Orthogonal coordinates ($d{s}^2={\sum }_i{h}_i^{2}d{q}_i^2$)

Orthogonal coordinates have pairwise orthogonal coordinate tangent directions. Their metric is diagonal, with scale factors of orthogonal coordinates $h_i=|\partial \mathbf{x}/\partial q_i|$. Spherical coordinates and cylindrical coordinates are examples.

This collects the gradient, divergence, curl and Laplacian formulas in cylindrical and spherical coordinates. Coordinate unit vectors vary with position, so taking their derivatives is essential when translating vector calculus operators from Cartesian coordinates. Curl in spherical coordinates is one of these formulas.

Hyperspherical coordinates extend polar coordinates to $n$ dimensions with one radius and $n-1$ angles.

Spherical coordinates $(r,\theta ,\varphi )$ in ${\mathbb{R}}^3$ use

At fixed angular coordinates, differentiating the Cartesian coordinates with respect to radial distance gives this unit vector. It is the local direction of increasing radius in the spherical coordinate system.

If orthogonal coordinates $q_i$ satisfythen $h_i=|\partial \mathbf{x}/\partial q_i|$ are their scale factors.

These factors determine the diagonal metric and volume element of orthogonal coordinates.

For $\mathbf{A}=A_r{\mathbf{e}}_r+A_{\theta }{\mathbf{e}}_{\theta }+A_{\varphi }{\mathbf{e}}_{\varphi }$, the spherical-coordinate curl follows from the orthogonal-coordinate scale factors $1,r,r\sin \theta$.

Its coordinate expression is part of del in cylindrical and spherical coordinates.

The radius-$R$ ball in ${\mathbb{R}}^n$ has volume ${\pi }^{n/2}{R}^n/\Gamma (n/2+1)$.

The boundary of the radius-$R$ ball has area $2{\pi }^{n/2}{R}^{n-1}/\Gamma (n/2)$.