\magnification 1200
\parskip 10pt
\parindent 0pt
\hoffset -0.3truein
\hsize 7truein
\voffset -0.5truein
\vsize 10truein

\def\N{{\bf N}}
\def\S{\Sigma_{i=1}^n}

\def\R{{\bf R}}
\def\C{{\bf C}}
\def\x{{\bf x}}
\def\y{{\bf y}}
\def\e{{\bf e}}
\def\a{{\bf a}}
\def\i{{\bf i}}
\def\j{{\bf j}}
\def\k{{\bf k}}
\def\ep{\epsilon}
\def\f{\vskip 10pt}
\def\u{\vskip -8pt}


HW 3:  

Let $F:  \R \rightarrow \R^2$, $F(x) = (2,3)x$ \hfil \break
$G:  \R^2 \rightarrow \R^3$, 
$G(x, y) = 
(xy, x^2, x + 2y + 5)$ \hfil \break 
$H:  \R \rightarrow \R^2$, $H(x) = (x^2, x^3)$ \hfil \break 
$k: \R^2 \rightarrow \R$, $k(x, y) = x^8 + 5xy$.    

1.) Use the chain rule to calculate $D( G \circ F)_2$

2.) Use the product rule to calculate $D(FH)_2$

3.)  Let $\a = (3, 4)$.  Let $X_a = 9E_{1\a} - E_{2\a}$. Then $X_a^*(k) =$ 


$F$ is a $C^r$-{\it diffeomorphism} if

(1)  $F$ is a homeomorphism

(2)  $F, F^{-1} \in C^r$


$F$ is a {\it diffeomorphism} if $F$ is a $C^\infty$-diffeomorphism.


4.)  Give an example of a homeomorphism which is analytic (ie $C^\infty$ and near $a$, $f(x) = f(a) + f'(a)(x-a) + {f''(a) \over 2!} (x - a)^2 + ...$ , its Taylor series) which is not a diffeomorphism.


5.)  Suppose $F'(\x) = {\bf 0}$ for all $\x \in U^{open} \subset \R^n$.  Show $F$ cannot be a homeomorphism.  What can you say about $F$ (hint:  MVT).


6.)  Suppose $f: R \rightarrow R$, $f'(x) \not= 0$ for all $x \in U$.  Show that the derivative of $f^{-1}$ exists for all $y \in f(U)$

7.)  Suppose $F$ is a $C^1$-diffeomorphism.  Show that  $DF_x$ is nonsingular (ie $det(DF_x) \not= 0) \forall x \in dom(F)$

8.)  Ex 1:  Show $F: \R^n \rightarrow \R^n$, $F(\x) = \x + \a$ is a diffeomorphism.

9.)  Ex 2:  Determine when $F: \R^n \rightarrow \R^m$, $F(\x) = Ax$, where $A$ is an $m \times n$ matrix, is a diffeomorphism.
$DF_x$ = .


Note that if $F$ and $G$ are diffeomorphism, then $F \circ G$ is a diffeomorphism (when $F \circ G$ is defined).
 
Thm 6.5 (Contracting mapping theorem):  Let $M$ be a complete metric space and let $T:  M \rightarrow M$.  Suppose there 
exists a constant $\lambda \in [0, 1)$ such that for all $ x, y \in M$, $d(T(x), T(y)) \leq \lambda d(x, y)$.  Then $T$ has 
a 
unique fixed point.

Proof:  See class notes (Recall $T^n(x_0)$ is a Cauchy sequence.  Since $M$ be a complete metric space, $T^n(x_0)$ converges, say to $a$.  Then $d(T(a), a) = 0$).

Thm 6.4 (Inverse Function Theorem):  Suppose $F: W^{open} \subset \R^n \rightarrow R^n \in C^r$.  Suppose for $a \in W$, $det(DF_a) \not=0$.  Then there exists $U$ such that $a \in U^{open}$, $V = F(U)$ is open, and
$F:  U \rightarrow V$ is a $C^r$-diffeomorphism.
Moreover, for $x \in U$ and $y = F(x)$,  $DF^{-1}_y = (DF_x)^{-1}$ 




\end