Spherical Coordinates

Three numbers locate any point in space

Spherical coordinates replace the three Cartesian numbers (x, y, z) with three numbers of a more geometric character:

• ρ (rho) — the radial distance from the origin. Always non-negative.
• θ (theta) — the azimuthal angle, measured counter-clockwise around the z-axis from the positive x-axis. Range: 0 to 2π.
• φ (phi) — the polar angle (also called colatitude or zenith angle), measured from the positive z-axis. Range: 0 to π. φ = 0 is the north pole; φ = π/2 is the equator; φ = π is the south pole.

The relationship to Cartesian coordinates is best read off a sketch. Drop a perpendicular from the point P to the xy-plane, hitting it at P'. The segment OP' has length ρ sin(φ) — the radius of the latitude circle. Its x and y components are ρ sin(φ) cos(θ) and ρ sin(φ) sin(θ). The vertical lift OP'P has height ρ cos(φ). So:

x = ρ sin(φ) cos(θ)
y = ρ sin(φ) sin(θ)
z = ρ cos(φ)

Inversely:

ρ = √(x² + y² + z²)
θ = atan2(y, x)
φ = arccos(z / ρ)

(The function atan2 returns θ in the right quadrant, unlike plain arctan, which has a 180° ambiguity.)

Deriving the volume element ρ²sin(φ) dρ dθ dφ

An infinitesimal cell in spherical coordinates is bounded by surfaces of constant ρ, θ, and φ. Pictured: a curved brick whose three sides are arcs of three different radii.

• The radial side has length dρ — this one is just a straight segment.
• The polar (north-south) side is an arc of the great circle of radius ρ, subtending angle dφ. Length: ρ dφ.
• The azimuthal (east-west) side is an arc of the latitude circle of radius ρ sin(φ), subtending angle dθ. Length: ρ sin(φ) dθ.

The infinitesimal volume is the product:

dV = dρ · ρ dφ · ρ sin(φ) dθ = ρ² sin(φ) dρ dθ dφ

The same result follows mechanically from the Jacobian determinant. With (u₁, u₂, u₃) = (ρ, θ, φ),

J = ∂(x, y, z)/∂(ρ, θ, φ) =
    | sin φ cos θ    −ρ sin φ sin θ    ρ cos φ cos θ |
    | sin φ sin θ     ρ sin φ cos θ    ρ cos φ sin θ |
    | cos φ                0          −ρ sin φ        |

|det J| = ρ² sin(φ)

Expanding the determinant along the third row and simplifying with cos²θ + sin²θ = 1 gives ρ² sin(φ). Either derivation is valid; the geometric one is more memorable.

Worked example — the volume of a ball

The volume of a solid ball of radius R is the integral of dV over the region 0 ≤ ρ ≤ R, 0 ≤ θ < 2π, 0 ≤ φ ≤ π:

V = ∫₀^R ∫₀^{2π} ∫₀^π ρ² sin(φ) dφ dθ dρ
  = ∫₀^R ρ² dρ · ∫₀^{2π} dθ · ∫₀^π sin(φ) dφ
  = (R³/3) · 2π · 2
  = (4/3) π R³

The integrals separated because the integrand factors and the limits are constants — the kind of clean factorisation only spherical coordinates allow. Try this in Cartesians and you will need ∫√(R² − x²) terms and the trigonometric substitution from hell.

Integrating a radial function

If f depends only on ρ — say f(ρ) — the angular integrals collapse:

∫∫∫ f(ρ) dV = ∫₀^∞ f(ρ) ρ² dρ · ∫₀^{2π} dθ · ∫₀^π sin(φ) dφ
            = 4π ∫₀^∞ f(ρ) ρ² dρ

This is the workhorse formula. The mass of a star with radial density profile ρ_m(r) is 4π ∫ ρ_m(r) r² dr. The total charge in a spherically symmetric cloud is 4π ∫ ρ_q(r) r² dr. The probability of finding an electron in a hydrogen 1s orbital between radii a and b is 4π ∫_a^b |ψ(r)|² r² dr. Same integral; different physics.

Gravity outside a uniform shell

One classic application — Newton's shell theorem. A spherically symmetric mass distribution with total mass M produces, at any external point at distance r from the centre, the same gravitational field as a point mass M at the centre.

The proof is a direct integral in spherical coordinates. Place the field point on the z-axis at distance r > R. The potential at that point from a thin shell of radius a (with surface density σ) is:

Φ = −G ∫∫ σ / d  dA

where d = √(r² + a² − 2ar cos φ) is the distance from a shell point to the field point and dA = a² sin(φ) dθ dφ is the shell area element. Substituting u = cos φ (so du = −sin φ dφ), the φ integral becomes elementary, and θ integrates to 2π. The total potential collapses to −GM/r — exactly the potential of a point mass at the origin. The shell theorem falls out of the geometry.

