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In vector calculus, divergence is a vector operator that produces a signed scalar field giving the quantity of a vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region. The divergence of the velocity field in that region would thus have a positive value. While the air is cooled and thus contracting, the divergence of the velocity has a negative value.
Physical interpretation 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 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, source and so on.)
More rigorously, the divergence of a vector field F at a point p can be 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 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 zero divergence everywhere 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.
Definition in Cartesian coordinates
Let x, y, z be a system of Cartesian coordinates in 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 = Ui + Vj + Wk is defined as the scalar-valued function:
Although expressed in terms of coordinates, the result is invariant under rotations, as the physical interpretation suggests. More generally, the trace of the Jacobian matrix of an N-dimensional vector field F in N-dimensional space is invariant under any invertible linear transformation.
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 the ∇ operator (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.
For a vector expressed in cylindrical coordinates as
where ea is the unit vector in direction a, the divergence is
It can be shown that any stationary flux v(r) that is at least twice 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 the source density div v by the circulation-density ∇ × v.
This "decomposition theorem" is 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.
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
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 topologically 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 as
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. Working with the current two-form 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.
In the case of one dimension, F reduces to a regular function, and the divergence reduces to the derivative.
For any n, the divergence is a linear operator, and it satisfies the "product rule"
for any scalar-valued function .
The divergence of a vector field extends naturally to any differentiable 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 an (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.
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 from Ricci calculus.
An equivalent expression without using connection is
where is the metric and denotes partial derivative with respect to coordinate .
where denotes the covariant derivative.
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). "DIVERGENCE of a Vector Field". musr.phas.ubc.ca. Retrieved 2016-08-09.
- Rudin, Walter (1976). Principles of mathematical analysis. McGraw-Hill. p. 281. ISBN 0-07-054235-X.
- Edwards, C. H. (1994). Advanced Calculus of Several Variables. Mineola, NY: Dover. p. 309. ISBN 0-486-68336-2.
- Gurtin, Morton (1981). An Introduction to Continuum Mechanics. Academic Press. p. 30. ISBN 0-12-309750-9.
- Cylindrical coordinates at Wolfram Mathworld
- Spherical coordinates at Wolfram Mathworld
- There is no natural choice of a "first" covariant index – that there is a "first index" at all is an artefact of the notation. Consequently, it is more natural to specify the divergence of T with respect to a specified index.
- 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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