1617-257/Homework Assignment 16

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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 30-32 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 section 33, just to get a feel for the future.

Doing

Solve all the problems in sections 30 and 31, but submit only your solutions of problem 2, 4, and 7 in section 30 and of problem 4 in section 31. In addition, ponder the following

Challenge Problem (do not submit). Make precise and prove Dror's assertion from class, that if \omega\in\Omega^k({\mathbb R}^n) and \xi_1,\ldots\xi_{k+1}\in T_x{\mathbb R}^n, then

d\omega(\xi_1,\ldots,\xi_{k+1}) = \lim_{\epsilon\to 0}\frac{1}{\epsilon^{k+1}}\omega(\partial(\epsilon P)),

where \partial(\epsilon P) denotes the boundary of the parallelepiped spanned by \epsilon\xi_1,\ldots,\epsilon\xi_{k+1}.

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 March 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).