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hw10s - H/wk 10 Due Friday April 13 NAME 1 Let be an r-form...

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H/wk 10. Due Friday, April 13. NAME: 1. Let ω be an r -form in a manifold M n such that for some r -chain σ in M with ∂σ = 0 we have σ ω = 0. Prove that ω is not exact. Solution. Suppose that ω is exact, so that ω = for some ( r - 1)-form α . Then by Stokes’ Theorem σ ω = σ = ∂σ α = 0 α = 0 , contradicting the assumption that σ ω = 0. 2. Let T 2 = { ( x, y, z, w ) R 4 : x 2 + y 2 = z 2 + w 2 = 1 } . Consider the following 1-forms on T 2 : ω 1 := - y x 2 + y 2 dx + x x 2 + y 2 dy and ω 2 := - w z 2 + w 2 dz + z z 2 + w 2 dw . Consider the following 1-cubes γ 1 : [0 , 1] T 2 and γ 2 : [0 , 1] T 2 in T 2 : γ 1 ( t ) = (cos 2 πt, sin 2 πt, 1 , 0) , γ 2 ( t ) = (1 , 0 , cos 2 πt, sin 2 πt ) , where t [0 , 1] . (a) Show that ∂γ 1 = ∂γ 2 = 0. (b) Compute γ 1 ω 1 and γ 2 ω 2 . Conclude that ω 1 and ω 2 are not exact on T 2 . (c) Consider the chart ( U, φ = ( θ 1 , θ 2 )) on T 2 where φ ( U ) = (0 , 2 π ) × (0 , 2 π ) and where φ - 1 ( θ 1 , θ 2 ) = (cos θ 1 , sin θ 1 , cos θ 2 , sin θ 2 ). Compute ω 1 and ω 2 in the chart ( U, φ ), that is, calculate ( φ - 1 ) * ω i for i = 1 , 2. Then verify that 1 = 0 and 2 = 0 in this chart hence, by continuity, 1 = 0 and 2 = 0 on T 2 . Thus ω 1 and ω 2 are closed on T 2 . (d) Let c 1 , c 2 R be such that c 2 1 + c 2 2 = 0. Show that there exists a linear combination σ = 1 + 2 such that σ [ c 1 ω 1 + c 2 ω 2 ] = 0 (and hence c 1 ω 1 + c 2 ω 2 is not exact since ∂σ = a∂γ 1 + b∂γ 2 = a · 0 + b · 0 = 0).
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