111 - Green's Theorem

Green's Theorem is all about the idea of fluid rotation around the boundary curve of a region. Suppose we go along a curve counter clockwise. In this case, we want to see if the flow has generally been with us(positive) or against us(negative).

We move along a vector as our direction as an infinitesimal step, with the circulation integral then being defined
In a sense, we split this circulation into infinitely small circulations, which inside the region all cancel eachother out, so we can sum them all up to find the total circulation along the curve

We know that each of these tiny circulations is interpreted as a curl, and this is precisely what Green's Theorem is useful for:

To find the total circulation around a boundary curve, we can take the surface integral of the curl of that vector field over this entire region

We find that Green's Theorem is defined in these two usual forms,

Both these forms are equivalent.

Note that Green's Theorem is assumed for counter-clockwise rotation, for which we say it's positively oriented. If the rotation is clockwise, we flip the sign on the integral and call it negatively oriented.

112 - The 2D Divergence Theorem