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So remember we left things with the statement
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of a divergence theorem.
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So the divergence theorem gives us a way
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to compute the flux of a vector field for a closed surface.
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It says if I have a closed surface,
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S bounding some region D, and I have a vector field
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defined in space.
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So then I can try to compute the flux of my vector field
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from my surface, double integral of f dot ds, or f dot n ds,
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if you want.
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And to set this up, of course, I need to use, well,
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the geometry of the surface, depending
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on what the surface is, we've seen various formulas
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for how to set up the double integral.
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But we've also seen that if it's a closed surface,
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and if the vector field is defined everywhere inside,
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then we can actually reduce that to a calculation
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of the triple integral of the divergence of f inside.
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So concretely, if I use coordinates,
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let's say that the coordinates of my vector field are,
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sorry, the components are pq and r dot n ds,
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then that will become the triple integral of, well,
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so divergence is p sub x plus q sub y plus r sub z.
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So by the way, how to remember this formula for divergence
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and over formulas for other things as well?
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So let me just tell you quickly about the del notation.
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So this guy usually pronounced del rather than as pointed
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triangle going downwards or something like that.
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It's a symbolic notation for an operator.
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So you're probably going to complain about my putting
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this guy into a vector, but let's think of partial respect
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to x with respect to y and respect to z
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as the components of some formal vector.
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Of course, it's not a real vector.
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These are not like anything.
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These are just symbols.
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So see, for example, the gradient of a function,
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well, if you multiply this vector by a scalar, which
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is a function, then you'll get partial partial x of f,
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partial partial y of f, partial partial z f.
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Well, that's the gradient.
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That seems to work.
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And so now the interesting thing about divergence
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is I can think of divergence as del dot vector field.
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See, if I do the dot product between this guy
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and my vector field pqr, well, it looks like I will indeed
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get partial partial x of p plus partial q partial y
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plus partial r partial z.
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That's the divergence.
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OK.
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And of course, similarly, when we had two variables only,
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x and y, we could have thought of the same notation
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just with a two component vector, partial x, partial partial y.
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So now, this is like slightly limited usefulness so far.
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It's going to become very handy pretty soon,
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because we're going to seek her.
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And the formula for curl in the plane
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was kind of complicated.
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But if you thought about it in terms of this,
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it was actually a determinant of del and f.
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And now in space, we're actually going to do del cos f.
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But I'm getting ahead of things.
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So let's not do anything with that.
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Curl will be for next week.
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Just getting you used to the notation,
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especially since you might be using it in physics already.
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So it might be worth doing.
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OK.
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So the other thing I wanted to say is, what does this theorem
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say physically?
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How should I think of this statement?
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So I think I said that very quickly at the end of last time,
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but not very carefully.
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So what's the physical interpretation
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of a divergence theorem?
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So I want to claim that the divergence of a vector field
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corresponds to what I'm going to call the source rate,
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which is somehow the amount of flux generated per unit volume.
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So to understand what that means,
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let's think of what's called an incompressible fluid.
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So an incompressible fluid is something like water,
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for example, where a fixed mass of it always occupies
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the same amount of volume.
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So gases are compressible.
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Liquid are incompressible, basically.
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So if you have an incompressible fluid flow, well,
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so again, what that means is really given mass
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occupies always a fixed volume.
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Then, well, let's say if we have such a flow with velocity
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given by our vector field f.
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So we are thinking of f as velocity in maybe something
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containing water, a pipe or something.
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So what does the divergence theorem say?
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It says that if I take a region in space, it's called d.
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So d is the inside.
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And s is the surface around it.
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So if I sum the divergence in d,
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then I'm going to get the flux going out
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through this surface as.
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I should have mentioned it earlier.
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The convention in the divergence theorem
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is that we orient the surface with a normal vector pointing
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always outward.
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So now, we know what flux means.
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Remember, we've been describing flux
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means how much fluid is passing through this surface.
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So that's the amount of fluid that's
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leaving the region d per unit time.
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And of course, when I'm saying that,
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means I'm counting, everything gets going out
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of the minus everything that's coming into d.
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That's what the flux measures.
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So now, if there's stuff coming into d
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or going out of d, well, it must come from somewhere.
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So one possibility would be that your fluid is actually
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being compressed or expanded.
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But I've said, no, I'm looking at something like water,
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that you cannot squish into smaller volume.
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So that case, the only explanation
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is that there's something in here that actually is sucking up
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water or producing more water.
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And so integrating the divergence gives you
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the total amount of sources minus the amount of sinks
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that are inside this region.
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So the divergence itself measures, basically,
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the amount of sources or sinks per unit volume
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in a given place.
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And now, if you think about it that way,
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well, so basically, the divergence theorem
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is just stating something completely obvious about all
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the matter that is leaving that region must come from somewhere.
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So that's basically how we think about it.
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Now, of course, if you're doing 802,
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then you might actually have seen the divergence theorem
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already being used for things that are more like force fields,
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say, electric fields and so on.
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Well, I'll try to say a few things about that
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during the last week of classes.
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But then this kind of interpretation doesn't quite work.
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OK, any questions generally speaking
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before we move on to the proof and over-applications?
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Yes?
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I'm not going to be very careful.
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Oh, not very great.
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So if the divergence, you have a divergence of f,
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measures the amount of sources or sinks in there.
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Well, what makes it happen if you want, in a way,
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it's this theorem or in another way,
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if you think about it, try to look at your favorite vector
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fields and compute their divergence.
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And if you take a vector field where maybe everything is rotating,
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a flow that's just rotating about some axis,
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then you'll find that its divergence is 0.
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If you, sorry?
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The divergence is 0.
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No, the divergence is not equal to the gradient.
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Sorry, there's a dot here, but maybe it's not very big,
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but it's very important.
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OK, so you take the divergence of a vector field
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while you take the gradient of a function.
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So the gradient of a function is a vector.
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The divergence of a vector field is a function.
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So somehow, these guys go back and forth between.
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So I should have said, with new notations
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comes new responsibility.
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I mean, now that we have this nice nifty notation that will
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let us do gradient divergence and later curl in a unified way,
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if you choose to use this notation,
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you have to be really, really careful what you put after it.
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Because otherwise, it's easy to get completely confused.
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So divergence and gradients are completely different things.
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The only thing they have in common is that both are what's
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called a first order differential operator.
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That means it involves the first partial derivatives
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of whatever you put into it.
