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

From Drorbn
Jump to: navigation, search
 
Line 1: Line 1:
 
{{0708-1300/Navigation}}
 
{{0708-1300/Navigation}}
{{In Preparation}}
 
  
 
==Reading==
 
==Reading==
Line 13: Line 12:
 
*#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 starred problems:
 
{|align=center border=1 cellspacing=0 cellpadding=5
 
{|align=center border=1 cellspacing=0 cellpadding=5
 
|- align=center
 
|- align=center
Line 19: Line 18:
 
!on page(s)
 
!on page(s)
 
|- align=center
 
|- align=center
|<u>1</u>, 2, 3, <u>4</u>, 5
+
|*1, 2, 3, *4, 5
 
|71
 
|71
 
|- align=center
 
|- align=center
|1, <u>2</u>
+
|1, *2
 
|75-76
 
|75-76
 
|- align=center
 
|- align=center

Revision as of 09:12, 20 September 2007

Announcements go here

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 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.