Polar, Cylindrical, and Spherical Coordinates

Choosing the right coordinate system simplifies integration dramatically.

Polar coordinates $(r, \theta)$ in $\mathbb{R}^{2}$:

$$x = r\cos\theta, \quad y = r\sin\theta, \quad dA = r\,dr\,d\theta$$

Cylindrical coordinates $(r, \theta, z)$ in $\mathbb{R}^{3}$:

$$x = r\cos\theta, \quad y = r\sin\theta, \quad z = z, \quad dV = r\,dr\,d\theta\,dz$$

Spherical coordinates $(\rho, \theta, \phi)$ in $\mathbb{R}^{3}$:

$$x = \rho\sin\phi\cos\theta, \quad y = \rho\sin\phi\sin\theta, \quad z = \rho\cos\phi$$

$$dV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta$$

When to use each

Example — Volume of a sphere of radius $R$:

$$V = \int_0^{2\pi}\int_0^{\pi}\int_0^R \rho^2\sin\phi\,d\rho\,d\phi\,d\theta = \frac{4}{3}\pi R^3$$

Physics link: Gravitational and electric potentials have spherical symmetry. The Laplacian $\nabla^{2}$ in spherical coordinates is essential for solving Laplace's and Poisson's equations.

Common pitfall: Spherical convention clash: mathematicians and physicists swap the names of $\theta$ and $\varphi$. Check which angle runs from the pole (0 to $\pi$) before copying any formula between sources.