Divergence

Template:Short description Template:About {{#invoke:sidebar|collapsible | class = plainlist | titlestyle = padding-bottom:0.25em; | pretitle = Part of a series of articles about | title = Calculus | image = <math>\int_{a}^{b} f'(t) \, dt = f(b) - f(a)</math> | listtitlestyle = text-align:center; | liststyle = border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa; | expanded = vector | abovestyle = padding:0.15em 0.25em 0.3em;font-weight:normal; | above =

| list2name = differential | list2titlestyle = display:block;margin-top:0.65em; | list2title = Template:Bigger | list2 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | heading1 = Definitions
 | content1 =

 | heading2 = Concepts
 | content2 =

 | heading3 = Rules and identities
 | content3 =

}}

| list3name = integral | list3title = Template:Bigger | list3 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | content1 =

| heading2 = Definitions

 | content2 =

 | heading3 = Integration by
 | content3 =

}}

| list4name = series | list4title = Template:Bigger | list4 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | content1 =

 | heading2 = Convergence tests
 | content2 =

}}

| list5name = vector | list5title = Template:Bigger | list5 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | content1 =

 | heading2 = Theorems
 | content2 =

}}

| list6name = multivariable | list6title = Template:Bigger | list6 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | heading1 = Formalisms
 | content1 =

 | heading2 = Definitions
 | content2 =

}}

| list7name = advanced | list7title = Template:Bigger | list7 ={{#invoke:sidebar|sidebar|child=yes

 |contentclass=hlist
 | content1 =

}}

| list8name = specialized | list8title = Template:Bigger | list8 =

| list9name = miscellanea | list9title = Template:Bigger | list9 =

}}

File:Divergence (captions).svg

The divergence of different vector fields. The divergence of vectors from point (x,y) equals the sum of the partial derivative-with-respect-to-x of the x-component and the partial derivative-with-respect-to-y of the y-component at that point: <math>\nabla\!\cdot(\mathbf{V}(x,y)) = \frac{\partial\, {V_x(x,y)}}{\partial{x}}+\frac{\partial\, {V_y(x,y)}}{\partial{y}}</math>

In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to area.) More precisely, the divergence at a point is the rate that the flow of the vector field modifies a volume about the point in the limit, as a small volume shrinks down to the 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

Template:See also In physical terms, the divergence of a vector field is the extent to which the vector field flux behaves like a source or a sink 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 of a fluid, a liquid or gas. A moving gas has a velocity, a speed and direction at each point, which can be represented by a vector, 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.

DefinitionEdit

File:Definition of divergence.svg

}</math>

The divergence of a vector field Template:Math at a point Template:Math is defined as the limit of the ratio of the surface integral of Template:Math out of the closed surface of a volume Template:Math enclosing Template:Math to the volume of Template:Math, as Template:Math shrinks to zero

Template:Oiint

where Template:Math is the volume of Template:Math, Template:Math is the boundary of Template:Math, and <math>\mathbf{\hat n}</math> 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 Template:Math and approach zero volume. The result, Template:Math, is a scalar function of Template:Math.

Since this definition is coordinate-free, it shows that the divergence is the same in any coordinate system. However the above definition 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 which case any closed surface has no net flux across it. This is the same as saying that the (flow of the) vector field preserves volume: The volume of any region does not change after it has been transported by the flow for any period of time.

Definition in coordinatesEdit

Cartesian coordinatesEdit

In three-dimensional Cartesian coordinates, the divergence of a continuously differentiable vector field <math>\mathbf{F} = F_x\mathbf{i} + F_y\mathbf{j} + F_z\mathbf{k}</math> is defined as the scalar-valued function:

<math display="block">\operatorname{div} \mathbf{F} = \nabla\cdot\mathbf{F} = \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \cdot (F_x,F_y,F_z) = \frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}.</math>

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 Template:Math-dimensional vector field Template:Math in Template:Mvar-dimensional space is invariant under any invertible linear transformationTemplate:Clarification needed.

