spherical coordinates

Spherical coordinates locate a point in space the way you might describe a star: how far away it is, and which two angles you must turn through to point at it. The triple (rho, phi, theta) gives the straight-line distance rho from the origin, the polar angle phi measured down from the positive z-axis (the colatitude, from 0 at the north pole to pi at the south), and the azimuthal angle theta swept around the z-axis exactly as in polar coordinates. A sphere of radius a, the bane of Cartesian integration, becomes the single equation rho = a.

The conversion (in the physics convention used here) is x = rho sin(phi) cos(theta), y = rho sin(phi) sin(theta), z = rho cos(phi). The volume element takes work to derive but is worth memorizing: dV = rho^2 sin(phi) d-rho d-phi d-theta. You can see where each factor comes from. A small spherical box has a radial edge of length d-rho; a north-south edge along the colatitude of length rho d-phi; and an east-west edge that is a circle of latitude of radius rho sin(phi), so its length is rho sin(phi) d-theta. Multiply the three: rho^2 sin(phi) d-rho d-phi d-theta. That rho^2 sin(phi) is the Jacobian of the spherical map, and like the polar r it is fatal to forget.

Spherical coordinates are indispensable wherever the geometry or the physics is centered on a point: the gravitational field of a planet, the electric field of a point charge, the hydrogen atom's wavefunctions, radiation from an antenna, the volume and moment of inertia of a ball. The Laplacian in spherical coordinates is the starting point of separation of variables for the wave, heat, and Laplace equations on spheres, leading to Legendre polynomials and spherical harmonics. The honest warning is that conventions clash: many mathematics texts swap the meanings (and even the letters) of phi and theta, so always state which angle is the colatitude before trusting a formula.