Divergence Theorem - Mathematical Statement

Mathematical Statement

Suppose V is a subset of Rn (in the case of n = 3, V represents a volume in 3D space) which is compact and has a piecewise smooth boundary S. If F is a continuously differentiable vector field defined on a neighborhood of V, then we have

The left side is a volume integral over the volume V, the right side is the surface integral over the boundary of the volume V. The closed manifold ∂V is quite generally the boundary of V oriented by outward-pointing normals, and n is the outward pointing unit normal field of the boundary ∂V. (dS may be used as a shorthand for n dS.) By the symbol within the two integrals it is stressed once more that ∂V is a closed surface. In terms of the intuitive description above, the left-hand side of the equation represents the total of the sources in the volume V, and the right-hand side represents the total flow across the boundary ∂V.