Divergence 2 The intuition of what the divergence of a vector field is.
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- So where we left off I was attempting to give you the
- intuition of divergence and then I ran out of time.
- But anyway, I had defined this fairly straightforward vector
- field that tells us the velocity of particles in a
- fluid at any given point.
- And let me clean it up a little bit.
- This one half I had all these scratch offs.
- The velocity, I'm just going to rewrite it, is equal to 1/2xi.
- So at any given point has no y component.
- So all of the velocity is only in the x direction -- there is
- no upwards movement in the xy plane.
- And I was drawing it out.
- I said OK, when x is equal to 1, the magnitude of the
- velocity is 1/2 maybe meters per second, if that's our unit.
- When x is equal to 2, the velocity to the right will
- be 1 meter per second, right -- 1/2 times 2.
- So the further we go to the right, or the more we go to the
- right, the faster the particles are moving to the right.
- So now let's try to get our handle on what
- divergence means.
- So first of all, let's take that the divergence
- of this function.
- So that divergence of v, of our velocity vector field -- you
- could also view that if you want to abuse some notation,
- is our del vector, dot v.
- But if we only have one dimension, so it's the
- partial derivative of the x magnitude with respect to x.
- So what's the partial derivative?
- So it's equal to the partial derivative with
- respect to x, of 1/2x.
- So it's equal to -- well the derivative of this with respect
- to x is just equal to 1/2 .
- So that divergence of this vector field at
- any point is 1/2.
- Now what does that tell us?
- Well, if you just look at the definition, right, we
- essentially just took the -- how much does the magnitude
- of the field increase in the x direction?
- And we see it visually.
- As we go, increase in the x direction, the field gets
- stronger and stronger.
- Or since we know that this is the velocity of particles, as
- we go in the x direction, the particles go faster and
- faster to the right.
- Now what this tells us, what this positive divergence tells
- us is if we were to take -- let's just take an
- arbitrarily small circle.
- I think it'll start to make sense once I draw the circle.
- If I take an arbitrarily -- I'm going to draw it in a different
- color -- and this circle could be arbitrarily small, but I'm
- drawing it pretty large so it can include some of our
- vectors that I've drawn.
- What's happening?
- On the right hand side, I have particles exiting really,
- really fast, right?
- And let's say in a given amount of time, let's say in one
- second, in one second out of the right side, since the
- particles are moving really fast, I'm going to have a
- bunch of particles leave the right hand side, right?
- And in the same amount of time, I will have some particles
- come in through the left hand side, but it's going to be a
- fewer number of particles.
- So the way you could think about it is in any given amount
- of time, what's happening?
- In this space, I have a few particles entering in through
- the left, and I have a much larger number of particle
- leaving through the right.
- So what's going to happen in this space?
- It's going to become less dense, right?
- Because in that space is going to be fewer particles after
- a certain amount of time.
- More are leaving than are coming in.
- So this positive divergence tells us that at that point, or
- really at any point in this vector field since the
- divergence is 1/2 everywhere, at any point in this vector
- field, the field is becoming less dense.
- Or you could say that more is flowing out of any
- point than flowing in.
- It makes sense, right?
- Because if as we move to the right, and it kind of gets
- funky if you go into the other quadrant, so we'll stick to the
- first quadrant while we're trying to get our intuition.
- But it makes sense, because as we move to the right
- our particles are getting faster and faster.
- And that kind of just falls out of the fact that our derivative
- with respect to x is positive.
- The slope of how much our x component is
- increasing is positive.
- So as we go to the right, our velocities are going getting
- faster and faster, which means if we were to draw a circle
- anywhere, we're always going to have more exiting the right
- than entering through the left.
- So we're going to be getting less dense at any given point.
- Or you could almost view it as any given point is almost a
- source of particles, or if you have a sphere, more particles
- are going to be coming out of the sphere through the right,
- than coming in through the sphere to the left.
- So you could view a positive divergence as you could kind of
- say well, the field is becoming less dense at that point, or
- the point is a source of the field, or it's a source of
- particles, depending on what model you want to use.
- Now, with that said, let's take the opposite situation.
