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

Takes a scalar field and gives a scalar output

In Cartesian

$$ \nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2} $$

The Laplace Equation is a common form in which the Laplacian appears

$$ \nabla^2 V=0 $$

→ source-free regions: the electric potential given a charge density of $0$ or the gravitational potential given a mass density of $0$

The Poisson Equation is the general case for $\nabla^2 V\ne0$

Schrödinger Equation

$$ -\frac{\hbar^2}{2m}\nabla^2\psi+U\psi=i\hbar\frac{\partial\psi}{\partial t} $$

Wave Equation

$$ \frac{\partial^2\phi}{\partial t^2}-v^2\nabla^2\phi=0 $$

Helmholtz Equation

$$ \nabla^2\phi+k^2\phi=0 $$

Heat/Diffusion Equation

$$ \frac{\partial\phi}{\partial t}-D\nabla^2\phi=0 $$

The Speed of Light

Consider a vacuum where $\rho=0,\vec J=0$

Then by Faraday’s law