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 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

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

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

\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

If D is the simple region so that x ∈ [a, b] and g₁(x) < y < g₂(x) and the boundary of D is divided into the curves C₁, C₂, C₃, C₄, we can demonstrate Green's theorem.

If it can be shown that

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

and

\int _{C}M\,dy=\iint _{D}\left({\frac {\partial M}{\partial x}}\right)\,dA\qquad \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=\{(x,y)|a\leq x\leq b,g_{1}(x)\leq y\leq g_{2}(x)\}

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

$\iint _{D}\left({\frac {\partial L}{\partial y}}\right)\,dA$	$=\int _{a}^{b}\!\!\int _{g_{1}(x)}^{g_{2}(x)}\left[{\frac {\partial L}{\partial y}}(x,y)\,dy\,dx\right]$
	$=\int _{a}^{b}{\Big \{}L[x,g_{2}(x)]-L[x,g_{1}(x)]{\Big \}}\,dx\qquad \mathrm {(3)}$

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:

\int _{C_{1}}L(x,y)\,dx=\int _{a}^{b}{\Big \{}L[x,g_{1}(x)]{\Big \}}\,dx

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

\int _{C_{3}}L(x,y)\,dx=-\int _{-C_{3}}L(x,y)\,dx=-\int _{a}^{b}[L(x,g_{2}(x))]\,dx

On C₂ and C₄, x remains constant, meaning

\int _{C_{4}}L(x,y)\,dx=\int _{C_{2}}L(x,y)\,dx=0

Therefore,

$\int _{C}L\,dx$	$=\int _{C_{1}}L(x,y)\,dx+\int _{C_{2}}L(x,y)\,dx+\int _{C_{3}}L(x,y)+\int _{C_{4}}L(x,y)\,dx$
	$=-\int _{a}^{b}[L(x,g_{2}(x))]\,dx+\int _{a}^{b}[L(x,g_{1}(x))]\,dx\qquad \mathrm {(4)}$

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

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

A similar proof can be employed on (2).

Proof of Green's theorem when D is a simple region

See also