Laplacian

The Laplacian ∇2\nabla^{2} is a differential operator defined as the Divergence of the Gradient of a scalar-valued function in Euclidean space. The canonical form is defined in Cartesian coordinates and, given a twice differentiable function f:R3→Rf:\mathbb{R}^{3}\to\mathbb{R}, is:

∇2f=∇⋅∇f=(∂2∂x2+∂2∂y2+∂2∂z2)f\nabla^{2}f=\nabla\cdot\nabla f=\left(\frac{\partial ^{2}}{\partial x^{2}} + \frac{\partial ^{2}}{\partial y^{2}} + \frac{\partial ^{2}}{\partial z^{2}}\right)f

which is the sum of all non-mixed second derivatives of the function ff. In an arbitrary dimensional space of (finite) dimension NN we have

∇2f=∑i=1N∂2f∂xi2\nabla^{2}f=\sum\limits_{i=1}^{N}\frac{\partial ^{2}f}{\partial x_{i}^{2}}

Polar coordinates

The Laplacian in polar coordinates (r,θ)(r,\theta) is

∇2f=∂2f∂r2+1r∂f∂r+1r2∂2f∂θ2\nabla^{2}f=\frac{\partial ^{2}f}{\partial r^{2}} + \frac{1}{r}\frac{\partial f}{\partial r}+ \frac{1}{r^{2}}\frac{\partial ^{2}f}{\partial \theta^{2}}

with rr the distance from the center and θ\theta the angle of rotation.

Cylindrical coordinates

The Laplacian in cylindrical coordinates (r,φ,z)(r,\varphi,z) is

∇2f=1r∂∂r(r∂f∂r)+1r2∂2f∂φ2+∂2f∂z2\nabla^{2}f=\frac{1}{r}\frac{\partial }{\partial r}\left(r \frac{\partial f}{\partial r}\right)+ \frac{1}{r^{2}}\frac{\partial ^{2}f}{\partial \varphi^{2}}+ \frac{\partial ^{2}f}{\partial z^{2}}

with rr the radial distance, φ\varphi the azimuthal angle and zz the height.

Spherical coordinates

The Laplacian in spherical polar coordinates (r,θ,ϕ)(r,\theta,\phi) is

∇2f=1r2∂∂r(r2∂f∂r)+1r2sin⁡θ∂∂θ(sin⁡θ∂f∂θ)+1r2sin⁡2θ(∂2f∂ϕ2)\nabla^{2}f=\frac{1}{r^{2}}\frac{\partial }{\partial r}\left(r^{2}\frac{\partial f}{\partial r}\right)+ \frac{1}{r^{2}\sin\theta}\frac{\partial }{\partial \theta}\left(\sin\theta \frac{\partial f}{\partial \theta}\right)+ \frac{1}{r^{2}\sin^{2}\theta}\left(\frac{\partial ^{2}f}{\partial \phi^{2}}\right)

with rr the radial distance, θ\theta the azimuthal angle and ϕ\phi the zenith angle.