Greens Theorem

Green's Theorem is the idea of relating what happens on the boundary of a curve to what happens inside of the curve❗

Proof

Imagine a curve C that looks like:

Well, you can actually turn this region into two regions that sum up to the original C:

The reason why this is the same even though the length of each curve is the same is because line integrals of a vector field have direction.

In fact, you can do this [[infinity|infinitely]] many times so, you will have a bunch of lines in the center, and you can imagine finding the divergence and curl of all of these little areas, and you will find that if you take the limit as the number of these goes to [[infinity]], they will equal the circulation and flux of the curve.

As n goes to [[infinity]], you will get greens theorm

This points to the idea that integrating along the boundary of the curve is the same as integrating the area of the curve. 🤯 This might be surprising, but it is the same thing that happens in normal [[calculus]] but 1 dimension higher. In normal [[Integration|integration]], you look at that the function F(x) is on the boundary, and that tells you everything about when you integrate the derivative of F(x).

Example

See Also

1. [[Multivariable calculus]]