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In
vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...
, divergence is a
vector operator A vector operator is a differential operator used in vector calculus. Vector operators are defined in terms of del, and include the gradient, divergence, and curl: :\begin \operatorname &\equiv \nabla \\ \operatorname &\equiv \nabla \cdot \\ \op ...
that operates on a vector field, producing a
scalar field In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward
flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
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 Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
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 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 are more of the field vectors exiting from an infinitesimal region of space than entering it. A point at which the flux is outgoing has positive divergence, and is often called a "source" of the field. A point at which the flux is directed inward has negative divergence, and is often called a "sink" of the field. The greater the flux of field through a small surface enclosing a given point, the greater the value of divergence at that point. A point at which there is zero flux through an enclosing surface has zero divergence. The divergence of a vector field is often illustrated using the simple example of the
velocity field In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
of a fluid, a liquid or gas. A moving gas has a
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
, a speed and direction at each point, which can be represented by a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
, so the velocity of the gas forms a vector field. If a gas is heated, it will expand. This will cause a net motion of gas particles outward in all directions. Any closed surface in the gas will enclose gas which is expanding, so there will be an outward flux of gas through the surface. So the velocity field will have positive divergence everywhere. Similarly, if the gas is cooled, it will contract. There will be more room for gas particles in any volume, so the external pressure of the fluid will cause a net flow of gas volume inward through any closed surface. Therefore the velocity field has negative divergence everywhere. In contrast, in a gas at a constant temperature and pressure, the net flux of gas out of any closed surface is zero. The gas may be moving, but the volume rate of gas flowing into any closed surface must equal the volume rate flowing out, so the ''net'' flux is zero. Thus the gas velocity has zero divergence everywhere. A field which has zero divergence everywhere is called solenoidal. If the gas is heated only at one point or small region, or a small tube is introduced which supplies a source of additional gas at one point, the gas there will expand, pushing fluid particles around it outward in all directions. This will cause an outward velocity field throughout the gas, centered on the heated point. Any closed surface enclosing the heated point will have a flux of gas particles passing out of it, so there is positive divergence at that point. However any closed surface ''not'' enclosing the point will have a constant density of gas inside, so just as many fluid particles are entering as leaving the volume, thus the net flux out of the volume is zero. Therefore the divergence at any other point is zero.


Definition

The divergence of a vector field at a point is defined as the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of the ratio of the
surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may ...
of out of the closed surface of a volume enclosing to the volume of , as shrinks to zero : where is the volume of , is the boundary of , and \mathbf is the outward unit normal to that surface. It can be shown that the above limit always converges to the same value for any sequence of volumes that contain and approach zero volume. The result, , is a scalar function of . Since this definition is coordinate-free, it shows that the divergence is the same in any
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
. However it is not often used practically to calculate divergence; when the vector field is given in a coordinate system the coordinate definitions below are much simpler to use. A vector field with zero divergence everywhere is called ''
solenoidal In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field: \nabla \cdot \mathbf ...
'' – in which case any closed surface has no net flux across it.


Definition in coordinates


Cartesian coordinates

In three-dimensional Cartesian coordinates, the divergence of a continuously differentiable vector field \mathbf = F_x\mathbf + F_y\mathbf + F_z\mathbf is defined as the
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
-valued function: :\operatorname \mathbf = \nabla\cdot\mathbf = \left(\frac, \frac, \frac \right) \cdot (F_x,F_y,F_z) = \frac+\frac+\frac. 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 In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as ...
of an -dimensional vector field in -dimensional space is invariant under any invertible linear transformation. The common notation for the divergence is a convenient mnemonic, where the dot denotes an operation reminiscent of the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
: take the components of the operator (see
del Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes ...
), apply them to the corresponding components of , and sum the results. Because applying an operator is different from multiplying the components, this is considered an
abuse of notation In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors a ...
.


