Cylindrical and Spherical Coordinates

Extend polar geometry into space

Cylindrical coordinates use $x=r\cos\theta$ , $y=r\sin\theta$ , and unchanged $z$ , with $dV=r\,dr\,d\theta\,dz$ in a compatible order. They suit cylinders and rotation about the vertical axis.

Here spherical coordinates mean $x=\rho\sin\phi\cos\theta$ , $y=\rho\sin\phi\sin\theta$ , $z=\rho\cos\phi$ , where $\rho\ge0$ , $0\le\phi\le\pi$ is measured down from the positive $z$ -axis, and $\theta$ is azimuth. Then $dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta$ . Other books may exchange angle names; the definitions determine the factor.

Visual guide

VISUAL GUIDEA meridian shows the spherical angle convention

Horizontal and vertical components
Angle φ

In a vertical meridian, ρ is distance from the origin and φ is measured down from the positive z axis. The horizontal distance is r = ρ sin φ and height is z = ρ cos φ. Rotation around the z axis supplies the azimuth θ.

Worked example: a paraboloid cap

The solid $0\le z\le4-x^2-y^2$ projects onto $r\le2$ . Cylindrical volume is

\int_0^{2\pi}\int_0^2\int_0^{4-r^2}r\,dz\,dr\,d\theta=8\pi.

The radial factor is from volume scaling, and the upper height is from the paraboloid. Neither can be omitted. Cartesian coordinates describe the same solid with less convenient disk bounds.

Worked example: a ball moment

For a ball of radius $R$ , integrate $x^2+y^2+z^2=\rho^2$ :

\int_0^{2\pi}\int_0^\pi\int_0^R\rho^4\sin\phi\,d\rho\,d\phi\,d\theta=\frac{4\pi R^5}{5}.

The average squared radius is this result divided by volume $4\pi R^3/3$ , giving $3R^2/5$ . Its units are squared length, as expected.

A cone from the origin often gives constant $\phi$ bounds; a sphere centered at the origin gives constant $\rho$ bounds. A shifted sphere may not. Sketch coordinate surfaces and their intersections before choosing an order. The axes and poles have coordinate degeneracies but form zero-volume boundaries in these standard integrals.

Practice

Write spherical bounds for the upper half of a ball of radius two.
Find the volume of a cylinder $r\le2$ , $0\le z\le3$ .
In this convention, which angle equals $\pi/2$ on the horizontal plane away from the origin?

Show worked solutions

$0\le\rho\le2$ , $0\le\phi\le\pi/2$ , $0\le\theta\le2\pi$ .
$\int_0^{2\pi}\int_0^2\int_0^3r dzdrd\theta=12\pi$ .
The polar angle $\phi$ , because $z=\rho\cos\phi=0$ .

Further study

MIT OpenCourseWare: Multivariable Calculus provides a full university course with additional lectures and exercises.