Green's Theorem
Let `R` be a simply connected region (i.e., no holes in it and
the edge does not cross itself) with a piecewise smooth boundary `C`
oriented counterclockwise (i.e., `C` is traversed once so that the region `R` always lies to the left).
If `M` and `N` have continuous first partial
derivatives in an open region containing `R`, then
`int_CM \quad dx + N \quad dy = int_Rint ((del N)/(del x) - (del M)/(del y) )\quad dA`.
Introducing additional paths as a way to take into account a hole in a region.
Alternative form for regions in the plane (one)
[just state the issue as a three dimensional issue with `0bbk` so we can
use the `curl bbF • bbk`]
`int_c bbF • dbbr = int_Rint (curl\quad bbF)•bbk\quad dA`
Alternative form (two)
For `s` as the arc length parameter for `C` (`bbr(s)=x(s)bbi+y(s)bbj`),
giving `bbT=x^'(s)bbi + y^'(s)bbj` as the unit tangent vector, so we can
look at the "outward" unit normal vector `bbN = y^'(s)bbi - x^'(s)bbj`,
and for `bbF(x,y) = Mbbi +Nbbj`
`int_C bbF•bbN\quad ds = int_Rint div\quad bbF\quad dA`