Divergence
Calculus 

Integral calculus

Specialized calculi

In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the moving air at a point. If air is heated in a region it will expand in all directions such that the velocity field points outward from that region. Therefore the divergence of the velocity field in that region would have a positive value, as the region is a source. If the air cools and contracts, the divergence has a negative value, as the region is a sink.
Contents
Definition of divergence
In physical terms, the divergence of a threedimensional vector field is the extent to which the vector field flow behaves like a source or a sink at a given point. It is a local measure of its "outgoingness"—the extent to which there is more exiting an infinitesimal region of space than entering it. If the divergence is nonzero at some point then there must be a source or sink at that position.^{1} (Note that we are imagining the vector field to be like the velocity vector field of a fluid (in motion) when we use the terms flow, sink and so on.)
More rigorously, the divergence of a vector field F at a point p is defined as the limit of the net flow of F across the smooth boundary of a threedimensional region V divided by the volume of V as V shrinks to p. Formally,
where V  is the volume of V, S(V) is the boundary of V, and the integral is a surface integral with n being the outward unit normal to that surface. The result, div F, is a function of p. From this definition it also becomes explicitly visible that div F can be seen as the source density of the flux of F.
In light of the physical interpretation, a vector field with constant zero divergence is called incompressible or solenoidal – in this case, no net flow can occur across any closed surface.
The intuition that the sum of all sources minus the sum of all sinks should give the net flow outwards of a region is made precise by the divergence theorem.
Application in Cartesian coordinates
Let x, y, z be a system of Cartesian coordinates in 3dimensional Euclidean space, and let i, j, k be the corresponding basis of unit vectors.
The divergence of a continuously differentiable vector field F = U i + V j + W k is equal to the scalarvalued function:
Although expressed in terms of coordinates, the result is invariant under orthogonal transformations, as the physical interpretation suggests.
The common notation for the divergence ∇ · F is a convenient mnemonic, where the dot denotes an operation reminiscent of the dot product: take the components of ∇ (see del), apply them to the components of F, and sum the results. Because applying an operator is different from multiplying the components, this is considered an abuse of notation.
The divergence of a continuously differentiable secondorder tensor field is a firstorder tensor field:^{2}
Cylindrical coordinates
For a vector expressed in cylindrical coordinates as
where e_{a} is the unit vector in direction a, the divergence is^{3}
Spherical coordinates
In spherical coordinates, with the angle with the z axis and the rotation around the z axis, the divergence reads^{4}
Decomposition theorem
It can be shown that any stationary flux v(r) which is at least two times continuously differentiable in and vanishes sufficiently fast for r → ∞ can be decomposed into an irrotational part E(r) and a sourcefree part B(r). Moreover, these parts are explicitly determined by the respective sourcedensities (see above) and circulation densities (see the article Curl):
For the irrotational part one has
with
The sourcefree part, B, can be similarly written: one only has to replace the scalar potential Φ(r) by a vector potential A(r) and the terms −∇Φ by +∇×A, and the sourcedensity div v by the circulationdensity ∇×v.
This "decomposition theorem" is in fact a byproduct of the stationary case of electrodynamics. It is a special case of the more general Helmholtz decomposition which works in dimensions greater than three as well.
Properties
The following properties can all be derived from the ordinary differentiation rules of calculus. Most importantly, the divergence is a linear operator, i.e.
for all vector fields F and G and all real numbers a and b.
There is a product rule of the following type: if is a scalar valued function and F is a vector field, then
or in more suggestive notation
Another product rule for the cross product of two vector fields F and G in three dimensions involves the curl and reads as follows:
or
The Laplacian of a scalar field is the divergence of the field's gradient:
The divergence of the curl of any vector field (in three dimensions) is equal to zero:
If a vector field F with zero divergence is defined on a ball in R^{3}, then there exists some vector field G on the ball with F = curl(G). For regions in R^{3} more complicated than this, the latter statement might be false (see Poincaré lemma). The degree of failure of the truth of the statement, measured by the homology of the chain complex
(where the first map is the gradient, the second is the curl, the third is the divergence) serves as a nice quantification of the complicatedness of the underlying region U. These are the beginnings and main motivations of de Rham cohomology.
Relation with the exterior derivative
One can express the divergence as a particular case of the exterior derivative, which takes a 2form to a 3form in R^{3}. Define the current two form
 .
It measures the amount of "stuff" flowing through a surface per unit time in a "stuff fluid" of density moving with local velocity F. Its exterior derivative is then given by
Thus, the divergence of the vector field F can be expressed as:
Here the superscript is one of the two musical isomorphisms, and is the Hodge dual. Note however that working with the current two form itself and the exterior derivative is usually easier than working with the vector field and divergence, because unlike the divergence, the exterior derivative commutes with a change of (curvilinear) coordinate system.
Generalizations
The divergence of a vector field can be defined in any number of dimensions. If
in a Euclidean coordinate system where and , define
The appropriate expression is more complicated in curvilinear coordinates.
In the case of one dimension, a "vector field" is simply a regular function, and the divergence is simply the derivative.
For any n, the divergence is a linear operator, and it satisfies the "product rule"
for any scalarvalued function .
The divergence can be defined on any manifold of dimension n with a volume form (or density) e.g. a Riemannian or Lorentzian manifold. Generalising the construction of a two form for a vector field on , on such a manifold a vector field X defines a n−1 form obtained by contracting X with . The divergence is then the function defined by
Standard formulas for the Lie derivative allow us to reformulate this as
This means that the divergence measures the rate of expansion of a volume element as we let it flow with the vector field.
On a Riemannian or Lorentzian manifold the divergence with respect to the metric volume form can be computed in terms of the Levi Civita connection
where the second expression is the contraction of the vector field valued 1form with itself and the last expression is the traditional coordinate expression used by physicists.
An equivalent expression without using connection is
where is the metric and denotes partial derivative with respect to coordinate .
Divergence can also be generalised to tensors. In Einstein notation, the divergence of a contravariant vector is given by
where is the covariant derivative.
Equivalently, some authors define the divergence of any mixed tensor by using the "musical notation #":
If T is a (p,q)tensor (p for the contravariant vector and q for the covariant one), then we define the divergence of T to be the (p,q−1)tensor
that is we trace the covariant derivative on the first two covariant indices.
See also
Notes
 ^ DIVERGENCE of a Vector Field
 ^ http://www.foamcfd.org/Nabla/guides/ProgrammersGuidese6.html#x10230002.1.2
 ^ Cylindrical coordinates at Wolfram Mathworld
 ^ Spherical coordinates at Wolfram Mathworld
References
 Brewer, Jess H. (19990407). "DIVERGENCE of a Vector Field". Vector Calculus. Retrieved 20070928.
 Theresa M. Korn; Korn, Granino Arthur. Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. New York: Dover Publications. pp. 157–160. ISBN 0486411478.
External links
 Hazewinkel, Michiel, ed. (2001), "Divergence", Encyclopedia of Mathematics, Springer, ISBN 9781556080104
 The idea of divergence of a vector field
 Khan Academy: Divergence video lesson
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