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But one of them goes from functions to vectors.
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That's gradient.
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Everyone goes from vectors to functions that the divergence.
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And curl later will go from vectors to vectors,
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but that will be later.
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Let's see, more questions?
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No?
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OK.
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So let's see.
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So how are we going to actually prove this theorem?
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Well, if you remember how we proved green sphere
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a while ago, the answer is we're
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going to do it exactly the same way.
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Now, if you don't remember, then I'm going to explain.
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So the first thing we need to do is actually a simplification.
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So instead of proving the divergence theorem,
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namely this equality of fair, I'm going
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to actually prove something easier.
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I'm going to prove that the flux of a vector field that
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has only a z component is actually equal to the triple
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integral of the divergence of this is just r sub z dv.
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Now, how do I go back to the general case?
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Well, I will just prove the same thing for a vector field
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that has only an x component or only a y component.
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And then I will add these things together.
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So if you think carefully about what happens when you evaluate
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this, you'll have some formula for NDS.
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And when you do the dot product, you'll
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end up with a sum, p times something plus q times something
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plus r times something.
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And basically, we are just dealing with the last term,
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r times something, and showing that it's equal to what it should
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be.
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And then we phrase such terms together,
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we'll get the general case.
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So then we get the general case by summing one such identity
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for each component.
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They should say, phrase such identities,
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one for each component.
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Now, let's make a second simplification
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because I'm still not feeling confident that I can prove
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this right away for any surface.
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I'm going to do it first for what's
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called a vertically simple region.
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So vertically simple means it will be something on which I can
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set up an integral of a z variable first, easily.
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So it's something that has a bottom face and a top face,
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and then some vertical sides.
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So let's look first at what happens if the given region
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d is vertically simple.
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So vertically simple means it looks like this.
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It has a top.
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It has a bottom.
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And it has some vertical sides.
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So if you want, if I look at it from above,
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it projects to some region in the xy plane.
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Let's call that r.
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And it lives between the top face and the bottom face.
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Let's say the top face is z equals z2 of xy.
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The bottom face is z equals z1 of xy.
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And I don't need to know actual formulas.
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I'm just going to work with these and proof things
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independently of what the formulas will be for these functions.
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OK, so anyway, a vertically simple region
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is something that lives above about the xy plane
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and is between two graphs of two functions.
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So let's see what we can do in that case.
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So the right hand side of this equality,
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so that's the triple integral.
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Let's start computing it.
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So of course, we will not be able to get a number out of it
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because we don't know actually formulas for anything.
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But at least we can start simplifying
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because the way this region looks like, should say this is d,
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tells me that I can start setting up the triple integral,
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at least in the order where I integrate first of z.
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So I can actually do it as a triple integral of rz dz dx dy,
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00:16:55.400 --> 00:16:57.320
dy dx doesn't matter.
261
00:16:57.320 --> 00:16:59.880
So what are the bounds on z?
262
00:16:59.880 --> 00:17:01.240
See, this is actually a good practice
263
00:17:01.240 --> 00:17:03.720
to remember how we set up triple integrals.
264
00:17:03.720 --> 00:17:06.240
So remember when we integrate first of z,
265
00:17:06.240 --> 00:17:09.200
we start by fixing a point x and y.
266
00:17:09.200 --> 00:17:12.520
And for that value of x and y, we look at a small vertical
267
00:17:12.520 --> 00:17:16.800
slice and see from where to where we have to go,
268
00:17:16.800 --> 00:17:21.000
where we start at z equals whatever the value is at the bottom,
269
00:17:21.000 --> 00:17:24.440
z1 of x and y.
270
00:17:24.440 --> 00:17:27.920
And we go up to the top phase, z2 of x and y.
271
00:17:30.680 --> 00:17:33.840
Now, for x and y, I'm not going to actually set up bounds
272
00:17:33.840 --> 00:17:38.800
because I, oh, sorry.
273
00:17:38.800 --> 00:17:41.880
I've already called r the quantity that I'm integrating.
274
00:17:41.880 --> 00:17:45.440
So let me change this to say u or something like that.
275
00:17:45.440 --> 00:17:47.040
If you already have an r, I mean,
276
00:17:47.040 --> 00:17:50.240
there's not much risk for confusion, but still.
277
00:17:50.240 --> 00:17:54.200
So I'm going to call u the shadow of my region instead.
278
00:17:57.720 --> 00:18:00.360
So now I want to integrate over all values of x and y
279
00:18:00.360 --> 00:18:01.680
that are in the shadow of my region.
280
00:18:01.680 --> 00:18:05.520
That means it's a double integral over this region, u,
281
00:18:05.520 --> 00:18:06.800
which I haven't described to you.
282
00:18:06.800 --> 00:18:09.200
So I can't actually set up bounds for x and y,
283
00:18:09.200 --> 00:18:13.520
but I'm going to just leave it like this.
284
00:18:13.520 --> 00:18:18.600
Now, you see, if you look at how you would start evaluating
285
00:18:18.600 --> 00:18:21.400
this, well, the inner integral certainly is not scary,
286
00:18:21.400 --> 00:18:23.680
because you're integrating the derivative of r
287
00:18:23.680 --> 00:18:28.040
with respect to z, integrating that with respect to z.
288
00:18:28.040 --> 00:18:29.240
So you forget our back.
289
00:18:29.240 --> 00:18:36.240
So triple integral over d of r z dv becomes, well,
290
00:18:36.240 --> 00:18:41.240
we'll have a double integral over u.
291
00:18:41.240 --> 00:18:46.240
So the inner integral becomes r at the point on the top.
292
00:18:46.240 --> 00:18:52.240
So that means, remember r is a function of x, y, and z.
293
00:18:52.240 --> 00:18:58.840
And in fact, I will plug into it the value of z at the top.
294
00:18:58.840 --> 00:19:09.600
So z of x, y minus the value of r at the point on the bottom,
295
00:19:09.600 --> 00:19:11.240
x, y, z, one of x, y.
296
00:19:19.920 --> 00:19:21.280
Any questions about this?
297
00:19:21.280 --> 00:19:29.280
Is that looking very believable?
298
00:19:29.280 --> 00:19:30.280
Yeah.
299
00:19:30.280 --> 00:19:31.280
OK.
300
00:19:31.280 --> 00:19:34.280
So now let's compute the other side, because here we are stuck.
301
00:19:34.280 --> 00:19:36.280
We won't be able to do anything else.
