The talk will start with a short introduction to the open-source Dedalus computational framework for solving PDEs. I'll briefly discuss some design motivations and interesting applied problems different groups are studying with the software. However, the true point of the talk will be to describe some of the mathematical aspects underlying Dedalus' flexible framework for curvilinear coordinates. Dedalus can compute arbitrary covariant calculus for scalar, vector and tensor fields on spheres and balls in one, two, and three dimensions. I plan to discuss some modest fraction of the rich mathematics used at the core of the code. In particular, Dedalus uses orthogonal polynomial bases on an infinite hierarchy of nested Hilbert spaces, with multiple derivative, multiplication and embedding operators acting between them. The structures become complicated quickly, but everything boils down to some simple guiding principles that I will demonstrate using freely available properties of meek little sine and cosine functions.