nLab polar coordinates

Idea

A polar coordinate system for Cartesian space $\mathbb{R}^{n+1}$ is a coordinate system adapted to the decomposition of the complement $\mathbb{R}^{n+1} \setminus \{0\}$ of the origin as a Cartesian product of the unit n-sphere times the positive real numbers inside the real line:

$$\mathbb{R}^{n+1} \setminus \{0\} \;\simeq\; S^n \times \mathbb{R}_{\gt 0} \,.$$

If $dvol_{S^n} \in \Omega^n(S^n)$ denotes the standard volume form on the unit n-sphere, then the standard volume form $dvol_{\mathbb{R}^{n+1}}$ of Cartesian space in polar coordinates is

$$dvol_{\mathbb{R}^{n+1}} \;=\; r^n\, d r \wedge dvol_{S^n} \,,$$

where $r \colon \mathbb{R}^{n+1} \to \mathbb{R}_{\gt 0}$ denotes the canonical coordinate function along the radial direction.

If a smooth function $\mathbb{R}^{n+1} \to \mathbb{R}$ depends at most on the radius coordinate $r$ and the angle $\theta$ of vectors in $\mathbb{R}^{n+1}$ to any fixed line through the origin, then

(1)$$\begin{aligned} f(\vec x)\, dvol_{\mathbb{R}^{n+1}} & = f(r,\theta) \,\, (r \sin(\theta))^{n-1}\, r d \theta \wedge dvol_{S^{n-1}} \,\wedge d r \end{aligned}$$

Spherical coordinates

One particular polar coordinate system on $\mathbb{R}^{n+1}$ goes by the name of spherical coordinates. This may be defined by induction (on $n$) as follows:

For $n=1$, we use the essentially unique polar coordinate system on the plane $\mathbb{R}^2$, that is $(r,\theta_1)$, where $r$ is the distance from the origin and $\theta_1$ is the signed angle from the positive first-coordinate axis, with orientation pointing towards the positive second-coordinate axis.

For $n=k+1$, we use $(r,\theta_2,\ldots,\theta_n,\theta_1)$, where $r$ gives the distance from the origin, $\theta_1,\ldots,\theta_{n-1}$ are given by projection onto $\mathbb{R}^n$ along the last coordinate axis, and $\theta_n$ is the unsigned angle from the last coordinate axis.

Notice that $\theta_1$ is treated differently from the other angles. For $k \gt 1$, we have $0 \leq \theta_k \leq \pi$, but we have $0 \leq \theta_1 \lt 2 \pi$ instead (or sometimes people use $-\pi \lt \theta_1 \leq \pi$). For $k \gt 1$, $\theta_k$ is determined relative to the $(k+1)$-th coordinate axis (if we number these from $1$ to $n+1$), but $\theta_1$ is determined relative to the first coordinate axis. To get the standard orientation on $\mathbb{R}^n$, $\theta_1$ also has to come after all of the other angles. And the system doesn't work at all for $\mathbb{R}^1$ ($n = 0$).

(The axis to measure spherical coordinates from is a matter of historical convention, and sometimes people use different conventions. But a polar coordinate system for $\mathbb{R}^1$ is technically impossible, since one coordinate must be the distance from the origin, which is not enough information on its own, and yet there isn't room for another coordinate. In terms of the decomposition $\mathbb{R}^1 \setminus \{0\} \simeq S^0 \times \mathbb{R}_{\gt0}$, the problem is that $S^0$ is a disconnected discrete space and so has no global coordinate system, even allowing for degeneracy.)

Related concepts

• coordinates

References

See also

• Wikipedia, Polar coordinate system

• Wikipedia, Spherical coordinate system