= Green theorem
{c}
{title2=$\oint_{\partial D}(P\,dx+Q\,dy)=\iint_D(Q_x-P_y)\,dA$}
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= Green's theorem
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{synonym}
Green’s theorem converts circulation around a positively oriented planar boundary into the area integral of scalar curl.
Let $D$ be a bounded region in the <Euclidean plane> whose <boundary> is a finite union of disjoint, piecewise <smooth> <simple closed curves>. Suppose the <vector field> $F=(P,Q)$ is <continuously differentiable> on an <open set> containing the whole closed region. With positive <boundary orientation>,
$$
\oint_{\partial D}F\cdot d\mathbf r
=\oint_{\partial D}(P\,dx+Q\,dy)
=\iint_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\,dA.
$$
The <partial derivatives> $Q_x-P_y$ give the scalar <curl>: the third component of the <curl> of $(P,Q,0)$. For a <parametric curve> $\gamma(t)=(x(t),y(t))$, the boundary <line integral> has <integrand> $P(\gamma(t))x'(t)+Q(\gamma(t))y'(t)$.<Calculus Volume 3 by Gilbert Strang and others/6.4 Green's Theorem>[Calculus Volume 3 by Gilbert Strang and others; §6.4, Theorem 6.12, equation (6.13); circulation form and its hypotheses]{ref}
Positive <boundary orientation> keeps $D$ on the left as one walks along each boundary component: the outer <curve> is counterclockwise, and curves around holes are clockwise. Every component belongs in the <line integral>. The <vector field> must be regular throughout $D$, but need not extend into a hole.<18.02SC Multivariable Calculus, Fall 2010/V5. Simply-Connected Regions>[18.02SC Multivariable Calculus, Fall 2010; V5. Simply-Connected Regions, p. 1, §1, Extended Green's Theorem, equation (1) and subdivision proof]{ref}
The useful tradeoff is between boundary and interior: integrate tangential components along a <curve>, or integrate a <partial derivative> over a region, whichever is simpler. The <boundary formula for planar area> uses this freedom in the opposite direction, recovering <area> from a <line integral>. As a planar case of <Stokes theorem>, <Green theorem> also explains why a local <curl> can determine a complete boundary measurement while leaving room for the global obstruction in <Green theorem with holes>.
A short proof starts with a <rectangle> $D=[a,b]\times[c,d]$. Add its four positively oriented edge <line integrals>:
$$
\oint_{\partial D}(P\,dx+Q\,dy)
=\int_a^b\bigl(P(x,c)-P(x,d)\bigr)\,dx
+\int_c^d\bigl(Q(b,y)-Q(a,y)\bigr)\,dy.
$$
The <fundamental theorem of calculus> turns the differences into <integrals> of $-P_y$ and $Q_x$; <Fubini's theorem> then gives the required <double integral>. On subdividing a region, adjacent edge <line integrals> cancel because their directions oppose. For curved boundaries, the proof uses pieces bounded by graphs and <double integrals>; cancellation works the same way. This is the geometric reason only the region's complete boundary survives.<Calculus Volume 3 by Gilbert Strang and others/6.4 Green's Theorem>[Calculus Volume 3 by Gilbert Strang and others; §6.4, proof following Theorem 6.12: rectangle edge integrals and the fundamental theorem of calculus]{ref}
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