Green's theorem is used to-

Explanation:

Green's theorem

• It converts the line integral to a double integral. 
• It transforms the line integral in xy - plane to a surface integral on the same xy - plane.

If M and N are functions of (x, y) defined in an open region then from Green's theorem

\(\oint \left( {{\rm{Mdx}} + {\rm{Ndy}}} \right) = \int\!\!\!\int \left( {\frac{{\partial {\rm{N}}}}{{\partial {\rm{x}}}} - \frac{{\partial {\rm{M}}}}{{\partial {\rm{y}}}}} \right){\rm{dxdy}}\)

Additional Information

Stokes theorem:

\(\oint \vec A \cdot d\vec l\) = \( \iint \left( {\vec \nabla \times \vec A} \right).d\vec s\)

Gauss Divergence theorem:

\( \oint A.ds=\iiint{\left( \nabla .A \right)dv}\)

\( \oint \overrightarrow{F.}\hat{n}ds=\iiint{\left( \nabla .F \right)dv}\)