Difference between revisions of "07081300/Homework Assignment 1"
From Drorbn
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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 the  +  *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  
−    +  S1, 2, 3, S4, 5 
71  71  
 align=center   align=center  
−  1,  +  1, S2 
7576  7576  
 align=center   align=center 
Revision as of 08:13, 20 September 2007

Reading
Read sections 15 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  7576 
14  80 
Due Date
This assignment is due in class on Thursday October 4, 2007.