1617-257/Homework Assignment 18

DBN: Classes: 2016-17: 1617-257 / Navigation Welcome to Math 257 Edits to the Math 257 web pages no longer count for the purpose of good deed points. # Week of... Notes and Links 1 Sep 12 About This Class; Day 1 Handout (pdf, html); Monday, Wednesday, Friday, Day 2 Handout (pdf, html); First Week Notes. 2 Sep 19 Monday, Wednesday, Tutorial 2 Handout, Friday, Second week notes, HW1, HW1 Solutions. 3 Sep 26 Monday, Wednesday, Tutorial 3 Handout, Friday, Third week notes, Class Photo, HW2, HW2 Solutions. 4 Oct 3 Monday, Wednesday, Friday, 4th week notes, HW3, HW3 Solutions. 5 Oct 10 Monday is Thanksgiving, no class; Wednesday, Friday, 5th week notes, HW4, HW4 Solutions. 6 Oct 17 Monday, Wednesday, Friday, 6th week notes, HW5, HW5 Solutions. 7 Oct 24 Monday, Wednesday, Friday, 7th week notes 8 Oct 31 Monday; Term test 1; Wednesday, HW6, HW6 pdf, HW6 Solutions, Friday, 8th week notes. 9 Nov 7 Monday is last day to switch to MAT 237; Monday-Tuesday is UofT Fall Break; Wednesday, HW7, HW7 Solutions, Friday, 9th week notes. 10 Nov 14 Monday, Wednesday, HW8, HW8 pdf, HW8 Solutions, Friday, Lecture recordings, 10th week notes. 11 Nov 21 Monday, Wednesday, HW9, HW9 pdf, HW9 Solutions, Friday, Lecture recordings, 11th week notes. 12 Nov 28 Monday, Wednesday, HW10, HW10 Solutions, makeup class on Thursday at GB 120 at 5PM, no class and no DBN office hours Friday! 12th week notes. 13 Dec 5 Monday, Wednesday, 13th week notes Semester ends on Wednesday - no class Friday. B Dec 12,19,26 No classes: other classes' finals, winter break. 14 Jan 2 Class resumes Friday at RS211, no tutorials or office hours this week, Friday, Friday notes. 15 Jan 9 Monday, Wednesday, Friday, Weekly notes 16 Jan 16 Monday, Term test 2; Wednesday, HW11, HW11 inline pdf,HW11 solutions, Friday, Weekly notes 17 Jan 23 Monday; Hour 44 Handout (pdf, html); Wednesday, HW12, HW12 inline pdf, HW12 Solutions, Friday, Weekly notes 18 Jan 30 Monday, Wednesday, HW13, HW13 inline pdf, HW13 Solutions,Friday, Weekly Notes 19 Feb 6 Monday, Wednesday, HW14, HW14 inline pdf, Friday, Weekly Notes 20 Feb 13 Monday, Wednesday, HW15, HW15 inline pdf, HW15 solutions, Friday,Weekly Notes, UofT examination table posted on Friday. R Feb 20 Reading week - no classes; Tuesday is the last day to drop this class. 21 Feb 27 Monday, Wednesday, HW16, HW16 inline pdf, Friday, Weekly Notes 22 Mar 6 Monday, Wednesday, Friday, Weekly Notes 23 Mar 13 Monday, Term test 3 on Tuesday at 5-7PM; Wednesday, HW17, HW17 inline pdf, HW17 Solutions, Friday, Weekly Notes,Orientation_notes 24 Mar 20 Monday, Wednesday, HW18, HW18 inline pdf, Friday 25 Mar 27 No Monday class, Wednesday, Thursday (makeup for Monday), HW19, HW19 inline pdf, Friday, Weekly Notes 26 Apr 3 Monday, Wednesday; Semester ends on Wednesday - no tutorials Wednesday and Thursday, no class Friday. F Apr 10-28 The Final Exam on Thursday April 20 (and some office hours sessions before). Register of Good Deeds Riddle Repository Add your name / see who's in! $\int _{M}d\omega =\int _{\partial M}\omega$

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 $S^{n-1}$ at the boundary of $D^{n}\subset {\mathbb {R} }^{n}$, taken with its standard orientation, and let $\iota \colon S^{n-1}\to {\mathbb {R} }^{n}$ be the inclusion map. Let $\omega =\iota ^{\ast }\left(\sum _{i}(-1)^{i-1}x_{i}dx_{1}\wedge \dots \wedge {\widehat {dx_{i}}}\wedge \dots \wedge dx_{n}\right)\in \Omega ^{\text{top}}(S^{n-1})$. Prove that if $(v_{1},\ldots ,v_{n-1})$ is a positively oriented basis of $T_{x}S^{n-1}$ for some $x\in S^{n-1}$, then $\omega (v_{1},\ldots ,v_{n-1})$ is the volume of the $(n-1)$-dimensional parallelepiped spanned by $v_{1},\ldots ,v_{n-1}$, and hence for any smooth function $f$ on $S^{n-1}$, $\int _{S^{n-1}}f\omega =\int _{S^{n-1}}fdV$, 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 $\omega =\iota ^{\ast }\left(\sum _{i}x_{i}dx_{1}\wedge \dots \wedge {\widehat {dx_{i}}}\wedge \dots \wedge dx_{n}\right)\in \Omega ^{\text{top}}(S^{n-1})$. 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 $V$ to be a function $\nu \colon \{{\text{ordered bases of }}V\}\to \{\pm 1\}$ which satisfies $\nu (v)=\operatorname {sign} (\det(C_{v}^{u}))\nu (u)$, whenever $u$ and $v$ are ordered bases of $V$ and $C_{v}^{u}$ is the change-of-basis matrix between them.

1. Explain how if $\dim(V)>1$, a norientation is equivalent to an orientation.
2. Come up with a reasonable definition of a norientation of a $k$-dimensional manifold.
3. Explain how a norientation of $M$ induces a norientation of $\partial M$.
4. What is a $0$-dimensional manifold? What is a norientation of a $0$-dimensional manifold?
5. What is the integral of a $0$-form on a $0$-dimensional noriented manifold?
6. What is $\partial [0,1]$ as a noriented $0$-manifold? (Assume that $[0,1]$ is endowed with its "positive" or "standard" orientation/norientation).

Problem C. Let $\omega =-ydx\in \Omega ^{1}({\mathbb {R} }_{x,y}^{2})$.

1. Let $\Gamma$ be the graph in ${\mathbb {R} }_{x,y}^{2}$ of some smooth function $f\colon [a,b]\to {\mathbb {R} }$. Using the inclusion of $\Gamma$ to ${\mathbb {R} }_{x,y}^{2}$, consider $\omega$ also as a 1-form on $\Gamma$. What is $\int _{\Gamma }\omega$?
2. Prove that if $E$ is an ellipse in ${\mathbb {R} }_{x,y}^{2}$ (of whatever major and minor axes, placed anywhere and tilted as you please), then $\int _{\partial E}\omega$ is the area of $E$.
3. Compute also $d\omega$ and $\int _{E}d\omega$.

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).