302
00:19:36.280 --> 00:19:39.280
So let's look at the flux integral.
303
00:19:39.280 --> 00:19:42.280
We have to look at the flux of this vector field
304
00:19:42.280 --> 00:19:47.280
for the entire surface, s, which is the whole boundary of d.
305
00:19:47.280 --> 00:19:50.280
So that consists of a lot of pieces, namely the top,
306
00:19:50.280 --> 00:19:56.280
the bottom, and the sides.
307
00:19:56.280 --> 00:19:56.280
OK.
308
00:19:56.280 --> 00:19:57.280
So the other side.
309
00:20:02.880 --> 00:20:10.280
So let me just remind you, s is bottom plus top plus sides
310
00:20:10.280 --> 00:20:16.760
of this vector field that m ds equals, OK.
311
00:20:16.760 --> 00:20:18.880
So what do we have?
312
00:20:18.880 --> 00:20:21.280
So first we have to look at the bottom.
313
00:20:21.280 --> 00:20:22.880
Now let's start with the top, actually.
314
00:20:22.880 --> 00:20:24.880
Sorry.
315
00:20:24.880 --> 00:20:24.880
OK.
316
00:20:24.880 --> 00:20:25.880
So let's start with the top.
317
00:20:34.880 --> 00:20:39.880
So just to remind you, well, OK.
318
00:20:39.880 --> 00:20:42.880
Let's do all of them.
319
00:20:42.880 --> 00:20:47.880
So let's look at the top first.
320
00:20:47.880 --> 00:20:51.840
So we need to set up the flux integral for vector field that m
321
00:20:51.840 --> 00:20:52.840
ds.
322
00:20:52.840 --> 00:20:54.880
We need to know what m ds is.
323
00:20:54.880 --> 00:20:57.880
Well, fortunately for us, we know that the top face is going
324
00:20:57.880 --> 00:21:00.880
to be the graph of some function of x and y.
325
00:21:00.880 --> 00:21:05.880
So we've seen a formula for m ds in this kind of situation.
326
00:21:05.880 --> 00:21:06.880
OK.
327
00:21:06.880 --> 00:21:09.880
We've seen that m ds, sorry, sorry.
328
00:21:09.880 --> 00:21:13.880
Just to remind you, this is the graph of a function,
329
00:21:13.880 --> 00:21:16.880
z equals z2 of xy.
330
00:21:16.880 --> 00:21:25.880
So we've seen m ds from that is negative partial derivative
331
00:21:25.880 --> 00:21:30.360
of this function with respect to x, negative partial z2
332
00:21:30.360 --> 00:21:33.880
with respect to y1 dx dy.
333
00:21:36.880 --> 00:21:41.400
And well, we can't compute this, guys,
334
00:21:41.400 --> 00:21:42.400
but it's not a big deal.
335
00:21:42.400 --> 00:21:49.160
Because if we do the dot product with 0, 0, r, dot m ds,
336
00:21:49.160 --> 00:21:53.800
that will give us, well, if you dot this with 0, 0, r,
337
00:21:53.800 --> 00:21:54.680
this times go away.
338
00:21:54.680 --> 00:21:57.200
You just have r dx dy.
339
00:22:02.800 --> 00:22:08.520
So that means that the double integral for flux
340
00:22:08.520 --> 00:22:15.520
from the top of our vector field dot n ds becomes double
341
00:22:15.520 --> 00:22:21.960
integral of the top of r dx dy.
342
00:22:21.960 --> 00:22:23.920
And now how do we evaluate that, actually?
343
00:22:23.920 --> 00:22:28.960
Well, so r is a function of x, y, z.
344
00:22:28.960 --> 00:22:32.880
But we said we have only two variables that we're going to use.
345
00:22:32.880 --> 00:22:34.360
We're going to use x and y.
346
00:22:34.360 --> 00:22:35.400
We're going to get rid of z.
347
00:22:35.400 --> 00:22:36.680
How do we get rid of z?
348
00:22:36.680 --> 00:22:39.880
Well, if we're on the top surface, z is given by this formula,
349
00:22:39.880 --> 00:22:41.800
z2 of x, y.
350
00:22:41.800 --> 00:22:48.840
So i plug z equals z2 of x, y into the formula for r, whatever
351
00:22:48.840 --> 00:22:50.280
it may be.
352
00:22:50.280 --> 00:22:52.040
Then I integrate dx dy.
353
00:22:52.040 --> 00:22:54.240
And what's the range for x and y?
354
00:22:54.240 --> 00:23:00.000
Well, my surface sets exactly above this region u
355
00:23:00.000 --> 00:23:01.240
in the x, y plane.
356
00:23:01.240 --> 00:23:07.920
So it's a double integral over u.
357
00:23:07.920 --> 00:23:12.200
Any questions about how I set up this flux integral?
358
00:23:18.440 --> 00:23:19.520
Let me close the doors, actually.
359
00:23:24.080 --> 00:23:29.200
So we've got one of the two terms that we had over there.
360
00:23:29.200 --> 00:23:33.200
Let's try to get the others.
361
00:23:33.200 --> 00:23:45.840
So no comment.
362
00:23:45.840 --> 00:23:54.040
OK, so we need to look also at the other parts of our surface
363
00:23:54.040 --> 00:23:56.720
for the flux integral.
364
00:23:56.720 --> 00:24:01.560
So the bottom, well, it will work pretty much the same way
365
00:24:01.560 --> 00:24:06.840
because it's the graph of a function z equals z1 of x, y.
366
00:24:06.840 --> 00:24:14.560
So we should be able to get NDS using the same method.
367
00:24:14.560 --> 00:24:16.960
Negative partial respect to x, negative partial
368
00:24:16.960 --> 00:24:24.720
with respect to y, 1 dx dy.
369
00:24:24.720 --> 00:24:26.520
Now there's a small catch.
370
00:24:26.520 --> 00:24:30.240
We have to think a bit carefully about orientations.
371
00:24:30.240 --> 00:24:35.720
So remember, when we set up the divergence theorem,
372
00:24:35.720 --> 00:24:40.880
we need the normal vectors to point out of our region,
373
00:24:40.880 --> 00:24:43.440
which means that on the top surface,
374
00:24:43.440 --> 00:24:46.440
we want N pointing up.
375
00:24:46.440 --> 00:24:50.280
But on the bottom face, we want N pointing down.
