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Unformatted text preview: 67 3.1. EXTERIOR ALGEBRA
• In components,
(iv α)i1 ...i p−1 = v j α ji1 ...i p−1 .
• The interior product is a map
iv : Λ p → Λ p−1 ,
or
iv : Λ → Λ .
• A map L : Λ → Λ is called an derivation if for any α ∈ Λ p , β ∈ Λq ,
L(α ∧ β) = (Lα) ∧ β + α ∧ Lβ . • A map L : Λ → Λ is called an antiderivation if for any α ∈ Λ p , β ∈ Λq ,
L(α ∧ β) = (Lα) ∧ β + (−1) p α ∧ Lβ .
• Theorem 3.1.10 Let v ∈ E be a vector. The interior product iv : Λ →
Λ is an antiderivation. diﬀgeom.tex; April 12, 2006; 17:59; p. 69 68 CHAPTER 3. DIFFERENTIAL FORMS 3.2
3.2.1 Orientation and the Volume Form
Orientation of a Vector Space • Let E be a vector space. Let {ei } = {e1 , . . . , en } and {e j } = {e1 , . . . , en } be two
diﬀerent bases in E related by
ei = Λ j i e j ,
where Λ = (Λi j ) is a transformation matrix.
• Note that the transformation matrix is nondegenerate
det Λ 0. • Since the transformation matrix Λ is invertible, then the determinant det Λ
is either positive or negative.
• If det Λ > 0 then we say that the bases {ei } and {ei } have the same orientation, and if det Λ < 0 then we say that the bases {ei } and {ei } have the
opposite orientation.
• If the basis {ei } is continuously deformed into the basis {e j }, then both bases
have the same orientation.
• Since det I = 1 > 0 and the function det : GL(n, R) → R is continuous, then
a oneparameter continuous transformation matrix Λ(t) such that Λ(0) = I
preserves the orientation.
• This deﬁnes an equivalence relation on the set of all bases on E called the
orientation of the vector space E .
• This equivalence relation divides the set of all bases in two equivalence
classes, called the positively oriented and negatively oriented bases.
• A vector space together with a choice of what equivalence class is positively
oriented is called an oriented vector space.
diﬀgeom.tex; April 12, 2006; 17:59; p. 70 ...
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 Spring '10
 Wong
 Algebra, Geometry

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