1617-257/Homework Assignment 18: Difference between revisions
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==Doing== |
==Doing== |
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Ponder the questions in sections 34 and 35, yet solve and submit only the following problems: |
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'''Solve''' ''all'' the problems in sections 32 and 33, but submit only your solutions of problem <u>3</u> and <u>5</u> in section 32 and of problems <u>2</u> and <u>3</u> in section 33. In addition, ponder the following |
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<u>'''Problem 1'''</u> Consider <math>S^{n-1}</math> at the boundary of <math>D^n\subset{\mathbb R}^n</math>, taken with its standard orientation, and let <math>\iota\colon S^{n-1}\to{\mathbb R}^n</math> be the inclusion map. Let <math>\omega=\iota^\ast\left(\sum_ix_idx_1\wedge\dots\wedge\widehat{dx_i}\wedge\dots\wedge dx_n\right)\in\Omega^{\text{top}}(S^{n-1})</math>. Prove that if <math>(v_1,\ldots,v_{n-1})</math> is a positively oriented basis of <math>T_xS^{n-1}</math> for some <math>x\in S^{n-1}</math>, then <math>\omega(v_1,\ldots,v_{n-1})</math> is the volume of the <math>(n-1)</math>-dimensional parallelepiped spanned by <math>v_1,\ldots,v_{n-1}</math>, and hence for any smooth function <math>f</math> on <math>S^{n-1}</math>, <math>\int_{S^{n-1}}f\omega = \int_{S^{n-1}}fdV</math>, where the latter integral is integration relative to the volume, as defined a long time ago. |
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'''Challenge Question''' (do not submit). What was it that you computed, in problem 3 of section 33? Could you have done it without any actual computation? |
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==Submission== |
==Submission== |
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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. |
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. |
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This assignment is due in class on <span style="color: blue;">Wednesday March |
This assignment is due in class on <span style="color: blue;">Wednesday March 29 by 2:10PM</span>. |
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===<span style="color: red;">Important</span>=== |
===<span style="color: red;">Important</span>=== |
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Revision as of 16:15, 20 March 2017
The information below is preliminary and cannot be trusted! (v)
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 33-38 (skip 36) 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 39, just to get a feel for the future.
Doing
Ponder the questions in sections 34 and 35, yet solve and submit only the following problems:
Problem 1 Consider at the boundary of , taken with its standard orientation, and let be the inclusion map. Let . Prove that if is a positively oriented basis of for some , then is the volume of the -dimensional parallelepiped spanned by , and hence for any smooth function on , , where the latter integral is integration relative to the volume, as defined a long time ago.
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 29 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).