376
00:24:50.280 --> 00:24:53.720
So in fact, we shouldn't use this formula here
377
00:24:53.720 --> 00:24:56.600
because that one corresponds to the other orientation.
378
00:24:56.600 --> 00:24:58.520
Well, we could use it and then subtract.
379
00:24:58.520 --> 00:25:01.560
But it's time's just going to say that NDS is actually
380
00:25:01.560 --> 00:25:03.360
the opposite of this.
381
00:25:03.360 --> 00:25:06.840
So I'm going to switch all my signs.
382
00:25:06.840 --> 00:25:08.960
That's the other side of the formula
383
00:25:08.960 --> 00:25:13.640
when you orient your graph with N pointing downwards.
384
00:25:13.640 --> 00:25:16.600
Now, if I do things the same way as before,
385
00:25:16.600 --> 00:25:25.440
I will get that 0, 0, r, dot NDS will be negative r dx dy.
386
00:25:25.440 --> 00:25:34.040
And so when I do the double integral of the bottom,
387
00:25:34.040 --> 00:25:39.040
I'm going to get the double integral of the bottom
388
00:25:39.040 --> 00:25:43.160
of negative r dx dy, which if I try to evaluate that,
389
00:25:43.160 --> 00:25:47.160
well, I actually will have to integrate,
390
00:25:47.160 --> 00:25:49.920
sorry, first I will have to substitute the value of z.
391
00:25:49.920 --> 00:25:52.760
The value of z that I will want to plug into r
392
00:25:52.760 --> 00:25:55.160
will be given by now 0, 1 of xy.
393
00:25:57.640 --> 00:26:01.320
And the bounds of integration will be given again
394
00:26:01.320 --> 00:26:04.840
by the shadow of our surface, which is again this guy u.
395
00:26:07.600 --> 00:26:10.240
OK, so we seem to be all set, except we're still missing
396
00:26:10.240 --> 00:26:13.360
one part of our surface s because we also
397
00:26:13.360 --> 00:26:14.760
need to look at the sides.
398
00:26:14.760 --> 00:26:16.040
Well, what about the sides?
399
00:26:18.320 --> 00:26:23.080
Well, our vector field, 0, 0, r, is actually vertical.
400
00:26:23.080 --> 00:26:25.360
It's parallel to the z axis.
401
00:26:25.360 --> 00:26:33.680
So my vector field does something like this everywhere.
402
00:26:33.680 --> 00:26:37.080
And why that makes it very interesting on the top and bottom?
403
00:26:37.080 --> 00:26:40.920
That means that on the sides really not much is going on.
404
00:26:40.920 --> 00:26:45.720
No matter is passing through the vertical side.
405
00:26:45.720 --> 00:26:57.120
So the sides are vertical.
406
00:26:57.120 --> 00:27:08.080
So 0, 0, r is tangent to the side.
407
00:27:08.080 --> 00:27:15.360
And therefore, the flux for the sides
408
00:27:15.360 --> 00:27:16.720
is just going to be 0.
409
00:27:20.440 --> 00:27:23.400
No calculation needed.
410
00:27:23.400 --> 00:27:27.920
That's because, of course, that's the reason why I
411
00:27:27.920 --> 00:27:31.200
simplified first things so that my vector field would only
412
00:27:31.200 --> 00:27:32.720
have a z component.
413
00:27:32.720 --> 00:27:35.240
Well, not just that, but that's the main place
414
00:27:35.240 --> 00:27:39.480
where it becomes very useful.
415
00:27:39.480 --> 00:27:45.640
So now if I compare my triple integral and my flux integral,
416
00:27:45.640 --> 00:27:53.760
I get that we are indeed the same.
417
00:27:53.760 --> 00:28:05.200
Well, that's the statement of a theorem we're trying to prove.
418
00:28:05.200 --> 00:28:06.160
I could not erase it.
419
00:28:06.160 --> 00:28:06.680
OK?
420
00:28:06.680 --> 00:28:14.680
OK.
421
00:28:14.680 --> 00:28:30.240
So just to recap, we got a formula for the triple integral
422
00:28:30.240 --> 00:28:32.360
of our sub z dv.
423
00:28:32.360 --> 00:28:35.040
It's a very top.
424
00:28:35.040 --> 00:28:38.600
And we got formulas for the flux through the top and the bottom
425
00:28:38.600 --> 00:28:39.120
and the sides.
426
00:28:39.120 --> 00:28:46.840
And when you add them together, you get indeed the same formula.
427
00:28:46.840 --> 00:29:03.680
Top plus bottom plus sides of.
428
00:29:03.680 --> 00:29:03.880
OK?
429
00:29:03.880 --> 00:29:08.280
And so we have actually completed the proof for this part.
430
00:29:08.280 --> 00:29:11.360
Now, well, that's only for a vertically simple region.
431
00:29:14.440 --> 00:29:24.120
So if d is not vertically simple, what do we do?
432
00:29:24.120 --> 00:29:39.600
Well, we cut it into vertically simple pieces.
433
00:29:39.600 --> 00:29:39.800
OK?
434
00:29:39.800 --> 00:29:44.320
So concretely, say that I wanted to integrate over a solid
435
00:29:44.320 --> 00:29:45.840
donut.
436
00:29:45.840 --> 00:29:49.720
Then that's not vertically simple because here in the middle,
437
00:29:49.720 --> 00:29:53.360
I have not only this top and this bottom,
438
00:29:53.360 --> 00:29:55.880
but I have like this middle face.
439
00:29:55.880 --> 00:29:59.360
So the way I would cut my donut would be probably
440
00:29:59.360 --> 00:30:02.080
I would slice it, not in the way that you would usually
441
00:30:02.080 --> 00:30:04.640
slice a donut or a big ol.
442
00:30:04.640 --> 00:30:08.120
But it's probably more spectacular if you think
443
00:30:08.120 --> 00:30:09.520
that it's a big ol.
444
00:30:09.520 --> 00:30:13.400
Then the mathematicians way of slicing it
445
00:30:13.400 --> 00:30:16.120
is like this into five pieces.
446
00:30:16.120 --> 00:30:16.880
OK?
447
00:30:16.880 --> 00:30:18.840
And that's not very convenient for eating,
448
00:30:18.840 --> 00:30:21.920
but that's convenient for integrating over it.
449
00:30:21.920 --> 00:30:25.200
Because now each of these pieces has a well-defined top
450
00:30:25.200 --> 00:30:26.480
and bottom face.
