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differential geometry w notes from teacher_Part_33

differential geometry w notes from teacher_Part_33 - 65 3.1...

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3.1. EXTERIOR ALGEBRA 65 1. The exterior square of any p -form α of odd degree p (in particular, for any 1-form) vanishes α α = 0 . The exterior algebra Λ (or Grassmann algebra ) is the set of all forms of all degrees, that is, Λ = Λ 0 ⊕ · · · ⊕ Λ n . The dimension of the exterior algebra is dim Λ = n p = 0 n p = 2 n . A basis of the space Λ p is σ i 1 ∧ · · · ∧ σ i p , (1 i 1 < · · · < i p n ) . An p -form α can be represented in one of the following ways α = α i 1 ... i p σ i 1 ⊗ · · · ⊗ σ i p = 1 p ! α i 1 ... i p σ i 1 ∧ · · · ∧ σ i p = i 1 < ··· < i p α i 1 ... i p σ i 1 ∧ · · · ∧ σ i p . The exterior product of a p -form α and a q -form β can be represented as α β = 1 p ! q ! α [ i 1 ... i p β i p + 1 ... i p + q ] σ i 1 ∧ · · · ∧ σ i p + q . Theorem 3.1.7 Let σ j Λ 1 , 1 j n, and α j Λ 1 , 1 j n, be two collections of n 1 -forms related by a linear transformation α j = n i = 1 A j i σ i , 1 j n , Then α 1 ∧ · · · ∧ α n = det A i j σ 1 ∧ · · · ∧ σ n . Proof : di ff geom.tex; April 12, 2006; 17:59; p. 67
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66 CHAPTER 3. DIFFERENTIAL FORMS 1. Theorem 3.1.8 Let α j Λ 1 = E * , 1 j p, be a collections of p 1 -forms and v i E, 1 i p, be a collection of p vectors. Let
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