06 - differential forms on manifolds

# 06 - differential forms on manifolds - FORMS AND...

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Unformatted text preview: FORMS AND INTEGRATION — I Differential forms: definitions Part I: Linear Theory Let V ’ R n be a linear space: we avoid the symbol R n since the latter implicitly implies some coordinates. ♥ Definition. An exterior k-form on V is a map ω : V × · × V | {z } k times → R , ( v 1 ,...,v k ) 7→ ω ( v 1 ,...,v k ) , which is: • linear in each argument, and • antisymmetric: if σ ∈ S k is a permutation on k symbols, and | σ | = ± 1 its parity, then ω ( v σ (1) ,...,v σ ( k ) ) = (- 1) | σ | ω ( v 1 ,...,v k ) . The space of all k-forms on V is denoted by ∧ k ( V * ): it is a linear space over R . / ♦ Example. Linear forms are 1-forms: ∧ 1 ( V * ) = V * . ♦ Example. If dim V = k and a coordinate system in V is chosen, and v j = ( v j 1 ,...,v jk ), then ω ( v 1 ,...,v k ) = det fl fl fl fl fl fl v 11 ... v k 1 . . . . . . . . . v 1 k ... v kk fl fl fl fl fl fl is a k-form. We denote it by det x , x explicitly indicating the coordinate system. ♣ Problem 1. Prove that for any u,v ∈ R 3 the two formulas, ω 2 = det x ( u, · , · ) , ω 1 = det x ( u,v, · ) define 2- and 1-forms respectively. In any coordinate system ( x 1 ,...,x n ) on V ’ R n a k-form can be associated with a tuple of reals: if α : { 1 ,...,k } → { 1 ,...,n } is an index map , and ( e 1 ,..., e n ) a basis in V , then we define a α = ω ( e α (1) ,..., e α ( k ) ) and consider the collection { a α } with α ranging over all possible index maps.possible index maps....
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06 - differential forms on manifolds - FORMS AND...

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