Editing Divergence (section)

== Definition in coordinates ==
===Cartesian coordinates===

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 (mathematics)|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 [[Rotation matrix|rotations]], as the physical interpretation suggests. This is because the trace of the [[Jacobian matrix and determinant|Jacobian matrix]] of an {{math|''N''}}-dimensional vector field {{math|'''F'''}} in {{mvar|N}}-dimensional space is invariant under any invertible linear transformation{{clarification needed|date=January 2024|reason=Presumably this means the trace of L^-1 J L, where J is the original Jacobian and L is the invertible linear function R^N to R^N? As opposed to the trace of LJ? It is true that the trace is preserved under the former, but obviously not the latter, e.g. take L = 2I, I the identity matrix.}}.

The common notation for the divergence {{math|∇ · '''F'''}} is a convenient mnemonic, where the dot denotes an operation reminiscent of the [[dot product]]: take the components of the {{math|∇}} operator (see [[del]]), apply them to the corresponding components of {{math|'''F'''}}, and sum the results. Because applying an operator is different from multiplying the components, this is considered an [[abuse of notation]].

=== Cylindrical coordinates ===
For a vector expressed in '''local''' unit [[Cylindrical coordinate system|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 {{math|'''e'''<sub>''a''</sub>}} is the unit vector in direction {{math|''a''}}, the divergence is{{refn|[http://mathworld.wolfram.com/CylindricalCoordinates.html Cylindrical coordinates] at Wolfram Mathworld}}
<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 {{math|'''x'''}} the position vector and the functions {{math|''r''('''x''')}}, {{math|''θ''('''x''')}}, and {{math|''z''('''x''')}}, which assign the corresponding '''global''' cylindrical coordinate to a vector, in general {{nowrap|<math>r(\mathbf{F}(\mathbf{x})) \neq F_r(\mathbf{x})</math>,}} {{nowrap|<math>\theta(\mathbf{F}(\mathbf{x})) \neq F_{\theta}(\mathbf{x})</math>,}} and {{nowrap|<math>z(\mathbf{F}(\mathbf{x})) \neq F_z(\mathbf{x})</math>.}} In particular, if we consider the identity function {{math|1='''F'''('''x''') = '''x'''}}, we find that:

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

=== Spherical coordinates ===
In [[spherical coordinates]], with {{mvar|θ}} the angle with the {{mvar|z}} axis and {{mvar|φ}} the rotation around the {{mvar|z}} axis, and {{math|'''F'''}} again written in local unit coordinates, the divergence is{{refn|[http://mathworld.wolfram.com/SphericalCoordinates.html Spherical coordinates] at Wolfram Mathworld}}
<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 field ===
Let {{math|'''A'''}} 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 field{{sfn|Gurtin|1981|p=30}} and can be defined in two ways:<ref>{{ cite web |title=1.14 Tensor Calculus I: Tensor Fields |work=Foundations of Continuum Mechanics |url=http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_1_Vectors_Tensors/Vectors_Tensors_14_Tensor_Calculus.pdf |archive-url=https://web.archive.org/web/20130108133336/http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_1_Vectors_Tensors/Vectors_Tensors_14_Tensor_Calculus.pdf |archive-date=2013-01-08 | url-status=live }}</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>
{{cite book 
|author=William M. Deen 
|title=Introduction to Chemical Engineering Fluid Mechanics 
|publisher= Cambridge University Press 
|date=2016 
|page=133 
|isbn=978-1-107-12377-9 
|url=https://books.google.com/books?id=H1CeDAAAQBAJ&q=cauchy+momentum+asymmetric&pg=PA146
}}</ref><ref>
{{cite book
|author=Tasos C. Papanastasiou |author2=Georgios C. Georgiou |author3=Andreas N. Alexandrou
|title=Viscous Fluid Flow
|date=2000
|page=66,68 
|publisher=CRC Press 
|isbn=0-8493-1606-5 
|url=https://www.mobt3ath.com/uplode/book/book-46462.pdf |archive-url=https://web.archive.org/web/20200220125401/https://www.mobt3ath.com/uplode/book/book-46462.pdf |archive-date=2020-02-20 |url-status=live 
}}</ref><ref>
{{cite web 
|author=Adam Powell 
|title=The Navier-Stokes Equations 
|date=12 April 2010 | url=http://texmex.mit.edu/pub/emanuel/CLASS/12.340/navier-stokes(2).pdf
}}</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 {{math|1=''A''<sub>''ij''</sub> = ''A''<sub>''ji''</sub>}} then {{nowrap|<math>\operatorname{div} (\mathbf{A}) = \nabla \cdot \mathbf A</math>.}} Because of this, often in the literature the two definitions (and symbols {{math|div}} 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 coordinates ===
Using [[Einstein notation]] we can consider the divergence in [[Curvilinear coordinates|general coordinates]], which we write as {{math|''x''<sup>1</sup>, …, ''x''<sup>''i''</sup>, …, ''x''<sup>''n''</sup>}}, where {{mvar|n}} is the number of dimensions of the domain. Here, the upper index refers to the number of the coordinate or component, so {{math|''x''<sup>2</sup>}} refers to the second component, and not the quantity {{mvar|x}} squared. The index variable {{mvar|i}} is used to refer to an arbitrary component, such as {{math|''x''<sup>''i''</sup>}}. The divergence can then be written via the [https://www.genealogy.math.ndsu.nodak.edu/id.php?id=59087 Voss]-[[Hermann Weyl|Weyl]] formula,<ref>{{cite web|last1=Grinfeld|first1=Pavel|title=The Voss-Weyl Formula (Youtube link)|website=[[YouTube]] |date=16 April 2014 |url=https://www.youtube.com/watch?v=BD2AiFk651E&list=PLlXfTHzgMRULkodlIEqfgTS-H1AY_bNtq&index=23| archive-url=https://ghostarchive.org/varchive/youtube/20211211/BD2AiFk651E| archive-date=2021-12-11 | url-status=live|access-date=9 January 2018|language=en}}{{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 {{math|''F<sup>i</sup>''}} are the components of <math>\mathbf{F} = F^i\mathbf{e}_i</math> with respect to the local '''unnormalized''' [[Curvilinear coordinates#Covariant and contravariant bases|covariant basis]] (sometimes written as {{nowrap|<math>\mathbf{e}_i = \partial\mathbf{x} / \partial x^i</math>).}} The Einstein notation implies summation over {{mvar|i}}, since it appears as both an upper and lower index.

The volume coefficient {{mvar|ρ}} 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 {{math|1=''ρ'' = 1}}, {{math|1=''ρ'' = ''r''}} and {{math|1=''ρ'' = ''r''<sup>2</sup> sin ''θ''}}, respectively. The volume can also be expressed as <math display="inline">\rho = \sqrt{\left|\det g_{ab}\right|}</math>, where {{math|''g<sub>ab</sub>''}} 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 {{nowrap|<math display="inline">\rho = \sqrt{\left|\det g\right|}</math>.}} 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 matrix and determinant|Jacobian]] of the transformation from Cartesian to curvilinear coordinates, which for {{math|1=''n'' = 3}} gives {{nowrap|<math display="inline">\rho = \left| \frac{\partial(x,y,z)}{\partial (x^1,x^2,x^3)}\right|</math>.}}

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 {{math|'''F'''}} 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 {{nowrap|<math>\hat{\mathbf{e}}^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 ''{{section link||In curvilinear coordinates}}'' for further discussion.