$$ \nabla^2 = \vec \nabla\cdot\vec \nabla $$

• We can simplify this operator to

$$ \nabla^2=\left( \frac{\partial}{\partial x}\,\hat i+ \frac{\partial}{\partial y}\,\hat j\frac{\partial}{\partial z}\,\hat k+\right)\cdot\left( \frac{\partial}{\partial x}\,\hat i+ \frac{\partial}{\partial y}\,\hat j+\frac{\partial}{\partial z}\,\hat k\right)=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2} $$

• For a vector field $\vec A=A_x\,\hat i+A_y\,\hat j+A_z\,\hat k$

$$ \nabla^2\vec A=(\vec\nabla^2 A_x)\,\hat i+(\vec\nabla^2 A_y)\,\hat j+(\vec\nabla^2 A_z)\,\hat k $$

Vector Identities

$$ \begin{aligned} \text{📐 Grad:}&\quad\left\{\;\begin{aligned} \vec\nabla(\psi+\phi)&=\vec\nabla\psi+\vec\nabla\phi \\ \vec\nabla(K\psi)&=K\nabla\psi \\ \vec\nabla(\psi\phi)&=\psi\vec\nabla\phi+\phi\vec\nabla\psi \end{aligned}\right. \\ \; \\ \text{👽 Div:}&\quad\left\{\;\begin{aligned} \vec\nabla\cdot(\vec A+\vec B)&=\vec\nabla\cdot \vec A+\vec\nabla\cdot \vec B \\ \vec\nabla\cdot(\phi\vec A)&=(\vec\nabla\phi)\cdot\vec A+\phi(\vec\nabla\cdot \vec A) \\ \vec\nabla\cdot(\vec A\times \vec B)&=\vec B\cdot(\vec\nabla\times\vec A)-\vec A\cdot(\vec\nabla\times\vec B) \end{aligned}\right. \\ \; \\ \text{🎈 Curl:}&\quad\left\{\;\begin{aligned} \vec\nabla\times(\vec A+\vec B)&=\vec\nabla\times \vec A+\vec\nabla\times \vec B \\ \vec\nabla\times(\phi\vec A)&=\phi(\vec\nabla\times \vec A)+(\vec\nabla\phi)\times \vec A \\ \vec\nabla\times(\vec A\times \vec B)&=\vec A(\vec\nabla\cdot \vec B)-\vec B(\vec\nabla\cdot\vec A)+(\vec B\cdot \vec\nabla)\vec A-(\vec A\cdot\vec\nabla)\vec B \end{aligned}\right. \\ \; \\ \text{🦜 Comb:}&\quad\left\{\;\begin{aligned} \vec\nabla\times\vec\nabla\phi&=0 \\ \vec\nabla\cdot(\vec\nabla\times \vec A)&=0 \\ \vec\nabla\times(\vec\nabla\times\vec A)&=\vec\nabla(\vec\nabla\cdot\vec A)-\vec\nabla^2\vec A \end{aligned}\right. \end{aligned} $$

Examples

• Maxwell’s equations:

$$ \begin{aligned} \text{Gauss' Law}\;\,\quad\vec\nabla\cdot \vec E&=\frac{\rho}{\epsilon_0} \\ \text{No magnetic monopoles}\;\,\quad \vec\nabla\cdot\vec B&=0 \\ \text{Faraday's Law} \quad\vec\nabla\times\vec E&=-\frac{\partial \vec B}{\partial t} \\ \text{Ampere's Law}\quad\vec\nabla\times\vec B&=\mu_0\vec J+\mu_0\epsilon_0\,\frac{\partial\vec E}{\partial t} \end{aligned} $$

• Wave equation for electromagnetic waves

$$ \nabla^2\vec E=\frac{1}{c^2}\frac{\partial^2 \vec E}{\partial t^2} $$

• Poisson’s equation

$$ \nabla^2V=-\frac{\rho}{\epsilon_0} $$

• generalisation of the Schrödinger equation with potential $V$

$$ i\hbar\frac{\partial\psi}{\partial t}=\left(-\frac{-\hbar^2}{2m}\vec\nabla^2+V\right)\psi $$

Operators in Cylindrical polar coordinates

$$ \vec A=A_r\,\hat r+A_\theta\,\hat \theta+A_z\,\hat z $$

• we get the following expressions

$$ \begin{aligned} \text{div}(\vec A)&=\vec\nabla\cdot \vec A=\frac{1}{r}\frac{\partial}{\partial r}(rA_r)+\frac{1}{r}\frac{\partial}{\partial\theta}(A_\theta)+\frac{\partial}{\partial z}(A_z) \\ \text{curl}(\vec A)&=\vec\nabla\times\vec A=\frac{1}{r}\begin{vmatrix} \hat r & r\,\hat\theta & \hat k \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial \theta} & \frac{\partial}{\partial z} \\ A_r & r A_\theta & A_z \end{vmatrix} \\ \vec\nabla^2\psi&=\frac{1}{r}\frac{\partial}{\partial r}\left( r\frac{\partial \psi}{\partial r} \right)+\frac{1}{r^2}\frac{\partial^2\psi}{\partial\theta^2}+\frac{\partial^2\psi}{\partial z^2} \end{aligned} $$