hw1-difftop - Math 343 Homework 1 Due Wednesday February 3...

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Unformatted text preview: Math 343 Homework 1 Due Wednesday February 3, 2010. When writing up your solutions, pay attention to what you write. I’m interested in seeing proofs written rigorously. What does this mean? Good proofs are: • Correct — ideally, every statement should follow from axioms or from what has been proved before. • Concise — a proof should not contain anything that is not necessary. • Readable — Human beings both write and read proofs. Don’t be afraid to explain in words what you are doing. For example, before embarking on a long computation, it is a good idea to explain what you are doing and why you are doing it. 1 To be handed in for grading 1. Suppose that X is a subset of R k and Z is a subset of X . Show that the restriction to Z of any smooth map on X is a smooth map on Z . (This exercise is designed to get you to use the definitions.) 2. Let X ⊂ R n , Y ⊂ R m , Z ⊂ R l be arbitrary subsets and let f : X → Y , g : Y → Z be smooth maps. (a) Show that the composite g ◦ f : X → Z is smooth. (This exercise is designed to get you to use the definitions.) (b) Show that if f and g are diffeomorphisms, then so is g ◦ f . (This is review.) 3. Prove that the paraboloid in R 3 defined by x 2 + y 2- z 2 = a is a manifold if a > 0. Why doesn’t x 2 + y 2- z 2 = 0 define a manifold? 4. Show that the union of the two coordinate axes in R 2 is not a manifold....
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hw1-difftop - Math 343 Homework 1 Due Wednesday February 3...

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