The common notation for the divergence Template:Math is a convenient mnemonic, where the dot denotes an operation reminiscent of the dot product: take the components of the Template:Math operator (see del), apply them to the corresponding components of Template:Math, and sum the results. Because applying an operator is different from multiplying the components, this is considered an abuse of notation.

Cylindrical coordinatesEdit

For a vector expressed in local unit cylindrical coordinates as <math display="block">\mathbf{F} = \mathbf{e}_r F_r + \mathbf{e}_\theta F_\theta + \mathbf{e}_z F_z,</math> where Template:Math is the unit vector in direction Template:Math, the divergence isTemplate:Refn <math display="block">\operatorname{div} \mathbf F = \nabla \cdot \mathbf{F} = \frac{1}{r} \frac{\partial}{\partial r} \left(r F_r\right) + \frac1r \frac{\partial F_\theta}{\partial\theta} + \frac{\partial F_z}{\partial z}. </math>

The use of local coordinates is vital for the validity of the expression. If we consider Template:Math the position vector and the functions Template:Math, Template:Math, and Template:Math, which assign the corresponding global cylindrical coordinate to a vector, in general Template:Nowrap Template:Nowrap and Template:Nowrap In particular, if we consider the identity function Template:Math, we find that:

<math display="block">\theta(\mathbf{F}(\mathbf{x})) = \theta \neq F_{\theta}(\mathbf{x}) = 0.</math>

Spherical coordinatesEdit

In spherical coordinates, with Template:Mvar the angle with the Template:Mvar axis and Template:Mvar the rotation around the Template:Mvar axis, and Template:Math again written in local unit coordinates, the divergence isTemplate:Refn <math>\operatorname{div}\mathbf{F} = \nabla \cdot \mathbf{F} = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 F_r\right) + \frac{1}{r\sin\theta} \frac{\partial}{\partial \theta} \left(\sin\theta\, F_\theta\right) + \frac{1}{r \sin\theta} \frac{\partial F_\varphi}{\partial \varphi}.</math>

Tensor fieldEdit

Let Template:Math be continuously differentiable second-order tensor field defined as follows:

<math display="block">\mathbf{A} = \begin{bmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{bmatrix}</math>

the divergence in cartesian coordinate system is a first-order tensor fieldTemplate:Sfn and can be defined in two ways:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<math display="block">\operatorname{div} (\mathbf{A}) = \frac{\partial A_{ik}}{\partial x_k}~\mathbf{e}_i = A_{ik,k}~\mathbf{e}_i = \begin{bmatrix} \dfrac{\partial A_{11}}{\partial x_1} +\dfrac{\partial A_{12}}{\partial x_2} +\dfrac{\partial A_{13}}{\partial x_3} \\ \dfrac{\partial A_{21}}{\partial x_1} +\dfrac{\partial A_{22}}{\partial x_2} +\dfrac{\partial A_{23}}{\partial x_3} \\ \dfrac{\partial A_{31}}{\partial x_1} +\dfrac{\partial A_{32}}{\partial x_2} +\dfrac{\partial A_{33}}{\partial x_3} \end{bmatrix}</math>

and<ref> Template:Cite book</ref><ref> Template:Cite book</ref><ref> {{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<math display="block"> \nabla \cdot \mathbf A = \frac{\partial A_{ki}}{\partial x_k} ~\mathbf{e}_i = A_{ki,k}~\mathbf{e}_i = \begin{bmatrix} \dfrac{\partial A_{11}}{\partial x_1} + \dfrac{\partial A_{21}}{\partial x_2} + \dfrac{\partial A_{31}}{\partial x_3} \\ \dfrac{\partial A_{12}}{\partial x_1} + \dfrac{\partial A_{22}}{\partial x_2} + \dfrac{\partial A_{32}}{\partial x_3} \\ \dfrac{\partial A_{13}}{\partial x_1} + \dfrac{\partial A_{23}}{\partial x_2} + \dfrac{\partial A_{33}}{\partial x_3} \\ \end{bmatrix} </math>

We have

<math display="block">\operatorname{div} {\left(\mathbf{A}^\mathsf{T}\right)} = \nabla \cdot \mathbf A</math>

If tensor is symmetric Template:Math then Template:Nowrap Because of this, often in the literature the two definitions (and symbols Template:Math and <math>\nabla \cdot</math>) are used interchangeably (especially in mechanics equations where tensor symmetry is assumed).

