Nabla Operator

$$\begin{array}{rl}& f(x_1,x_2\dots x_n)\\ & \nabla f=\left(\begin{array}{c}\frac{\delta f}{\delta x_1}\\ \frac{\delta f}{\delta x_2}\\ \dots \\ \frac{\delta f}{\delta x_n}\end{array}\right)\end{array}$$

Cylindrical $(r,\theta ,z)$:

$$\nabla f=\frac{\delta f}{\delta r}{\vec{u}}_r+\frac{1}{r}\frac{\delta f}{\delta \theta }{\vec{u}}_{\theta }+\frac{\delta f}{\delta z}\vec{k}$$

Spherical $(r,\theta ,\varphi )$:

$$\nabla f=\frac{\delta f}{\delta r}{\vec{u}}_r+\frac{1}{r}\frac{\delta f}{\delta \theta }{\vec{u}}_{\theta }+\frac{1}{r\sin \theta }\frac{\delta f}{\delta \varphi }{\vec{u}}_{\varphi }$$

Curl

$$\nabla \times \vec{\mathbf{A}}=\left\{\begin{array}{c}\frac{\partial }{\partial x}\\ \frac{\partial }{\partial y}\\ \frac{\partial }{\partial z}\end{array}\right\}\times \left\{\begin{array}{c}{A}_x\\ A_y\\ A_z\end{array}\right\}=\left\{\begin{array}{c}\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z}\\ \frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x}\\ \frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y}\end{array}\right\}$$

Divergence

$$div(\vec{A})=\nabla \cdot \vec{A}=\frac{\delta A_{x_1}}{\delta x_1}+\frac{\delta A_{x_2}}{\delta x_2}+\cdots +\frac{\delta A_{x_n}}{\delta x_n}$$

Gauss’ Divergence Theorem

$${\oint }_S\vec{A}\cdot {\vec{u}}_{n}dS={\int }_V(\nabla \cdot \vec{A})dV$$

Gauss’ Law (Differential Form)

$$\nabla \cdot \vec{E}=\frac{\rho }{{\epsilon }_0}$$