\magnification 1600
\parskip 10pt
\parindent 0pt

\def\R{\bf R}
\def\C{\bf C}
\def\x{\bf x}
\def\y{\bf y}
\def\e{\bf e}
\def\ep{\epsilon}
\def\f{\phi}

$\R^n$ a vector space over $\R$ (or $\C$) with canonical basis $\{ {\bf e_1}, ..., {\bf e_n}\}$  where ${\bf e_i} = (0, ., 0, 1, 0, ..., 0)$

%%An $n$-dimensional vector space over $\R$ with a positive definite inner product is 
%%called {\it Euclidean} 

Inner product on $\R^n$: $(\x, \y) = \Sigma_{i=1}^n x_iy_i$

The basis is orthonormal:  $({\bf e_i}, {\bf e_j},) = \delta_{ij} = \cases{ 0 & $i \not= j$ \cr 1 & $i = j$}$
 
$d(\x, \y) = || \x - \y || = (\x, \y)^{1 \over 2}$

The {\it norm} of $\x = ||\x|| = d(\x, {\bf 0})$ 

$B_{\ep}^n(\x) = \{\y \in \R^n ~|~ d(\x, \y) <  \ep \}$ = ball of radius $\ep$ centered at 
$\x$.



$C_{\ep}^n(\x)  = \{ \y \in \R^n ~|~  |x_i - y_i| < \ep, i = 1, ..., n \}$ = cube of side $2\ep$ centered at $\x$.  

$\R^1 = \R$, $\R^0 = \{0\}$ .

I.2

$\R^n = {\bf E^n}$ where a coordinate system is defined on ${\bf E^n}$ 

A property is {\it Euclidean} if is does not depend on the choice of an orthonormal coordinate system.

%%analytic

I.3 Topological Manifolds

Defn:  $M$ is {\it locally Euclidean of dimension $n$} if for all $p \in M$, there exists an open set $U_p$ such that $p \in U_p$ and there exists a homeomorphism $f_p:  U_p \rightarrow V_p$ where $V_p \subset \R^n$.

\vfill
\eject

Defn 3.1:  An {\it n-manifold}, $M$, is a topological space with the following properties:

1.)  $M$ is locally Euclidean of dimension $n$.

2.)  $M$ is Hausdorff.

3.)  $M$ has a countable basis.


Give an example of a locally Euclidean space which is not Hausdorff:

\vskip 1in

Ex 3.2:  If $U$ is an open subset of an $n-$manifold, then $U$ is also an $n-$manifold.

\vskip 0.5in

Ex 3.3:  $S^{n} = \{ \x \in \R^{n+1} ~|~ ||x|| = 1 \}$ is an $\underline{\hskip .5in} $manifold

Proof. stereographic projection:


\vskip 1in
\eject
projection: 

\vskip 1in


Remark 3.5.  For a ``smooth'' manifold, $M \subset \R^n$, can choose a projection by using the fact that for all $p \in M$ there exists a unit normal vector $N_p$ and tangent plane $T_p(M)$ which varies continuously with $p$.  

Example:  smooth and non-smooth curve.
\vskip 1in

Example 3.4:  The product of two manifolds is also a manifold.

Example:  Torus = $S^1 \times S^1$.

Theorem 3.6:  A manifold is 

1.) locally connected, 
2.) locally compact, 
3.) a union of a countable collection of compact subsets, 
4.) normal, and 
5.) metrizable.


Defn: $X$ is {\bf locally connected at $x$} if for every neighborhood $U$ of $x$, there exists connected
open set
$V$ such that $x \in V \subset U$.
$X$ is {\bf locally connected} if $x$ is locally connected at each of its points.

Defn:  $X$ is {\bf locally compact at $x$} is there exists a compact set $C \subset X$ and a set $V$ open in
$X$ such that $x \in V \subset C$.
$X$ is {\bf locally compact} if it is locally compact at each of its points.


\vskip 1.7in
\vfill

Defn:  $X$ is {\bf regular} if one-point sets are closed in $X$ and if for all closed sets $B$ and for all points $x \not\in B$, there exist disjoint open sets, U, V, such that $x \in U$ and $B \subset V$.


Defn:  $X$ is {\bf normal} if one-point sets are closed in $X$ and if for all pairs of disjoint closed sets $A, ~B$, there exist disjoint open sets, U, V, such that $A \subset U$ and $B \subset V$.



\vskip 5pt
\hrule

Brouwer's Theorem on Invariance of Domain (1911).  If $\R^n = \R^m$, then $n = m$.
\vskip 5pt
\hrule

Recall:  $M$ is {\it locally Euclidean of dimension $n$} if for all $p \in M$, there exists an open set $U_p$ such that $p \in U_p$ and there exists a homeomorphism $f:  U_p \rightarrow V_p$ where $V_p \subset \R^n$.

$(U_p, f)$ is a {\it coordinate nbhd} of $p$.  

Given $(U_p, f)$ Let $q \in U \subset M$. $f(q) = (f_1(q), f_2(q), ...., f_n(q)) \in \R^n$ are the  {\it coordinates} of $q$.

\eject
I.4  Manifolds with boundary and Cutting and Pasting


If $dim M = 0$, then $M = $

If $M$ is connected and $dim M = 1$, then 


Thm 4.1:  Every compact, connected, orientable 2-manifold is homeomorphic 
to a sphere, or to a connected sum of tori, or to a connected sum of 
projective planes

\eject

Let upper half-space, $H^n = \{(x_1, x_2, ..., x_n) \in \R^n ~|~ x_n \geq 0 \}$, 

$\partial H^n = \{(x_1, x_2, ..., x_{n-1}, 0) \in \R^n  \} \sim \R^{n-1}$  


$M$ is a manifold with boundary if it is Hausdorff, has a countable basis, and if for all $p \in U$, there exists an open set $U_p$ such that $p \in U_p$ and there exists a homeomorphism $f:  U_p \rightarrow V_p$ one of the following holds:

i.)  $V_p \subset H^n - \partial H^n$ ($p$ is an interior point) or

ii.)  $V_p \subset H^n$ and $f(p) \in \partial H^n$ ($p$ is a boundary point).


$\partial M$ = set of all boundary points of $M$ is an (n-1)-dimensional manifold.
\vskip 10pt


 
\vskip 1in

\end

1.2 Parametric Equations

$x = f_1(t)$ \break
$y = f_2(t)$

$x = cos(t)$ \break
$y = sin(t)$

$x = cos(2t)$ \break
$y = sin(2t)$


Lines:


Find an equation of the line containing the point (1, 2, 3) and is parallel to the vector (5, 3, 4)





Find an equation of the line containing the points (1, 2, 3) and  (6, 1, 3) 



Symmetric form

Determine if the following lines intersect:


Other parametric equations:

cycloid


1.2:  25, 26, 29, 34, 36, 38

1.3  Dot product


8, 10, 14, 17