Contents

# Contents

## 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 $r \colon \mathbb{R}_{\gt 0} \to \mathbb{R}$ 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} dvol_{S^{n-1}} \,\wedge r d\theta\, \wedge d r \end{aligned}

sign correct?