Divergence

Illustration of the divergence of a vector field. This is a field of speed convergent on the left and divergent on the right.

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 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 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 flow 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,

\left.\operatorname {div} \mathbf {F} \right|_{p}=\lim _{V\rightarrow \{p\}}\iint _{S(V)}{\frac {\mathbf {F} \cdot \mathbf {\hat {n}} }{|V|}}\,dS,

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 flow across it.

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

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 $F = U i + V j + W k$ is defined as the scalar-valued function:^[1]^[2]

\operatorname {div} \mathbf {F} =\nabla \cdot \mathbf {F} =\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)\cdot (U,V,W)={\frac {\partial U}{\partial x}}+{\frac {\partial V}{\partial y}}+{\frac {\partial W}{\partial z}}.

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 $\nabla \cdot F$ is a convenient mnemonic, where the dot denotes an operation reminiscent of the dot product: take the components of the $\nabla$ 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.

The divergence of a continuously differentiable second-order tensor field $ε$ is a first-order tensor field:^[3]

{\overrightarrow {\operatorname {div} }}(\mathbf {\varepsilon } )={\begin{bmatrix}{\dfrac {\partial \varepsilon _{xx}}{\partial x}}+{\dfrac {\partial \varepsilon _{yx}}{\partial y}}+{\dfrac {\partial \varepsilon _{zx}}{\partial z}}\\{\dfrac {\partial \varepsilon _{xy}}{\partial x}}+{\dfrac {\partial \varepsilon _{yy}}{\partial y}}+{\dfrac {\partial \varepsilon _{zy}}{\partial z}}\\{\dfrac {\partial \varepsilon _{xz}}{\partial x}}+{\dfrac {\partial \varepsilon _{yz}}{\partial y}}+{\dfrac {\partial \varepsilon _{zz}}{\partial z}}\end{bmatrix}}.

Cylindrical coordinates

For a vector expressed in local unit cylindrical coordinates as

\mathbf {F} =\mathbf {e} _{r}F_{r}+\mathbf {e} _{\theta }F_{\theta }+\mathbf {e} _{z}F_{z},

where $e a$ is the unit vector in direction $a$ , the divergence is^[4]

\operatorname {div} \mathbf {F} =\nabla \cdot \mathbf {F} ={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(rF_{r}\right)+{\frac {1}{r}}{\frac {\partial F_{\theta }}{\partial \theta }}+{\frac {\partial F_{z}}{\partial z}}.

The use of local coordinates is vital for the validity of the expression. If we consider $x$ the position vector and the functions $r(\mathbf {x} )$ , $\theta (\mathbf {x} )$ , and $z(\mathbf {x} )$ , which assign the corresponding global cylindrical coordinate to a vector, in general $r(\mathbf {F} (\mathbf {x} ))\neq F_{r}(\mathbf {x} )$ , $\theta (\mathbf {F} (\mathbf {x} ))\neq F_{\theta }(\mathbf {x} )$ , and $z(\mathbf {F} (\mathbf {x} ))\neq F_{z}(\mathbf {x} )$ . In particular, if we consider the identity function $\mathbf {F} (\mathbf {x} )=\mathbf {x}$ , we find that:

\theta (\mathbf {F} (\mathbf {x} ))=\theta \neq F_{\theta }(\mathbf {x} )=0

.

Spherical coordinates

In spherical coordinates, with $θ$ the angle with the $z$ axis and $φ$ the rotation around the $z$ axis, and $\mathbf {F}$ again written in local unit coordinates, the divergence is^[5]

\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 }}(\sin \theta \,F_{\theta })+{\frac {1}{r\sin \theta }}{\frac {\partial F_{\varphi }}{\partial \varphi }}.

