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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, General Edition

Proof of Green's Theorem when D is a simple region

If we show Equations 1 and 2


are true, we would prove Green's Theorem.

If we express D as a region such that:

where g1 and g2 are continuous functions, we can compute the double integral of equation 1:

Now we break up C as the union of four curves: C1, C2, C3, C4.

With C1, use the parametric equations, x = x, y = g1(x), a ≤ x ≤ b. Therefore:

With -C3, use the parametric equations, x = x, y = g2(x), a ≤ x ≤ b. Therefore:

With C2 and C4, x is a constant, meaning:


Combining this with equation 4, we get:

\\int_{C} P(x,y) dx = \\int\\!\\!\\!\\int_{D} \\left(- \\frac{\\partial P}{\\partial y}\\right) dA

A similar proof can be employed on Eq.2.