451
00:30:26.480 --> 00:30:30.720
And of course, you've introduced some vertical sides.
452
00:30:30.720 --> 00:30:33.600
But we don't really care what vertical sides for two reasons.
453
00:30:33.600 --> 00:30:36.960
One is that we've said the flux for them is zero anyway.
454
00:30:36.960 --> 00:30:38.480
So who bothers?
455
00:30:38.480 --> 00:30:38.960
Who cares?
456
00:30:38.960 --> 00:30:39.840
Sorry.
457
00:30:39.840 --> 00:30:41.000
Why bother?
458
00:30:41.000 --> 00:30:45.680
But also, if you sum the flux through the surface
459
00:30:45.680 --> 00:30:47.760
of each little piece, well, you see
460
00:30:47.760 --> 00:30:49.160
that you will be integrating twice
461
00:30:49.160 --> 00:30:51.320
over each of these vertical cuts.
462
00:30:51.320 --> 00:30:52.840
Once when you do this piece, you'll
463
00:30:52.840 --> 00:30:55.920
be taking the flux through this right guy
464
00:30:55.920 --> 00:30:58.480
with normal vector pointing to the right.
465
00:30:58.480 --> 00:31:01.400
And once when you take this middle little piece,
466
00:31:01.400 --> 00:31:05.080
you'll be taking the flux through that cut surface again,
467
00:31:05.080 --> 00:31:08.880
but now with normal vector pointing the other way around.
468
00:31:08.880 --> 00:31:11.600
So in fact, you'll be summing the flux for this guy
469
00:31:11.600 --> 00:31:15.640
twice with opposite orientations, the cancel out.
470
00:31:15.640 --> 00:31:18.440
Well, and again, because of what we were doing,
471
00:31:18.440 --> 00:31:20.840
actually, the integral was just zero anyway.
472
00:31:20.840 --> 00:31:21.800
So it didn't matter.
473
00:31:21.800 --> 00:31:26.960
But even if it hadn't simplified, that would do it for us.
474
00:31:30.080 --> 00:31:32.320
So that's how we do it with a general region.
475
00:31:32.320 --> 00:31:34.680
And then, as I say, at the beginning,
476
00:31:34.680 --> 00:31:37.080
when we can do it for a vector field that has only a z
477
00:31:37.080 --> 00:31:39.800
component, well, we can also do it for a vector field that
478
00:31:39.800 --> 00:31:42.120
has only an x or only a y component.
479
00:31:42.120 --> 00:31:44.960
And then we sum together, and we get the general case.
480
00:31:44.960 --> 00:31:51.480
So that's the end of the proof.
481
00:31:51.480 --> 00:31:53.960
So you see the idea is really the same as for Green's
482
00:31:53.960 --> 00:31:55.440
theorem.
483
00:31:55.440 --> 00:31:55.960
Yes?
484
00:31:55.960 --> 00:31:59.040
The second you've used the other five pieces.
485
00:31:59.040 --> 00:32:00.480
Oh, there's only four pieces.
486
00:32:00.480 --> 00:32:00.800
Thank you.
487
00:32:00.800 --> 00:32:02.200
Yes.
488
00:32:02.200 --> 00:32:03.960
There's three kinds of mathematicians.
489
00:32:03.960 --> 00:32:05.960
Those who know how to count and those who don't.
490
00:32:05.960 --> 00:32:15.960
Well.
491
00:32:15.960 --> 00:32:22.960
Okay.
492
00:32:22.960 --> 00:32:30.960
So, okay.
493
00:32:30.960 --> 00:32:34.960
So I hope that you can see already V
494
00:32:34.960 --> 00:32:37.960
and the interest of this theorem for the deviatrons theorem
495
00:32:37.960 --> 00:32:40.960
for computing flux integrals, just for the sake of computing
496
00:32:40.960 --> 00:32:41.720
flux integrals.
497
00:32:41.720 --> 00:32:46.080
Like, might happen on the problem set or on the next test.
498
00:32:46.080 --> 00:32:48.920
But let me tell you also why it's important physically
499
00:32:48.920 --> 00:32:52.960
to understand equations that had been observed empirically
500
00:32:52.960 --> 00:32:55.480
well before they were actually understood
501
00:32:55.480 --> 00:32:58.520
in terms of how things go.
502
00:32:58.520 --> 00:33:09.720
So let's look at something called the diffusion equation.
503
00:33:09.720 --> 00:33:11.360
So let me explain to you what it does.
504
00:33:11.360 --> 00:33:14.720
So the diffusion equation is something that governs,
505
00:33:14.720 --> 00:33:16.120
well, what's called diffusion.
506
00:33:16.120 --> 00:33:20.560
Diffusion is when you have a fluid in which you're
507
00:33:20.560 --> 00:33:22.000
introducing some substance.
508
00:33:22.000 --> 00:33:26.920
And you won't figure out how that thing is going to spread out
509
00:33:26.920 --> 00:33:30.320
to the technical term is diffuse into the ambient fluid.
510
00:33:30.320 --> 00:33:36.920
So for example, that governs the motion of, say, smoke in the air.
511
00:33:41.920 --> 00:33:47.920
Or if you put dye in a solution or things like that,
512
00:33:47.920 --> 00:33:51.920
that will tell you, say that you actually drop some ink
513
00:33:51.920 --> 00:33:53.720
into a glass of water.
514
00:33:53.720 --> 00:33:55.920
Well, you can imagine that obviously it will get diluted
515
00:33:55.920 --> 00:33:57.520
into there.
516
00:33:57.520 --> 00:34:01.720
And that equation will tell you how exactly over time this thing
517
00:34:01.720 --> 00:34:06.800
is going to spread out and start filling the entire glass.
518
00:34:06.800 --> 00:34:09.600
So what's the equation?
519
00:34:09.600 --> 00:34:12.320
Well, we need first to know what the unknown will be.
520
00:34:12.320 --> 00:34:14.240
So it's a partial differential equation.
521
00:34:14.240 --> 00:34:16.560
So the unknown is a function.
522
00:34:16.560 --> 00:34:19.480
And the equation will relate the partial derivatives
523
00:34:19.480 --> 00:34:21.560
of that function to each other.
524
00:34:21.560 --> 00:34:30.400
So you, the unknown, will be the concentration at a given point.
525
00:34:35.760 --> 00:34:38.800
And of course, that depends on the point where you are.
