Editing Divergence (section)

=== 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]].