We've now reached the pinnacle of our journey through Multivariable Land. Learning about partial derivatives was like 'meh'. Optimisation was like 'aha'. And now we'll learn about a few multivariable calc theorems - the really cool stuff.
But you'll need a solid foundation to understand these theorems. So if you feel a bit shaky on line integrals, curl and divergence, do have a look at those lecture notes again.

As we've seen, line integrals can be really nasty. It takes quite a bit of time to compute in itself! And then you've got to plug in into and compute the entire integral. Ewww...

This is where Green's theorem comes into the picture. Green's transforms a line integral into a normal double integral. It says that

Here, is a closed curve, which is piecewise continuously differentiable. The curve encloses the region . In addition, we require that and be continuously differentiable.

The thing on the right hand side is just the 2D curl. And indeed, if you'll take a look at the following pictures, doesn't it seem natural that the line integral increases as the curl over the region increases?

Here are a few of examples, so you can see Green's theorem in action.

Calculate

where is the positively oriented boundary of the quarter disk described by

Solution: In this example, we have that

We can use Green's theorem to calculate

We change to polar coordinates

Calculate

where is the positively oriented boundary of the ellipse described by

Solution: In this example, we have that

We can use Green's theorem to calculate

The last integral is the area of , which of course is . Thus,