In vector calculus, the divergence is an operator that measures the magnitude of a vector field's source or sink at a given point; the divergence of a vector field is a (signed) scalar. For a vector field that denotes the velocity of air expanding as it is heated, the divergence of the velocity field would have a positive value because the air expands. If the air cools and contracts, the divergence is negative. The divergence could be thought of as a measure of the change in density. Look up divergence in Wiktionary, the free dictionary. ... Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... Vector field given by vectors of the form (−y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a (locally) Euclidean space. ... Vector field given by vectors of the form (−y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a (locally) Euclidean space. ... This article is about velocity in physics. ...

A vector field that has zero divergence everywhere is called solenoidal. In vector calculus a solenoidal vector field is a vector field v with divergence zero: This condition is clearly satisfied whenever v has a vector potential, because if then The converse holds: for any solenoidal v there exists a vector potential A such that . ...

Definition

Let x, y, z be a system of Cartesian coordinates on a 3-dimensional Euclidean space, and let i, j, k be the corresponding basis of unit vectors. Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ... In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ...

The divergence of a continuously differentiable vector field F = F_x i + F_y j + F_z k is defined to be the scalar-valued function: In mathematics, a smooth function is one that is infinitely differentiable, i. ... Vector field given by vectors of the form (−y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a (locally) Euclidean space. ... In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ...

Although expressed in terms of coordinates, the result is invariant under orthogonal transformations, as the physical interpretation suggests. In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ...

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. As a result, this is considered an abuse of notation. In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... In vector calculus, del is a vector differential operator represented by the nabla symbol: ∇. Del is a mathematical tool serving primarily as a convention for mathematical notation; it makes many equations easier to comprehend, write, and remember. ... In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition (while being unlikely to introduce errors or cause confusion). ...

Physical interpretation as source density

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 [1].

An alternate but equivalent definition, gives the divergence as the derivative of the net flow of the vector field across the surface of a small sphere relative to the volume of the sphere. (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.) Formally, For other uses, see Derivative (disambiguation). ... A net flow network is a mere simplification notation over the standard positive flow network. ... For other uses, see Sphere (disambiguation). ... For other uses, see Volume (disambiguation). ...

where S(r) denotes the sphere of radius r about a point p in R³, and the integral is a surface integral taken with respect to n, the normal to that sphere. In mathematics, a surface integral is a definite integral taken over some surface that may be a curved set in space; it can be thought of as the double integral analog of the path integral. ...

Instead of a sphere, any other volume ΔV is possible, if instead of one writes From this definition it also becomes explicity visible that can be seen as the source density of the flux

In light of the physical interpretation, a vector field with constant zero divergence is called incompressible – 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. In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Gauss-Ostrogradsky theorem is a result that relates the flow (that is, flux) of a vector field through a surface to the behaviour of the vector field inside the surface. ...

Decomposition theorem

Main article: Helmholtz decomposition

It can be shown that any stationary flux which is at least two times continuously differentiable in and vanishes sufficiently fast for can be decomposed into an irrotational part and a source-free part Moreover, these parts are explicitly determined by the respective source-densities (see above) and 'circulation densities (see the article Curl): In mathematics, in the area of vector calculus, Helmholtzs theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field can be resolved into irrotational (curl-free) and solenoidal (divergence-free) component vector fields. ... For other uses, see Curl (disambiguation). ...

For the irrotational part one has

with  

The source-free part, , can be simillarly written: one only has to replace the scalar potential by a vector potential and the terms by , and finally the source-density by the circulation-density

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. Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, composed of the electric field and the magnetic field. ... In mathematics, in the area of vector calculus, Helmholtzs theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field can be resolved into irrotational (curl-free) and solenoidal (divergence-free) component vector fields. ...

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 other uses, see Calculus (disambiguation). ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

for all vector fields F and G and all real numbers a and b. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

There is a product rule of the following type: if φ is a scalar valued function and F is a vector field, then In calculus, the product rule also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ...

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: For the cross product in algebraic topology, see Künneth theorem. ... For other uses, see Curl (disambiguation). ...

or

The Laplacian of a scalar field is the divergence of the field's gradient. In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ... In mathematics and physics, a scalar field associates a scalar to every point in space. ...

The divergence of the curl of any vector field (in three dimensions) is constant and 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 balls, this 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 In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations dα = 0 for a given form α to be a closed form, and α = dβ for an exact form, with α given and β unknown. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... In mathematics, a chain complex is a construct originally used in the field of algebraic topology. ...

(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. In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. ...

Relation with the exterior derivative

One can establish a parallel between the divergence and a particular case of the exterior derivative, when it takes a 2-form to a 3-form in R³. If we define:

its exterior derivative dα is given by In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ...

See also Hodge star operator. In mathematics, the Hodge star operator on an oriented inner product space V is a linear operator on the exterior algebra of V, interchanging the subspaces of k-vectors and n−k-vectors where n = dim V, for 0 ≤ k ≤ n. ...

Generalizations

The divergence of a vector field can be defined in any number of dimensions. If

define

For any n, the divergence is a linear operator, and it satisfies the "product rule"

for any scalar-valued function φ.

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 vectorfield on , on such a manifold a vectorfield X defines a n-1 form j = i_Xμ obtained by contracting X with μ. The divergence is then the function defined by In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ...

Standard formulas for the Lie derivative allow us to reformulate this as In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented...

This means that the divergence measures the rate of expansion of a volume element as we let it flow with the vectorfield.

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 In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). ...

where the second expression is the contraction of the vectorfield valued 1 -form with itself and the last expression is the traditional coordinate expression used by physicists.

See also

• Divergence theorem
• Curl
• Gradient
• Nabla in cylindrical and spherical coordinates

In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Gauss-Ostrogradsky theorem is a result that relates the flow (that is, flux) of a vector field through a surface to the behaviour of the vector field inside the surface. ... For other uses, see Curl (disambiguation). ... For other uses, see Gradient (disambiguation). ... This is a list of some vector calculus formulae of general use in working with standard coordinate systems. ...

References

1. Brewer, Jess H. (1999-04-07). DIVERGENCE of a Vector Field. Vector Calculus. Retrieved on 2007-09-28.
2. Theresa M. Korn; Korn, Granino Arthur. Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. New York: Dover Publications, 157-160. ISBN 0-486-41147-8. 

Events of 2008: (EMILY) Me Lesley and MIley are going to China! This article is about the year. ... April 7 is the 97th day of the year in the Gregorian calendar (98th in leap years). ... Year 2007 (MMVII) was a common year starting on Monday of the Gregorian calendar in the 21st century. ... is the 271st day of the year (272nd in leap years) in the Gregorian calendar. ...

External links

• The idea of divergence and curl