General coordinates

Using Einstein notation we can consider the divergence in general coordinates, which we write as $x 1, ..., x i, ..., x n$ , where $n$ is the number of dimensions of the domain. Here, the upper index refers to the number of the coordinate or component, so $x 2$ 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 $x i$ . The divergence can then be written via the Voss- Weyl formula^[6], as:

\operatorname {div} (\mathbf {F} )={\frac {1}{\rho }}{\frac {\partial \left(\rho \,F^{i}\right)}{\partial x^{i}}},

where $\rho$ is the local coefficient of the volume element and $F i$ are the components of $F$ with respect to the local unnormalized covariant basis (sometimes written as $\mathbf {e} _{i}=\partial \mathbf {x} /\partial x^{i}$ ). The Einstein notation implies summation over $i$ , since it appears as both an upper and lower index.

The volume coefficient $\rho$ is a function of position which depends on the coordinate system. In Cartesian, cylindrical and polar coordinates, $\rho =1,r,$ and $r^{2}\sin {\theta }$ respectively, using the same conventions as above. It can also be expressed as $\rho ={\sqrt {\operatorname {det} g_{ab}}}$ , where $g_{ab}$ 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 $\rho ={\sqrt {\operatorname {det} g}}$ . Another expression comes from computing the determinant of the Jacobian for transforming from Cartesian coordinates, which for $n = 3$ gives $\rho =\left|{\frac {\partial (x,y,z)}{\partial (x^{1},x^{2},x^{3})}}\right|.$

Some conventions expect all local basis elements to be normalized to unit length, as was done in the previous sections. If we write ${\hat {\mathbf {e} }}_{i}$ for the normalized basis, and ${\hat {F}}^{i}$ for the components of $F$ with respect to it, we have that

\mathbf {F} =F^{i}\mathbf {e} _{i}=F^{i}{\lVert {\mathbf {e} _{i}}\rVert }{\frac {\mathbf {e} _{i}}{\lVert {\mathbf {e} _{i}}\rVert }}=F^{i}{\sqrt {g_{ii}}}\,{\hat {\mathbf {e} }}_{i}={\hat {F}}^{i}{\hat {\mathbf {e} }}_{i},

using one of the properties of the metric tensor. By dotting both sides of the last equality with the contravariant element ${\hat {\mathbf {e} }}^{i}$ , we can conclude that $F^{i}={\hat {F}}^{i}/{\sqrt {g_{ii}}}$ . After substituting, the formula becomes:

\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 {\operatorname {det} g}}}{\frac {\partial \left({\sqrt {\frac {\operatorname {det} g}{g_{ii}}}}\,{\hat {F}}^{i}\right)}{\partial x^{i}}}

.

See § Generalizations for further discussion.

Decomposition theorem

It can be shown that any stationary flux $v (r)$ that is at least twice continuously differentiable in $R 3$ and vanishes sufficiently fast for $| r | \to \infty$ 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

\mathbf {E} =-\nabla \Phi (\mathbf {r} ),

with

\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|}}.

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 $-\nablaΦ$ by $+\nabla \times A$ , and the source density $div v$ by the circulation density $\nabla \times v$ .

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.

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.,

\operatorname {div} (a\mathbf {F} +b\mathbf {G} )=a\operatorname {div} \mathbf {F} +b\operatorname {div} \mathbf {G}

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

\operatorname {div} (\varphi \mathbf {F} )=\operatorname {grad} \varphi \cdot \mathbf {F} +\varphi \operatorname {div} \mathbf {F} ,

or in more suggestive notation

\nabla \cdot (\varphi \mathbf {F} )=(\nabla \varphi )\cdot \mathbf {F} +\varphi (\nabla \cdot \mathbf {F} ).

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

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

or

\nabla \cdot (\mathbf {F} \times \mathbf {G} )=(\nabla \times \mathbf {F} )\cdot \mathbf {G} -\mathbf {F} \cdot (\nabla \times \mathbf {G} ).

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

\operatorname {div} (\nabla \varphi )=\Delta \varphi .

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

\nabla \cdot (\nabla \times \mathbf {F} )=0.