Cylindrical coordinates

For a vector expressed in local unit
cylindrical coordinates A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference d ...
as :\mathbf= \mathbf_r F_r + \mathbf_\theta F_\theta + \mathbf_z F_z, where is the unit vector in direction , the divergence is :\operatorname \mathbf F = \nabla \cdot \mathbf = \frac1r \frac \left(rF_r\right) + \frac1r \frac + \frac. The use of local coordinates is vital for the validity of the expression. If we consider the position vector and the functions , , and , which assign the corresponding global cylindrical coordinate to a vector, in general r(\mathbf(\mathbf))\neq F_r(\mathbf), \theta(\mathbf(\mathbf))\neq F_(\mathbf), and z(\mathbf(\mathbf))\neq F_z(\mathbf). In particular, if we consider the identity function , we find that: :\theta(\mathbf(\mathbf)) = \theta \neq F_(\mathbf) = 0.


Spherical coordinates

In
spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
, with the angle with the axis and the rotation around the axis, and again written in local unit coordinates, the divergence is :\operatorname\mathbf = \nabla \cdot \mathbf = \frac1 \frac\left(r^2 F_r\right) + \frac1 \frac (\sin\theta\, F_\theta) + \frac1 \frac.


Tensor field

Let be continuously differentiable second-order
tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
defined as follows: :\mathbf = \begin A_ & A_ & A_ \\ A_ & A_ & A_ \\ A_ & A_ & A_ \end the divergence in cartesian coordinate system is a first-order tensor field and can be defined in two ways: :\operatorname (\mathbf) = \cfrac~\mathbf_i = A_~\mathbf_i = \begin \dfrac +\dfrac +\dfrac \\ \dfrac +\dfrac +\dfrac \\ \dfrac +\dfrac +\dfrac \end and : \nabla\cdot \mathbf A = \cfrac~\mathbf_i = A_~\mathbf_i = \begin \dfrac + \dfrac + \dfrac \\ \dfrac + \dfrac + \dfrac \\ \dfrac + \dfrac + \dfrac \\ \end We have :\operatorname (\mathbf) = \nabla\cdot\mathbf A If tensor is symmetric then \operatorname (\mathbf) = \nabla\cdot\mathbf A. Because of this, often in the literature the two definitions (and symbols and \nabla\cdot) are used interchangeably (especially in mechanics equations where tensor symmetry is assumed). Expressions of \nabla\cdot\mathbf A in cylindrical and spherical coordinates are given in the article
del in cylindrical and spherical coordinates This is a list of some vector calculus formulae for working with common curvilinear coordinate systems. Notes * This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reve ...
.


General coordinates

Using Einstein notation we can consider the divergence in general coordinates, which we write as , where is the number of dimensions of the domain. Here, the upper index refers to the number of the coordinate or component, so refers to the second component, and not the quantity squared. The index variable is used to refer to an arbitrary component, such as . The divergence can then be written via th
Voss
Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is ass ...
formula, as: :\operatorname(\mathbf) = \frac \frac, where \rho is the local coefficient of the
volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form :dV ...
and are the components of with respect to the local unnormalized covariant basis (sometimes written as . The Einstein notation implies summation over , 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 spherical coordinates, using the same conventions as before, we have , and , respectively. The volume can also be expressed as \rho = \sqrt, where is the metric tensor. The
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
appears because it provides the appropriate invariant definition of the volume, given a set of vectors. Since the determinant is a scalar quantity which doesn't depend on the indices, these can be suppressed, writing \rho=\sqrt. The absolute value is taken in order to handle the general case where the determinant might be negative, such as in pseudo-Riemannian spaces. The reason for the square-root is a bit subtle: it effectively avoids double-counting as one goes from curved to Cartesian coordinates, and back. The volume (the determinant) can also be understood as the Jacobian of the transformation from Cartesian to curvilinear coordinates, which for gives Some conventions expect all local basis elements to be normalized to unit length, as was done in the previous sections. If we write \hat_i for the normalized basis, and \hat^i for the components of with respect to it, we have that :\mathbf=F^i \mathbf_i = F^i \, \, \frac = F^i \sqrt \, \hat_i = \hat^i \hat_i, using one of the properties of the metric tensor. By dotting both sides of the last equality with the contravariant element \hat^i, we can conclude that F^i = \hat^i / \sqrt. After substituting, the formula becomes: :\operatorname(\mathbf) = \frac 1 \frac = \frac 1 \frac. See ' for further discussion.