Expressions of <math>\nabla\cdot\mathbf A</math> in cylindrical and spherical coordinates are given in the article del in cylindrical and spherical coordinates.

General coordinatesEdit

Using Einstein notation we can consider the divergence in general coordinates, which we write as Template:Math, where Template:Mvar is the number of dimensions of the domain. Here, the upper index refers to the number of the coordinate or component, so Template:Math refers to the second component, and not the quantity Template:Mvar squared. The index variable Template:Mvar is used to refer to an arbitrary component, such as Template:Math. The divergence can then be written via the Voss-Weyl formula,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}Template:Cbignore</ref> as:

<math display="block">\operatorname{div}(\mathbf{F}) = \frac{1}{\rho} \frac{\partial {\left(\rho \, F^i\right)}}{\partial x^i},</math>

where <math>\rho</math> is the local coefficient of the volume element and Template:Math are the components of <math>\mathbf{F} = F^i\mathbf{e}_i</math> with respect to the local unnormalized covariant basis (sometimes written as Template:Nowrap The Einstein notation implies summation over Template:Mvar, since it appears as both an upper and lower index.

The volume coefficient Template:Mvar 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 Template:Math, Template:Math and Template:Math, respectively. The volume can also be expressed as <math display="inline">\rho = \sqrt{\left|\det g_{ab}\right|}</math>, where Template:Math is the metric tensor. The determinant 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 Template:Nowrap 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 Template:Math gives Template:Nowrap

Some conventions expect all local basis elements to be normalized to unit length, as was done in the previous sections. If we write <math>\hat{\mathbf{e}}_i</math> for the normalized basis, and <math>\hat{F}^i</math> for the components of Template:Math with respect to it, we have that <math display="block">\mathbf{F} = F^i \mathbf{e}_i = F^i \|{\mathbf{e}_i }\| \frac{\mathbf{e}_i}{\| \mathbf{e}_i \|} = F^i \sqrt{g_{ii}} \, \hat{\mathbf{e}}_i = \hat{F}^i \hat{\mathbf{e}}_i,</math> using one of the properties of the metric tensor. By dotting both sides of the last equality with the contravariant element Template:Nowrap^i</math>,}} we can conclude that <math display="inline">F^i = \hat{F}^i / \sqrt{g_{ii}}</math>. After substituting, the formula becomes:

<math display="block">\operatorname{div}(\mathbf{F}) = \frac 1{\rho} \frac{\partial \left(\frac{\rho}{\sqrt{g_{ii}}}\hat{F}^i\right)}{\partial x^i} =

\frac 1{\sqrt{\det g}} \frac{\partial \left(\sqrt{\frac{\det g}{g_{ii}}}\,\hat{F}^i\right)}{\partial x^i}.</math>

See Template:Section link for further discussion.

PropertiesEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

The following properties can all be derived from the ordinary differentiation rules of calculus. Most importantly, the divergence is a linear operator, i.e.,

<math>\operatorname{div}(a\mathbf{F} + b\mathbf{G}) = a \operatorname{div} \mathbf{F} + b \operatorname{div} \mathbf{G}</math>

for all vector fields Template:Math and Template:Math and all real numbers Template:Math and Template:Math.

