\magnification 1800
\parskip 10pt
\parindent 0pt
\hoffset -0.3truein
\hsize 7truein

\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\p{{\bf p}}
\def\ep{\epsilon}
\def\f{\vskip 10pt}
\def\u{\vskip -8pt}

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



2.5

 Let $U \subset R^n$ 




Defn: A {\it vector field} is a function, ${\cal V}:  U \rightarrow \cup_{\a \in U}T_\a(\R^n)$, such that ${\cal V}:  (\a) \in T_\a(\R^n)$


 



Defn:  A vector field is {\it smooth} if its components relative to the canonical basis $\{E_{i\a} ~|~ i = 1, ..., n \}$ are smooth.

Ex:  ${\cal V}:  \R^2 \rightarrow \cup_{\a \in U}T_\a(\R^2)$, 

${\cal V}(x, y) = (2x, -y) = 2x E_{1\a} - y E_{2\a}$.

Defn:  A field of frames is a set of vector fields $\{ {\cal V}_1, ..., {\cal V}_2 \}$ such that $\{ {\cal 
V}_1(\a), ..., {\cal V}_2(\a) \}$ forms a basis for  $T_\a(\R^n)$ for all $\a$.


Ex:  $\{ E_{1\a}, ..., E_{n\a}\}$ on $\R^n$.



Ex: $\{ x  E_{1\a} + y E_{2\a}, y E_{1\a} - x E_{2\a} \}$ on $\R^n - \{{\bf 0}\}$.

\vskip 10pt

\hrule

Let $ {\cal V} = \S \alpha_i (\a) E_{i\a}$ be a smooth vector field on $U$. 


$ {\cal V}:  C^{\infty} \rightarrow C^{\infty}$ 

$ {\cal V}(f) = \S \alpha_i (\a) {\partial f \over \partial x_i} (\a)$ is a derivation.


\vskip 10pt

%%\hrule
\eject

Thm 5.1:  Let $F^{closed} \subset \R^n$, $K^{compact} \subset \R^n$, $F \cap K = \emptyset$.  There there is a $C^\infty$ function $\sigma:  \R^n \rightarrow [0, 1]$ such that $\sigma(K) = \{1\}$, $\sigma(F) = \{0\}$


Show that $h(t) = \cases{0 & $t \leq 0$ \cr e^{-1 \over t}  & $t > 0$}$ is $C^\infty$, (but not $C^\omega$). 


Let $\overline{g} (x) = { h(\epsilon^2 - ||\x||^2) \over h(\epsilon^2 - ||\x||^2)  + h(||\x||^2 -  { \epsilon^2 \over 4} )}$ 

$\overline{g}(\x) = \cases{ 1 & $0 \leq ||x || \leq {\epsilon \over 2}$ \cr
positive &  ${\epsilon \over 2} \leq ||x || < {\epsilon }$ \cr
0 & $||x || \geq \epsilon$ }$ 

Let $g(\x) = \overline{g}(\x - \a )$
$$\sigma(\x) = 1 - \Pi_{i = 1}^k (1 - g_i)$$ where $K \subset \cup_{i=1}^k B_{{\ep \over 2}}(\a_i)$ and $B_{\ep}(\a_i) \subset \R^n - F$.


%%If there is time, do problem 1 in 2.5 in discussion section





\end