| Abstract: This will be an introductory talk on ``exterior calculus,'' which is the proper framework for understanding the ``complex'' of operators grad, curl and div on $\mathbb{R}^3$ as well as the corresponding integral theorems (Stokes' Theorem, the Divergence Theorem, and, in the plane, Green's Theorem). Our eventual goal is to cover exterior calculus on ``manifolds'' (such as the unit sphere in $\mathbb{R}^3$, relevant, e.g., in the study of symmetry reductions of, say, the Navier-Stokes equations) including the Hodge decomposition. Later, we may potentially discuss the work of Douglas Arnold and others on finite element exterior calculus and applications (see, e.g., arxiv.org/abs/0906.4325). |