nLab divergence theorem

Idea

The Divergence Theorem is a generalization of the classical Ostrogradsky–Gauss Theorem to arbitrary dimensions. As such, it is also a generalization of the (second) Fundamental Theorem of Calculus and a special case of the Stokes Theorem. The value of picking out this particular case is its formulation using vector fields in any number of dimensions.

Statement

Let $n$ be a natural number, and let $S$ be a continuously differentiable simple closed hypersurface in $\mathbb{R}^n$, in other words the image of a continuously differentiable immersion of the $(n-1)$-sphere. By the Jordan–Brouwer separation theorem, $S$ is the boundary of some bounded open region $U$ in $\mathbb{R}^n$.

Let $F$ be a continuously differentiable vector field defined on a neighbourhood of $cl(U)$ (so defined on both $U$ and $S$). We can integrate $F$ outwards across $S$ by taking the dot product $F \cdot \mathbf{n}$, where $\mathbf{n}$ is the unit normal vector field on $S$ (perpendicular to $S$ and pointing outwards, that is away from $U$) since $S$ is continuously differentiable, and integrating this with respect to hypersurface area on $S$. Equivalently, form an exterior differential pseudoform of rank $n - 1$ by taking the dot product of $F$ with the Hodge dual of the identity vector-valued $1$-form $\mathrm{d}\mathbf{x}$ and integrate that outwards across $S$. We can also form the divergence of $F$, a scalar field, by differentiating each component of $F$ with respect to the corresponding coordinate and adding these, and then integrate this with respect to volume on $U$; equivalently, form an exterior differential pseudoform of rank $n$ by multiplying the divergence of $F$ by the volume pseudoform.

The Divergence Theorem states that these two integrals are equal:

$$\int_S F \cdot \mathbf{n} = \int_U div F ,$$

or

$$\int_S F \cdot *\mathrm{d}\mathbf{x} = \int_U div F \vol .$$

This is a special case of the generalized Stokes Theorem, since $div F \vol$ is the exterior differential of $F \cdot *\mathrm{d}\mathbf{x}$.