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Let's try to get our heads around the idea of divergence.

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So first, like I did with gradients, I'll show you the

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mechanics, which are actually pretty straightforward.

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And then I'll try to give you the intuition.

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And once you have the intuition, at first it

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will seemed very, I don't know, unintuitive, maybe.

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But it once you get it, you're like oh, that's it.

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So let's see what divergence is.

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Let's say I have a vector field.

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And let's say this vector field, just for the purposes

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of visualization it could be anything, but let's say it

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represents the velocity of particles of fluid of any

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point in two dimensions.

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So it's going to be a two-dimensional vector field.

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It's going to be a function of x and y, so the velocity at any

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point-- it's a vector field -- let's say it is, and I'm just

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going to make up something.

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Let's say it's x squared, yi.

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So at any point in the x-direction, at any point x

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comma y, its velocity in the x-direction will

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be x squared, y.

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And then its velocity in the y-direction, I don't know

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maybe it's just 3y, j.

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That's its velocity in the x-direction.

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So its velocity in the x-direction is actually

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a function of x and y.

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its velocity in the y-direction is just a function of y.

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So what is the divergence?

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So a couple of ways we can write it.

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The correct way to write it is the divergence of

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our vector field, v.

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But a common mnemonic to remember the operation of

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diverge and is to write the upside down triangle, which was

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the same notation we used for gradient, but take the dot

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product of that and the vector.

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And if you remember from the gradient discussion, we said

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that you can view, although it's kind of an abuse of

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notation, but you could view this upside down triangle as

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being equal to the partial derivative with respect to x in

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the x-direction plus the partial derivative with respect

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to y in the y-direction, which is the j-unit vector.

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And then if we went to three dimensions, the partial

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derivative with respect to z and the k-direction,

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et cetera, et cetera.

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But we're dealing with a two-dimensional vector here,

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so let's just stick with two dimensions, x and y.

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So what would this turn out to be?

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If you took the dot product of this, which is this upside down

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triangle, with this vector field, what would you get?

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Well, you would just get the partial derivative of the x

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dimension with respect to x, so you would get-- it's actually

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pretty straight forward to memorize; you might not even

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need this mnemonic right here, this abuse of notation; you

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might just know it off hand --the x component, you take the

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partial derivative with respect to x, and the y component, you

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take the partial derivative with respect to y.

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But I'll show you why it looks like the dot product.

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So if you took the dot product of that and that, it would be

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the partial derivative with respect to x of that

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expression, of x squared, y and then plus the partial

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derivative with respect to y of that second expression, the y

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component of 3y, and then you would evaluate it.

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What's the partial derivative of this with respect to x?

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We just pretended y is a constant, just a number, so

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the derivative of this with respect to x, would be

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2x times the constant.

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So it'll be 2xy plus-- what's the partial derivative

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of 3y with respect to y?

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Well, there's nothing else to hold constant, so it's just

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like taking the derivative with respect to y

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--so it's 2y plus 3.

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So this is the divergence at a point x, y.

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You could almost view it as a function of x and y.

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So you could almost say you know, that the divergence of

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v-- I'm going to make up some notation here --as long as you

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get the point across, you can say that the divergence

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of v, that this is a function of x and y.

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That we just have an expression that if you give me a point

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anywhere in this vector field, I can tell you the

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divergence at that point.

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So I think you'll find that the computation of divergence

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isn't too difficult.

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You just take the partial derivative of the x component

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with respect to x, and you add that to the partial derivative

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to the y component with respect to y.

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And if you had the z, you would do the same thing,

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so on and so forth.

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Actually, let me do just do one more just hit the point home,

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and then we'll work on intuition.

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So if I said that I had, I don't know, let's say, my

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vector field is cosine of yi plus-- so it's interesting; my

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x-direction is dependent on my y-coordinate --plus, I

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don't know, e to the xyj.

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So then oh, that's difficult because I have these

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e's and these cosines.

