Question

State Green’s Theorem.

Short answer

Green’s Theorem, \(\int_C {Pdx + Qdy = \iint_D {\left( {\frac{{\partial Q}}{{\partial x}} - \frac{{\partial P}}{{\partial y}}} \right)}} dA\)

Step-by-step solution

Define line integral

The function to be integrated is determined along a curve in the coordinate system for a line integral. It doesn't matter if the function to be integrated is a scalar or a vector field.

Explanation

Let \({\rm{C}}\) be any piecewise-smooth, simple closed curve in the plane with a positive inclination. Let the area \({\rm{D}}\) bordered by \({\rm{C}}\). On an open area containing \({\rm{D}}\), if \({\rm{P}}\)and \({\rm{Q}}\) have continuous partial derivatives, then \(\int_C {Pdx + Qdy = \iint_D {\left( {\frac{{\partial Q}}{{\partial x}} - \frac{{\partial P}}{{\partial y}}} \right)}} dA\)

Therefore, \(\int_C {Pdx + Qdy = \iint_D {\left( {\frac{{\partial Q}}{{\partial x}} - \frac{{\partial P}}{{\partial y}}} \right)}} dA\)