Nonlinear Equations - UFRJ

[SEC. 4.1: MULTILINEAR ALGEBRA OVER R 43
with 1 ≤ i 1 < i 2 < · · · < i k ≤ n, defined by
dx i1 ∧ · · · ∧ dx ik (u 1 , . . . , u k ) = ∑
σ∈S k
(−1) |σ| u σ(1)i1 u σ(2)i2 · · · u σ(k)ik .
The wedge product ∧ : A k × A l → A k+l is defined by
α ∧ β (u 1 , . . . , u k+l ) =
= 1 ∑
(−1) |σ| α(u σ(1) , . . . , u σ(k) )β(u σ(k+1) , . . . , u σ(k+l) )
k!l!
σ∈S k+l
( )
1
k + l
The coefficient
k!l!
above may be replaced by if one replaces
the sum by the anti-symmetric average over S k+l . This con-
k
vention makes the wedge product associative, in the sense that
(α ∧ β) ∧ γ = α ∧ (β ∧ γ). (4.1)
so we just write α ∧ β ∧ γ. This is also compatible with the notation
dx i1 ∧ · · · ∧ dx in .
Another important property of the wedge product is the following:
if α ∈ A k and β ∈ A l , then
α ∧ β = (−1) kl β ∧ α. (4.2)
Let U ⊆ R n be an open set (in the usual topology), and let C ∞ (U)
denote the space of all smooth real valued functions defined on U.
The fact that a linear k-form takes values in R is immaterial in all
the definitions above.
Definition 4.1. The space of differential k-forms in U, denoted by
A k (U), is the space of linear k-forms defined in R n with values in
C ∞ (U).
This is equivalent to smoothly assigning to each point x on U, a
linear k-form with values in R. If α ∈ A k , we can therefore write
∑
α x =
α i1,...,i k
(x) dx i1 ∧ · · · ∧ dx ik .
1≤i 1