- Let's say that the vector field is equal to is
- minus 1/2x times i.
- And so the divergence -- I'll use this notation -- the
- divergence of our vector field is just a partial derivative
- with respect to x, which is just minus 1/2.
- If I were to graph it -- this is my y-axis,
- this is my x-axis.
- So here at like, say, the point 1, my velocity is
- going to be the left 1/2.
- At the point 2, my velocity is going to be the left
- 1 meter per second.
- At the velocity 3, it's going to be 3/2.
- You know it doesn't depend on y.
- It only depends on x.
- So now let's draw a little circle and see
- what's happening.
- Let's draw it here.
- It could be anywhere.
- It's infinitely small, but we're just trying
- to get some intuition.
- So after a certain amount of time what's happening?
- Let's say after a second.
- Well, I'm having a few particles leave through the
- left hand side, right, but I have many more particles
- entering this little region that I've defined, this little
- circle, I'm having many more particles enter through the
- right in a given amount of time.
- So in any given amount of time, in my defined space, it's going
- to get denser and denser.
- There's going to be more and more particles in
- that space over time.
- So it's getting denser or you could almost view it as this
- space is sucking up particles.
- In the previous example it was a source of particles -- more
- were coming out than going in.
- Now more going in through the right than coming out.
- And that's what a negative divergence.
- You could almost say -- let's think about the
- word, divergence.
- When it's positive, if I have a positive divergence, the
- particles or the field is diverting out of that point.
- If I have a negative divergence -- maybe
- let's define a new term.
- I've never actually heard it this way, but maybe a negative
- divergence we view as a convergence, right?
- Converge is the opposite of diverge.
- So here, even though some particles are leaving through
- the left, many more particles are coming through the
- right, so it's getting denser and denser.
- And that's this example here.
- And actually at every point in this field we have
- a negative divergence.
- So every point is getting denser and denser actually
- everywhere in this field.
- And then the classic example of a divergence, although I wanted
- to show you that what matters is the net that's coming
- in to a certain area.
- But the classic example of a divergence is a field that
- looks something like this.
- Where maybe that's the x -- that's the y, this is the x.
- If you have a field that looks something like this, this is
- the classical example of a negative divergence, right?
- Where from every direction you have particles entering,
- nothing's leaving.
- So obviously, in any given amount of time, that point is
- getting more and more dense.
- And the classic example of a positive divergence is a point
- where from every direction things are leaving it.
- So clearly this area is going to become less dense.
- If we're talking about velocity of particles, after any moment
- in time, more particles are leaving than coming in because
- no particles are coming in.
- Now what does it mean if we have a 0 divergence?
- So let's try to create a vector field that has a 0 divergence.
- And we'll just stay at a one-dimension just
- for the intuition.
- So that means that the partial derivative with
- respect to x is 0.
- So let's say my vector field is 5i.
- So the magnitude is always 5 in the i direction.
- So let me draw that.
- Vector field is always 5.
- Another way to think of it if you have a constant
- vector field.
- So the magnitude of the vectors, no matter what
- my value of x, is always going to be the same.
- It's always going to be 5.
- So if I were to draw a region, what's happening here?
- Are more particles entering than leaving or
- leaving than entering?
- For any amount that's coming in, an equal amount are coming
- out in a certain amount of time, if we use velocity
- as our example.
- So when you have a divergence of 0, that means that that part
- of the field is not becoming any more or less dense.
- And you could have done it -- let me show you another.
- If my function was, let's say it equals 2i plus 2j.
- It's still a constant, right?
- So this velocity field or vector field will look
- something like this.
- All the points would be, the vectors would
- have a slope of 1.
- But I just wanted you to see something in two dimensions.
- I'll do a fancier example in the next video.
- But even here, if I were to draw some region, the same
- amount is entering as exiting.
- So it's not getting any denser at any point.
- And that makes sense because the divergence of this vector
- field -- well, both of them actually, the divergence
- of that vector field.
- The partial derivative of 2 with respect to
- x, well that's 0.
- Plus the partial derivative of 2 with respect to y.
- Well, that's also 0.
- Anyway, I've run out of time again.
- I will see you in the next video.
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