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Green's theorem

Green's theorem relates a line integral around a closed curve $C$ to a double integral over the region $D$ inside. It is a shortcut when the line integral is hard but the double integral is simple.

\oint_C P\,dx+Q\,dy = \iint_D (Q_x - P_y)\,dA

Green's theorem

Symbols

$C$	closed curve, counterclockwise around D
$D$	the region inside C

Example

$\vec F=(0,x)$ around the unit square: $\oint = \iint_D (1-0)\,dA = 1$ .

The curve

C

must go counterclockwise around

D

, otherwise the sign of the result flips.

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Part of Multivariable Calculus: Vector calculus.