If a vector field $F$ with zero divergence is defined on a ball in $R 3$ , then there exists some vector field $G$ on the ball with $F = curl G$ . For regions in $R 3$ 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

\{{\text{scalar fields on }}U\}~{\overset {\operatorname {grad} }{\rightarrow }}~\{{\text{vector fields on }}U\}~{\overset {\operatorname {curl} }{\rightarrow }}~\{{\text{vector fields on }}U\}~{\overset {\operatorname {div} }{\rightarrow }}~\{{\text{scalar fields on }}U\}

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 $R 3$ . 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 $ρ = 1 dx \land dy \land dz$ moving with local velocity $F$ . Its exterior derivative $dj$ is then given by

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 .

Thus, the divergence of the vector field $F$ can be expressed as:

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

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.

Generalizations

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

\mathbf {F} =(F_{1},F_{2},\ldots F_{n}),

in a Euclidean coordinate system with coordinates $x 1, x 2, ..., x n$ , define

\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}}}.

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"

\nabla \cdot (\varphi \mathbf {F} )=(\nabla \varphi )\cdot \mathbf {F} +\varphi (\nabla \cdot \mathbf {F} )

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 $R 3$ , on such a manifold a vector field $X$ defines an $(n - 1)$ -form $j = i X μ$ obtained by contracting $X$ with $μ$ . The divergence is then the function defined by

dj=(\operatorname {div} X)\mu .

Standard formulas for the Lie derivative allow us to reformulate this as

{\mathcal {L}}_{X}\mu =(\operatorname {div} X)\mu .

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

On a pseudo-Riemannian manifold, the divergence with respect to the metric volume form can be computed in terms of the Levi-Civita connection $\nabla$ :

\operatorname {div} X=\nabla \cdot X={X^{a}}_{;a},

where the second expression is the contraction of the vector field valued 1-form $\nabla X$ with itself and the last expression is the traditional coordinate expression from Ricci calculus.

An equivalent expression without using connection is

\operatorname {div} (X)={\frac {1}{\sqrt {\operatorname {det} g}}}\partial _{a}\left({\sqrt {\operatorname {det} g}}\,X^{a}\right),

where $g$ is the metric and $\partial a$ denotes the partial derivative with respect to coordinate $x a$ .

Divergence can also be generalised to tensors. In Einstein notation, the divergence of a contravariant vector $F μ$ is given by

\nabla \cdot \mathbf {F} =\nabla _{\mu }F^{\mu },

where $\nabla μ$ 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

(\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 )};

that is, we take the trace over the first two covariant indices of the covariant derivative^{[lower-alpha 1]}

Notes

↑ 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 $V \times V \times ... \times V \to R$ . 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.

Citations

↑ Rudin 1976, p. 281.
↑ Edwards 1994, p. 309.
↑ Gurtin 1981, p. 30.
↑ Cylindrical coordinates at Wolfram Mathworld
↑ Spherical coordinates at Wolfram Mathworld
↑ Grinfeld, Pavel. "The Voss-Weyl Formula". Retrieved 9 January 2018.

References

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. ISBN 0-07-054235-X.
Edwards, C. H. (1994). Advanced Calculus of Several Variables. Mineola, NY: Dover. ISBN 0-486-68336-2.
Gurtin, Morton (1981). An Introduction to Continuum Mechanics. Academic Press. ISBN 0-12-309750-9.
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.

External links

Hazewinkel, Michiel, ed. (2001) [1994], "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.

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[7] 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 $V \times V \times ... \times V \to R$ . 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.

[FOOTNOTERudin1976p._281-1] Rudin 1976, p. 281.

[FOOTNOTEEdwards1994p._309-2] Edwards 1994, p. 309.

[FOOTNOTEGurtin1981p._30-3] Gurtin 1981, p. 30.

[4] Cylindrical coordinates at Wolfram Mathworld

[5] Spherical coordinates at Wolfram Mathworld

[6] Grinfeld, Pavel. "The Voss-Weyl Formula". Retrieved 9 January 2018.