There is a product rule of the following type: if Template:Mvar is a scalar-valued function and Template:Math is a vector field, then

<math>\operatorname{div}(\varphi \mathbf{F}) = \operatorname{grad} \varphi \cdot \mathbf{F} + \varphi \operatorname{div} \mathbf{F},</math>

or in more suggestive notation

<math>\nabla\cdot(\varphi \mathbf{F}) = (\nabla\varphi) \cdot \mathbf{F} + \varphi (\nabla\cdot\mathbf{F}).</math>

Another product rule for the cross product of two vector fields Template:Math and Template:Math in three dimensions involves the curl and reads as follows:

<math>\operatorname{div}(\mathbf{F}\times\mathbf{G}) = \operatorname{curl} \mathbf{F} \cdot\mathbf{G} - \mathbf{F} \cdot \operatorname{curl} \mathbf{G},</math>

or

<math>\nabla\cdot(\mathbf{F}\times\mathbf{G}) = (\nabla\times\mathbf{F})\cdot\mathbf{G} - \mathbf{F}\cdot(\nabla\times\mathbf{G}).</math>

The Laplacian of a scalar field is the divergence of the field's gradient:

<math>\operatorname{div}(\operatorname{grad}\varphi) = \Delta\varphi.</math>

The divergence of the curl of any vector field (in three dimensions) is equal to zero:

<math>\nabla\cdot(\nabla\times\mathbf{F})=0.</math>

If a vector field Template:Math with zero divergence is defined on a ball in Template:Math, then there exists some vector field Template:Math on the ball with Template:Math. For regions in Template:Math 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

<math>\{ \text{scalar fields on } U \} ~ \overset{\operatorname{grad}}{\rarr} ~ \{ \text{vector fields on } U \} ~ \overset{\operatorname{curl}}{\rarr} ~ \{ \text{vector fields on } U \} ~ \overset{\operatorname{div}}{\rarr} ~ \{ \text{scalar fields on } U \}</math>

serves as a nice quantification of the complicatedness of the underlying region Template:Math. These are the beginnings and main motivations of de Rham cohomology.

Decomposition theoremEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} It can be shown that any stationary flux Template:Math that is twice continuously differentiable in Template:Math and vanishes sufficiently fast for Template:Math can be decomposed uniquely into an irrotational part Template:Math and a source-free part Template:Math. 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

<math>\mathbf E=-\nabla \Phi(\mathbf r),</math>

with

<math>\Phi (\mathbf{r})=\int_{\mathbb R^3}\,d^3\mathbf r'\;\frac{\operatorname{div} \mathbf{v}(\mathbf{r}')}{4\pi\left|\mathbf{r}-\mathbf{r}'\right|}.</math>

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

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.

In arbitrary finite dimensionsEdit

The divergence of a vector field can be defined in any finite number <math>n</math> of dimensions. If

<math>\mathbf{F} = (F_1 , F_2 , \ldots F_n) ,</math>

in a Euclidean coordinate system with coordinates Template:Math, define

<math>\operatorname{div} \mathbf{F} = \nabla\cdot\mathbf{F} = \frac{\partial F_1}{\partial x_1} + \frac{\partial F_2}{\partial x_2} + \cdots + \frac{\partial F_n}{\partial x_n}.</math>

In the 1D case, Template:Math reduces to a regular function, and the divergence reduces to the derivative.

For any Template:Math, the divergence is a linear operator, and it satisfies the "product rule"

<math>\nabla\cdot(\varphi \mathbf{F}) = (\nabla\varphi) \cdot \mathbf{F} + \varphi (\nabla\cdot\mathbf{F})</math>

for any scalar-valued function Template:Mvar.

Relation to 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 Template:Math. Define the current two-form as

<math>j = F_1 \, dy \wedge dz + F_2 \, dz \wedge dx + F_3 \, dx \wedge dy .</math>

It measures the amount of "stuff" flowing through a surface per unit time in a "stuff fluid" of density Template:Math moving with local velocity Template:Math. Its exterior derivative Template:Math is then given by

<math>dj = \left(\frac{\partial F_1}{\partial x} +\frac{\partial F_2}{\partial y} +\frac{\partial F_3}{\partial z} \right) dx \wedge dy \wedge dz = (\nabla \cdot {\mathbf F}) \rho </math>

where <math>\wedge</math> is the wedge product.

