Divergence

 ( Differential Operators ) 

[[File:Divergence (captions).svg|500px|thumb|upright=1.75|alt= A vector field with diverging vectors, a vector field with converging vectors, and a vector field with parallel vectors that neither diverge nor converge|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: $\backslash nabla\backslash !\backslash cdot(\backslash mathbf\{V\}(x,y))\; =\; \backslash frac\{\backslash partial\backslash ,\; \{V\_x(x,y)\}\}\{\backslash partial\{x\}\}+\backslash frac\{\backslash partial\backslash ,\; \{V\_y(x,y)\}\}\{\backslash partial\{y\}\}$]]

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.

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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 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.

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Definition The divergence of a vector field at a point is defined as the limit of the ratio of the surface integral 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 $\backslash mathbf\{\backslash hat\; n\}$ is the outward normal vector 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. 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.

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Definition in coordinates

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Cartesian coordinates In three-dimensional Cartesian coordinates, the divergence of a continuously differentiable vector field $\backslash mathbf\{F\}\; =\; F\_x\backslash mathbf\{i\}\; +\; F\_y\backslash mathbf\{j\}\; +\; F\_z\backslash mathbf\{k\}$ is defined as the scalar-valued function:

$$\backslash operatorname\{div\}\; \backslash mathbf\{F\}\; =\; \backslash nabla\backslash cdot\backslash mathbf\{F\}\; =\; \backslash left(\backslash frac\{\backslash partial\}\{\backslash partial\; x\},\; \backslash frac\{\backslash partial\}\{\backslash partial\; y\},\; \backslash frac\{\backslash partial\}\{\backslash partial\; z\}\; \backslash right)\; \backslash cdot\; (F\_x,F\_y,F\_z)\; =\; \backslash frac\{\backslash partial\; F\_x\}\{\backslash partial\; x\}+\backslash frac\{\backslash partial\; F\_y\}\{\backslash partial\; y\}+\backslash frac\{\backslash partial\; F\_z\}\{\backslash partial\; z\}.$$

Although expressed in terms of coordinates, the result is invariant under Rotation matrix, as the physical interpretation suggests. This is because the trace of the Jacobian matrix 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: take the components of the operator (see del), 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.

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Cylindrical coordinates For a vector expressed in local unit cylindrical coordinates as $$\backslash mathbf\{F\}\; =\; \backslash mathbf\{e\}\_r\; F\_r\; +\; \backslash mathbf\{e\}\_\backslash theta\; F\_\backslash theta\; +\; \backslash mathbf\{e\}\_z\; F\_z,$$ where is the unit vector in direction , the divergence is $$\backslash operatorname\{div\}\; \backslash mathbf\; F\; =\; \backslash nabla\; \backslash cdot\; \backslash mathbf\{F\}\; =\; \backslash frac\{1\}\{r\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; r\}\; \backslash left(r\; F\_r\backslash right)\; +\; \backslash frac1r\; \backslash frac\{\backslash partial\; F\_\backslash theta\}\{\backslash partial\backslash theta\}\; +\; \backslash frac\{\backslash partial\; F\_z\}\{\backslash partial\; z\}.$$

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 and In particular, if we consider the identity function , we find that:

$$\backslash theta(\backslash mathbf\{F\}(\backslash mathbf\{x\}))\; =\; \backslash theta\; \backslash neq\; F\_\{\backslash theta\}(\backslash mathbf\{x\})\; =\; 0.$$

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Spherical coordinates In spherical coordinates, with the angle with the axis and the rotation around the axis, and again written in local unit coordinates, the divergence is $\backslash operatorname\{div\}\backslash mathbf\{F\}\; =\; \backslash nabla\; \backslash cdot\; \backslash mathbf\{F\}\; =\; \backslash frac\{1\}\{r^2\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; r\}\; \backslash left(r^2\; F\_r\backslash right)\; +\; \backslash frac\{1\}\{r\backslash sin\backslash theta\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; \backslash theta\}\; \backslash left(\backslash sin\backslash theta\backslash ,\; F\_\backslash theta\backslash right)\; +\; \backslash frac\{1\}\{r\; \backslash sin\backslash theta\}\; \backslash frac\{\backslash partial\; F\_\backslash varphi\}\{\backslash partial\; \backslash varphi\}.$

