I’ve set a tentative goal of heading towards the solution of the Dirichlet problem on compact manifoldw-with-boundary and Hodge theory; these will require various preliminaries, since of course it will be more fun to do it this way than to restrict to open sets in . Today I will go through some of the basics of how the well-known operators from multivariable calculus work more generally on a Riemannian manifold, which will be necessary in the sequel. (I shamelessly took the title of the post from a book I haven’t read.)

Recall the well-known operator on functions of -variables, the **Laplacian**

The problem is, doesn’t transform nicely with respect to changes in coordinates, and since we want to define the Laplacian on manifolds, this causes a problem. However, it can be done on a Riemannian manifold. The idea is to use the formula

which is immediate from the definitions on . The key point is that and make sense on any Riemannian manifold—from, respectively, vector fields to functions and from functions to vector fields. (more…)