Divergence

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

$\nabla \cdot \vec{a}={\partial }_i{a}_i$

$(\nabla \cdot \mathbf{A}{)}_j={\partial }_j{A}_{ij}$

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

$\nabla \cdot \vec{F}=\frac{{\partial }_r}{r^2}(r^2{F}_r)+\frac{{\partial }_{\theta }}{r\sin \theta }(F_{\theta }\sin \theta )+\frac{{\partial }_{\varphi }}{r\sin \theta }{F}_{\varphi }$

• Mnemonic for remembering

Properties

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

• $k$ is a constant

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

• Effectively a chain rule
• $f$ 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

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