Hyperspherical coordinates

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

With polar angle $\theta$ measured from positive $z$ and azimuth $\varphi$, the Cartesian formulas $x=r\sin \theta \cos \varphi$, $y=r\sin \theta \sin \varphi$, $z=r\cos \theta$ have Jacobian determinant $r^2\sin \theta$. For $r\ge 0$ and $0\le \theta \le \pi$ this gives the unsigned volume element shown. The inverse determinant is $1/(r^2\sin \theta )$ off the origin and polar axis.

In the usual physics convention for spherical coordinates, the polar angle is the angle between the positive $z$ axis and the position vector. For $r=\sqrt{x^2+y^2+z^2}>0$, it satisfies $\cos \theta =z/r$ with $0\le \theta \le \pi$. The north and south axes have $\theta =0$ and $\theta =\pi$, respectively; the plane $z=0$ has $\theta =\pi /2$. The angle is undefined at the origin. In plane polar coordinates, the same phrase usually denotes the oriented in-plane angle instead, so the reference axis and convention should always be stated. The azimuth supplies the other angular coordinate in three dimensions.

For radial mass density $\rho (r)=A{r}^{-2}(1-e^{-r/r_0})$ within radius $R>0$, integration against the spherical volume element gives $M=4\pi ARf(r_0/R)$. The dimensionless quantity $q=r_0/R>0$ hasThus $f$ is positive, strictly decreasing and convex, with $f(0^{+})=1$ and $f(\infty )=0$. Its asymptotic forms are $f(q)=1-q+q{e}^{-1/q}$ near zero and $f(q)=1/(2q)-1/(6q^2)+O(q^{-3})$ at infinity. The density's central $1/r$ divergence is integrable against the factor $r^{2}dr$.

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 ${\pi }^{n/2}{R}^n/\Gamma (n/2+1)$.

Let $V_d$ be the volume of an n-ball of unit radius in ${\mathbb{R}}^d$. A radial shell has volume $d{V}_d{r}^{d-1}dr$, by differentiating $V_d{r}^d$. Integrating shells, or using hyperspherical coordinates, therefore reduces a radial integrand over a Euclidean ball to the displayed one-dimensional integral. The constant $d{V}_d$ is the surface area of an n-sphere of unit radius, with sphere dimension $d-1$.

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