0708-1300/Homework Assignment 1: Difference between revisions
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*#Show explicitly that the restricted implicit function theorem, with <math>x_0=y_0=0</math> and <math>\partial_yg=I</math>, is equivalent to general implicit function theorem, in which <math>x_0</math> and <math>y_0</math> are arbitrary and <math>\partial_yg</math> is an arbitrary invertible matrix. |
*#Show explicitly that the restricted implicit function theorem, with <math>x_0=y_0=0</math> and <math>\partial_yg=I</math>, is equivalent to general implicit function theorem, in which <math>x_0</math> and <math>y_0</math> are arbitrary and <math>\partial_yg</math> is an arbitrary invertible matrix. |
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*#Show that the definition <math>f\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}x\\g(x,y)\end{pmatrix}</math> reduces the implicit function theorem to the inverse function theorem. A key fact to verify is that differential of <math>f</math> at the relevant point is invertible. |
*#Show that the definition <math>f\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}x\\g(x,y)\end{pmatrix}</math> reduces the implicit function theorem to the inverse function theorem. A key fact to verify is that differential of <math>f</math> at the relevant point is invertible. |
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*Solve the following problems from Bredon's book, but submit only the solutions of the |
*Solve the following problems from Bredon's book, but submit only the solutions of the problems marked with an "S": |
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|S1, 2, 3, S4, 5 |
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|71 |
|71 |
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|1, |
|1, S2 |
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|75-76 |
|75-76 |
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Revision as of 08:13, 20 September 2007
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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:
- Show explicitly that the restricted implicit function theorem, with and , is equivalent to general implicit function theorem, in which and 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 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.
