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 is negative and the region is called a sink.
Definition of divergence
In physical terms, the divergence of a three dimensional 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. (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 three dimensional 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 the location 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
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 tensor field is:
For a vector expressed in cylindrical coordinates as
where ea is the unit vector in direction a, the divergence is
In spherical coordinates, with θ the angle with the z axis and ϕ the rotation around the z axis, the divergence reads
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 source-free part B(r). Moreover, these parts are explicitly determined by the respective source-densities (see above) and circulation densities (see the article Curl):
For the irrotational part one has
The source-free 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 finally the source-density div v by the circulation-density ∇×v.
This "decomposition theorem" is in fact a by-product 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.
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 φ is a scalar valued function and F is a vector field, then
or in more suggestive notation
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 R3, then there exists some vector field G on the ball with F = curl(G). For regions in R3 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 2-form to a 3-form in R3. 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 dj 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.
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.
For any n, the divergence is a linear operator, and it satisfies the "product rule"
for any scalar-valued function .
The divergence can be defined on any manifold of dimension n with a volume form (or density) μ 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 j = iXμ obtained by contracting X with μ. 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 1 -form with itself and the last expression is the traditional coordinate expression used by physicists.
Divergence can also be generalised to tensors. In Einstein notation, the divergence of a contravariant vector Fμ is given by
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.
- Brewer, Jess H. (1999-04-07). "DIVERGENCE of a Vector Field". Vector Calculus. http://musr.phas.ubc.ca/~jess/hr/skept/Gradient/node4.html. Retrieved 2007-09-28.
- 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 0-486-41147-8.
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