In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D (surface in R 2 {\displaystyle \mathbb {R} ^{2}} ) bounded by C. It is the two-dimensional special case of Stokes' theorem (surface in R 3 {\displaystyle \mathbb {R} ^{3}} ). In one dimension, it is equivalent to the fundamental theorem of calculus. In two dimensions, it is equivalent to the divergence theorem. It is named after mathematical physicist George Green.
Theorem Let C be a positively oriented, piecewise smooth, simple closed curve in a plane, and let D be the region bounded by C. If L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there, then
∮ C ( L d x + M d y ) = ∬ D ( ∂ M ∂ x − ∂ L ∂ y ) d A {\displaystyle \oint _{C}(L\,dx+M\,dy)=\iint _{D}\left({\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}\right)dA}
where the path of integration along C is counterclockwise.
Application Green's theorem in the plane relates line integrals around a simple closed curve to double integrals over the regions it encloses. It has two equivalent forms: the circulation form, which says the tangential line integral of a vector field F = ( P , Q ) {\displaystyle \mathbf {F} =(P,Q)} around a positively oriented simple closed curve C {\displaystyle C} equals the double integral of the scalar curl ∂ Q ∂ x − ∂ P ∂ y {\displaystyle {\frac {\partial Q}{\partial x}}-{\frac {\partial P}{\partial y}}} over the region D {\displaystyle D} bounded by C {\displaystyle C} ; and the flux (divergence) form, which says that the normal line integral of F {\displaystyle \mathbf {F} } around C {\displaystyle C} equals the double integral of the divergence ∇ ⋅ F {\displaystyle \nabla \cdot \mathbf {F} } over D {\displaystyle D} . In applications, the circulation form is used for two-dimensional circulation and rotational flow calculations, while the flux form measures the net outflow across a closed boundary. Green's theorem also yields practical boundary-integral formulas for the area and centroid of a plane region.
Proof when D is a simple region
The following is a proof of half of the theorem for the simplified area D {\displaystyle D} , a type I region where C 1 {\displaystyle C_{1}} and C 3 {\displaystyle C_{3}} are curves connected by vertical lines (possibly of zero length). A similar proof exists for the other half of the theorem when D {\displaystyle D} is a type II region where C 2 {\displaystyle C_{2}} and C 4 {\displaystyle C_{4}} are curves connected by horizontal lines (again, possibly of zero length). Putting these two parts together, the theorem is thus proven for regions of type III (defined as regions which are both type I and type II). The general case can then be deduced from this special case by decomposing D {\displaystyle D} into a set of type III regions. If it can be shown that
and
are true, then Green's theorem follows immediately for the region D {\displaystyle D} . We can prove (1) easily for regions of type I, and (2) for regions of type II. Green's theorem then follows for regions of type III. Assume region D {\displaystyle D} is a type I region and can thus be characterized, as pictured on the right, by
D = { ( x , y ) ∣ a ≤ x ≤ b , g 1 ( x ) ≤ y ≤ g 2 ( x ) } {\displaystyle D=\{(x,y)\mid a\leq x\leq b,g_{1}(x)\leq y\leq g_{2}(x)\}}
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