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\newtheorem{lem}[thm]{��������}
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\begin{document}
\parindent=0cm
\section*{Tangent vector.}

{\bf 1.} In $\rb^n$, we have an obvious notion of tangent vectors
as a {\bf velocity vector of a curve.} We want a notion of tangent
vector on a manifold.\\

\begin{figure}[htb]
\centerline{\includegraphics*[scale=0.5,clip=true]{grp13.eps}}

\end{figure}

$\hspace{15em}$ �׸� 13\\

���� manifold�� curve $\sig : I \subset \rb \rightarrow M $��
�����ϰ� �װ��� �ӵ� vector�� �������� �� curve�� coordinate chart
$\varphi(U)$���� �������� $C: x= x(t)$��, �Ǵٸ� coordinate chart
$\psi(U)$���� �������� $y=y(t)$�� �־����ٸ� ������ tangent vector
$X=\frac{dx}{dt}$, $Y=\frac{dy}{dt}$�� �־����� �� ���� coordinate
transition map $y=f(x):=\psi\varphi^{-1}(x)$�� ���� $Df(x)=Y$ ��
�־�����. ������ $Y=\frac{d}{dt}(f\circ
x)=Df\cdot\frac{dx}{dt}=Df\cdot X$ �Ǵ� �̰��� ��ȣ
$f'=Df=\frac{\pa y}{\pa
x}=\frac{\pa(y_1,\cdot\cdot,y_n)}{\pa(x_1,\cdot\cdot,x_m)}$ ��
�̿��Ͽ� $\frac{dy}{dt}=\frac{\pa y}{\pa x}\cdot\frac{dx}{dt}$��
���� �� �� �ִ�.

���� manifold $M$�� tangent vector�� �����ϴ� �� ���� �����
coordinate transition�� differential�� ���� coordinate chart������
tangent vector���� identify�� ������ ���� ���̴�.\\

{\bf 2. Intrinsic way of defining tangent vector.}\\

$\rb$���� tangent vector $X$�� characterize�� �� �ִ� �Ѱ���
����� ����̺�(directional derivative)�� �̿��Ͽ� $X$��
differential operator�� ���� ����̴�. �ٽø��� $f: \rb^n
\rightarrow \rb$�� $p$�� �ٹ濡�� ���ǵ� $\ccn$ �Լ��� �ξ��� ��
$Xf := D_X f = \nabla f \cdot X = \frac{d}{dt}\mid_0 (f \circ
\sig(t))$(���⿡�� $\sig(t)$�� $X$�� "fitting"�ϴ� ��̴�.
�ٽø��ϸ� $\sig(0)=p$, $\sig'(0)=X$�� �����ϴ� ��̴�.)��
��������. �׷��� $X$��
"directional derivative"�� ���� �������� �����Ѵ�.\\

$<Properties \,\,of\,\, directional\,\, derivative.>$

\[ (*) \left[
\begin{array}{ll}
& X(f+g)=Xf+Xg$ , $X(af)=a(Xf)$, $\forall\,a\in \rb. (linearity) \\
& X(f\cdot g)(p)=(Xf)(p)g(p)+f(p)(X\cdot g)(p): (derivation
\,\,property)
\end{array} \right. \]


$(*)$�� �����ϴ� differential operator�� linear derivation�̶��
�Ѵ�.\\

\begin{defn}
{\it X is a {\bf tangent vector }at $p\in M$ if $X$ is a linear
derivation defined on $\ccn(p)$ ($=$ the space of $\ccn$ functions
defind on some neighborhood of $p$) , i.e.,
$X:\ccn(p)\rightarrow\rb $ with property $(*)$}
\end{defn}

���� ���ǰ�� $f, g \in \ccn(p)$�� ���� $f+g$, $fg$�� $domain(f)
\cap domain(g)$�󿡼� ���ǵȴ�( ���� $f+g$, $fg$ $\in \ccn(p)$��
�����ϰ� ���� ���ǰ� well-defined�ȴ�.).\\

$T_pM :=$ the set of all tangent vector at $p$.

