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Divergence theorem

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.

Definition of divergence

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 = F1i + F2j + F3k

is defined to be the function

div F = ∂F1/∂x + ∂F2/∂y + ∂F3/∂z

Another common notation for the divergence is ·F, a convenient mnemonic. See del for more details.

Physical interpretation of the divergence

In physical terms, the divergence of a vector field is the extent to which the vector field flow behaves like a source or a sink at a given point. Indeed, an alternative, but logically equivalent definition, gives the divergence as the derivative of the net flow of the vector field across the surface of a small sphere relative to the surface area of the sphere. To wit,

where S denotes the sphere of radius r about a point p in R3, and the integral is a surface integral taken with respect to N, the normal to that sphere.

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

where V, a subset of R3, is a compact region with a smooth boundary, and S = ∂V is that boundary oriented by outward-pointing normals. We note that Gauss's theorem follows from the more general
Stokes' theorem, which itself generalizes the fundamental theorem of calculus.

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