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

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

\def\p{{\bf p}}
\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}

Method 1:

$T_\a(\R^n) = \{(\a,\x ) ~|~ \x \in \R^n \}$

$\phi(\a\x) = \x- \a$

canonical basis $\{\phi^{-1}(e_i) ~|~ i = 1, ..., n \}$


Method 2:

Let $x(t):  \R \rightarrow \R^n$, a $C^1$ curve such that $x(0) = \a$

$x(t) \sim y(t)$ if $x_i'(t) = y_i'(t)$ for $t \in (-\epsilon, \epsilon) $


Let $f([x(t)]) = \x'(0) = (x_1'(0), ..., x_n'(0))$

Let $T_\a(\R^n) = \{ [x(t)] ~|~ x \in C^1, x(0) = \a \}$

$[x(t)] + [y(t)] = f^{-1}(x'(0) + y'(0))$

$\alpha[x(t)]  = f^{-1}(\alpha x'(0))$
\vskip 10pt
\hrule

Let $f: \R^n \rightarrow \R$ and let ${\bf v} \in R^n$ such that $||{\bf v}|| = 1$

The directional derivative of $f$ at $\a$ in the direction of ${\bf v}$ is

$D_{\bf v} f(\a) = lim_{h \rightarrow 0} {f(\a + h{\bf v}) - f(\a) \over h}$

= $D[f(\a + t{\bf v})]_0 = Df_\a {\bf v} =Df_\a \cdot {\bf v} = 
( {\partial f \over \partial x_1}, ...,   {\partial f \over \partial x_n})|_a \cdot {\bf v}  = $ 

\end

2.5

 Let $U \subset R^n$

Let $E_{i\p} = \phi^{-1}_\p(\e_i)$ where $\phi_\p:  T_\p(\R^n) \rightarrow \R^n$, 
$\phi_p(\p\x) = $


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




Suppose

Defn:  A vector field is {\it smooth} if its components relative to the canonical basis are 
$$



\end