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

the divergence in cartesian coordinate system is a first-order tensor field and can be defined in two ways:

$$\backslash operatorname\{div\}\; (\backslash mathbf\{A\})\; =\; \backslash frac\{\backslash partial\; A\_\{ik\}\}\{\backslash partial\; x\_k\}~\backslash mathbf\{e\}\_i\; =\; A\_\{ik,k\}~\backslash mathbf\{e\}\_i\; =\; \backslash begin\{bmatrix\}\; \backslash dfrac\{\backslash partial\; A\_\{11\}\}\{\backslash partial\; x\_1\}\; +\backslash dfrac\{\backslash partial\; A\_\{12\}\}\{\backslash partial\; x\_2\}\; +\backslash dfrac\{\backslash partial\; A\_\{13\}\}\{\backslash partial\; x\_3\}\; \backslash \backslash \; \backslash dfrac\{\backslash partial\; A\_\{21\}\}\{\backslash partial\; x\_1\}\; +\backslash dfrac\{\backslash partial\; A\_\{22\}\}\{\backslash partial\; x\_2\}\; +\backslash dfrac\{\backslash partial\; A\_\{23\}\}\{\backslash partial\; x\_3\}\; \backslash \backslash \; \backslash dfrac\{\backslash partial\; A\_\{31\}\}\{\backslash partial\; x\_1\}\; +\backslash dfrac\{\backslash partial\; A\_\{32\}\}\{\backslash partial\; x\_2\}\; +\backslash dfrac\{\backslash partial\; A\_\{33\}\}\{\backslash partial\; x\_3\}\; \backslash end\{bmatrix\}$$

and

Tasos C. Papanastasiou (2026). 9780849316067, CRC Press. . ISBN 9780849316067

$$\backslash nabla\; \backslash cdot\; \backslash mathbf\; A\; =\; \backslash frac\{\backslash partial\; A\_\{ki\}\}\{\backslash partial\; x\_k\}\; ~\backslash mathbf\{e\}\_i\; =\; A\_\{ki,k\}~\backslash mathbf\{e\}\_i\; =\; \backslash begin\{bmatrix\}\; \backslash dfrac\{\backslash partial\; A\_\{11\}\}\{\backslash partial\; x\_1\}\; +\; \backslash dfrac\{\backslash partial\; A\_\{21\}\}\{\backslash partial\; x\_2\}\; +\; \backslash dfrac\{\backslash partial\; A\_\{31\}\}\{\backslash partial\; x\_3\}\; \backslash \backslash \; \backslash dfrac\{\backslash partial\; A\_\{12\}\}\{\backslash partial\; x\_1\}\; +\; \backslash dfrac\{\backslash partial\; A\_\{22\}\}\{\backslash partial\; x\_2\}\; +\; \backslash dfrac\{\backslash partial\; A\_\{32\}\}\{\backslash partial\; x\_3\}\; \backslash \backslash \; \backslash dfrac\{\backslash partial\; A\_\{13\}\}\{\backslash partial\; x\_1\}\; +\; \backslash dfrac\{\backslash partial\; A\_\{23\}\}\{\backslash partial\; x\_2\}\; +\; \backslash dfrac\{\backslash partial\; A\_\{33\}\}\{\backslash partial\; x\_3\}\; \backslash \backslash \; \backslash end\{bmatrix\}$$

We have

$$\backslash operatorname\{div\}\; \{\backslash left(\backslash mathbf\{A\}^\backslash mathsf\{T\}\backslash right)\}\; =\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; A$$

If tensor is symmetric then Because of this, often in the literature the two definitions (and symbols and $\backslash nabla\; \backslash cdot$) are used interchangeably (especially in mechanics equations where tensor symmetry is assumed).

Expressions of $\backslash nabla\backslash cdot\backslash mathbf\; A$ in cylindrical and spherical coordinates are given in the article del in cylindrical and spherical coordinates.

