A Geometric Approach to Differential Forms - download pdf or read online

By David Bachman

ISBN-10: 0817644997

ISBN-13: 9780817644994

ISBN-10: 0817645209

ISBN-13: 9780817645205

Don't buy the Kindle version of this ebook. you'll be wasting precious funds. The mathematical fonts are bitmapped and nearly unreadable. Amazon must repair this challenge. purchase the print version.

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1. A differential 2-form, ω, acts on a pair of vector fields, and returns a function from Rn to R. Example 19. V1 = 2y, 0, −x (x,y,z) is a vector field on R3 . For example, it contains the vector 4, 0, −1 ∈ T(1,2,3) R3 . If V2 = z, 1, xy (x,y,z) and ω is the differential 2-form, x 2 y dx ∧ dy − xz dy ∧ dz, then ω(V1 , V2 ) = x 2 y dx ∧ dy − xz dy ∧ dz( 2y, 0, x = x2y (x,y,z) , z, 1, xy (x,y,z) ) 2y z 0 1 − xz = 2x 2 y 2 − x 2 z, 01 −x xy which is a function from R3 to R. Notice that V2 contains the vector 3, 1, 2 (1,2,3) .

Then we can rotate α and β to α and β so that c α = V , for some scalar c. We can then replace the pair ( α , β ) with the pair (c α , 1/c β ) = (V , 1/c β ). To complete the proof, let W = 1/c β . Lemma 2. If ω1 = α1 ∧ β1 and ω2 = α2 ∧ β2 are 2-forms on Tp R3 , then there exist 1-forms, α3 and β3 , such that ω1 + ω2 = α3 ∧ β3 . Proof. Let’s examine the sum, α1 ∧ β1 + α2 ∧ β2 . Our first case is that the plane spanned by the pair ( α1 , β1 ) is the same as the plane spanned by the pair, ( α2 , β2 ).

In other words, 1 a more general integral would be −1 1 f (φ1 (a))ω( dφ da )da, where f is a function of points and ω is a function of vectors. It is not the purpose of the present work to undertake a study of integrating with respect to all possible functions, ω. However, as with the study of functions of real variables, a natural place to start is with linear functions. This is the study of differential forms. A differential form is precisely a linear function which eats vectors, spits out numbers and is used in integration.

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A Geometric Approach to Differential Forms by David Bachman

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