Properties

The following properties can all be derived from the ordinary differentiation rules of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
. Most importantly, the divergence is a
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
, i.e., :\operatorname(a\mathbf + b\mathbf) = a \operatorname \mathbf + b \operatorname \mathbf for all vector fields and and all
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s and . There is a
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
of the following type: if is a scalar-valued function and is a vector field, then :\operatorname(\varphi \mathbf) = \operatorname \varphi \cdot \mathbf + \varphi \operatorname \mathbf, or in more suggestive notation :\nabla\cdot(\varphi \mathbf) = (\nabla\varphi) \cdot \mathbf + \varphi (\nabla\cdot\mathbf). Another product rule for the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
of two vector fields and in three dimensions involves the
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was fi ...
and reads as follows: :\operatorname(\mathbf\times\mathbf) = \operatorname \mathbf \cdot\mathbf - \mathbf \cdot \operatorname \mathbf, or :\nabla\cdot(\mathbf\times\mathbf) = (\nabla\times\mathbf)\cdot\mathbf - \mathbf\cdot(\nabla\times\mathbf). The
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
of a
scalar field In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
is the divergence of the field's
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
: :\operatorname(\operatorname\varphi) = \Delta\varphi. The divergence of the
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was fi ...
of any vector field (in three dimensions) is equal to zero: :\nabla\cdot(\nabla\times\mathbf)=0. If a vector field with zero divergence is defined on a ball in , then there exists some vector field on the ball with . For regions in more topologically complicated than this, the latter statement might be false (see
Poincaré lemma In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another ...
). The degree of ''failure'' of the truth of the statement, measured by the
homology Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...
of the
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of t ...
:\ ~ \overset ~ \ ~ \overset ~ \ ~ \overset ~ \ serves as a nice quantification of the complicatedness of the underlying region . These are the beginnings and main motivations of de Rham cohomology.


Decomposition theorem

It can be shown that any stationary flux that is twice continuously differentiable in and vanishes sufficiently fast for can be decomposed uniquely 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 cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was fi ...
): For the irrotational part one has :\mathbf E=-\nabla \Phi(\mathbf r), with :\Phi (\mathbf)=\int_\,d^3\mathbf r'\;\frac. The source-free part, , can be similarly written: one only has to replace the ''scalar potential'' by a ''vector potential'' and the terms by , and the source density by the circulation density . This "decomposition theorem" is a by-product of the stationary case of
electrodynamics In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
. It is a special case of the more general
Helmholtz decomposition In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into ...
, which works in dimensions greater than three as well.


In arbitrary finite dimensions

The divergence of a vector field can be defined in any finite number n of dimensions. If :\mathbf = (F_1 , F_2 , \ldots F_n) , in a Euclidean coordinate system with coordinates , define :\operatorname \mathbf = \nabla\cdot\mathbf = \frac + \frac + \cdots + \frac. In the 1D case, reduces to a regular function, and the divergence reduces to the derivative. For any , the divergence is a linear operator, and it satisfies the "product rule" :\nabla\cdot(\varphi \mathbf) = (\nabla\varphi) \cdot \mathbf + \varphi (\nabla\cdot\mathbf) for any scalar-valued function .


Relation to the exterior derivative

One can express the divergence as a particular case of the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
, which takes a 2-form to a 3-form in . Define the current two-form as :j = F_1 \, dy \wedge dz + F_2 \, dz \wedge dx + F_3 \, dx \wedge dy . It measures the amount of "stuff" flowing through a surface per unit time in a "stuff fluid" of density moving with local velocity . Its exterior derivative is then given by :dj = \left(\frac +\frac +\frac \right) dx \wedge dy \wedge dz = (\nabla \cdot ) \rho where \wedge is the
wedge product A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by convert ...
. Thus, the divergence of the vector field can be expressed as: :\nabla \cdot = d \big(^\flat \big) . Here the superscript is one of the two
musical isomorphism In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle \mathrmM and the cotangent bundle \mathrm^* M of a pseudo-Riemannian manifold induced by ...
s, and is the
Hodge star operator In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the ...
. When the divergence is written in this way, the operator d is referred to as the codifferential. 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.


