Explicit formulas for gradients and the divergence in n-dimensional spherical coordinates

Abstract

We use the Laplacian in n-dimensional spherical coordinates (n>1) to write the divergence of a vector field defined on radially symmetric domains in the context of vector calculus. We apply straightforward equations of vector calculus with the nabla operator and the transformation matrices from Cartesian to spherical polar coordinates. One needs the divergence of a vector field e.g. to prove that vector fields are eigenfunctions of the Stokes operator on n-dimensional annuli and balls. Our divergence formula in partial derivatives in n-dimensional spherical polar coordinates is an important step in a future verification of further Stokes eigenfunctions on those domains.