526
00:34:38.800 --> 00:34:42.640
So that depends on x, y, z, the location where you are.
527
00:34:42.640 --> 00:34:44.920
But since the goal is to also understand how things spread
528
00:34:44.920 --> 00:34:46.840
over time, it would also depend on time.
529
00:34:46.840 --> 00:34:51.040
Otherwise, we are not going to get very far.
530
00:34:51.040 --> 00:34:53.240
And in fact, what the equation will give us
531
00:34:53.240 --> 00:34:55.360
is the derivative of u with respect to t.
532
00:34:55.360 --> 00:34:58.920
It will tell us how the concentration at a given point varies
533
00:34:58.920 --> 00:35:04.080
over time in terms of how the concentration varied in space.
534
00:35:04.080 --> 00:35:07.360
So it will relate partial u, partial t
535
00:35:07.360 --> 00:35:23.800
to partial derivatives with respect to x, y, and z.
536
00:35:23.800 --> 00:35:45.520
So what's the equation?
537
00:35:45.520 --> 00:35:50.760
The equation is partial u, partial t equals
538
00:35:50.760 --> 00:35:54.120
some constant, let me call that constant k,
539
00:35:54.120 --> 00:35:57.720
times something I will call del square d u, which
540
00:35:57.720 --> 00:36:02.200
is also called the Laplacian of u.
541
00:36:02.200 --> 00:36:05.200
And what is that?
542
00:36:05.200 --> 00:36:12.880
Well, that means, OK, so just to scare you, del square
543
00:36:12.880 --> 00:36:20.000
is this, which means it's the divergence of gradient u
544
00:36:20.000 --> 00:36:23.520
that we've used this notation for gradient.
545
00:36:23.520 --> 00:36:27.000
So if you actually expand it in terms of the variables,
546
00:36:27.000 --> 00:36:31.760
that becomes partial u of a partial x squared plus partial
547
00:36:31.760 --> 00:36:37.360
squared u of a partial y squared plus partial squared u
548
00:36:37.360 --> 00:36:41.240
of a partial z squared.
549
00:36:41.240 --> 00:36:50.080
So the equation is this equals that.
550
00:36:50.080 --> 00:36:51.800
So that's a weird looking equation.
551
00:36:51.800 --> 00:36:55.600
And I'm going to have to explain to you where does it come from.
552
00:36:55.600 --> 00:37:02.200
But before I do that, well, let me point out, actually,
553
00:37:02.200 --> 00:37:04.920
that this equation is not just the diffusion equation.
554
00:37:04.920 --> 00:37:11.920
It's also known as the heat equation.
555
00:37:11.920 --> 00:37:16.960
And that's because exactly the same equation governs
556
00:37:16.960 --> 00:37:24.040
how temperature changes over time when you have gain.
557
00:37:24.040 --> 00:37:26.480
So I should have been actually more careful.
558
00:37:26.480 --> 00:37:31.000
I should have said this is in air that's not moving.
559
00:37:31.000 --> 00:37:32.640
And same thing with the solution.
560
00:37:32.640 --> 00:37:35.480
If you drop some ink into your glass of water,
561
00:37:35.480 --> 00:37:37.040
well, if you start stirring, obviously,
562
00:37:37.040 --> 00:37:40.200
it will mix much faster than if you don't do anything.
563
00:37:40.200 --> 00:37:42.560
So that's the case where we don't actually,
564
00:37:42.560 --> 00:37:45.400
the fluid is not moving.
565
00:37:45.400 --> 00:37:48.320
And the heat equation now does the same bit for temperature
566
00:37:48.320 --> 00:37:52.200
in the fluid that's at rest, that's not moving.
567
00:37:52.200 --> 00:37:56.040
It tells you how the heat goes from the warmest parts
568
00:37:56.040 --> 00:37:58.200
to the coldest parts, and eventually,
569
00:37:58.200 --> 00:38:03.440
temperature should somehow even out.
570
00:38:03.440 --> 00:38:10.000
So in the heat equation, this u would just
571
00:38:10.000 --> 00:38:18.560
measure the temperature of your fluid at a given point.
572
00:38:18.560 --> 00:38:19.760
Actually, it doesn't have to be a fluid.
573
00:38:19.760 --> 00:38:22.240
It could be a solid for a heat equation.
574
00:38:22.240 --> 00:38:29.440
So, for example, say that you have a big box made of metal
575
00:38:29.440 --> 00:38:34.000
or something, and you do some heating at one side,
576
00:38:34.000 --> 00:38:36.800
you want to know how quickly is the other side going
577
00:38:36.800 --> 00:38:38.080
to get hot?
578
00:38:38.080 --> 00:38:41.480
Well, you can use the equation to figure out, say
579
00:38:41.480 --> 00:38:44.120
that you have a metal bar, and you keep one side that,
580
00:38:44.120 --> 00:38:47.000
I don't know, 400 degrees because it's in your oven.
581
00:38:47.000 --> 00:38:52.760
How quickly will the other side get warm?
582
00:38:52.760 --> 00:38:55.040
So it's the same equation for both phenomena,
583
00:38:55.040 --> 00:38:56.880
even for, of course, different phenomena.
584
00:39:00.000 --> 00:39:01.520
Well, the physical reason why they're the same
585
00:39:01.520 --> 00:39:03.640
is actually that phenomena are different,
586
00:39:03.640 --> 00:39:05.080
but not all that much.
587
00:39:05.080 --> 00:39:07.480
They involve actually how the atoms and molecules
588
00:39:07.480 --> 00:39:12.560
are actually moving and hitting each other inside this medium.
589
00:39:12.560 --> 00:39:17.160
OK, so anyway, what's the explanation for this?
590
00:39:17.160 --> 00:39:20.360
So to understand the explanation, and given what we've been doing,
591
00:39:20.360 --> 00:39:23.560
we should have a vector field in there.
592
00:39:23.560 --> 00:39:30.480
So I'm going to think of the flow of, well, let's
593
00:39:30.480 --> 00:39:33.360
imagine that we're doing more fun of smoke in air.
594
00:39:33.360 --> 00:39:37.080
So that's the flow of the smoke.
595
00:39:37.080 --> 00:39:39.640
That means, at every point, it's a vector,
596
00:39:39.640 --> 00:39:43.120
which direction tells me in which direction the smoke is actually
597
00:39:43.120 --> 00:39:48.000
moving, and its magnitude tells me how fast it's moving,
598
00:39:48.000 --> 00:39:49.880
and also what amount of smoke is moving.
