Divergence Theorem

A.k.a. Gauss's Theorem. Let $V$ be a region in space with boundary $\partial V$. Then

$$\int_V (\nabla\cdot{\bf F})\,dV = \int_{\partial V} {\bf F}\cdot d{\bf a}.$$ (1)

Let $S$ be a region in the plane with boundary $\partial S$.

$$\int_S \nabla\cdot{\bf F}\, dA = \int_{\partial S}{\bf F}\cdot {\bf n} \,ds.$$ (2)

If the Vector Field ${\bf F}$ satisfies certain constraints, simplified forms can be used. If ${\bf F}(x,y,z) = v(x,y,z){\bf c}$ where ${\bf c}$ is a constant vector $\not = {\bf0}$, then

$$\int_S {\bf F}\cdot d{\bf a} = {\bf c}\cdot \int_S v\,d{\bf a}.$$ (3)

But

$$\nabla \cdot(f{\bf v}) = (\nabla f)\cdot {\bf v}+f(\nabla\cdot {\bf v}),$$ (4)

so

$$\int_V \nabla \cdot({\bf c}v)\,dV = {\bf c}\cdot\int_V(\nabla v+v\nabla\cdot {\bf c})\,dV = {\bf c}\cdot\int_V \nabla v\,dV$$ (5)

$${\bf c}\cdot\left({\int_S v\,d{\bf a}-\int_V\nabla v\,dV}\right)= 0.$$ (6)

But ${\bf c} \not = \bf {0}$, and ${\bf c}\cdot {\bf f}(v)$ must vary with $v$ so that ${\bf c}\cdot {\bf f}(v)$ cannot always equal zero. Therefore,

$$\int_S v\,d{\bf a} = \int_V \nabla v\,dV.$$ (7)

If ${\bf F}(x,y,z) = {\bf c}\times P(x,y,z)$, where ${\bf c}$ is a constant vector $\not = {\bf0}$, then

$$\int_S d{\bf a}\times {\bf P} = \int_V \nabla \times {\bf P}\,dV.$$ (8)

See also Curl Theorem, Gradient, Green's Theorem

References

Arfken, G. ``Gauss's Theorem.'' §1.11 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 57-61, 1985.