Green’s Theorem gives you a relationship between the line integral of a 2D vector field over a closed path in a plane and the double integral over the region that it encloses. However, the integral of a 2D conservative field over a closed path is zero is a type of special case in Green’s Theorem.
Green’s Theorem is commonly used for the integration of lines when combined with a curved plane. It is used to integrate the derivatives in a plane. If the line integral is given, it is converted into the surface integral or the double integral or vice versa with the help of this theorem.
Green’s Theorem Statement
Green’s Theorem states that a line integral around the boundary of the plane region D can be computed as the double integral over the region D.
Let C be a positively oriented, smooth and closed curve in a plane, and let D to be the region that is bounded by the region C. Consider P and Q to be the functions of (x, y) that are defined on the open region that contains D, and have continuous partial derivatives, then,
∮c(Pdx+Qdy) = ∫∫D(∂Q/∂x - ∂P/∂y)dxdy
where the path integral is traversed anti-clockwise.
