1617-257/Homework Assignment 18: Difference between revisions
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Ponder the questions in sections 34 and 35, yet solve and submit only the following problems: |
Ponder the questions in sections 34 and 35, yet solve and submit only the following problems: |
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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. |
<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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<u>'''Problem 2'''</u> (an alternative definition for "orientation"). Define a norientation ("new orientation") of a vector space <math>V</math> to be a function <math>\nu\colon\{\text{ordered bases of }V\}\to\{\pm 1\}</math> which satisfies <math>\nu(v)=\operatorname{sign}(\det(C^u_v))\nu(u)</math>, whenever <math>u</math> and <math>v</math> are ordered bases of <math>V</math> and <math>C^u_v</math> is the change-of-basis matrix between them. |
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<ol> |
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<li> Explain how if <math>\dim(V)>1</math>, a norientation is equivalent to an orientation. |
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<li> Come up with a reasonable definition of a norientation of a <math>k</math>-dimensional manifold. |
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<li> Explain how a norientation of <math>M</math> induces a norientation of <math>\partial M</math>. |
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<li> What is a <math>0</math>-dimensional manifold? What is a norientation of a <math>0</math>-dimensional manifold? |
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<li> What is <math>\partial[0,1]</math> as a noriented <math>0</math>-manifold? (Assume that <math>[0,1]</math> is endowed with its "positive" or "standard" orientation/norientation). |
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</ol> |
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==Submission== |
==Submission== |
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Revision as of 16:29, 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.
Problem 2 (an alternative definition for "orientation"). Define a norientation ("new orientation") of a vector space to be a function which satisfies , whenever and are ordered bases of and is the change-of-basis matrix between them.
- Explain how if , a norientation is equivalent to an orientation.
- Come up with a reasonable definition of a norientation of a -dimensional manifold.
- Explain how a norientation of induces a norientation of .
- What is a -dimensional manifold? What is a norientation of a -dimensional manifold?
- What is as a noriented -manifold? (Assume that is endowed with its "positive" or "standard" orientation/norientation).
![{\displaystyle \partial [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73fea761a2ffb7a40e89d7e025c7f5f154a712f5)
![{\displaystyle [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d)
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).
