Divergence theorem

From Glossary of Meteorology

divergence theorem

(Also called Gauss's theorem.) The statement that the volume integral of the divergence of a vector, such as the velocity V, over a volume V is equal to the surface integral of the normal component of V over the surface s of the volume (often called the "export" through the closed surface), provided that V and its derivatives are continuous and single-valued throughout V and s.

This may be written
where n is a unit vector normal to the element of surface ds, and the symbol ∮ ∮S indicates that the integration is to be carried out over a closed surface. This theorem is sometimes called Green's theorem in the plane for the case of two-dimensional flow, and Green's theorem in space for the three-dimensional case described above. The divergence theorem is used extensively in manipulating the meteorological equations of motion.