Cylindrical and Spherical Coordinates — Mathematics Knowledge System

Overview

Cartesian coordinates (x, y, z) describe every point in space with three perpendicular distances, but many solids — cylinders, cones, spheres, planetary orbits — have symmetry that Cartesian coordinates hide and complicate. Cylindrical coordinates (r, θ, z) replace the x and y axes with polar coordinates in the horizontal plane while keeping z unchanged, ideal for objects with an axis of rotational symmetry. Spherical coordinates (ρ, θ, φ) instead describe a point by its distance ρ from the origin, an azimuthal angle θ around the z-axis, and a polar angle φ down from the positive z-axis — ideal for objects with symmetry about a single point, like spheres and radiating fields. Both systems trade some of Cartesian coordinates' simplicity for equations that become dramatically simpler once the right symmetry is exploited.

Intuition

Cylindrical coordinates are just polar coordinates with height added on: to locate a point, walk out a distance r from the z-axis at angle θ, then go straight up (or down) a height z — think of specifying a location in a parking garage by which spiral ramp position and which floor. Spherical coordinates instead work like a GPS or a globe: ρ is how far you are from the center of the Earth, φ is your angle down from the North Pole (colatitude), and θ is your longitude-like angle around the equator. A sphere of radius R becomes the ridiculously simple equation ρ = R in spherical coordinates, the same way a circle becomes r = R in polar coordinates.

Formal Definition

Definition

Cylindrical coordinates (r, θ, z) relate to Cartesian coordinates exactly as polar coordinates do in the xy-plane, with z left alone:

x = r\cos\theta, \qquad y = r\sin\theta, \qquad z = z

Cylindrical to Cartesian

r^2 = x^2+y^2, \qquad \tan\theta = \frac{y}{x}, \qquad z=z

Cartesian to cylindrical

x = \rho\sin\varphi\cos\theta, \quad y = \rho\sin\varphi\sin\theta, \quad z = \rho\cos\varphi

Spherical to Cartesian

\rho^2 = x^2+y^2+z^2, \qquad \varphi = \arccos\!\left(\frac{z}{\rho}\right), \qquad \tan\theta = \frac{y}{x}

ρ ≥ 0, 0 ≤ φ ≤ π (measured from the positive z-axis), 0 ≤ θ < 2π

Cartesian to spherical

Notation

Notation	Meaning
$(r,\theta,z)$	Cylindrical coordinates: polar radius, polar angle, and height
$(\rho,\theta,\varphi)$	Spherical coordinates: distance from origin, azimuthal angle, polar (colatitude) angle
$dV = r\, dz\, dr\, d\theta$	Volume element in cylindrical coordinates
$dV = \rho^2 \sin\varphi\, d\rho\, d\varphi\, d\theta$	Volume element in spherical coordinates

Derivation

The spherical volume element ρ²sinφ comes from the determinant of the Jacobian matrix of partial derivatives of (x,y,z) with respect to (ρ,θ,φ):

\frac{\partial(x,y,z)}{\partial(\rho,\theta,\varphi)} = \det\begin{pmatrix} \sin\varphi\cos\theta & -\rho\sin\varphi\sin\theta & \rho\cos\varphi\cos\theta \\ \sin\varphi\sin\theta & \rho\sin\varphi\cos\theta & \rho\cos\varphi\sin\theta \\ \cos\varphi & 0 & -\rho\sin\varphi \end{pmatrix}

Differentiate each Cartesian coordinate with respect to each spherical coordinate

= -\rho^2\sin\varphi \quad \text{so} \quad |J| = \rho^2\sin\varphi \ (0\le\varphi\le\pi \implies \sin\varphi\ge 0)

Expanding the determinant and taking the absolute value, since sinφ ≥ 0 on [0, π]

Properties

Sphere equation

\rho = R \text{ (constant) describes a sphere of radius } R \text{ centered at the origin}

Cylinder equation

r = R \text{ (constant) describes an infinite cylinder of radius } R \text{ around the } z\text{-axis}

Cone equation

\varphi = \varphi_0 \text{ (constant, } 0<\varphi_0<\pi/2\text{) describes a cone opening around the positive } z\text{-axis}

Theorems

Theorem 1: Change of variables to cylindrical/spherical coordinates

\iiint_E f\, dV = \iiint f(r\cos\theta, r\sin\theta, z)\, r\, dz\, dr\, d\theta = \iiint f(\rho\sin\varphi\cos\theta, \rho\sin\varphi\sin\theta, \rho\cos\varphi)\, \rho^2\sin\varphi\, d\rho\, d\varphi\, d\theta

Applications

Gravitational and electrostatic potentials from a point mass/charge depend only on ρ, so spherical coordinates reduce these 3D problems to single-variable calculus in ρ.

3D Visualization

Worked Examples

Apply x = r cos θ with r=4, θ=π/3 (cos(π/3) = 1/2).
$x = 4\cos(\pi/3) = 4\cdot\tfrac12 = 2$
Apply y = r sin θ (sin(π/3) = √3/2).
$y = 4\sin(\pi/3) = 4\cdot\tfrac{\sqrt3}{2} = 2\sqrt3$
z carries over unchanged.
$z = 5$

Answer: Cartesian point: (2, 2√3, 5).

Practice Problems

Difficulty 4/10

Convert the cylindrical point (r, θ, z) = (2, π/2, -3) to Cartesian coordinates.

Difficulty 5/10

Convert the Cartesian point (0, 0, 5) to spherical coordinates.

Difficulty 6/10

A cylindrical water tank has radius 2 m and height 5 m, standing with its axis along the z-axis from z=0 to z=5. Using cylindrical coordinates, set up (and evaluate) the triple integral for its volume.

Common Mistakes

Common Mistake

Confusing the two common conventions for spherical angle names, e.g. treating φ as the azimuthal angle (like in some physics textbooks) instead of the polar/colatitude angle used here (the math convention).

In the math convention used throughout this course, θ is the azimuthal angle (around the z-axis, same as polar θ) and φ is measured DOWN from the positive z-axis, with 0 ≤ φ ≤ π. Always check which convention a source uses before combining formulas.

Common Mistake

Forgetting the extra ρ² sinφ factor (or the single r factor in cylindrical) when converting a triple integral.

dV = r dz dr dθ in cylindrical coordinates and dV = ρ²sinφ dρ dφ dθ in spherical coordinates — omitting these Jacobian factors is the single most common setup error.

Quiz

Flashcards

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Summary

Cylindrical coordinates (r, θ, z) are polar coordinates in the xy-plane with height z unchanged; ideal for axis-symmetric solids.
Spherical coordinates (ρ, θ, φ) use distance from the origin ρ, azimuthal angle θ, and polar angle φ from the positive z-axis; ideal for solids symmetric about a point.
Volume elements pick up Jacobian factors: dV = r dz dr dθ (cylindrical), dV = ρ²sinφ dρ dφ dθ (spherical).
Simple equations result from matching symmetry: ρ=R is a sphere, r=R is a cylinder, φ=φ₀ is a cone.
Choosing the coordinate system that matches a solid's symmetry is often the single biggest simplification in a multivariable integration problem.

References

BookStewart, J. Calculus: Early Transcendentals, 8th ed. Ch. 15.7-15.8.
WebsiteWikipedia — Spherical coordinate system
WebsiteWikipedia — Cylindrical coordinate system