Question 1 (12 points). Pick your favourite strictly increasing sequence of odd positive integers n =...
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Question 1 (12 points). Pick your favourite strictly increasing sequence of odd positive integers n = (n₁,...,n) starting with n 1 and of length /> 2. = 1. Prove that the polynomial function Pn: R→R:t+thi is a diffeomorphism of R. Hint: Follow the steps 1-3 discussed in lecture 3+4. Why pn is surjective? 2. Consider the sequence 7 = (n₂,..., n) obtained from n by deleting the first entry n₁ = 1. Explain why the polynomial function Pr: R→R:t> - 2 is not a diffeomorphism of R. 3. Use the following example to explain why T-periodicity (for T> 0) is not preserved under reparametriza- tion. Example: Let r>0 and consider the regular circular curve : R → R² :t+ [cos(t) Recall sin(t) that is a 27-periodic, non-constant parametrized curve in R2 and therefore defines a closed curve [r] in R². Explain why the reparametrization P(1,3) is not T-periodic for any T > 0? If you = are unable to find a short proof, it suffices if you describe "the reason" informally in a few sentences. Question 1 (12 points). Pick your favourite strictly increasing sequence of odd positive integers n = (n₁,...,n) starting with n 1 and of length /> 2. = 1. Prove that the polynomial function Pn: R→R:t+thi is a diffeomorphism of R. Hint: Follow the steps 1-3 discussed in lecture 3+4. Why pn is surjective? 2. Consider the sequence 7 = (n₂,..., n) obtained from n by deleting the first entry n₁ = 1. Explain why the polynomial function Pr: R→R:t> - 2 is not a diffeomorphism of R. 3. Use the following example to explain why T-periodicity (for T> 0) is not preserved under reparametriza- tion. Example: Let r>0 and consider the regular circular curve : R → R² :t+ [cos(t) Recall sin(t) that is a 27-periodic, non-constant parametrized curve in R2 and therefore defines a closed curve [r] in R². Explain why the reparametrization P(1,3) is not T-periodic for any T > 0? If you = are unable to find a short proof, it suffices if you describe "the reason" informally in a few sentences.
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Related Book For
Income Tax Fundamentals 2013
ISBN: 9781285586618
31st Edition
Authors: Gerald E. Whittenburg, Martha Altus Buller, Steven L Gill
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