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. Im 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. Dont 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 doesnt 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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This note was uploaded on 03/06/2010 for the course DEPARTMENT math434 taught by Professor Elizabethdenne during the Winter '10 term at Smith.

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

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