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 a special case of the more general Stokes' theorem.

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

$\backslash int\_\{C\}\; L\backslash ,\; dx\; +\; M\backslash ,\; dy\; =\; \backslash iint\_\{D\}\; \backslash left(\backslash frac\{\backslash partial\; M\}\{\backslash partial\; x\}\; -\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; y\}\backslash right)\backslash ,\; dA.$

Sometimes a small circle is placed on top of the integral symbol:

$\backslash oint\_\{C\}$

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

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We will prove the theorem for the simplified area D where C₂ and C₄ are vertical lines, however the theorem remains valid for any area D as defined above.

If it can be shown that

$\backslash int\_\{C\}\; L\backslash ,\; dx\; =\; \backslash iint\_\{D\}\; \backslash left(-\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; y\}\backslash right)\; dA\backslash qquad\backslash mathrm\{(1)\}$

and

$\backslash int\_\{C\}\; M\backslash ,\; dy\; =\; \backslash iint\_\{D\}\; \backslash left(\backslash frac\{\backslash partial\; M\}\{\backslash partial\; x\}\backslash right)\backslash ,\; dA\backslash qquad\backslash mathrm\{(2)\}$

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:

$D\; =\; \backslash \{(x,y)|a\backslash le\; x\backslash le\; b,\; g\_1(x)\; \backslash le\; y\; \backslash le\; g\_2(x)\backslash \}$

where g₁ and g₂ are continuous functions, the double integral in (1) can be computed:

{| $\backslash iint\_\{D\}\; \backslash left(\backslash frac\{\backslash partial\; L\}\{\backslash partial\; y\}\backslash right)\backslash ,\; dA$ $=\backslash int\_a^b\backslash !\backslash !\backslash int\_\{g\_1(x)\}^\{g\_2(x)\}\; \backslash leftL\}\{\backslash partial\; y\}\; (x,y)\backslash ,\; dy\backslash ,\; dx\; \backslash right$ $=\; \backslash int\_a^b\; \backslash Big\backslash \{L-\; L[x,g\_1(x)\backslash Big\backslash \}\; \backslash ,\; dx\backslash qquad\backslash mathrm\{(3)\}$

Now C can be rewritten 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:

$\backslash int\_\{C\_1\}\; L(x,y)\backslash ,\; dx\; =\; \backslash int\_a^b\; \backslash Big\backslash \{L*\backslash Big\backslash \}\backslash ,\; dx$

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

$\backslash int\_\{C\_3\}\; L(x,y)\backslash ,\; dx\; =\; -\backslash int\_\{-C\_3\}\; L(x,y)\backslash ,\; dx\; =\; -\; \backslash int\_a^b*\backslash ,\; dx$

On C₂ and C₄, x remains constant, meaning

$\backslash int\_\{C\_4\}\; L(x,y)\backslash ,\; dx\; =\; \backslash int\_\{C\_2\}\; L(x,y)\backslash ,\; dx\; =\; 0$

Therefore,

{| $\backslash int\_\{C\}\; L\backslash ,\; dx$ $=\; \backslash int\_\{C\_1\}\; L(x,y)\backslash ,\; dx\; +\; \backslash int\_\{C\_2\}\; L(x,y)\backslash ,\; dx\; +\; \backslash int\_\{C\_3\}\; L(x,y)\; +\; \backslash int\_\{C\_4\}\; L(x,y)\backslash ,\; dx$ $=\; -\backslash int\_a^bdx\; +\; \backslash int\_a^b\; [L(x,g\_1(x))\backslash ,\; dx\backslash qquad\backslash mathrm\{(4)\}$

Combining (3) with (4), we get:

$\backslash int\_\{C\}\; L(x,y)\backslash ,\; dx\; =\; \backslash iint\_\{D\}\; \backslash left(-\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; y\}\backslash right)\backslash ,\; dA$

A similar proof can be employed on (2).

See also

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• Planimeter
• Method of image charges - A method used in electro statics that takes strong advantage of the uniqueness theorem (derived from Green's theorem)

Vector calculus | Mathematical theorems

Satz von Green | Teorema de Green | Théorème de Green | Teorema di Green | グリーンの定理 | Twierdzenie Greena | Greens sats | 格林公式