Difference between revisions of "0708-1300/Homework Assignment 1"

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{{0708-1300/Navigation}}
 
{{0708-1300/Navigation}}
{{In Preparation}}
 
  
 
==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.
 
*#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.
*Solve the following problems from Bredon's book, but submit only the solutions of underlined problems:
+
*Solve the following problems from Bredon's book, but submit only the solutions of the problems marked with an "S":
 
{|align=center border=1 cellspacing=0 cellpadding=5
 
{|align=center border=1 cellspacing=0 cellpadding=5
 
|- align=center
 
|- align=center
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!on page(s)
 
!on page(s)
 
|- align=center
 
|- align=center
|<u>1</u>, 2, 3, <u>4</u>, 5
+
|2, S3, S4, 5
 
|71
 
|71
 
|- align=center
 
|- align=center
|1, <u>2</u>
+
|1, S2
 
|75-76
 
|75-76
 
|- align=center
 
|- align=center
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|80
 
|80
 
|}
 
|}
 +
 +
<span style="color: red;">Important.</span> 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.
  
 
==Due Date==
 
==Due Date==
 
This assignment is due in class on Thursday October 4, 2007.
 
This assignment is due in class on Thursday October 4, 2007.

Latest revision as of 21:11, 24 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:
    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)
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.

Due Date

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