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Spherical coordinates

Spherical coordinates describe a point by its distance ρ\rho from the origin, a polar angle φ\varphi from the z-axis, and an azimuthal angle θ\theta in the xy-plane. They are useful for integrals over spheres and shells.

dV=ρ2sin⁡φ dρ dφ dθdV = \rho^2\sin\varphi\,d\rho\,d\varphi\,d\thetavolume element in spherical coordinates

Symbols

ρ\rhodistance from the origin
φ\varphipolar angle from the z-axisrad
θ\thetaazimuthal anglerad

Example

Volume of a sphere with radius 2: ∫02π ⁣ ⁣∫0π ⁣ ⁣∫02ρ2sin⁡φ dρ dφ dθ=43π⋅23≈33.51\displaystyle\int_0^{2\pi}\!\!\int_0^{\pi}\!\!\int_0^2 \rho^2\sin\varphi\,d\rho\,d\varphi\,d\theta = \tfrac43\pi\cdot 2^3 \approx 33.51.

φ\varphi is measured from the z-axis, not from the xy-plane - easy to confuse with latitude.
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← Polar coordinates · Divergence and curl →

Part of Multivariable Calculus: Multiple integrals.