Polar, Cylindrical, and Spherical Coordinates

Coordinate Systems

Choosing the right coordinate system simplifies integration dramatically. Match your choice to the region's symmetry.

Here, $\rho \ge 0$ is the radius, $0 \le \theta \le 2\pi$ is the azimuth, and $0 \le \phi \le \pi$ is the polar angle.

Integration & Pitfalls

The volume element for spherical coordinates is $dV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta$.

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

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

Common pitfall: Mathematicians and physicists swap the names of $\theta$ and $\phi$. Always check which angle runs from the pole ($0$ to $\pi$) before using a formula.

Physics link: The Laplacian $\nabla^{2}$ in spherical coordinates uses this geometry to solve Laplace's equations for potentials.