divergence theorem

The Divergence Theorem


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.


Let nn be a natural number, and let SS be a continuously differentiable simple closed hypersurface in n\mathbb{R}^n, in other words the image of a continuously differentiable immersion of the (n1)(n-1)-sphere. By the Jordan–Brouwer separation theorem, SS is the boundary of some bounded open region UU in n\mathbb{R}^n.

Let FF be a continuously differentiable vector field defined on a neighbourhood of cl(U)cl(U) (so defined on both UU and SS). We can integrate FF outwards across SS by taking the dot product FnF \cdot \mathbf{n}, where n\mathbf{n} is the unit normal vector field on SS (perpendicular to SS and pointing outwards, that is away from UU) since SS is continuously differentiable, and integrating this with respect to hypersurface area on SS. Equivalently, form an exterior differential pseudoform of rank n1n - 1 by taking the dot product of FF with the Hodge dual of the identity vector-valued 11-form dx\mathrm{d}\mathbf{x} and integrate that outwards across SS. We can also form the divergence of FF, a scalar field, by differentiating each component of FF with respect to the corresponding coordinate and adding these, and then integrate this with respect to volume on UU; equivalently, form an exterior differential pseudoform of rank nn by multiplying the divergence of FF by the volume pseudoform.

The Divergence Theorem states that these two integrals are equal:

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


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

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