differential geometry w notes from teacher_Part_32

differential geometry w notes from teacher_Part_32 - 3.1...

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Unformatted text preview: 3.1. EXTERIOR ALGEBRA 63 • Notation. • To deal with forms it is convenient to introduce multi-indices. We will denote a p-tuple of integers from 1 to n by a capital letter I = ( i 1 , . . . , i p ) , where 1 ≤ i 1 , . . . , i p ≤ n . For a p-tuple of the same integers ordered in an increasing order we define ˆ I = ( i 1 , . . . , i p ) . where 1 ≤ i 1 < i 2 < ··· < i p ≤ n . We call ˆ I an increasing p-tuple associated with I . • Therefore, a collection of p-forms p !Alt σ i 1 ⊗ ··· ⊗ σ i p , where 1 ≤ i 1 < i 2 < ··· < i p ≤ n , forms a basis in the space Λ p . • Thus, every p-form α ∈ Λ p has the form α = X 1 ≤ i 1 < ··· < i p ≤ n α i 1 ··· i p p !Alt σ i 1 ⊗ ··· ⊗ σ i p . • Therefore, the dimension of the space Λ p is equal to the number of distinct increasing p-tuples of integers from 1 to n . • Theorem 3.1.5 The dimension of the space Λ p of p-forms is dim Λ p = n p !...
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

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differential geometry w notes from teacher_Part_32 - 3.1...

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