 
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 higherdimensional generalization thereof. We will often shorten the phrase
"manifold of dimension k" to "kmanifold." When we integrate over a kmanifold 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 1manifolds. 2manifolds 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, higherdimensional integrals will never be written as multiple integrals.
Definition 1. A tensor of degree k, or a ktensor, on Rn
is a realvalued function of k variables, where
each input variable is a vector in Rn
