0708-1300/Homework Assignment 1: Difference between revisions
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{{In Preparation}} |
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==Reading== |
==Reading== |
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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 |
*Solve the following problems from Bredon's book, but submit only the solutions of the starred problems: |
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!on page(s) |
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|*1, 2, 3, *4, 5 |
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|71 |
|71 |
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|1, |
|1, *2 |
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|75-76 |
|75-76 |
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Revision as of 08:12, 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 [math]\displaystyle{ x_0=y_0=0 }[/math] and [math]\displaystyle{ \partial_yg=I }[/math], is equivalent to general implicit function theorem, in which [math]\displaystyle{ x_0 }[/math] and [math]\displaystyle{ y_0 }[/math] are arbitrary and [math]\displaystyle{ \partial_yg }[/math] is an arbitrary invertible matrix.
- Show that the definition [math]\displaystyle{ 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]\displaystyle{ f }[/math] at the relevant point is invertible.
- Solve the following problems from Bredon's book, but submit only the solutions of the starred problems:
| problems | on page(s) |
|---|---|
| *1, 2, 3, *4, 5 | 71 |
| 1, *2 | 75-76 |
| 1-4 | 80 |
Due Date
This assignment is due in class on Thursday October 4, 2007.
