Green's, Stoke's and Divergence Theorem Concepts

The Divergence Theorem states that the flux of a vector field through a closed surface is equal to the volume integral of the divergence of the field over the region enclosed by the surface.

Stokes' Theorem relates a surface integral of a vector field over a surface to a line integral of the vector field over the boundary curve of the surface.

Green's Theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by the curve.

To convert surface integrals into volume integrals, allowing for easier computation of flux.

It allows the evaluation of a line integral of a vector field around a closed curve by relating it to a surface integral over the surface bounded by the curve.

Green's Theorem is a special case of Stokes' Theorem in two dimensions.

∮_S **F** • d**S** = ∫∫∫_V div(**F**) dV, where S is the closed surface and V is the volume.

∮_C **F** • d**r** = ∬_S (curl **F**) • d**S**, where C is the boundary curve and S is the surface.

∮_C (P dx + Q dy) = ∬_R (∂Q/∂x - ∂P/∂y) dA, where C is the closed curve and R is the region it encloses.

A vector field is a function that assigns a vector to every point in a subset of space.