188宝金博页面版

FORMS AND INTEGRATION — IDifferential forms: definitionsPart I: Linear TheoryLet V Rnbe a linear space: we avoid the symbol Rnsince the latter implicitlyimplies some coordinates.♥ Definition. An exterior k-form on V is a mapω: V × · × Vk times→ R,(v1, . . . , vk) → ω(v1, . . . , vk),which is:• linear in each argument, and• antisymmetric: if σ ∈ Sk is a permutation on k symbols, and |σ| = ±1 itsparity, thenω(vσ(1), . . . , vσ(k)) = (−1)|σ|ω(v1, . . . , vk).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 vj =(vj1, . . . , vjk), thenω(v1, . . . , vk) = detv11...v1k. . ..... . .vk1...vkkis a k-form. We denote it by detx, x explicitly indicating the coordinate system.Prove that for any u, v ∈ R3the two formulas,♣ Problem 1.ω2= detx(u, ·, ·),ω1= detx(u, v, ·)define 2- and 1-forms respectively.In any coordinate system (x1, . . . , xn) on V Rna k-formcan be associated with a tuple of reals: if α: { 1, . . . , k } →{ 1, . . . , n } is an index map, and (e1, . . . , en) a basis in V, thenwe defineaα= ω(eα(1), . . . , eα(k))and consider the collection { aα} with α ranging over allpossible index maps.Typeset by AMS-TEX1