Green's theorem
In physics and mathematics, Green's Theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. Green's Theorem was named after British scientist George Green, and is based on Stokes' theorem. The theorem states:
 Let C be a positively oriented, piecewise smooth, simple closed curve in the plane and let D be the region bounded by C. If P and Q have continuous partial deriviatives on an open region containing D, then
Sometimes the notation

is used to indicate the line integral is calculuated using the positive orientation of the closed curve
C.
Proof of Green's Theorem when D is a simple region
If we show Equations 1 and 2

and
are true, we would prove Green's Theorem.
If we express D as a region such that:

where g_{1} and g_{2} are continuous functions, we can compute the double integral of equation 1:
Now we break up C as the union of four curves: C_{1}, C_{2}, C_{3}, C_{4}.
 (Pic could be added here to see how C could be broken up and help explain following proof)
With C_{1}, use the parametric equations, x = x, y = g_{1}(x), a ≤ x ≤ b. Therefore:
With C_{3}, use the parametric equations, x = x, y = g_{2}(x), a ≤ x ≤ b. Therefore:
With C_{2} and C_{4}, x is a constant, meaning:
Therefore,

Combining this with equation 4, we get:
 $\backslash \backslash int\_\{C\}\; P(x,y)\; dx\; =\; \backslash \backslash int\backslash \backslash !\backslash \backslash !\backslash \backslash !\backslash \backslash int\_\{D\}\; \backslash \backslash left(\; \backslash \backslash frac\{\backslash \backslash partial\; P\}\{\backslash \backslash partial\; y\}\backslash \backslash right)\; dA$
A similar proof can be employed on Eq.2.