$\forall X,Y\in T_pM\,,\,\,(X+Y)f:=Xf+Yf, \,\,(aX)f:=a(Xf)$��
�����ϸ�

$T_pM$�� vector space�� �ȴ�.\\

���� �ڿ������� $T_p M$�� basis�� �����ΰ��ϴ� ������
�����Եȴ�. ���������� $\frac{\partial}{\partial x_i}$���� �ٷ�
$T_p M$�� basis�� �ȴٴ� ���� �����ش�. �� ������ $X \in T_p M$ ��
$\frac{\partial}{\partial x_i}$���� ��������(linear
combination)���� ǥ���ǰ� �̶� �� ������� �ٷ� $X x_i$�� �ȴ�.\\


\begin{thm}
Let $(u, \phi)$ be a coordinate chart about $p \in M$, and let
$x_i = u_i \circ \phi$ ($u_i = i$ th coordinate function on
$\rb^n$). \\
Define "coordinate tangent vectors" at $p$,
\,\,$\frac{\partial}{\partial x_i}\mid _p$ (or
$\frac{\partial}{\partial x_i}(x)$) by $\frac{\partial}{\partial
x_i}\mid_p f = \frac{\partial (f \circ
\phi ^{-1} ) }{\partial u_i} \mid_{\phi(p)} $\\
Then $\{ \frac{\partial}{\partial x_1}(p), \cdots
,\frac{\partial}{\partial x_n}(p) \}$ forms a basis for the vector
space $T_p M$ and each $X \in T_p M$ can be represented uniquely
as $$ X = \sum_{i=1}^{n} (X x_i) \frac{\partial}{\partial
x_i}(p)$$

\end{thm}

\vspace{1em}

������ ������ �ϱ����� ��� ����� ���캸��. ������ ���캻
��ó�� ���� $X$�� operator�� ������ �� �ִµ� (i.e $X f = D_X f $)
�̿� ���������� $\frac{\partial}{\partial x_i}(p)$�� operator��
������ �� �ִ�. Ư�� �� ��� ������ $\frac{\partial}{\partial
x_i}\mid _p f$�� ǥ���Ѵ�.\\


{\bf Note} {\bf 1.} ���� ���� ������ $\frac{\partial}{\partial
x_i}$ �� �翬�ϰ� tangent vector�� �ȴ�.

$\forall f, g \in \ccn(p)$,

(1). linearity

\begin{eqnarray*}
\frac{\pa}{\pa x_i}\mid_{p}(f + g) &=& \frac{\pa ((f + g) \circ \varphi^{-1})}{\pa u_i}\mid_{\varphi(p)} \\
&=& \frac{\pa ((f \circ \varphi^{-1} + g \circ \varphi^{-1})}{\pa u_i}\mid_{\varphi(p)} \\
&=& \frac{\pa (f \circ \varphi^{-1})}{\pa u_i}\mid_{\varphi(p)} + \frac{\pa (g \circ \varphi^{-1})}{\pa u_i}\mid_{\varphi(p)}(\because linearity )\\
&=& \frac{\pa}{\pa x_i}\mid_{p}f + \frac{\pa}{\pa x_i}\mid_{p}g
\end{eqnarray*}

(2). derivation property

\begin{eqnarray*}
\frac{\pa}{\pa x_i}\mid_{p}(f \cdot g) &=& \frac{\pa ((f \cdot g) \circ \varphi^{-1})}{\pa u_i}\mid_{\varphi(p)} \\
&=& \frac{\pa (f \circ \varphi^{-1})\cdot (g \circ \varphi^{-1}))}{\pa u_i}\mid_{\varphi(p)} \\
&=& \frac{\pa (f \circ \varphi^{-1})}{\pa u_i}\mid_{\varphi(p)}(g\circ\varphi^{-1})\mid_{\varphi(p)} + (f\circ\varphi^{-1})\mid_{\varphi(p)} \frac{\pa (g \circ \varphi^{-1})}{\pa u_i}\mid_{\varphi(p)}\\
&=& \frac{\pa (f \circ \varphi^{-1})}{\pa u_i}\mid_{\varphi(p)}g(p) + f(p) \frac{\pa (g \circ \varphi^{-1})}{\pa u_i}\mid_{\varphi(p)}\\
&=& (\frac{\pa}{\pa x_i}\mid_{p}f)g(p) + f(p) (\frac{\pa}{\pa
x_i}\mid_{p}g)
\end{eqnarray*}

$\therefore \frac{\pa}{\pa x_i} \in T_p M$.\\


{\bf 2.} Ư�� $M \subset \rb^n$�� ���, "���������� $\frac{\pa f}{\pa x_i}$" �� ���� ���غ���.\\

$\frac{\partial}{\partial x_i}\mid _p f$ ; $f$�� $x_i$-��ǥ
���������� ����̺�(directional derivative)�� �ȴ�.\\

�� "$\frac{\partial}{\partial x_i}$" $=
\phi^{-1}_*(\frac{\partial}{\partial u_i})$�̶�� ���̰� �̰���
���� ��ü������ ���� ������ ����.\\

$"\frac{\partial}{\partial x_i}" f =
\phi^{-1}_*(\frac{\partial}{\partial u_i}) f$

