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So a mathematical concept that is very important in physics, especially, for example, in electrodynamics,

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is multi-dimensional derivatives.

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So this includes the so-called Nublar operator, which contains the partial derivatives with respect

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to the individual coordinates.

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So with the help of this novel operator, we can calculate the gradient and the curl and the divergence,

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and we can calculate those things for multidimensional functions.

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So this means, for example, that a function is itself still single valued.

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So it's not a vector, but just a scalar function.

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But it depends on multiple arguments.

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For example, X, Y and Z.

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Or it could also mean that the function has multiple dimensions itself, so it has an X, Y and Z component,

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and all of them depend on X, Y and Z themselves.

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So now we can use to novel operator to basically calculate what happens when we apply this to a scalar

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function.

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You see this the gradient where we apply the partial derivatives now to the function for each of these

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components.

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For the curl, we calculate the cross product and for the divergence we calculate the dot product of

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the novel operator with the function.

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So for example, here we see the typical cross product shape y z minus z y z x minus XY and x y minus

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y x.

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So you just have to consider these partial derivatives, even though they are just operators, just

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like you would consider a number of variables.

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And then you just let them act on the function.

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So we we don't want to really deal so much here with the mathematics, but we will just implemented

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and we will consider a few examples.

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So as the functions that we are using here, we have these examples.

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The first one is an exponential function, basically a Gaussian function, which is a bit modulated

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in the white direction.

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And I have programmed this already.

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I have written down F of R R as a vector that has X, Y and Z.

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So three components, it is basically an exponential function and then X component square and here y

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component for the total power of for.

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And then I have used the plot prediction command to plot this, and you see it looks a bit like a mountain.

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This is really the physical example that we will consider here.

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Let's consider a mountain that looks like this.

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We have the X and Y coordinate and the values to Z coordinate.

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And then the question is what is the gradient at?

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The gradient always tells us the direction of the steepest slope of the mountain.

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So when we are at some point and we calculate the gradient, it will always point in to the direction

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where the slope is the steepest.

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And so the other example of a vector valued function is Geof River G is a vector and it is Vector R,

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so basically a vector with X, Y and Z as two components divided by the absolute value of art.

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So square root of x squared plus y squared plus c square.

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This looks very simple, but actually to calculate the divergence and the curl of this is not so easy

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to do analytically.

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But here we will do that just numerically, and we will implement these functions here.

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And in terms of physics, this expression is very relevant, for example, in electoral dynamics.

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And there we even have to calculate the curl and the divergence of this term to determine, for example,

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the electric field of some charged distribution or vice versa.

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So now let me show you what this function looks like.

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Vector are divided by arm.

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So you see if you're interested, how to do this and please just post a video.

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I have used here a quiver plot, but actually it's not so important right now.

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I will use this later on to calls several times, and there I will explain to you in detail how this

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works.

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Basically, you just provide X, Y and Z coordinates by using a mesh grid command and then you provide

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the values of the function.

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Also, X, Y and Z components, another will show you the direction of G at every point in space.

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And you see, since it is vector are divided by R, it always points along the radial director.

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All right.

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So the next thing that we will implement is we will implement the gradient, so the gradient of.

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So this means, as I told you, the direction of the steepest slope of this mountain here.

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So let's go ahead and do this.

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So the gradient is defined.

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And it is the fight for function, if it is to find for position arc, which is a vector of three components

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and it is defined with a step size because it is still a derivative, after all.

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So we need the step size.

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And here, just for simplicity, I will use the central difference as method.

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You could also apply any other method that we have encountered previously.

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It will just increase or decrease the accuracy of the method.

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But here central differences will do.

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It's just just an exercise to show you how it works.

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So just to make our lives a bit more easy, we can define something like this where the individual components

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of arm are labeled X, Y and Z, and then we can write partial x has given as.

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But now we must use central differences with respect to the variable of X.

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So this would be f off.

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Now we must apply the array to vector array.

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X, Y and Z.

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And as I told you, central differences, we will change X by the value of H, and then we will just

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subtract the value of the function where H is reduced from X and we divide by two.

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H, this is really exactly as we did before, but we need to be careful now that we used to correct

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argument here.

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Then when you remember how the gradient was defined, we do know the same thing for the other components.

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But we we calculate the derivative respect to Y and with respect to see.

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So we will just write basically the same thing again, partial y and there the age will be added to

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Y and subtracted from Y and for Z, the same thing with the C component.

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And then the last thing is we must, of course, return something, and this will be a because now it's

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the vector gradient is always a vector because it shows you an orientation of the steep slope, so it

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must be a vector.

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And those will be partial acts, comma, partial light and partial z.

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And then we can just give us some random reference point.

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For example, are is given as an P dot array

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0.5 minus one point two.

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And you see, our function doesn't even depend on C, so it can take it and evaluate just to zero.

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But I could also take minus eight.

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Doesn't matter.

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And then we must specify the step size, which of course, gives us a better result the smaller it is.

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So I take it quite small and then I call gradient of f petition are steps each, and the value is now

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something like this.

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So you see we are to position 0.5 and minus one point two.

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And the direction of the gradient is a bit in the negative x direction and along the positive y direction.

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So let's look at the figure we go to 0.5 and minus one point two.

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So, for example, 0.5 minus one point two is something like here.

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So the gradient would point a bit to the left.

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Negative x direction and strongly along.

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Y yeah, positive y direction.

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Let's see if this is correct.

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Yes, this is exactly what we have calculated, and even we can check this with the analytical solution.

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Because if you know a bit about the mathematics and the derivatives, it's pretty easy to calculate

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the gradient.

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You just have the outer derivative with it, which is the exponential function of minus x squared minus

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Y to the power of four.

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And then you multiply the derivative, which for the X component will be minus two x and for the Y component

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will be minus four times Y two to pull off three.

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So this is exactly what I've written down here and now I can just copy what I have prepared.

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This is just an array which basically includes these exact components that I have just written down,

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and the value is exactly what we have here.

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And so you see within it with the step size of zero point zero zero zero one, we get really exactly

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the same result as we should get from analytics.

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So in this case, especially the central differences method was totally sufficient and we get a nice

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result for the gradient.