The theorem statement is the following. Let C be a positively oriented, piecewise smooth, simple closed curve in the plane and let D be the region bounded by C. If L and M have continuous partial derivatives on an open region containing D, then
Sometimes a small circle is placed on top of the integral symbol:
This indicates that the curve C is closed. To indicate positive orientation, an arrow pointing in the counter-clockwise direction is sometimes drawn in the circle over the integral symbol.
Proof of Green's theorem when D is a simple region
If D is the simple region so that x ∈ [a, b] and g1(x) < y < g2(x) and the boundary of D is divided into the curves C1, C2, C3, C4, we can demonstrate Green's theorem.
If it can be shown that
and
are true, then Green's theorem is proven.
We define a region D that is simple enough for our purposes. If region D is expressed such that:
where g1 and g2 are continuous functions, the double integral in (1) can be computed:
Now C can be rewritten as the union of four curves: C1, C2, C3, C4.