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)
S1, 2, 3, S4, 5	71
1, S2	75-76
1-4	80

Due Date

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