1617-257/Homework Assignment 18: Difference between revisions
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{{In Preparation}} |
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==Reading== |
==Reading== |
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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. |
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. |
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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 A.'''</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(\ |
<u>'''Problem A.'''</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_i(-1)^{i-1}x_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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'''Note.''' Earlier I made a sign mistake in the definition of and wrote <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>. I'd like to thank the students who emailed me the correction. |
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<u>'''Problem B'''</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. |
<u>'''Problem B'''</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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<u>'''Problem C'''.</u> Let <math>\omega=ydx\in\Omega^1({\mathbb R}^2_{x,y})</math>. |
<u>'''Problem C'''.</u> Let <math>\omega=-ydx\in\Omega^1({\mathbb R}^2_{x,y})</math>. |
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<li> Let <math>\Gamma</math> be the graph in <math>{\mathbb R}^2_{x,y}</math> of some smooth function <math>f\colon[a,b]\to{\mathbb R}</math>. Using the inclusion of <math>\Gamma</math> to <math>{\mathbb R}^2_{x,y}</math>, consider <math>\omega</math> also as a 1-form on <math>\Gamma</math>. What is <math>\int_\Gamma\omega</math>? |
<li> Let <math>\Gamma</math> be the graph in <math>{\mathbb R}^2_{x,y}</math> of some smooth function <math>f\colon[a,b]\to{\mathbb R}</math>. Using the inclusion of <math>\Gamma</math> to <math>{\mathbb R}^2_{x,y}</math>, consider <math>\omega</math> also as a 1-form on <math>\Gamma</math>. What is <math>\int_\Gamma\omega</math>? |
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Latest revision as of 09:41, 20 April 2017
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 A. 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.
Note. Earlier I made a sign mistake in the definition of and wrote . I'd like to thank the students who emailed me the correction.
Problem B (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 the integral of a -form on a -dimensional noriented 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)
Problem C. Let .
- Let be the graph in of some smooth function . Using the inclusion of to , consider also as a 1-form on . What is ?
- Prove that if is an ellipse in (of whatever major and minor axes, placed anywhere and tilted as you please), then is the area of .
- Compute also and .
![{\displaystyle f\colon [a,b]\to {\mathbb {R} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c486a52b15a997c1cfbc14dd25a02238e1997e9)
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).
