18.1. THE WEDGE PRODUCT 475

18.1 The Wedge ProductNext is the definition of the wedge product.

Definition 18.1.1 Denote differential forms, the functionals defined above whichact on C1 ([a,b] ,Rp) as Ωk. Then letting ω ∈Ωk and η ∈Ωl , let ω ∧η ∈Ωk+l be definedas follows. Letting

ω ≡ ∑I ascending

aI (x)dxI , η ≡ ∑J ascending

bJ (x)dxJ

thenω ∧η ≡ ∑

I ascending∑

J ascendingaI (x)bJ (x)dxI ∧dxJ

where for I = (i1, · · · , ik) ,J ≡ ( j1, · · · , jl) ,

dxI ∧dxJ ≡ dxi1 ∧·· ·∧dxik ∧dx j1 ∧·· ·∧dx jl

This is well defined thanks to the above lemma which shows that there is only one way towrite a differential form like the above sums in which I is ascending in each term.

What if I and J are not ascending? Does it still work? Let I = (i1, · · · , ik) ,J ≡( j1, · · · , jl) and let σ (I) be ascending and let η (J) be ascending, σ ,η being two permuta-tions. Then from Lemma 18.0.7, the second claim,

∈Ωk

dxI ∧∈Ωl

dxJ =writing each in ascending form

sgn(σ)dxσ(I)∧ sgn(η)dxη(J)

above definition= sgn(σ)sgn(η)dxσ(I)∧dxη(J)

=definition of the wedge product dxσ(I)∧dxη(J)

sgn(σ)sgn(η)dxσ(i1)∧·· ·∧dxσ(ik)∧dxη( j1)∧·· ·∧dxη( jl)

= dxi1 ∧·· ·∧dxik ∧dx j1 ∧·· ·∧dx jl

Thus it is correct to write dxI ∧ dxJ = dxi1 ∧ ·· · ∧ dxik ∧ dx j1 ∧ ·· · ∧ dx jl even if I =(i1, · · · , ik) ,J ≡ ( j1, · · · , jl) are not ascending.

This is the main idea behind the following fundamental result. In this lemma, I,J arenot necessarily ascending.

Lemma 18.1.2 Let ω ≡ ∑I aI (x)dxI ,η ≡ ∑J bJ (x)dxJ be in Ωk and Ωl respectively.Then

ω ∧η = ∑I,J

aI (x)bJ (x)dxI ∧dxJ

Proof: For each I, let σ I (I) be ascending where σ I is a permutation, similar for J.Then, as noted above in the proof of Lemma 18.0.7 and denoting by Î, Ĵ the ascending listsof indices.

ω = ∑Î

 ∑{I:σ I(I)=Î}

aI (x)sgn(σ I)

dxÎ , η = ∑Ĵ

 ∑{I:σ J(J)=Ĵ}

bJ (x)sgn(σ J)

dxĴ