 
Summary: Green's Theorem.
Suppose
(1)
r : [a, b] × [0, 1] R2
is one to one and continuously differentiable and
det
xs xt
ys yt
> 0
where x, y are such that r(s, t) = (x(s, t), y(s, t));
(2)
F = P i + Q j
is a continuously differentiable vector field defined on D where D is the range of r and
(3) C is the closed curve in R2
obtained by applying r to the boundary of [a, b] × [0, 1] traversed in
the counterclockwise sense: First apply r to the points (s, 0) as s increases from a to b; then apply r to the
points (b, t) as t increases from 0 to 1; then apply r to the points (s, 1) as s decreases from b to a; finally,
apply r to the points (a, t) as t decreases from 1 to 0.
Then
(4)
