# Divergence theorem

From AMS Glossary

## 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.