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

\def\R{{\bf R}}
\def\C{{\bf C}}
\def\N{{\bf N}}
\def\r{{\bf r}}
\def\c{{\bf c}}
\def\p{{\bf p}}
\def\x{{\bf x}}
\def\y{{\bf y}}
\def\e{{\bf e}}
\def\i{{\bf i}}
\def\j{{\bf j}}
\def\k{{\bf k}}
\def\0{{\bf 0}}
\def\a{{\bf a}}
\def\emph{}

\def\S{\Sigma_{i=1}^n}
\def\Sm{\Sigma_{i=1}^m}

\def\s{{\sigma}}
\def\ep{\epsilon}
\def\f{\vskip 10pt}
\def\h{\hskip 10pt}
\def\u{\vskip -8pt}
\def\bh{\hfil \break}

$T_p(M) = \{v:  G(p) \rightarrow \R ~|~ v $ is linear and satisfies the Leibniz 
rule $ \}$ 


$G(p) = \{[g] ~|~ g^{smooth}:  U \rightarrow \R$, for some $U^{open}$ such that $p 
\in U
\subset M \}$ is an algebra over $\R$.

 $v \in T_p(M)$ is called a {\it derivation}
\vskip 5pt
\hrule
\vskip -5pt

The directional derivative of $[g]$ in direction $[\alpha]$ = 
$$D_{\alpha} g = {d(g \circ 
\alpha) \over dt}|_{t = 0} = g'(\alpha(0))\alpha'(0) 
\in \R$$

Properties:

$D_\alpha$ is linear and satisfies the Liebniz rule.  i.e,

1a.) $D_{\alpha }(g+h)=D_{\alpha }g+D_{\alpha }h$
\hfill  1b.) $D_{\alpha }(cg)=cD_{\alpha }g$

2.) $D_{\alpha }(g\cdot h)=D_{\alpha }g\cdot h(p)+g(p)\cdot D_{\alpha }h$

Thus $D_\alpha \in T_p(M)$  

\vskip 5pt
\hrule
\vskip -5pt

Thm:  Let $M$ be an $m$-manifold, then  $T_p(M)$ is an $m$-dimensional real vector space.  
[$(c_1v + c_2w)(f) = c_1v(f) + c_2 w(f)$].

Take a chart  $(U, \phi)$  at $p$ where $\phi(p) = \0$, 

The {\it standard  basis} for $T_p(M)$ w.r.t.$(U, \phi)$ =  
$\{v_1, ..., 
v_m\}$, 

where  $v_i = D_{\alpha_i}$ and

 $\alpha_i: (-\ep, \ep) 
\rightarrow M$, 
$\alpha_i(t) = \phi^{-1}(0, ..., t, ..., 0) $  for some $\ep > 0$. 

\vfill
\eject

Prop:  $\{v_1, ..., 
v_m\}$ are linearly independent.

Proof:  Evaluate $v_i$ at a ``projection map".

Thm: $\{v_1, ..., 
v_m\}$ span $T_p(M)$. 

If $v \in T_p(M)$, then $v = \Sm a_i v_i$ where $a_i = v([\pi_i \circ \phi])$

Prop:  If $v$ is a derivation, and $f$ is constant, then $v(f) = 0$.

 \vskip 10pt
\hrule


Suppose $f^{smooth}: M \rightarrow N$, $f(p) = q$. 

The {\it tangent (or 
differential) map},
$$df_p:T_pM \rightarrow T_qN$$
$df_p(v) =$ the derivation which takes  $[g] \in G(q)$ to the real number $v([g \circ f])$ 


I.e., $df_p$ takes the derivation $(v:G(p) \rightarrow \R) \in T_pM$ 
\vskip 10pt
\centerline{to the 
derivation in  $T_qN$ which takes ~~~~~~}

\vskip 10pt
\rightline
{the germ $[g] \in G(q)$ to the real number $v([g 
\circ f])$.}

 \vskip 10pt
\hrule

Suppose $f:  M \rightarrow N$ is smooth.  Then $f^*: G(q) \rightarrow G(p)$ is a homomorphism.

Let $V$ be a real vector space.  \hfil \break
Then the dual space, $V^* = \{ f: V \rightarrow \R ~|~  
f$ is linear $\}$

Proposition:  With respect to this choice of basis, the matrix of $%
d_{p}f$:~$T_{p}M\rightarrow T_{q}N$ is $({\partial F_{i} \over \partial x_{j}})_{\varphi _{(p)}}$, where $x_{j}$ are coordinates
in $\R^{m}$ and $F_{i}=(F_{1},\cdots ,F_{n})$ in coordinates for $%
\R^{n}$.

\vfill
\eject


Here are some more properties of $d_{p}f$ and $T_{p}M$:
Let $f$\emph{:}~$M\rightarrow N$ be smooth.

1.)  If $f${:}~$M\rightarrow N$ is a
diffeomorphism, the $d_{p}f$ is an isomorphism, for all $p\in M$.

2.)  If $d_{p}f=0$ for all $p\in M$ iff $f$ is
a constant map.

3.)  If $id: M \rightarrow M$, $id(x) = x$, then $d_{p}(id) = I_m$ 

4.)  $d_r(f \circ h) = d_p(f) \circ d_r (h)$ where $p = h(r)$.

5.)  If $N\cong M/G,$ where $G$ is a discrete
Lie group acting properly discontinuously on $M$, and $f$:~$M\rightarrow M/G$
is the orbit map, the $d_{p}f$ is an isomorphism for all $p$.

6.)  $T_{(p,q)}(M\times N)\cong T_{p}(M)\times
T_{q}(N)$.



\vfill
\eject

Rank Theorem:  Suppose $A_0 \subset \R^n$, $B_0 \subset \R^m$, $F: A_0 \rightarrow B_0 \in C^1$ 
$a \in A_0, b \in B_0$.  Suppose rank F = $k$.

Then there exists $A^{open} \subset A_0$ such that $a \in A$ and  $B^{open} \subset B_0$ such that $b \in B$ and $G, H$, $C^r$ diffeomorphisms

such that $G:  A \rightarrow U^{open} \subset \R^n$, $H:  B \rightarrow V^{open} \subset \R^m$ and
$$H \circ F \circ G^{-1} (x_1, ..., x_n) = (x_1, ..., x_k, 0, ..., 0)$$

\vskip 5pt
\hrule
\vskip -5pt


Proposition:  rank $DF(a) = k$  implies there exists $V$ open such that $a \in V$ and $DF(x) \geq k$ for all $x \in V$

Proof:  

Rank of $A$ = $dim(span\{\r_1, ..., \r_m\}) = dim(span\{\c_1, ..., \c_m\})$ 

\rightline{= maximum order of any nonvanishing minor determinant.}

Use determinant is a continuous function.

\vskip 5pt
\hrule
\vskip -5pt

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


3.4  1, 4, 8, 10, 12, 13, 21, 24