More generally, see also the Laplace-Beltrami and Laplace-de Rham operators, both defined on (pseudo-)Riemannian manifolds not necessarily embedded in ℝ³. Even

More generally, see also the Laplace-Beltrami and Laplace-de Rham operators, both defined on (pseudo-)Riemannian manifolds not necessarily embedded in ℝ³. Even more generally, see the huge body of beautiful mathematics that has arisen from the study of elliptic differential operators in general (de Rham cohomology, Hodge theory, the Atiyah-Singer index theorem, …).