18.024ESG Notes 3 Pramod N. Achar Summary: 18.024­ESG Notes 3 Pramod N. Achar Spring 2000 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 Rk . If M Rn , then we say the codimension of M is n - k. Thus, curves are 1-manifolds. 2-manifolds sitting in R3 have codimension 1, but if they sit in R4 , 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. Definition 1. A tensor of degree k, or a k-tensor, on Rn is a real-valued function of k variables, where each input variable is a vector in Rn Collections: Mathematics