1617-257/Homework Assignment 13

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Contents

Reading

Read, reread and rereread your notes to this point, and make sure that you really, really really, really really really understand everything in them. Do the same every week! Also, read, reread and rereread sections 25-28 of Munkres' book to the same standard of understanding. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, preread sections 29-30, just to get a feel for the future.

Doing

Solve all the problems in sections 25-26, but submit only your solutions of problems 4 and 8 in section 25 and problems 5 and 6 in section 26. In addition, solve the following problem, though submit only your solutions of parts d and e:

Problem A. Let \alpha\colon\{(u,v)\in{\mathbb R}^2\colon u^2+v^2\leq 1\}\to{\mathbb R}^3 be given by \alpha(u,v)=\left(u-v,\,u+v,\,2(u^2+v^2)\right). Let M be the image of \alpha.

a. Describe M.

b. Show that M is a manifold.

c. Find the boundary \partial M of M.

d. Find the volume V(M) of M.

e. Find \int_MzdV (where z denotes the third coordinate of {\mathbb R}^3).

Submission

Here and everywhere, neatness counts!! You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

This assignment is due in class on Wednesday February 8 by 2:10PM.

Important

Please write on your assignment the day of the tutorial when you'd like to pick it up once it is marked (Wednesday or Thursday).