:43,030

It's a special point of the curl of this vector field.



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00:24:43,400 --> 00:24:47,120

And this shows also in this plot here where we have plotted the kernel.



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00:24:48,170 --> 00:24:50,840

So we have learned now about the divergence and the curl.



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00:24:51,080 --> 00:24:56,530

And both of these are very useful features to characterize difficult looking vector fields.



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00:24:57,380 --> 00:25:06,310

And for example, the kernel can be used to characterize the magnetic field that arises from here from



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00:25:06,320 --> 00:25:07,460

from a current.



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00:25:07,910 --> 00:25:12,710

For example, if you have a wire and you let the current flow through this wire, then you will get



292

00:25:12,710 --> 00:25:21,590

a magnetic field that rotates around the wire and then the to the current or the wire itself then acts



293

00:25:21,740 --> 00:25:23,800

like the curl of this magnetic field.



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00:25:25,680 --> 00:25:30,480

Now, the very last thing that I would just want to mention briefly, because we will not really use



295

00:25:30,480 --> 00:25:39,480

it that often is to lab class operator Plata operator is a second order differential operator.



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00:25:39,780 --> 00:25:44,520

And it is basically just as some of the second order partial derivatives.



297

00:25:45,900 --> 00:25:51,870

So you can let it act on a scale or function and then you have to calculate the second order, partial



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00:25:51,870 --> 00:25:52,550

derivatives.



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00:25:53,490 --> 00:26:00,600

And when you look at this, you will see that this is nothing different than the divergence of the gradient



300

00:26:00,600 --> 00:26:00,900

of.



301

00:26:01,620 --> 00:26:07,080

So when you want to calculate the A plus operator acting on an F, you can do it like this.



302

00:26:07,440 --> 00:26:12,360

Or you can first calculate the gradient and then calculate the divergence of this gradient.



303

00:26:13,680 --> 00:26:21,780

And the cool thing is that this lab class operator can also act on vector functions because this is



304

00:26:21,780 --> 00:26:24,650

just some of the scale of operation.



305

00:26:24,670 --> 00:26:28,020

So it doesn't matter if you have a scalar here or a vector here.



306

00:26:28,410 --> 00:26:32,730

So you can also calculate this lab class operator acting on this function here.



307

00:26:33,630 --> 00:26:42,840

And the operator is used, for example, in the wave equation or even in the equation in quantum mechanics.



308

00:26:44,520 --> 00:26:50,670

So now you know about the operator and you know about gradients, divergences and curls, and we are



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00:26:50,670 --> 00:26:53,540

now ready to apply them to our problems.