PX275 - C2b - examples of using the laplacian

laplace's equation

\nabla^{2} ϕ = 0

where, $ϕ$ is a scalar

heat transfer in a steady state

poisson's equation

\nabla^{2} ϕ = k

where, $ϕ$ is a scalar, and $k$ is a constant

potential field from a charge/mass
wave equation:

\nabla^{2} ϕ = \frac{1}{c^{2}} \frac{\partial^{2} ϕ}{\partial t^{2}}

where, $ϕ$ is a scalar, and $c =$ phase speed

it can be rearranged and written as $◻ ϕ = 0 :$ d'alembert's equation with applications in electrodynamics

on vectors

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

in cartesians (only):

\begin{aligned} \nabla^{2} \vec{E} & = (\nabla^{2} E_{x}) \hat{i} + (\nabla^{2} E_{y}) \hat{j} + (\nabla^{2} E_{z}) \hat{k} \\ \nabla^{2} E_{x} & = \frac{\partial^{2} E_{x}}{\partial x^{2}} + \frac{\partial^{2} E_{x}}{\partial y^{2}} + \frac{\partial^{2} E_{x}}{\partial z^{2}} \\ \nabla^{2} E_{y} & = \frac{\partial^{2} E_{y}}{\partial x^{2}} + \frac{\partial^{2} E_{y}}{\partial y^{2}} + \frac{\partial^{2} E_{y}}{\partial z^{2}} \\ \nabla^{2} E_{z} & = \frac{\partial^{2} E_{z}}{\partial x^{2}} + \frac{\partial^{2} E_{z}}{\partial y^{2}} + \frac{\partial^{2} E_{z}}{\partial z^{2}} \end{aligned}