Divergence

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

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

Define

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

Then in arbitrary orthogonal Curvilinear Coordinates,

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

$$\nabla\cdot {{\hbox{\sf A}}{\bf x}\over \vert{\bf x}\vert} =... ...f x}^{\rm T}({\hbox{\sf A}}{\bf x})\over \vert{\bf x}\vert^3}.$$ (4)

The divergence of a Tensor $A$ is

$$\nabla\cdot A \equiv A^\alpha_{;\alpha} = A^k_{,k}+\Gamma^k_{jk}A^j,$$ (5)

where $A^\alpha_{;\alpha}$ is the Covariant Derivative and $A^k_{,k}$ is the Comma Derivative. Expanding the terms gives

$$A^\alpha_{;\alpha } = A^\alpha_{,\alpha} + (\Gamma^\alpha_{\alpha \alpha }A^\alpha+\Gamma^\alpha_{\beta\alpha }A^\beta+\Gamma^\alpha_{\gamma\alpha }A^\gamma)$$

$$\phantom{=} \mathop{+} A^\beta _{,\beta }+\left({\Gamma^\beta _{\alpha \beta ... ...amma^\beta _{\beta \beta }A^\beta +\Gamma^\beta _{\gamma\beta }A^\gamma}\right)$$

$$\phantom{=} \mathop{+} A^\gamma_{,\gamma } + \left({\Gamma^\gamma _{\alpha \g... ...\gamma _{\beta \gamma }A^\beta +\Gamma^\gamma _{\gamma\gamma }A^\gamma}\right).$$ (6)

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

References

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