In curvilinear coordinates

The appropriate expression is more complicated in
curvilinear coordinates In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inve ...
. The divergence of a vector field extends naturally to any
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
of dimension that has a
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of th ...
(or
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
) , e.g. a Riemannian or
Lorentzian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
. Generalising the construction of a
two-form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
for a vector field on , on such a manifold a vector field defines an -form obtained by contracting with . The divergence is then the function defined by :dj = (\operatorname X) \mu . The divergence can be defined in terms of the
Lie derivative In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
as :_X \mu = (\operatorname X) \mu . This means that the divergence measures the rate of expansion of a unit of volume (a
volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form :dV ...
) as it flows with the vector field. On a
pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
, the divergence with respect to the volume can be expressed in terms of the Levi-Civita connection : :\operatorname X = \nabla \cdot X = _ , 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 a connection is :\operatorname(X) = \frac \, \partial_a \left(\sqrt \, X^a\right), where is the
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
and \partial_a denotes the partial derivative with respect to coordinate . The square-root of the (absolute value of the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the) metric appears because the divergence must be written with the correct conception of the
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
. In curvilinear coordinates, the basis vectors are no longer orthonormal; the determinant encodes the correct idea of volume in this case. It appears twice, here, once, so that the X^a can be transformed into "flat space" (where coordinates are actually orthonormal), and once again so that \partial_a is also transformed into "flat space", so that finally, the "ordinary" divergence can be written with the "ordinary" concept of volume in flat space (''i.e.'' unit volume, ''i.e.'' one, ''i.e.'' not written down). The square-root appears in the denominator, because the derivative transforms in the opposite way ( contravariantly) to the vector (which is covariant). This idea of getting to a "flat coordinate system" where local computations can be done in a conventional way is called a
vielbein The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independe ...
. A different way to see this is to note that the divergence is the codifferential in disguise. That is, the divergence corresponds to the expression \star d\star with d the differential and \star the
Hodge star In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the ...
. The Hodge star, by its construction, causes the
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of th ...
to appear in all of the right places.


The divergence of tensors

Divergence can also be generalised to
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
s. In Einstein notation, the divergence of a
contravariant vector In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. In modern mathematical notat ...
is given by :\nabla \cdot \mathbf = \nabla_\mu F^\mu , where denotes the covariant derivative. In this general setting, the correct formulation of the divergence is to recognize that it is a codifferential; the appropriate properties follow from there. Equivalently, some authors define the divergence of a
mixed tensor In tensor analysis, a mixed tensor is a tensor which is neither strictly covariant nor strictly contravariant; at least one of the indices of a mixed tensor will be a subscript (covariant) and at least one of the indices will be a superscript ( ...
by using the
musical isomorphism In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle \mathrmM and the cotangent bundle \mathrm^* M of a pseudo-Riemannian manifold induced by ...
: if is a -
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
( for the contravariant vector and for the covariant one), then we define the ''divergence of '' to be the -tensor :(\operatorname T) (Y_1 , \ldots , Y_) = \Big(X \mapsto \sharp (\nabla T) (X , \cdot , Y_1 , \ldots , Y_) \Big); that is, we take the trace over the ''first two'' covariant indices of the covariant derivative. The \sharp symbol refers to the
musical isomorphism In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle \mathrmM and the cotangent bundle \mathrm^* M of a pseudo-Riemannian manifold induced by ...
.


See also

*
Curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was fi ...
*
Del in cylindrical and spherical coordinates This is a list of some vector calculus formulae for working with common curvilinear coordinate systems. Notes * This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reve ...
*
Divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
*
Gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...


Notes


Citations


References

* * * * *


External links

*
The idea of divergence of a vector field

Khan Academy: Divergence video lesson
* {{Authority control Differential operators Linear operators in calculus Vector calculus