polar coordinates







A polar coordinate system for Cartesian space ℝ n+1\mathbb{R}^{n+1} is a coordinate system adapted to the decomposition of the complement ℝ n+1βˆ–{0}\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:

ℝ n+1βˆ–{0}≃S n×ℝ >0. \mathbb{R}^{n+1} \setminus \{0\} \;\simeq\; S^n \times \mathbb{R}_{\gt 0} \,.

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

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

where r:ℝ >0→ℝr \colon \mathbb{R}_{\gt 0} \to \mathbb{R} denotes the canonical coordinate function along the radial direction.

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

(1)f(xβ†’)dvol ℝ n+1 =f(r,ΞΈ)(rsin(ΞΈ)) nβˆ’1dvol S nβˆ’1∧rdθ∧dr \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?


See also

Last revised on December 1, 2019 at 14:09:58. See the history of this page for a list of all contributions to it.