differential geometry w notes from teacher_Part_34

differential geometry w notes from teacher_Part_34 - 67...

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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 anti-derivation 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 anti-derivation. 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 non-degenerate 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 one-parameter 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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