In vector calculus, the **divergence theorem**, also known as **Gauss' theorem** or **Ostrogradsky-Gauss 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. It was first discovered by Joseph Louis Lagrange (1736-1813) in 1762, then later independently rediscovered by Carl Friedrich Gauss (1777-1855) in 1813, by George Green (1793-1841) in 1825 and by Mikhail Vasilievich Ostrogradsky (1801-1862) in 1831, who also gave the first proof of the theorem.

Let x,y,z be a system of Cartesian coordinates on a 3-dimensional Euclidean space, and let **i**,**j**,**k** be the corresponding basis of unit vectors.

The divergence of a continuously differentiable vector field

**F**= F^{1}**i**+ F^{2}**j**+ F^{3}**k**

- div
**F**= ∂F^{1}/∂x + ∂F^{2}/∂y + ∂F^{3}/∂z

The non-infinitesimal interpretation of divergence is given by Gauss's Theorem. This theorem is 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. In symbols,

- ∫
_{V}div**F**∂V = ∫_{S}(**F**·**N**) ∂S

In light of the physical interpretation, a vector field with constant zero divergence is called incompressible - in this case, no net flow can occur across any closed surface.

*This article was originally based on the GFDL article from PlanetMath at http://planetmath.org/encyclopedia/Divergence.html *