Spherical Coordinates

Summary

Spherical coordinates $(\rho ,\theta ,\varphi )$ locate a point by distance from the origin, azimuth angle in the $xy$ -plane, and polar angle down from the positive $z$ -axis. They are ideal for spheres and cones about the $z$ -axis. The volume element is ${\rho }^2\sin \varphi d\rho d\varphi d\theta$ .

Prerequisites

Polar Coordinates, Cylindrical Coordinates, basic trigonometry

Definition / Notation

Standard calculus convention (as in OpenStax):

Conversion to Cartesian:

Inverse (principal values):

Formula

Volume element:

Geometric Sets

• $\rho =\text{const}$ : sphere centered at the origin.
• $\varphi =\text{const}$ ( $0<\varphi <\pi /2$ ): cone opening from the positive $z$ -axis with opening angle $\varphi$ (not “angle with the $x$ -axis”).
• $\theta =\text{const}$ : half-plane hinged on the $z$ -axis.

Conditions / Assumptions

• $\sin \varphi \ge 0$ on $[0,\pi ]$ , so the Jacobian factor ${\rho }^2\sin \varphi$ is nonnegative.
• Physics texts sometimes swap the names of $\theta$ and $\varphi$ ; keep the conversion formulas consistent with your convention.

Worked Example

Point $(\rho ,\theta ,\varphi )=(3,\pi /4,\pi /3)$ :

Constant $\varphi =\pi /3$ is the cone making angle $\pi /3$ with the positive $z$ -axis.

Common Mistakes

• Defining $\varphi$ as the angle with the $xy$ -plane (that would be $\pi /2-\varphi$ in this convention).
• Forgetting $\sin \varphi$ in $dV$ .
• Allowing $\varphi >\pi$ or negative $\rho$ without a careful identification scheme.

Connections

• Triple integrals: Triple Integrals
• Related systems: Cylindrical Coordinates

References

Spherical coordinates and the Jacobian are standard in OpenStax Calculus Volume 3.^[1]