文档介绍：Page 337
6
Differential Forms
Differential k-forms are tensor fields of type (0,k) that pletely antisymmetric. Such tensor fields arise
in many applications in physics, engineering, and mathematics. A hint at why this is so is the fact that the
classical operations of grad, div, and curl and the theorems of Green, Gauss, and Stokes can all be expressed
concisely in terms of differential forms and an operator on differential forms to be studied in this chapter,
the exterior derivative d. However, identities like ∇×(∇f)=0and∇·(∇×X) = 0 are elegantly phrased
as the single identity d2 = 0. However, the examples of Hamiltonian mechanics and Maxwell’s equations
(see Chapter 8) show that their applicability goes well beyond this.
The goal of the chapter is to develop the calculus of differential forms, due largely to Cartan [1945]. The
exterior derivative operator d plays a central role; its properties and the expression of the Lie derivative in
terms of it will be developed.
Exterior Algebra
We begin with the exterior algebra of a vector space and extend this fiberwise to a vector bundle. As with
tensor fields, the most important case is the tangent bundle of a manifold, which is considered in the next
section.
We first recall a few facts about the permutation group on k elements; some of these facts have already been
discussed in §. Proofs of the results that we cite are obtainable from virtually any elementary algebra book.
The permutation group on k elements, denoted Sk, consists of all bijections σ: {1,...,k}→{1,...,k}
usually given in the form of a table
1 ··· k
,
σ(1) ··· σ(k)
together with the structure of a group position of maps. Clearly, Sk has order k!. Letting {−1, 1}
have its natural multiplicative group structure, there is a homomorphism denoted sign : Sk →{−1, 1}; that
is, for σ, τ∈ Sk, sign(σ◦τ) = (sign σ)(sign τ). A permutation σ is called even when sign σ= +1 and odd
when sign σ= −1. This homomorphism can be described a