# Divergence

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of a vector field at a point The scalar field where are the components of the vector field .

The divergence is denoted by or by the inner product of the Hamilton operator and the vector .

If the vector field is the field of velocities of a stationary flow of a non-compressible liquid, coincides with the intensity of the source ( ) or the sink ( ) at the point .

The integral where is the density of the liquid computed for the -dimensional domain , is equal to the amount of the liquid "issuing" from in unit time. This amount (cf. Ostrogradski formula) coincides with the magnitude where is the unit exterior normal vector to , and is the area element of . The divergence is the derivative with respect to the rate of the flow across the closed surface: Thus, the divergence is invariant with respect to the choice of the coordinate system.

In curvilinear coordinates , , , (*) where and is the unit tangent vector to the -th coordinate line at the point : The divergence of a tensor field of type defined in a domain of an -dimensional manifold with an affine connection, is defined with the aid of the corresponding absolute (covariant) derivatives of the components of , with subsequent convolution (contraction), and is a tensor of type with components  In tensor analysis and differential geometry a differential operator operating on the space of differential forms and connected with the operator of exterior differentiation is also called a divergence.

How to Cite This Entry:
Divergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Divergence&oldid=15062
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article