Thus, the divergence of the vector field Template:Math can be expressed as:

<math>\nabla \cdot {\mathbf F} = {\star} d{\star} \big({\mathbf F}^\flat \big) .</math>

Here the superscript Template:Music is one of the two musical isomorphisms, and Template:Math is the Hodge star operator. When the divergence is written in this way, the operator <math>{\star} d{\star}</math> 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 coordinatesEdit

The appropriate expression is more complicated in curvilinear coordinates. The divergence of a vector field extends naturally to any differentiable manifold of dimension Template:Math that has a volume form (or density) Template:Mvar, e.g. a Riemannian or Lorentzian manifold. Generalising the construction of a two-form for a vector field on Template:Math, on such a manifold a vector field Template:Math defines an Template:Math-form Template:Math obtained by contracting Template:Math with Template:Mvar. The divergence is then the function defined by

<math>dj = (\operatorname{div} X) \mu .</math>

The divergence can be defined in terms of the Lie derivative as

<math>{\mathcal L}_X \mu = (\operatorname{div} X) \mu .</math>

This means that the divergence measures the rate of expansion of a unit of volume (a volume element) as it flows with the vector field.

On a pseudo-Riemannian manifold, the divergence with respect to the volume can be expressed in terms of the Levi-Civita connection Template:Math:

<math>\operatorname{div} X = \nabla \cdot X = {X^a}_{;a} ,</math>

where the second expression is the contraction of the vector field valued 1-form Template:Math with itself and the last expression is the traditional coordinate expression from Ricci calculus.

An equivalent expression without using a connection is

<math>\operatorname{div}(X) = \frac{1}{\sqrt{\left|\det g \right|}} \, \partial_a \left(\sqrt{\left|\det g \right|} \, X^a\right),</math>

where Template:Mvar is the metric and <math>\partial_a</math> denotes the partial derivative with respect to coordinate Template:Math. The square-root of the (absolute value of the determinant of the) metric appears because the divergence must be written with the correct conception of the volume. 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 <math>X^a</math> can be transformed into "flat space" (where coordinates are actually orthonormal), and once again so that <math>\partial_a</math> 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. 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 <math>\star d\star</math> with <math>d</math> the differential and <math>\star</math> the Hodge star. The Hodge star, by its construction, causes the volume form to appear in all of the right places.

The divergence of tensorsEdit

Divergence can also be generalised to tensors. In Einstein notation, the divergence of a contravariant vector Template:Mvar is given by

<math>\nabla \cdot \mathbf{F} = \nabla_\mu F^\mu ,</math>

where Template:Math 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 by using the musical isomorphism Template:Music: if Template:Math is a Template:Math-tensor (Template:Math for the contravariant vector and Template:Math for the covariant one), then we define the divergence of Template:Mvar to be the Template:Math-tensor

<math>(\operatorname{div} T) (Y_1 , \ldots , Y_{q-1}) = {\operatorname{trace}} \Big(X \mapsto \sharp (\nabla T) (X , \cdot , Y_1 , \ldots , Y_{q-1}) \Big);</math>

that is, we take the trace over the first two covariant indices of the covariant derivative.Template:Efn The <math>\sharp</math> symbol refers to the musical isomorphism.

NotesEdit

Template:Notelist

CitationsEdit

Template:Reflist

ReferencesEdit

{{#invoke:citation/CS1|citation

|CitationClass=web }}

External linksEdit

Template:Sister project

Template:Springer
The idea of divergence of a vector field
Khan Academy: Divergence video lesson
{{#invoke:citation/CS1|citation

|CitationClass=web }}Template:Cbignore

Template:Calculus topics Template:Authority control