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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 Voss-Hermann Weyl formula, which allows the divergence to be determined using simply partial coordinate derivatives, is as follows:

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

where $\backslash rho$ is the local coefficient of the volume element and are the components of $\backslash mathbf\{F\}\; =\; F^i\backslash mathbf\{e\}\_i$ 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 $\backslash rho\; =\; \backslash sqrt\{\backslash left|\backslash det\; g\_\{ab\}\backslash right|\}$, where 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 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 $\backslash hat\{\backslash mathbf\{e\}\}\_i$ for the normalized basis, and $\backslash hat\{F\}^i$ for the components of with respect to it, we have that $$\backslash mathbf\{F\}\; =\; F^i\; \backslash mathbf\{e\}\_i\; =\; F^i\; \backslash |\{\backslash mathbf\{e\}\_i\; \}\backslash |\; \backslash frac\{\backslash mathbf\{e\}\_i\}\{\backslash |\; \backslash mathbf\{e\}\_i\; \backslash |\}\; =\; F^i\; \backslash sqrt\{g\_\{ii\}\}\; \backslash ,\; \backslash hat\{\backslash mathbf\{e\}\}\_i\; =\; \backslash hat\{F\}^i\; \backslash hat\{\backslash mathbf\{e\}\}\_i,$$ using one of the properties of the metric tensor. By dotting both sides of the last equality with the contravariant element ^i,}} we can conclude that $F^i\; =\; \backslash hat\{F\}^i\; /\; \backslash sqrt\{g\_\{ii\}\}$. After substituting, the formula becomes:

$$\backslash operatorname\{div\}(\backslash mathbf\{F\})\; =\; \backslash frac\; 1\{\backslash rho\}\; \backslash frac\{\backslash partial\; \backslash left(\backslash frac\{\backslash rho\}\{\backslash sqrt\{g\_\{ii\}\}\}\backslash hat\{F\}^i\backslash right)\}\{\backslash partial\; x^i\}\; =$$

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

See for further discussion.

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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.,

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

for all vector fields and and all and .

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

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

or in more suggestive notation

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

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

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

or

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

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

$\backslash operatorname\{div\}(\backslash operatorname\{grad\}\backslash varphi)\; =\; \backslash Delta\backslash varphi.$

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

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

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). The degree of failure of the truth of the statement, measured by the homology of the chain complex

$\backslash \{\; \backslash text\{scalar\; fields\; on\; \}\; U\; \backslash \}\; ~\; \backslash overset\{\backslash operatorname\{grad\}\}\{\backslash rarr\}\; ~\; \backslash \{\; \backslash text\{vector\; fields\; on\; \}\; U\; \backslash \}\; ~\; \backslash overset\{\backslash operatorname\{curl\}\}\{\backslash rarr\}\; ~\; \backslash \{\; \backslash text\{vector\; fields\; on\; \}\; U\; \backslash \}\; ~\; \backslash overset\{\backslash operatorname\{div\}\}\{\backslash rarr\}\; ~\; \backslash \{\; \backslash text\{scalar\; fields\; on\; \}\; U\; \backslash \}$

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

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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):

For the irrotational part one has

$\backslash mathbf\; E=-\backslash nabla\; \backslash Phi(\backslash mathbf\; r),$

with

$\backslash Phi\; (\backslash mathbf\{r\})=\backslash int\_\{\backslash mathbb\; R^3\}\backslash ,d^3\backslash mathbf\; r\text{'}\backslash ;\backslash frac\{\backslash operatorname\{div\}\; \backslash mathbf\{v\}(\backslash mathbf\{r\}\text{'})\}\{4\backslash pi\backslash left|\backslash mathbf\{r\}-\backslash mathbf\{r\}\text{'}\backslash right|\}.$

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. It is a special case of the more general Helmholtz decomposition, which works in dimensions greater than three as well.

