Divergence theorem

In vector calculus, the divergence theorem, also known as Gauss' theorem, Ostrogradsky's theorem, or OstrogradskyGauss theorem is a result that links the divergence of a vector field to the value of surface integrals of the flow defined by the field. The divergence theorem is an important result for the mathematics of physics, in particular in electrostatics and fluid dynamics.
Contents 

Intuition
The intuitive content is simple: if there's water flowing in some area, and you are interested in how much water flows out of a certain region within that area, then you need to add up the sources inside the region and subtract the sinks. The water flow is represented by a vector field, and the vector field's divergence at a given point describes the strength of the source or sink there. So integrating the field's divergence over the interior of the region should equal the integral of the vector field over the region's boundary. The divergence theorem says that this is in fact the case.
The divergence theorem is thus a conservation law, stating that the volume total of all sinks and sources, i.e. the volume integral of the divergence, is equal to the net flow across the volume's boundary.
Formal statement
Suppose V is a subset of R^{n} (think of the case n=3 for now) which is compact and has a piecewise smooth boundary. If F is a continuously differentiable vector field defined on a neighborhood of V, then we have
 <math>\iiint_V\left(\nabla\cdot\mathbf{F}\right)dV=\iint_S\mathbf{F}\cdot d\mathbf{S}<math>
where S = ∂V is the boundary of V oriented by outwardpointing normals, and dS is shorthand for NdS, the outward pointing normal of the boundary ∂V.
We note that Gauss' theorem follows from the more general Stokes' theorem, which generalizes the fundamental theorem of calculus.
Applications
Electrostatics
Applied to an electrostatic field we get Gauss's law: the divergence is a constant times the volume charge density.
Gravity
Applied to a gravitational field we get that the surface integral is 4πG times the mass inside, regardless of how the mass is distributed, and regardless of any masses outside.
Spherically symmetric mass distribution
In the case of a spherically symmetric mass distribution we can conclude from this that the field strength at a distance r from the center is inward with a magnitude of G/r² times the total mass at a smaller distance, regardless of any masses at a larger distance.
For example, a hollow sphere does not produce any gravity inside. The gravitational field inside is the same as if the hollow sphere were not there (i.e. the field is that of any masses inside and outside the sphere only).
Cylindrically symmetric mass distribution
In the case of an infinite cylindrically symmetric mass distribution we can conclude that the field strength at a distance r from the center is inward with a magnitude of 2G/r times the total mass per unit length at a smaller distance, regardless of any masses at a larger distance.
For example, an infinite hollow cylinder does not produce any gravity inside.
Bouguer plate
We can conclude that for an infinite, flat plate (Bouguer plate) of thickness H gravity outside the plate is perpendicular to the plate, towards it, with magnitude 2πG times the mass per unit area, independent of the distance to the plate (see also gravity anomalies).
More generally, for a mass distribution with the density depending on one Cartesian coordinate z only, gravity for any z is 2πG times the difference in mass per unit area on either side of this z value.
In particular, a combination of two equal parallel infinite plates does not produce any gravity inside.
History
The theorem was first discovered by Joseph Louis Lagrange in 1762, then later independently rediscovered by Carl Friedrich Gauss in 1813, by George Green in 1825 and in 1831 by Mikhail Vasilievich Ostrogradsky, who also gave the first proof of the theorem. Subsequently, variations on the Divergence theorem are called Gauss's Theorem, Green's theorem, and Ostrogradsky's theorem.
This article was originally based on the GFDL article from PlanetMath at http://planetmath.org/encyclopedia/Divergence.html de:Gaußscher Integralsatz fr:Théorème de fluxdivergence ja:発散定理 es:Teorema de Gauss