Summary: 18.024ESG Notes 3
Pramod N. Achar
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