0708-1300/Homework Assignment 1

0708-1300/Navigation Panel   Add your name / see who's in! # Week of... Links Fall Semester 1 Sep 10 About, Tue, Thu 2 Sep 17 Tue, HW1, Thu 3 Sep 24 Tue, Photo, Thu 4 Oct 1 Questionnaire, Tue, HW2, Thu 5 Oct 8 Thanksgiving, Tue, Thu 6 Oct 15 Tue, HW3, Thu 7 Oct 22 Tue, Thu 8 Oct 29 Tue, HW4, Thu, Hilbert sphere 9 Nov 5 Tue,Thu, TE1 10 Nov 12 Tue, Thu 11 Nov 19 Tue, Thu, HW5 12 Nov 26 Tue, Thu 13 Dec 3 Tue, Thu, HW6 Spring Semester 14 Jan 7 Tue, Thu, HW7 15 Jan 14 Tue, Thu 16 Jan 21 Tue, Thu, HW8 17 Jan 28 Tue, Thu 18 Feb 4 Tue 19 Feb 11 TE2, Tue, HW9, Thu, Feb 17: last chance to drop class R Feb 18 Reading week 20 Feb 25 Tue, Thu, HW10 21 Mar 3 Tue, Thu 22 Mar 10 Tue, Thu, HW11 23 Mar 17 Tue, Thu 24 Mar 24 Tue, HW12, Thu 25 Mar 31 Referendum,Tue, Thu 26 Apr 7 Tue, Thu R Apr 14 Office hours R Apr 21 Office hours F Apr 28 Office hours, Final (Fri, May 2) Register of Good Deeds Errata to Bredon's Book

Announcements go here

Reading

Read sections 1-5 of chapter II of Bredon's book three times:

• First time as if you were reading a novel - quickly and without too much attention to detail, just to learn what the main keywords and concepts and goals are.
• Second time like you were studying for an exam on the subject - slowly and not skipping anything, verifying every little detail.
• And then a third time, again at a quicker pace, to remind yourself of the bigger picture all those little details are there to paint.

Doing

• Solve and submit the following two problems:
  1. Show explicitly that the restricted implicit function theorem, with $x_{0}=y_{0}=0$ and $\partial _{y}g=I$, is equivalent to general implicit function theorem, in which $x_{0}$ and $y_{0}$ are arbitrary and $\partial _{y}g$ is an arbitrary invertible matrix.
  2. Show that the definition $f{\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}x\\g(x,y)\end{pmatrix}}$ reduces the implicit function theorem to the inverse function theorem. A key fact to verify is that differential of $f$ at the relevant point is invertible.
• Solve the following problems from Bredon's book, but submit only the solutions of the problems marked with an "S":

problems	on page(s)
2, S3, S4, 5	71
1, S2	75-76
1-4	80

Important. Note the change from an earlier version of this assignment - Franklin Vera and Damir Kinzebulatov found that exercise 1 on page 71 of Bredon's book is wrong, so it was removed from the assignment and replaced by exercise 3.

Due Date

This assignment is due in class on Thursday October 4, 2007.