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

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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 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.

• (Pic could be added here to see how C could be broken up and help explain following proof)

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:

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.