Divergence 3 Analyzing a vector field using its divergence.
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- Now let's do a slightly fancier example, then we'll try to
- analyze the vector field.
- And hopefully this will make everything a little
- bit more tangible.
- So let's say that the velocity of the fluid, or the particles
- in the fluid, at any given point in the x-y plane, let's
- say in the x-direction, it is x squared, x squared minus 3x
- plus 2 in the x-direction, plus y squared minus 3y plus
- 2 in the y-direction.
- Make it simple so that we only have one thing to factor.
- So let's just do the math first.
- Let's figure out the divergence of our vector field, the
- divergence of our field.
- And going to show you a graph of this field soon, so we'll
- get an intuition of what it actually looks like, instead of
- my not-so-accurate drawings.
- So what's the divergence?
- We take the partial derivative of the x-component
- with respect to x.
- So that's just, there's only an x-variable here, so we don't
- really have to worry about keeping y or z constant.
- It's really just a derivative of this expression
- with respect to x.
- So it's 2x minus 3.
- And then we add that to the partial derivative of the
- y-component, or the y-function, with respect to y.
- There's only y's in the y-component, so we just
- take the derivative with respect to y.
- So it's plus 2y minus 3.
- Or we could just say that the divergence of v, at any point
- xy, so this is a function of x and y, is 2x plus 2y minus 3.
- Now before I show you the graph, let's analyze this
- function a little bit.
- First of all, let's just look at the original vector field,
- and think about when does that vector field have some
- interesting points?
- Well, I think some interesting points are when either the x-
- or the y-components are equal to 0.
- So when is the x-component equal to 0?
- Well, if we factor the x-component, that's the
- same thing as, we could rewrite our vector field.
- If we just factored that, that's x minus 1 times x minus
- 2i plus, and it's the same polynomial, just with y, for
- the y-component, so y minus 1 times y minus 2 times j.
- So the x-component is 0 when x is equal to 1, these are just
- the roots of this polynomial, when x is equal to
- 1 or 2, right?
- And the y-component is 0 when y is equal to 1 or 2.
- And they're both equal to 0 if we have any combination
- of these points.
- So the points where they're both equal to 0 are 1, 1, x's
- is 1, y is 2, right, because then both components
- are 0, 2, 1, or 2, 2.
- So these are the points where the magnitude of the velocity
- of our fluid, or the particles of the fluid, are 0.
- And we'll see that on our graph in a second.
- And let me ask another question.
- At what points are, well, let's first decide, what point is
- the divergence equal to 0?
- Let's say, at what point is a divergence equal to 0.
- Let me click clear up some space.
- I think I can delete this.
- We got our points, we figured out what coordinates are the
- magnitude of the vector field 0.
- So let's try to figure out, when is the divergence
- equal to 0?
- So this is divergence.
- So if we set that equal to 0, 2x plus 2y, 2x
- plus 2y, oh, sorry.
- You know what, this is 2x minus 3 plus 2y minus 3.
- So this is minus 6, right?
- Minus 3 minus 3, that's minus 6.
- That's my major flaw, adding and subtracting.
- Anyway, so the divergence.
- 2x plus 2y, minus 6.
- And we want to know, when does that equal 0?
- So let's set it equal to 0.
- And we can simplify this a little bit.
- We can divide both sides of the equation by 2, and you get x
- plus y minus 3 is equal to 0.
- You get x plus y is equal to 3.
- We could be finished there, or we could just put it in our
- traditional mx plus b form, that's the way I find it
- easier to visualize a line.
- We could say y is equal to 3 minus x.
- So along this line, the divergence of the vector fields
- b is equal to 0, along the line y is equal to 3 minus x.
- And if we're above that line, the divergence is going to be
- positive, right, because if you just made this a greater than
- sign, that would carry over.
- You'd have y is greater than 3-x.
- So y greater than 3-x, the divergence is positive.
- And y is less than 3 minus x, the divergence is negative.
- And you could just make this a less than sign then solve, and
- you'll get y is less than 3-x, you want to know when the
- divergence is negative.
- So I think we've done all of the analyzing we can do.
- So let's take a look at the graph and see if it if it's
- consistent with our intuition of what a divergence is,
- and the numbers we found.
- I hope you can see this.
- So this is the vector field.
- I don't have space to show it, but I think you remember, this
- is, you know, x squared minus 3x plus 2.
- This is the definition of our vector field.
