0708-1300/Homework Assignment 1
From Drorbn
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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.
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Doing
- Solve and submit the following two problems:
- Show explicitly that the restricted implicit function theorem, with x0 = y0 = 0 and
, is equivalent to general implicit function theorem, in which x0 and y0 are arbitrary and 
is an arbitrary invertible matrix.
- Show that the definition
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.
- Show explicitly that the restricted implicit function theorem, with x0 = y0 = 0 and
- 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.
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Due Date
This assignment is due in class on Thursday October 4, 2007.
