\magnification 1400
\parskip 10pt
\parindent 0pt
\hoffset -0.3truein
\hsize 7truein

\voffset -0.4truein
\vsize 10truein

\def\emph{}

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

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

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

\def\0{{\bf 0}}
\def\Z{{\bf Z}}
\def\R{{\bf R}}
\def\C{{\bf C}}
\def\x{{\bf x}}
\def\a{{\bf a}}
\def\y{{\bf y}}
\def\e{{\bf e}}
\def\i{{\bf i}}
\def\j{{\bf j}}
\def\k{{\bf k}}
\def\p{{\bf p}}
\def\f{\phi}
\def\s{\sigma}
\def\ep{\epsilon}
\def\f{\vskip 10pt}
\def\u{ \vskip -5pt }
\def\h{\vskip -5pt \hskip 20pt}
\def\grad{ \nabla }

\def\tens{ \otimes }
\def\l{\vskip 5pt
\hrule
\vskip -5pt}

\l

Let $\Lambda$ be an algebra over $\R$


The exterior algebra $\Lambda_n^*$ is the algebra with product $\wedge$ generated by
$v_1, ..., v_n$ with unit such that $v_i \wedge v_j = - v_j \wedge v_i$





\l


Let $V$ be a finite-dimensional vector space over $\R$.

The dual of $V$ = $V^* = \{f:  V \rightarrow \R ~|~ f$ linear $\}$

Note $V^*$ is a vector space.  The elements of $V^*$ are called {\it covectors}.


If $e_1, ..., e_n$ basis for $V$, then $w_1, ..., w_n$ basis for $V^*$ where 
$w_i: V \rightarrow \R$ where $w_i(e_j) = \delta_{ij} = \cases{1 & $i = j$ \cr 0 & $i \not= j$}$


Let $F_*:  V \rightarrow W$ be a linear map between vector spaces \bh
The dual map map is $F^*:  W^* \rightarrow V^*$, $F(g) = g \circ F$.

$F_*$ is injective implies $F^*$ injective 

$F_*$ is surjective implies $F^*$ surjective 


$(G_* \circ F_*)^* = F^* \circ G^*$.


$d:  V \rightarrow (V^*)^*$, $d(v) = h$ where $h: V^* \rightarrow R$, $h(f) = f(v)$.

Thus $ (V^*)^*$ is naturally isomorphic to $V$.

\l

Defn:  The dual of $T_pM= T_p^*M$ is the {\it cotangent space} to $M$ at $p$.

If $ {\partial \over \partial x_1}, ..., {\partial \over \partial x_m}$ is a basis for $T_pM$, then the  dual basis will be denoted $dx_1, ..., dx_m$. 

%%Defn:  The cotangent bundle of M 




\end