The divergence theorem is a generalisation of Green’s theorem (or more precisely of Green’s theorem in normal form). This theorem states that the flux of a vector field out of a closed surface equals to the integral of the divergence of that vector field over the volume enclosed by the surface (recall that Green’s theorem states that the flux out of a simple closed curve equals to the integral of the divergence over the region enclosed by the curve).
Gauss’ Divergence theorem
Let be a closed smooth surface and let be continuously differentiable. Let be the region enclosed by . Then
where the surface is oriented such that the normal vector points outward.
Find the flux of the vector field out of the cylinder and
Evaluate where and is the surface of the region bounded by and the planes and