18 . 024 – ESG Notes 3

In this set of notes, we will develop the basic theory of tensors and differential forms; we will learn what it means to integrate a differential form; and we will state the generalized Stokes’ Theorem in terms of differential forms. Caveat lector: there are several closely related meanings of the word tensor. We begin introducing some convenient terminology for which we will not give precise definitions. A manifold is a curve, surface, or higher-dimensional generalization thereof. We will often shorten the phrase “manifold of dimension k” to “k-manifold.” When we integrate over a k-manifold M , we will need to make use of a parametrization r : D → M , where D ⊂ R. If M ⊂ R, then we say the codimension of M is n− k. Thus, curves are 1-manifolds. 2-manifolds sitting in R have codimension 1, but if they sit in R, they have codimension 2. Note that it makes sense to speak of “normal vectors” to manifolds of codimension 1. In this set of notes, higher-dimensional integrals will never be written as multiple integrals.