0708-1300/Homework Assignment 1

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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_yg=I, is equivalent to general implicit function theorem, in which x_0 and y_0 are arbitrary and \partial_yg 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.