Cartesian vs cylindrical vs spherical — comparison

	Cartesian	Cylindrical	Spherical
Coordinates	(x, y, z)	(r, θ, z)	(ρ, θ, φ)
Volume element	dx dy dz	r dr dθ dz	ρ² sin(φ) dρ dθ dφ
Jacobian	1	r	ρ² sin(φ)
Best symmetry	Boxes, planes	Cylinders, pipes, vortex flow	Balls, central forces, atoms
Singular points	None	r = 0 (axis)	ρ = 0; φ = 0, π
Gradient ∂/∂(coord 1)	∂/∂x	∂/∂r	∂/∂ρ
Laplacian radial part	∂²/∂x² + ...	(1/r) ∂/∂r(r ∂/∂r)	(1/ρ²) ∂/∂ρ(ρ² ∂/∂ρ)
Natural for	Cubes, lattices	Pipes, cans, wires	Planets, atoms, antennas

The Jacobians explain everything. The lone "1" for Cartesians says boxes don't shrink as you move through them. The factor "r" for cylindricals says the azimuthal arc length r dθ shrinks as you approach the axis. The factor "ρ² sin(φ)" for sphericals says you simultaneously shrink in the polar arc and in the latitude circle as you approach the poles.

Gradient, divergence, and Laplacian

In spherical coordinates the differential operators acquire metric factors:

∇f = (∂f/∂ρ) ρ̂ + (1/ρ)(∂f/∂φ) φ̂ + (1/(ρ sin φ))(∂f/∂θ) θ̂

∇·F = (1/ρ²) ∂(ρ² F_ρ)/∂ρ
     + (1/(ρ sin φ)) ∂(sin φ · F_φ)/∂φ
     + (1/(ρ sin φ)) ∂F_θ/∂θ

∇²f = (1/ρ²) ∂/∂ρ (ρ² ∂f/∂ρ)
    + (1/(ρ²sin φ)) ∂/∂φ (sin φ ∂f/∂φ)
    + (1/(ρ²sin²φ)) ∂²f/∂θ²

The 1/ρ and 1/(ρ sin φ) factors come from the same arc-length scales that determine the volume element. Each angular derivative ∂/∂(angle) needs to be divided by the corresponding arc length so that ∇f has units of (units of f) per metre, not (units of f) per radian.

For a radially symmetric function f(ρ), the angular derivatives vanish and ∇²f reduces to (1/ρ²) d/dρ (ρ² df/dρ) — equivalently, (1/ρ) d²(ρf)/dρ². This is why Coulomb's potential −1/ρ has Laplacian zero everywhere except the origin.

Multipole expansion and spherical harmonics

Any function on the sphere f(θ, φ) can be expanded as a sum of spherical harmonics Yₗᵐ(θ, φ):

f(θ, φ) = Σ_{ℓ=0}^∞ Σ_{m=−ℓ}^{ℓ} c_{ℓm} Y_{ℓm}(θ, φ)

The Yₗᵐ are eigenfunctions of the angular Laplacian with eigenvalue −ℓ(ℓ+1). Combined with radial functions Rₙₗ(ρ), they form a complete basis for solving the Laplace and Helmholtz equations in spherical regions. The hydrogen atom's wavefunctions ψ_{n,ℓ,m}(ρ, θ, φ) = Rₙₗ(ρ) Yₗᵐ(θ, φ) are the canonical example.

The same spherical harmonics describe the cosmic microwave background's temperature anisotropy (the famous "power spectrum" Cₗ comes from squaring its harmonic coefficients), the geomagnetic field, gravitational fields of non-spherical planets, and acoustic radiation patterns. They are the angular Fourier basis for the universe.

Where spherical coordinates earn their keep

• Gravitational and electrostatic fields. Around a point mass or point charge, the field has spherical symmetry. Spherical coordinates make Gauss's law trivial — the flux through a sphere equals the enclosed mass or charge times a constant.
• The hydrogen atom. The Schrödinger equation separates into radial and angular pieces; the angular part is solved by spherical harmonics, the radial part by Laguerre polynomials. Quantum numbers (n, ℓ, m) are labels in this spherical decomposition.
• Geophysics and atmospheres. Earth's gravitational field, the geoid, atmospheric pressure profiles, tides — all naturally written in spherical coordinates with the planet's centre at ρ = 0.
• GPS and astronomy. Right ascension and declination are spherical coordinates of the celestial sphere. Positions on Earth use latitude (related to π/2 − φ) and longitude (θ).
• Antenna radiation patterns. The far field of an antenna is a function on the sphere; gain patterns are usually plotted as G(θ, φ).
• Cosmology. Maps of the cosmic microwave background and large-scale galaxy distributions live on the celestial sphere; their power spectra are spherical-harmonic decompositions.

Common mistakes

• θ–φ swap. The single most common error. Always confirm whether your textbook's φ is from z (physics convention) or from the equator. The volume element looks the same but the formulas for x, y, z change.
• Forgetting sin(φ) in the volume element. The naive ρ² dρ dθ dφ misses the latitude factor — your "volume of a ball" comes out to π² R³ instead of (4/3)π R³.
• arctan instead of atan2. Plain arctan returns θ in (−π/2, π/2), losing quadrant information. Always use atan2(y, x), which uses the signs of x and y to return θ in the full (−π, π].
• Using radians vs degrees inconsistently. Mixing degree-mode trig with radian formulas is a classic source of factor-of-(180/π) errors.
• Treating ρ̂, θ̂, φ̂ as constant. The unit vectors of spherical coordinates rotate as you move through space — they are functions of position. ∂ρ̂/∂φ ≠ 0, which is why the divergence formula has those metric factors.
• Forgetting metric factors in ∇ and ∇². The angular derivatives must be divided by the corresponding arc-length scales (ρ for φ, ρ sin φ for θ). Otherwise units don't match and the answer is wrong by powers of ρ.