Triple Integrals in Cylindrical and Spherical Coordinates

The multivariate normal density — the workhorse of multi-factor risk models — lives on spherical symmetry when the factors are uncorrelated: the density contours are spheres $\{\mathbf{x}:\|\mathbf{x}\|^2=c\}$. Integrating it over a spherical region in three dimensions requires spherical coordinates, not Cartesian, for the same reason that the Gaussian integral over a disk required polar coordinates: the domain's symmetry matches the coordinate system, and the limits become constants. This lesson introduces cylindrical coordinates (the 3D extension of polar) and spherical coordinates (the optimal system for balls and shells), derives both volume elements from the Jacobian determinant, and illustrates when to use each system.

Learning objectives. By the end of this note you should be able to: