Divergence 1 Introduction to the divergence of a vector field.
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- Let's try to get our heads around the idea of divergence.
- So first, like I did with gradients, I'll show you the
- mechanics, which are actually pretty straightforward.
- And then I'll try to give you the intuition.
- And once you have the intuition, at first it
- will seemed very, I don't know, unintuitive, maybe.
- But it once you get it, you're like oh, that's it.
- So let's see what divergence is.
- Let's say I have a vector field.
- And let's say this vector field, just for the purposes
- of visualization it could be anything, but let's say it
- represents the velocity of particles of fluid of any
- point in two dimensions.
- So it's going to be a two-dimensional vector field.
- It's going to be a function of x and y, so the velocity at any
- point-- it's a vector field -- let's say it is, and I'm just
- going to make up something.
- Let's say it's x squared, yi.
- So at any point in the x-direction, at any point x
- comma y, its velocity in the x-direction will
- be x squared, y.
- And then its velocity in the y-direction, I don't know
- maybe it's just 3y, j.
- That's its velocity in the x-direction.
- So its velocity in the x-direction is actually
- a function of x and y.
- its velocity in the y-direction is just a function of y.
- So what is the divergence?
- So a couple of ways we can write it.
- The correct way to write it is the divergence of
- our vector field, v.
- But a common mnemonic to remember the operation of
- diverge and is to write the upside down triangle, which was
- the same notation we used for gradient, but take the dot
- product of that and the vector.
- And if you remember from the gradient discussion, we said
- that you can view, although it's kind of an abuse of
- notation, but you could view this upside down triangle as
- being equal to the partial derivative with respect to x in
- the x-direction plus the partial derivative with respect
- to y in the y-direction, which is the j-unit vector.
- And then if we went to three dimensions, the partial
- derivative with respect to z and the k-direction,
- et cetera, et cetera.
- But we're dealing with a two-dimensional vector here,
- so let's just stick with two dimensions, x and y.
- So what would this turn out to be?
- If you took the dot product of this, which is this upside down
- triangle, with this vector field, what would you get?
- Well, you would just get the partial derivative of the x
- dimension with respect to x, so you would get-- it's actually
- pretty straight forward to memorize; you might not even
- need this mnemonic right here, this abuse of notation; you
- might just know it off hand --the x component, you take the
- partial derivative with respect to x, and the y component, you
- take the partial derivative with respect to y.
- But I'll show you why it looks like the dot product.
- So if you took the dot product of that and that, it would be
- the partial derivative with respect to x of that
- expression, of x squared, y and then plus the partial
- derivative with respect to y of that second expression, the y
- component of 3y, and then you would evaluate it.
- What's the partial derivative of this with respect to x?
- We just pretended y is a constant, just a number, so
- the derivative of this with respect to x, would be
- 2x times the constant.
- So it'll be 2xy plus-- what's the partial derivative
- of 3y with respect to y?
- Well, there's nothing else to hold constant, so it's just
- like taking the derivative with respect to y
- --so it's 2y plus 3.
- So this is the divergence at a point x, y.
- You could almost view it as a function of x and y.
- So you could almost say you know, that the divergence of
- v-- I'm going to make up some notation here --as long as you
- get the point across, you can say that the divergence
- of v, that this is a function of x and y.
- That we just have an expression that if you give me a point
- anywhere in this vector field, I can tell you the
- divergence at that point.
- So I think you'll find that the computation of divergence
- isn't too difficult.
- You just take the partial derivative of the x component
- with respect to x, and you add that to the partial derivative
- to the y component with respect to y.
- And if you had the z, you would do the same thing,
- so on and so forth.
- Actually, let me do just do one more just hit the point home,
- and then we'll work on intuition.
- So if I said that I had, I don't know, let's say, my
- vector field is cosine of yi plus-- so it's interesting; my
- x-direction is dependent on my y-coordinate --plus, I
- don't know, e to the xyj.
