Green’s theorem is a fundamental result in vector calculus that relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. It is named after the British mathematician George Green.
Green’s theorem states that for a continuously differentiable vector field
F(x, y) = P(x, y)i + Q(x, y)j defined over a region D and its boundary C, the following relationship holds:
∮C F · dr = ∬D (Qx - Py) dA
where ∮C F · dr is the line integral of the vector field F along the curve C, ∬D (Qx – Py) dA is the double integral of the scalar field (Qx – Py) over the region D, and Py and Qx denote the partial derivatives of P and Q with respect to y and x, respectively.
Green’s theorem has various applications in mathematics, physics, and engineering, such as calculating the circulation and flux of a vector field, finding the area of a region in the plane, and verifying the Stokes’ theorem and the divergence theorem in certain cases.