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But we'll see; if you just keep your head straight on

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what's constant and what's not, it's not too bad.

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So the divergence of v is equal to the partial derivative

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of this expression with respect to x.

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Well, what's the derivative of this with respect to x?

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If y is just a constant, cosine of y is just a number.

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So the derivative of this with respect to x is just 0 plus--

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what's the derivative of this with respect to y?

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Well, you could just do x, since it's a constant,

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as the coefficient on y.

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So the derivative of x, y with respect to y is just x.

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And then the derivative of e to anything is e to anything.

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I just did the chain rule.

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e to the x, y.

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And so that is the divergence.

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So you could just ignore this.

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It's x, e to the x, y.

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One thing to immediately realize, even before we work on

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the intuition, is when we did gradient I gave you a surface

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and it gave us a vector field.

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Or I gave you a scalar field and you got a vector field.

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When you take the divergence of something, you're going in the

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opposite direction, in some ways.

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You start with the vector field, right?

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And what's a factor field?

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It's something that if you give me any point x and y,

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I'll give you a vector.

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So if you wanted to graph it, in the x, y plane you'd have a

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bunch of vectors, and I'll show you how that looks in a second

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when we go over to intuition.

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Well, when you take the divergence of it, you get a

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value for any point x, y.

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So even though a vector field has all these vectors on it,

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the divergence tells you an actual scalar number at

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any point in the field.

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So let's get a little bit of intuition of what a

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divergence actually is.

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Let me do it in one dimension.

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Or we can even, let's do it in two dimensions, but I'll

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make it constant in the y.

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So let's say that my-- let me erase this; I'll probably

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need some space.

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OK, oh, I didn't want to do that dot.

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OK let's say the velocity of fluid, or the particles in

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fluid, at any point in the x, y plane, let's say it is equal to

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5xi plus, I don't know, 0y-- there's never any, sorry --0j,

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right? j is the unit vector in the y-direction.

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So there's never a y component to the velocity vector.

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So what would that look like?

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I don't need a computer to draw this.

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I can handle this one myself I think.

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So if that's the y-axis, that's my x-axis.

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So when x is equal-- I'll just sample some points and draw

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some vectors --when x is equal to 1-- let's say x is 1 there

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--what's the magnitude of this vector?

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It'll be 5, right?

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Actually, let me make this a different number, because

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it'll make it hard to do.

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Let's make this 1/2 x.

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So when x is 1, the magnitude of my vector is 1/2.

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Only in the x-direction.

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It has no y component; ignore this right here.

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It's 1/2xi plus 0j.

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Or you could just say 1/2xi.

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And when x is equal to 2-- I could have picked any points,

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but I'm just picking the numbers that's easy to

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calculate --when x is equal to 2, what is the magnitude

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of the vector?

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It's 1/2 times 2, which is 1.

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So it's going to be twice as big.

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And remember, if I have a particle right here in my

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fluid, if this is a particle, its velocity in the x-direction

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is going to be 1 meter per second to the right.

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If I have a particle here, it's velocity in the x-direction

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is going to be 1/2 a meter per second to the right.

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Let's just do one more point.

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So let's say that x is equal to 3.

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What's my velocity to the right?

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I'll do it in a different color just so that you

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don't get confused.

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There's going to be 3/2; it's going to be even longer.

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But the general idea here, and as we move up in x it

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doesn't change much, right?

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It doesn't change at all.

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Our x value doesn't--

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[COUGHS].

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So for any y, the magnitude of the vector doesn't

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change, right?

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It's only dependent on x.

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And then for example, here, it'll be even longer.

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If we draw the vector here it'll be even longer, right?

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If you do it here.

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I think you get the point.

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The further you go to the right, the faster the particles

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are moving towards the right.

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So now let's try to get a little bit of intuition.

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Oh, I just realized that I ran out of time, so I will continue

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this in the next video.

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