Green's theorem

So, there's one place in real life where Green's theorem used to be extremely useful.

Well, if you remember how we prove Green's theorem a while ago, the answer is we're going to do it exactly the same way.

Green's theorem and applications, triple integrals, line and surface integrals in space, Divergence theorem, Stokes' theorem; applications.

Of course, I made up the example in that way so that it simplifies when you use Green's theorem.

Probably if you met an alien, I'm not sure if you would be discussing Green's theorem first, but just in case.

And I know from yesterday what this is equal to, namely using the tangential form of Green's theorem.

OK, so basically that tells you, you can still play tricks with Green's theorem when the region has holes in it.

Actually, I should say This is called Green's theorem in normal form also.

So, we've seen the statement of Green's theorem in two forms.

But, you can't use Green's theorem directly if the curve is not closed.

I can't apply Green's theorem inside it, but let's cut out the smaller thing.

It will be Green's theorem not in x, y, z coordinates but in some funny rotated coordinate systems.

And for each of them, you will apply Green's theorem.

This is really one of the features of Green's theorem.

That's a closed curve. So, I would like to use Green's theorem.

One is direct calculation, and the other one is Green's theorem.

When we add things together, we get Green's theorem in its full generality.

But let's see what Green's theorem does for us here.

What does Green's theorem for flux say?

So, using Green's theorem, the way we'll do it is I will, instead, compute a double integral.

There is a version of Green's theorem for flux.

So, there is an extended version of Green's theorem that tells you the following thing.

Green's theorem for flux says I have a closed curve that goes counterclockwise around some region.

That's why it is also called Green's theorem.

And, the argument to prove that is basically to use Green's theorem.

And so that's why you have this line integral that makes perfect sense, but you can't apply Green's theorem to it.

So, Green's theorem is another way to avoid calculating line integrals if we don't want to.