In vector calculus, divergence is a vector operator that produces a 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 divergenceEdit
In physical terms, the divergence of a three-dimensional vector field is the extent to which the vector field flux behaves like a source at a given point. It is a local measure of its "outgoingness" – the extent to which there is more of some quantity exiting an infinitesimal region of space than entering it. If the divergence is nonzero at some point then there is compression or expansion at that point. (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 flux 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 flux 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 n̂ being the outward unit normal to that surface. The result, div F, is a function of p. From this definition it also becomes obvious 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 which case any closed surface has no net flux across it.
The intuition that the sum of all sources minus the sum of all sinks should give the net flux outwards of a region is made precise by the divergence theorem.
Although expressed in terms of coordinates, the result is invariant under rotations, as the physical interpretation suggests. This is because 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 corresponding 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 local unit cylindrical coordinates as
where ea is the unit vector in direction a, the divergence is
The use of local coordinates is vital for the validity of the expression. If we consider x the position vector and the functions , , and , which assign the corresponding global cylindrical coordinate to a vector, in general , , and . In particular, if we consider the identity function , we find that:
Using Einstein notation we can consider the divergence in general coordinates, which we write as x1, ..., xi, ...,xn, where n is the number of dimensions of the domain. Here, the upper index refers to the number of the coordinate or component, so x2 refers to the second component, and not the quantity x squared. The index variable i is used to refer to an arbitrary element, such as xi. The divergence can then be written via the Voss- Weyl formula, as:
where is the local coefficient of the volume element and Fi are the components of F with respect to the local unnormalized covariant basis (sometimes written as ). The Einstein notation implies summation over i, since it appears as both an upper and lower index.
The volume coefficient is a function of position which depends on the coordinate system. In Cartesian, cylindrical and polar coordinates, and respectively, using the same conventions as above. It can also be expressed as , where is the metric tensor. Since the determinant is a scalar quantity which doesn't depend on the indices, we can suppress them and simply write . Another expression comes from computing the determinant of the Jacobian for transforming from Cartesian coordinates, which for n = 3 gives
Some conventions expect all local basis elements to be normalized to unit length, as was done in the previous sections. If we write for the normalized basis, and for the components of F with respect to it, we have that
using one of the properties of the metric tensor. By dotting both sides of the last equality with the contravariant element , we can conclude that . After substituting, the formula becomes:
See § Generalizations for further discussion.
It can be shown that any stationary flux v(r) that is at least twice continuously differentiable in R3 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 φ 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
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 derivativeEdit
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 ρ = 1 dx ∧ dy ∧ dz 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 star operator. 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 with coordinates x1, x2, ..., xn, 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 φ.
The divergence of a vector field extends naturally to any differentiable manifold of dimension n that has 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 R3, on such a manifold a vector field X defines an (n − 1)-form j = iX μ 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.
where the second expression is the contraction of the vector field valued 1-form ∇X with itself and the last expression is the traditional coordinate expression from Ricci calculus.
An equivalent expression without using connection is
where g is the metric and ∂a denotes the partial derivative with respect to coordinate xa.
where ∇μ denotes the covariant derivative.
Equivalently, some authors define the divergence of a mixed tensor by using the musical isomorphism ♯: 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 take the trace over the first two covariant indices of the covariant derivative[a]
- The choice of "first" covariant index of a tensor is intrinsic and depends on the ordering of the terms of the Cartesian product of vector spaces on which the tensor is given as a multilinear map V × V × ... × V → R. But equally well defined choices for the divergence could be made by using other indices. Consequently, it is more natural to specify the divergence of T with respect to a specified index. There are however two important special cases where this choice is essentially irrelevant: with a totally symmetric contravariant tensor, when every choice is equivalent, and with a totally antisymmetric contravariant tensor (a.k.a. a k-vector), when the choice affects only the sign.
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- Rudin, Walter (1976). Principles of mathematical analysis. McGraw-Hill. ISBN 0-07-054235-X.
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|Wikimedia Commons has media related to Divergence.|
- Hazewinkel, Michiel, ed. (2001) , "Divergence", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- The idea of divergence of a vector field
- Khan Academy: Divergence video lesson
- Sanderson, Grant (June 21, 2018). "Divergence and curl: The language of Maxwell's equations, fluid flow, and more". 3Blue1Brown – via YouTube.