Divergence

#Math

$\displaystyle \nabla\cdot \vec{F}: \text{vector}\rightarrow \text{scalar}$

$\displaystyle \nabla\cdot \vec{a}=\partial_{i}a_{i}$

$\displaystyle (\nabla \cdot \mathbf{A}){j}=\partial{j}A_{ij}$

For a matrix $\displaystyle \mathbf{A}$
Final dimension of $\displaystyle \nabla \cdot \mathbf{A}$ is $\displaystyle n\times 1$

$\displaystyle \nabla \cdot \vec{F}=\frac{\partial_{r}}{r^{2}}(r^{2}F_{r})+\frac{\partial_{\theta}}{r\sin \theta}(F_{\theta}\sin \theta)+\frac{\partial_{\phi}}{r\sin \theta}F_{\phi}$

Mnemonic for remembering

Properties

$\displaystyle \nabla\cdot(k\vec{a})=k(\nabla\cdot \vec{a})$

$\displaystyle k$ is a constant

$\displaystyle \nabla \cdot (f\vec{a})=f(\nabla \cdot \vec{a})+\vec{a}\cdot (\nabla f)$

Effectively a chain rule
$\displaystyle f$ is a scalar function
The mnemonic to remember is that the value has to be scalar, and it's still high d low, low d high

$\displaystyle \nabla \cdot \left(\frac{\hat{\mathscr{r}}}{\mathscr{r}^{2}}\right)=4\pi \delta^{3}(\vec{\mathscr{r}})$