|Welcome to Math 257
|Edits to the Math 257 web pages no longer count for the purpose of good deed points.
||Notes and Links
||About This Class; Day 1 Handout (pdf, html); Monday, Wednesday, Friday, Day 2 Handout (pdf, html); First Week Notes.
||Monday, Wednesday, Tutorial 2 Handout, Friday, Second week notes, HW1, HW1 Solutions.
||Monday, Wednesday, Tutorial 3 Handout, Friday, Third week notes, Class Photo, HW2, HW2 Solutions.
||Monday, Wednesday, Friday, 4th week notes, HW3, HW3 Solutions.
||Monday is Thanksgiving, no class; Wednesday, Friday, 5th week notes, HW4, HW4 Solutions.
||Monday, Wednesday, Friday, 6th week notes, HW5, HW5 Solutions.
||Monday, Wednesday, Friday, 7th week notes
||Monday; Term test 1; Wednesday, HW6, HW6 pdf, HW6 Solutions, Friday, 8th week notes.
||Monday is last day to switch to MAT 237; Monday-Tuesday is UofT Fall Break; Wednesday, HW7, HW7 Solutions, Friday, 9th week notes.
||Monday, Wednesday, HW8, HW8 pdf, HW8 Solutions, Friday, Lecture recordings, 10th week notes.
||Monday, Wednesday, HW9, HW9 pdf, HW9 Solutions, Friday, Lecture recordings, 11th week notes.
||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.
||Monday, Wednesday, 13th week notes Semester ends on Wednesday - no class Friday.
||No classes: other classes' finals, winter break.
||Class resumes Friday at RS211, no tutorials or office hours this week, Friday, Friday notes.
||Monday, Wednesday, Friday, Weekly notes
||Monday, Term test 2; Wednesday, HW11, HW11 inline pdf,HW11 solutions, Friday, Weekly notes
||Monday; Hour 44 Handout (pdf, html); Wednesday, HW12, HW12 inline pdf, HW12 Solutions, Friday, Weekly notes
||Monday, Wednesday, HW13, HW13 inline pdf, HW13 Solutions,Friday, Weekly Notes
||Monday, Wednesday, HW14, HW14 inline pdf, Friday, Weekly Notes
||Monday, Wednesday, HW15, HW15 inline pdf, HW15 solutions, Friday,Weekly Notes, UofT examination table posted on Friday.
||Reading week - no classes; Tuesday is the last day to drop this class.
||Monday, Wednesday, HW16, HW16 inline pdf, Friday, Weekly Notes
||Monday, Wednesday, Friday, Weekly Notes
||Monday, Term test 3 on Tuesday at 5-7PM; Wednesday, HW17, HW17 inline pdf, HW17 Solutions, Friday, Weekly Notes,Orientation_notes
||Monday, Wednesday, HW18, HW18 inline pdf, Friday
||No Monday class, Wednesday, Thursday (makeup for Monday), HW19, HW19 inline pdf, Friday, Weekly Notes
||Monday, Wednesday; Semester ends on Wednesday - no tutorials Wednesday and Thursday, no class Friday.
||The Final Exam on Thursday April 20 (and some office hours sessions before).
|Register of Good Deeds
Add your name / see who's in!
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.
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).
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 .
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.
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).