599
00:39:53.200 --> 00:39:57.800
So there's two things to understand.
600
00:39:57.800 --> 00:40:03.520
One is how the disparities in the concentration
601
00:40:03.520 --> 00:40:08.160
between different points cause the flow to be there.
602
00:40:08.160 --> 00:40:11.520
How smoke will flow from the regions
603
00:40:11.520 --> 00:40:15.640
where there's more smoke to the regions where there's less smoke?
604
00:40:15.640 --> 00:40:17.280
And that's actually physics.
605
00:40:17.280 --> 00:40:19.560
But in a way, it's also common sense.
606
00:40:24.080 --> 00:40:25.680
So physics and common sense.
607
00:40:25.680 --> 00:40:39.360
Tell us that the smoke will flow from high concentration
608
00:40:39.360 --> 00:40:57.200
towards low concentration regions.
609
00:40:57.200 --> 00:41:00.760
So if we think of this function
610
00:41:00.760 --> 00:41:03.160
new that measures concentration, that means we're always
611
00:41:03.160 --> 00:41:06.600
going to go in the direction where the concentration decreases
612
00:41:06.600 --> 00:41:07.960
the fastest.
613
00:41:07.960 --> 00:41:09.600
Well, what's that?
614
00:41:09.600 --> 00:41:10.600
Negative gradient.
615
00:41:18.680 --> 00:41:28.280
So F is directed along minus gradient view.
616
00:41:28.280 --> 00:41:32.000
Now, how big is F going to be?
617
00:41:32.000 --> 00:41:34.480
Well, there you have to come up with some intuition
618
00:41:34.480 --> 00:41:39.320
for how the two are related.
619
00:41:39.320 --> 00:41:40.920
And the easiest relation you can think of
620
00:41:40.920 --> 00:41:43.000
is that they might be just proportional.
621
00:41:43.000 --> 00:41:45.280
So the steeper of the differences in concentration,
622
00:41:45.280 --> 00:41:46.840
the faster the flow will be.
623
00:41:46.840 --> 00:41:49.000
The more there will be flow.
624
00:41:49.000 --> 00:41:51.560
And if you try to think about it as molecules moving
625
00:41:51.560 --> 00:41:53.400
in random directions, you will see it makes actually
626
00:41:53.400 --> 00:41:54.640
a complete sense.
627
00:41:54.640 --> 00:41:57.800
But anyway, it should be part of your physics class,
628
00:41:57.800 --> 00:42:00.440
not part of what I'm telling you.
629
00:42:00.440 --> 00:42:08.240
So I'm just going to accept that the flow is just
630
00:42:08.240 --> 00:42:12.720
proportional to the gradient of view.
631
00:42:12.720 --> 00:42:15.560
So if you want, if the differences between concentration
632
00:42:15.560 --> 00:42:17.800
at different points are very small, then the flow will
633
00:42:17.800 --> 00:42:19.200
be very gentle.
634
00:42:19.200 --> 00:42:21.880
And if on the other hand, you have huge disparities,
635
00:42:21.880 --> 00:42:24.320
then things will try to even out faster.
636
00:42:24.320 --> 00:42:31.160
OK, so that's the first part.
637
00:42:31.160 --> 00:42:35.120
Now, we need to understand the second part, which is, once we
638
00:42:35.120 --> 00:42:38.120
know how the flow is going, how does that affect the
639
00:42:38.120 --> 00:42:39.440
concentration?
640
00:42:39.440 --> 00:42:41.880
If smoke is going that way, then it means the concentration
641
00:42:41.880 --> 00:42:45.280
here should be decreasing, and there it should be increasing.
642
00:42:45.280 --> 00:42:57.640
So that's the relation between f and partial u partial t.
643
00:42:57.640 --> 00:43:03.120
And that part is actually math, namely the divergence
644
00:43:03.120 --> 00:43:03.120
theorem.
645
00:43:03.120 --> 00:43:11.480
So let's try to understand that part more carefully.
646
00:43:11.480 --> 00:43:24.040
So let's take a small piece of a small region in space, d,
647
00:43:24.040 --> 00:43:28.080
bounded by a surface s.
648
00:43:28.080 --> 00:43:33.080
OK, so I want to figure out what's going on in here.
649
00:43:33.080 --> 00:43:38.880
So let's look at the flux out of the OS.
650
00:43:42.280 --> 00:43:49.880
Well, we say that this flux would be given by double
651
00:43:49.880 --> 00:43:53.480
integral on s of f dot n ds.
652
00:43:53.480 --> 00:43:56.120
So this flux measures how much smoke is passing
653
00:43:56.120 --> 00:44:09.160
through s per unit time, that's the amount of smoke through
654
00:44:09.160 --> 00:44:14.120
s per unit time.
655
00:44:14.120 --> 00:44:19.640
But now, how can I compute that differently?
656
00:44:19.640 --> 00:44:22.960
Well, I can compute it just by looking at the total amount
657
00:44:22.960 --> 00:44:26.560
of smoke in this region and saying how much it changes.
658
00:44:26.560 --> 00:44:28.480
If I'm gaining or losing smoke, it means it's
659
00:44:28.480 --> 00:44:30.120
must be going out there.
660
00:44:30.120 --> 00:44:32.040
Well, unless I have a smoker in here,
661
00:44:32.040 --> 00:44:35.440
but that's not part of the data.
662
00:44:35.440 --> 00:44:41.680
So that should be, sorry, that's the same thing
663
00:44:41.680 --> 00:44:44.480
as the derivative with respect to t
664
00:44:44.480 --> 00:44:47.200
of the total amount of smoke in the region, which
665
00:44:47.200 --> 00:44:50.080
is given by the triple integral of u.
666
00:44:50.080 --> 00:44:53.560
If I integrate the concentration of smoke, which means the amount
667
00:44:53.560 --> 00:44:57.080
of smoke in the volume over d, I will get the total amount
668
00:44:57.080 --> 00:44:59.800
of smoke in d.
669
00:44:59.800 --> 00:45:04.320
Except, well, let's see, this flux is counted positively
670
00:45:04.320 --> 00:45:06.440
if we go out of d.
671
00:45:06.440 --> 00:45:09.400
So actually, it's minus the derivative.
