Sort: 2D polar coordinate or 3D coordinate system?

Groups: 2D polar · 3D coordinate system

(r, $\theta$ ) with $dA = r$ dr dθ
Useful for circles and disks in ℝ²
( $\rho$ , $\theta$ , $\varphi$ ) with $dV = \rho^{2} \sin \varphi d\rho d\varphi d\theta$
Useful for spheres and cones in ℝ³
$x = r \cos \theta, y = r \sin \theta$
(r, $\theta$ , z) with $dV = r$ dr d $\theta dz$

Hints

Count how many coordinates each system uses.
Two coordinates describe a plane; three describe space.

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2D polar: (r, $\theta$ ) with $dA = r$ dr dθ, $x = r \cos \theta, y = r \sin \theta$ , Useful for circles and disks in ℝ²

3D coordinate system: (r, $\theta$ , z) with $dV = r$ dr d $\theta dz$ , ( $\rho$ , $\theta$ , $\varphi$ ) with $dV = \rho^{2} \sin \varphi d\rho d\varphi d\theta$ , Useful for spheres and cones in ℝ³

Why

Polar coordinates are 2D; cylindrical and spherical are 3D extensions.

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Sort: 2D polar coordinate or 3D coordinate system?

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