# Green's Theorem

Green's theorem brings in a relationship between a line integral around a simple closed curve C and a double integral in the plane region D, whose boundary is C. The statement of Green's Theorem is as follows:

Suppose C is a positively oriented, piecewise smooth, simple closed curve in the plane and D is the region bounded by C. If P and Q have continuous partial derivatives on an open region containing D, then

$\int_{C}(Pdx+Qdy) = \iint_{D}(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dA$

The notation $\oint _{C}$ is used to indicate that the curve C satisfies the conditions stated in Green's theorem and hence we have the equivalent theorem statement

$\oint _{C}(Pdx+Qdy) = \iint_{D}(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dA$

If C is traversed clockwise that is in the negative direction, then -C is taken to be the positive direction.

Thus

$\oint _{C}(Pdx+Qdy) = -\oint _{-C}(Pdx + Qdy) =-\iint_{D}(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dA$

If we consider P and Q as components of vector field, that is when F = Pi + Qj, then the line integral in Green's theorem can be rewritten as

$\oint _{C}F.dr$ = $\oint _{C}(Pdx+Qdy)$

Green's theorem can then be written in vector form using curl of F as,

$\oint _{C}F.dr=\iint_{D}(curlF).k dA$As F.dr = F.Tds, the above formula involves the Tangent component of F and re written as

$\oint _{C}F.Tds=\iint_{D}(curlF).k dA$

Here the the tangential form of Green's theorem includes the curl of the vector Filed, while the normal form of the statements contains divergence of F.

$\oint _{C}F.nds=\iint_{D}divF(x,y)dA$

Green's theorem also gives a method to calculate the area of the bounded region D using a line integral.

Area of D = $\frac{1}{2}$ $\oint _{C}(xdy-ydx)$