672
00:45:09.400 --> 00:45:13.160
It's the amount of smoke that we are losing per unit time.
673
00:45:13.160 --> 00:45:20.160
So now we are almost there.
674
00:45:29.640 --> 00:45:30.640
Well, let me actually.
675
00:45:30.640 --> 00:45:43.560
Because we know yet another way to compute this guy using
676
00:45:43.560 --> 00:45:47.920
the divergence theorem.
677
00:45:47.920 --> 00:45:51.400
So this part here is just like common sense and thinking
678
00:45:51.400 --> 00:45:53.400
about what it means.
679
00:45:53.400 --> 00:45:56.880
The divergence theorem tells me this is also equal to the
680
00:45:56.880 --> 00:46:03.000
triple integral d of df dv.
681
00:46:03.000 --> 00:46:12.680
So what I got is that the triple integral of our d of dv
682
00:46:12.680 --> 00:46:14.880
equals this derivative.
683
00:46:14.880 --> 00:46:17.680
Well, let's think a bit about this derivative.
684
00:46:17.680 --> 00:46:21.400
So see you're integrating a function over x, y, and z,
685
00:46:21.400 --> 00:46:24.000
and then you're differentiating with respect to t.
686
00:46:24.000 --> 00:46:26.720
I claim that you can actually switch the order in which you do
687
00:46:26.720 --> 00:46:29.200
things.
688
00:46:29.200 --> 00:46:32.440
So what we think about it is here you're taking the total
689
00:46:32.440 --> 00:46:35.560
amount of smoke and then seeing how that changes over time.
690
00:46:35.560 --> 00:46:38.880
So you're taking the derivative of the sum of all the small
691
00:46:38.880 --> 00:46:40.880
amounts of smoke everywhere.
692
00:46:40.880 --> 00:46:43.360
Well, that would be the sum of the derivatives of the
693
00:46:43.360 --> 00:46:45.600
amounts of smoke into each little box inside each little
694
00:46:45.600 --> 00:46:47.000
box.
695
00:46:47.000 --> 00:46:55.560
So we can actually move the derivative into the integral.
696
00:46:55.560 --> 00:46:56.480
So let's see.
697
00:46:56.480 --> 00:47:03.440
I said minus ddt is triple integral over d u dv.
698
00:47:03.440 --> 00:47:09.160
And I'm now saying this is the same as the triple integral
699
00:47:09.160 --> 00:47:13.960
in d of d u dt dv.
700
00:47:13.960 --> 00:47:16.880
And the reason why this is going to work is you should
701
00:47:16.880 --> 00:47:25.240
think of it as ddt of a sum of u, of some values you plug in
702
00:47:25.240 --> 00:47:26.080
the values of your points.
703
00:47:29.040 --> 00:47:31.600
And that given time, time is the small volume.
704
00:47:31.600 --> 00:47:32.360
You sum them.
705
00:47:32.360 --> 00:47:33.600
And then you take the derivative.
706
00:47:33.600 --> 00:47:36.320
And now you see the derivative of this sum is the sum of
707
00:47:36.320 --> 00:47:48.400
the derivatives, oops, yi, zit.
708
00:47:52.800 --> 00:47:54.920
So if you think about what the integral measures, that's
709
00:47:54.920 --> 00:47:56.440
actually pretty easy.
710
00:47:56.440 --> 00:47:59.640
And it's because this variable here is not the same as the
711
00:47:59.640 --> 00:48:01.600
variables on which we're integrating.
712
00:48:01.600 --> 00:48:03.800
That's why we can do it.
713
00:48:03.800 --> 00:48:07.200
OK, so now if we have this for any region d.
714
00:48:13.440 --> 00:48:16.760
So we have a function of x, y, z, t.
715
00:48:16.760 --> 00:48:19.520
And we have another function here.
716
00:48:19.520 --> 00:48:23.360
And whenever we integrate them anywhere, we get the same answer.
717
00:48:23.360 --> 00:48:26.760
Well, that must mean they're the same.
718
00:48:26.760 --> 00:48:29.360
Just think of what happens if you just take d to be a tiny
719
00:48:29.360 --> 00:48:32.640
little box, you'll get roughly the value of dv at that point
720
00:48:32.640 --> 00:48:33.880
times the volume of a box.
721
00:48:33.880 --> 00:48:36.480
Or you'll get roughly the value of d udt at that point times
722
00:48:36.480 --> 00:48:38.640
the volume of a little box.
723
00:48:38.640 --> 00:48:41.320
So the values must be the same.
724
00:48:41.320 --> 00:48:45.840
Well, let me actually explain that a tiny bit better.
725
00:48:45.840 --> 00:48:51.400
So what I get is that let me divide by the volume of d.
726
00:48:51.400 --> 00:48:52.400
Sorry.
727
00:48:52.400 --> 00:48:54.560
I promise I'm done in a minute.
728
00:48:54.560 --> 00:49:09.120
So the same thing as one of our volume d of negative d udt dv.
729
00:49:09.120 --> 00:49:11.160
So that means the average value.
730
00:49:11.160 --> 00:49:12.240
Maybe that's the best way.
731
00:49:12.240 --> 00:49:12.760
I think it.
732
00:49:12.760 --> 00:49:24.480
The average of dv in d is equal to the average of minus partial
733
00:49:24.480 --> 00:49:27.120
t in d.
734
00:49:27.120 --> 00:49:31.520
And that's true for any region d, not just for some regions,
735
00:49:31.520 --> 00:49:34.120
but for really any region I can think of.
736
00:49:34.120 --> 00:49:40.640
So the outcome is that actually the divergence of f is equal
737
00:49:40.640 --> 00:49:43.720
to minus d udt.
738
00:49:43.720 --> 00:49:47.640
And that's another way to think about what divergence means.
739
00:49:47.640 --> 00:49:50.080
The divergence we say is how much stuff is actually
740
00:49:50.080 --> 00:49:51.320
expanding flowing out.
741
00:49:51.320 --> 00:49:53.920
That's how much we are losing.
742
00:49:53.920 --> 00:49:57.160
And so now if you combine this with that,
743
00:49:57.160 --> 00:50:00.520
you'll get that d udt is minus divergence f, which
744
00:50:00.520 --> 00:50:03.120
is plus k del square u.
745
00:50:06.280 --> 00:50:09.480
So you combine this guy with that guy,
746
00:50:09.480 --> 00:50:33.520
and you get the diffusion equations.