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The divergence of a Vector Field ${\bf F}$ is given by

{\rm div}({\bf F})\equiv\nabla\cdot{\bf F} \equiv \lim_{V\to 0} {\oint_S {\bf F}\cdot d{\bf a}\over V}.
\end{displaymath} (1)

{\bf F}\equiv F_1\hat {\bf u}_1+F_2\hat {\bf u}_2+F_3\hat {\bf u}_3.
\end{displaymath} (2)

Then in arbitrary orthogonal Curvilinear Coordinates,

{\rm div}(F) \equiv \nabla \cdot {\bf F} \equiv {1\over h_1h...
...h_3h_1F_2) + {\partial \over \partial u_3}(h_1h_2F_3)}\right].
\end{displaymath} (3)

If $\nabla \cdot {\bf F} = 0$, then the field is said to be a Divergenceless Field. For divergence in individual coordinate systems, see Curvilinear Coordinates.
\nabla\cdot {{\hbox{\sf A}}{\bf x}\over \vert{\bf x}\vert} =...
...f x}^{\rm T}({\hbox{\sf A}}{\bf x})\over \vert{\bf x}\vert^3}.
\end{displaymath} (4)

The divergence of a Tensor $A$ is
\nabla\cdot A \equiv A^\alpha_{;\alpha} = A^k_{,k}+\Gamma^k_{jk}A^j,
\end{displaymath} (5)

where $A^\alpha_{;\alpha}$ is the Covariant Derivative and $A^k_{,k}$ is the Comma Derivative. Expanding the terms gives
$\displaystyle A^\alpha_{;\alpha }$ $\textstyle =$ $\displaystyle A^\alpha_{,\alpha}
+ (\Gamma^\alpha_{\alpha \alpha }A^\alpha+\Gamma^\alpha_{\beta\alpha }A^\beta+\Gamma^\alpha_{\gamma\alpha }A^\gamma)$  
  $\textstyle \phantom{=}$ $\displaystyle \mathop{+} A^\beta _{,\beta }+\left({\Gamma^\beta _{\alpha \beta ...
...amma^\beta _{\beta \beta }A^\beta
+\Gamma^\beta _{\gamma\beta }A^\gamma}\right)$  
  $\textstyle \phantom{=}$ $\displaystyle \mathop{+} A^\gamma_{,\gamma }
+ \left({\Gamma^\gamma _{\alpha \g...
...\gamma _{\beta \gamma }A^\beta
+\Gamma^\gamma _{\gamma\gamma }A^\gamma}\right).$ (6)

See also Curl, Curl Theorem, Gradient, Green's Theorem, Divergence Theorem, Vector Derivative


Arfken, G. ``Divergence, $\nabla\cdot$.'' §1.7 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 37-42, 1985.

© 1996-9 Eric W. Weisstein