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In arbitrary finite dimensions The divergence of a vector field can be defined in any finite number $n$ of dimensions. If

$\backslash mathbf\{F\}\; =\; (F\_1\; ,\; F\_2\; ,\; \backslash ldots\; F\_n)\; ,$

in a Euclidean coordinate system with coordinates , define

$\backslash operatorname\{div\}\; \backslash mathbf\{F\}\; =\; \backslash nabla\backslash cdot\backslash mathbf\{F\}\; =\; \backslash frac\{\backslash partial\; F\_1\}\{\backslash partial\; x\_1\}\; +\; \backslash frac\{\backslash partial\; F\_2\}\{\backslash partial\; x\_2\}\; +\; \backslash cdots\; +\; \backslash frac\{\backslash partial\; F\_n\}\{\backslash partial\; x\_n\}.$

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"

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

for any scalar-valued function .

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Relation to 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 . Define the current two-form as

$j\; =\; F\_1\; \backslash ,\; dy\; \backslash wedge\; dz\; +\; F\_2\; \backslash ,\; dz\; \backslash wedge\; dx\; +\; F\_3\; \backslash ,\; dx\; \backslash wedge\; dy\; .$

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

$dj\; =\; \backslash left(\backslash frac\{\backslash partial\; F\_1\}\{\backslash partial\; x\}\; +\backslash frac\{\backslash partial\; F\_2\}\{\backslash partial\; y\}\; +\backslash frac\{\backslash partial\; F\_3\}\{\backslash partial\; z\}\; \backslash right)\; dx\; \backslash wedge\; dy\; \backslash wedge\; dz\; =\; (\backslash nabla\; \backslash cdot\; \{\backslash mathbf\; F\})\; \backslash rho$

where $\backslash wedge$ is the wedge product.

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

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

Here the superscript is one of the two musical isomorphisms, and is the Hodge star operator. When the divergence is written in this way, the operator $\{\backslash star\}\; d\{\backslash star\}$ 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.

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In curvilinear coordinates The appropriate expression is more complicated in curvilinear coordinates. The divergence of a vector field extends naturally to any differentiable manifold of dimension that has a volume form (or density) , e.g. a Riemannian or Lorentzian manifold. Generalising the construction of a two-form 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

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

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

$\{\backslash mathcal\; L\}\_X\; \backslash mu\; =\; (\backslash operatorname\{div\}\; X)\; \backslash mu\; .$

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 :

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

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

$\backslash operatorname\{div\}(X)\; =\; \backslash frac\{1\}\{\backslash sqrt\{\backslash left|\backslash det\; g\; \backslash right|\}\}\; \backslash ,\; \backslash partial\_a\; \backslash left(\backslash sqrt\{\backslash left|\backslash det\; g\; \backslash right|\}\; \backslash ,\; X^a\backslash right),$

where is the metric tensor and $\backslash partial\_a$ denotes the partial derivative with respect to coordinate . 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 $X^a$ can be transformed into "flat space" (where coordinates are actually orthonormal), and once again so that $\backslash partial\_a$ 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. 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 $\backslash star\; d\backslash star$ with $d$ the differential and $\backslash star$ the Hodge star. The Hodge star, by its construction, causes the volume form to appear in all of the right places.

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

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

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

$(\backslash operatorname\{div\}\; T)\; (Y\_1\; ,\; \backslash ldots\; ,\; Y\_\{q-1\})\; =\; \{\backslash operatorname\{trace\}\}\; \backslash Big(X\; \backslash mapsto\; \backslash sharp\; (\backslash nabla\; T)\; (X\; ,\; \backslash cdot\; ,\; Y\_1\; ,\; \backslash ldots\; ,\; Y\_\{q-1\})\; \backslash Big);$

that is, we take the trace over the first two covariant indices of the covariant derivative. The $\backslash sharp$ symbol refers to the musical isomorphism.

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See also

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

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Notes

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Citations

• Walter Rudin (1976). 007054235X, McGraw-Hill. . 007054235X
• C. H. Edwards (1994). 9780486683362, Dover. ISBN 9780486683362
• Morton Gurtin (1981). 9780123097507, Academic Press. ISBN 9780123097507
• (2000). 9780486411477, Dover Publications. ISBN 9780486411477

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External links

• The idea of divergence of a vector field
• Khan Academy: Divergence video lesson

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