- Have it graphed here.
- And we figured out, we just figured out when the
- x-components and the y-components are equal to 0,
- and then we said, when are both of them equal to 0?
- And we said, oh, well at the point 1, 1.
- Well, this is the point 1, 1, and we that the magnitudes of
- the vectors are 0 at that point.
- And actually, I could zoom in a little bit.
- Right around here, they're all pointing inwards,, but they get
- smaller and smaller as you approach the point 1, 1, right?
- We also said at the point 1, 2. x is 1, y is 2.
- And here, too, we see that the magnitude of the vectors
- get very, very, very small.
- We could zoom in again.
- And we see, the magnitude gets very small.
- The other point, 2, 1, once again, we see the magnitude
- get small, and then 2, 2.
- So that's consistent with what we found out, that the vector
- field gets very small at this point.
- And the other interesting thing we said, OK, when does
- the divergence equal 0?
- Well, the divergence equaled 0 along the line y is
- equal to 3 minus x.
- So the line y is equal to 3 minus x starts, the y-intercept
- is going to be 3, and it's going to come down like this.
- So anything along, at any point along that line,
- the divergence is 0.
- And if we actually look at the graph, it makes sense.
- Because I can't draw on this graph, but if we drew a circle
- right there, let's assume that that's on that, the line y is
- equal to 3 minus x, we would see that in a given amount of
- time, just as many particles are entering through the top
- right as leaving through the bottom left, right?
- A lot of entering through the top right and a lot of leaving
- through the bottom left.
- And the vectors, though, are about the same.
- And if we go to the bottom, if we go here on the line, it
- looks like maybe there are less entering, but there's
- also less leaving.
- I know it's hard to see, but anywhere along the line,
- you see just as much entering as leaving.
- And that's why the divergence is 0.
- Now, let's look at some of the other points.
- Up here, we figure the diverge is positive.
- And does that make [UNINTELLIGIBLE]
- an arbitrary point.
- If we were to draw a circle around here, we see the vectors
- on the left-hand side of that circle that you can't see,
- because I can't draw on this graph.
- But actually, let's just say the square.
- Let's say the square is my region, right?
- This one right here.
- If that square is my region, we see the vectors on the
- left-hand side are larger, than factors leaving are larger than
- the vectors entering it, right?
- So if, in a given amount of time, more is leaving than
- entering, then I'm becoming less dense, or you could say
- that the particles are diverging.
- And that makes sense, because I have a positive divergence.
- And if we go here, where the divergence is negative, let's
- pick an arbitrary spot.
- Let's say this square right here.
- We see that the vectors entering it, the magnitude of
- the vectors entering it, are larger than the magnitude of
- the vectors exiting it.
- So in any given amount of time, more is coming in than leaving.
- So it's getting denser, or it's converging.
- So negative divergence, you can view it as getting denser,
- or it's actually converging.
- Actually, something interesting is happening
- at these two points.
- So we said that at the point 2, 1, we see that in the
- y-direction it's actually converging, right?
- Above y is equal to 1, the arrows are pointing down,
- and below it the arrows are pointing up, right?
- So in the y-direction, we're actually converging, or we
- have a negative divergence.
- Things are entering any given spot.
- But in the x-direction, things are getting pushed out, right?
- So the reason why the divergence is 0 here, you might
- have particles entering above and below a certain space, but
- you have just as many particles exiting to the left
- and the right.
- So it's kind of like particles are getting deflected out.
- So on a net basis, between both dimensions, you have no
- increase or decrease in density along that line y is
- equal to 3 minus x.
- And before I run out of time, I just want to give you that core
- intuition again, of why the divergence is positive, and why
- that means that things are flowing out, when the rate
- of change is positive.
- So we said the divergence is positive.
- Let's say at the spot, right?
- So it makes sense, if our partial derivatives are
- positive, that means that the magnitude of our vector is
- getting larger and larger for larger values of our
- x's and y's, right?
- So if the magnitude of our vectors are getting larger and
- larger for larger values of x and y, the vectors on the
- right are going to have a larger magnitude than
- the factors on the left.
- They're increasing in magnitude.
- And so if I were to draw a boundary, more is going to
- be coming out of the right than entering to the left.
- And so you have a positive diverge, or you're
- getting less dense.
- Anyway, I hope I haven't confused you too much,
- but I have run out of time once again.
- I'll see you in the next video.
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