- So then oh, that's difficult because I have these
- e's and these cosines.
- But we'll see; if you just keep your head straight on
- what's constant and what's not, it's not too bad.
- So the divergence of v is equal to the partial derivative
- of this expression with respect to x.
- Well, what's the derivative of this with respect to x?
- If y is just a constant, cosine of y is just a number.
- So the derivative of this with respect to x is just 0 plus--
- what's the derivative of this with respect to y?
- Well, you could just do x, since it's a constant,
- as the coefficient on y.
- So the derivative of x, y with respect to y is just x.
- And then the derivative of e to anything is e to anything.
- I just did the chain rule.
- e to the x, y.
- And so that is the divergence.
- So you could just ignore this.
- It's x, e to the x, y.
- One thing to immediately realize, even before we work on
- the intuition, is when we did gradient I gave you a surface
- and it gave us a vector field.
- Or I gave you a scalar field and you got a vector field.
- When you take the divergence of something, you're going in the
- opposite direction, in some ways.
- You start with the vector field, right?
- And what's a factor field?
- It's something that if you give me any point x and y,
- I'll give you a vector.
- So if you wanted to graph it, in the x, y plane you'd have a
- bunch of vectors, and I'll show you how that looks in a second
- when we go over to intuition.
- Well, when you take the divergence of it, you get a
- value for any point x, y.
- So even though a vector field has all these vectors on it,
- the divergence tells you an actual scalar number at
- any point in the field.
- So let's get a little bit of intuition of what a
- divergence actually is.
- Let me do it in one dimension.
- Or we can even, let's do it in two dimensions, but I'll
- make it constant in the y.
- So let's say that my-- let me erase this; I'll probably
- need some space.
- OK, oh, I didn't want to do that dot.
- OK let's say the velocity of fluid, or the particles in
- fluid, at any point in the x, y plane, let's say it is equal to
- 5xi plus, I don't know, 0y-- there's never any, sorry --0j,
- right? j is the unit vector in the y-direction.
- So there's never a y component to the velocity vector.
- So what would that look like?
- I don't need a computer to draw this.
- I can handle this one myself I think.
- So if that's the y-axis, that's my x-axis.
- So when x is equal-- I'll just sample some points and draw
- some vectors --when x is equal to 1-- let's say x is 1 there
- --what's the magnitude of this vector?
- It'll be 5, right?
- Actually, let me make this a different number, because
- it'll make it hard to do.
- Let's make this 1/2 x.
- So when x is 1, the magnitude of my vector is 1/2.
- Only in the x-direction.
- It has no y component; ignore this right here.
- It's 1/2xi plus 0j.
- Or you could just say 1/2xi.
- And when x is equal to 2-- I could have picked any points,
- but I'm just picking the numbers that's easy to
- calculate --when x is equal to 2, what is the magnitude
- of the vector?
- It's 1/2 times 2, which is 1.
- So it's going to be twice as big.
- And remember, if I have a particle right here in my
- fluid, if this is a particle, its velocity in the x-direction
- is going to be 1 meter per second to the right.
- If I have a particle here, it's velocity in the x-direction
- is going to be 1/2 a meter per second to the right.
- Let's just do one more point.
- So let's say that x is equal to 3.
- What's my velocity to the right?
- I'll do it in a different color just so that you
- don't get confused.
- There's going to be 3/2; it's going to be even longer.
- But the general idea here, and as we move up in x it
- doesn't change much, right?
- It doesn't change at all.
- Our x value doesn't--
- So for any y, the magnitude of the vector doesn't
- change, right?
- It's only dependent on x.
- And then for example, here, it'll be even longer.
- If we draw the vector here it'll be even longer, right?
- If you do it here.
- I think you get the point.
- The further you go to the right, the faster the particles
- are moving towards the right.
- So now let's try to get a little bit of intuition.
- Oh, I just realized that I ran out of time, so I will continue
- this in the next video.
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