0708-1300/Homework Assignment 1 - Revision history

Drorbn at 01:11, 25 September 2007

2007-09-25T01:11:25Z

← Older revision		Revision as of 21:11, 24 September 2007
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	!on page(s)		!on page(s)
	\|- align=center		\|- align=center
	\|~~S1,~~ 2, 3, S4, 5		\|2, S3, S4, 5
	\|71		\|71
	\|- 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==

Drorbn at 13:13, 20 September 2007

2007-09-20T13:13:47Z

← Older revision		Revision as of 09:13, 20 September 2007
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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 ~~starred~~ 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
Line 18:		Line 18:
	!on page(s)		!on page(s)
	\|- align=center		\|- align=center
	\|1, 2, 3, 4, 5		\|S1, 2, 3, S4, 5
	\|71		\|71
	\|- align=center		\|- align=center
	\|1, *2		\|1, S2
	\|75-76		\|75-76
	\|- align=center		\|- align=center

Drorbn at 13:12, 20 September 2007

2007-09-20T13:12:48Z

← Older revision		Revision as of 09:12, 20 September 2007
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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 starred problems:
	{\|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		\|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

Drorbn at 23:42, 19 September 2007

2007-09-19T23:42:55Z

New page

{{0708-1300/Navigation}}
{{In Preparation}}

==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>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.
*Solve the following problems from Bredon's book, but submit only the solutions of underlined problems:
{|align=center border=1 cellspacing=0 cellpadding=5
|- align=center
!problems
!on page(s)
|- align=center
|1, 2, 3, 4, 5
|71
|- align=center
|1, 2
|75-76
|- align=center
|1-4
|80
|}

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