$M$�󿡼� ���ǵ� $f$�� �̺��� $p \in M$�� ������ � $\sig$��
�����ؼ� �� $\sig_i(t)=\varphi^{-1}(a_1, \cdots, t, \cdots, a_n)$; $i$-th\\

$=\frac{d}{d t}\mid _{t=a_i}(f \circ \sig_i)$

$=\frac{d}{d t}\mid _{t=a_i}(f \circ \varphi^{-1})(a_1, \cdots, t,
\cdots, a_n)$

$=\frac{\pa(f \circ \varphi^{-1})}{\pa u_i}\mid _{\varphi(p)}$

$=\frac{\pa}{\pa x_i}\mid _p f$

��, $\rb^n$�������� �������Ͽ� abstract�� ���ǰ� ������ ��ġ�Ѵ�.


���� ����1 �� �����ϱ� ���� ������ ���� ������ ���� ������.\\

\begin{lem}
$p\in M,\forall \,f\in \cc^{\infty}(p),$ $f$ can be represented as
of the form

$f=f(p)+\displaystyle{\sum^n_{i=1}}(x_i-x_i(p))g_i\,$ , where
$g_i\in\cc^{\infty}(p)$ and $g_i(p)=\frac{\partial f}{\partial
x_i}(p)$.
\end{lem}

\begin{proof}
$(U,\phi)$�� $p$������ coordinate chart��� ����. $\phi(U)$����
$\phi(p) = a$�� �ٹ� $V$�� ���� $F$��
$F=f\circ\phi^{-1}|_V:V(\subset\rb^n)\rightarrow\rb$ , $\phi(p)=a$
�� �����ϸ�\\

$\displaystyle{F(x)-F(a)=F(a+t(x-a))|_{t=0}^{t=1}\hspace{2em},\hspace{2em}let
\,\,\,F(a+t(x-a))=G(t)}$

$\displaystyle{\hspace{5.8em}=\int_o^1\frac{d}{dt}G(t)dt=\int_0^1\frac{d}{dt}F(a+t(x-a))dt}$

$\hspace{5.8em} =\displaystyle{\int_0^1\sum_{i=1}^n\frac{\pa F
}{\pa u_i}(a+t(x-a))(x_i-a_i)dt}$

$\displaystyle{\hspace{5.8em}
=\sum_{i=1}^n(x_i-a_i)\int_o^1\frac{\pa F }{\pa
u_i}(a+t(x-a))dt}$\\

$\displaystyle{h_i(x) = \int_o^1\frac{\pa F }{\pa
u_i}(a+t(x-a))dt}$�� �θ�\\

$\therefore
F(x)=F(a)+\displaystyle{\sum_{i=1}^n(u_i(x)-u_i(a))h_i(x)}$ ,
$h_i(a)=\frac{\pa F}{\pa u_i}(a)$.

���� $x=\phi(q)
\,,\,a=\phi(p)\,,\,u_i\circ\phi=x_i\,,\,F\circ\phi=f$ �� �θ�

$f(q)=f(p)+\displaystyle{\sum^n_{i=1}}(x_i(q)-x_i(p))g_i(q)\,$(���⿡��
$g_i = h_i \circ \phi$ ) �̰�,

$g_i(p)=(h_i\circ\phi)(p)=h_i(a)=\frac{\pa F }{\pa u_i
}(a)=\frac{\pa (f \circ \phi^{-1})}{\pa u_i}(a)=\frac{\pa f}{\pa
x_i}(a)$.
\end{proof}\\


���� ���������� ���� 1�� ��������.\\

\begin{proof}(proof of the theorem1)

���������� ���ϸ�
$f=f(p)+\displaystyle{\sum^n_{i=1}}(x_i-x_i(p))g_i$ ���� ǥ���ǰ�
����

$Xf=Xf(p)+\displaystyle{\sum^n_{i=1}}\{X(x_i-x_i(p))g_i(p)+(x_i-x_i(p))(p)Xg_i\}$

$Xf(p)\,,\,X(x_i(p))\,,\,(x_i-x_i(p))(p)$�� ��� 0�� �ǰ�, ����

$Xf=\displaystyle{\sum^n_{i=1}}(Xx_i)g_i(p)=\displaystyle{\sum^n_{i=1}}(Xx_i)\frac{\partial
f }{\partial x_i}(p)$.

���� $\{\frac{\partial}{\partial x_i}(p)\}$�� $T_p M$�� span����
������ linearly independent���� ���̸� �ȴ�. ���� $\disj
c_j\frac{\pa}{\pa x_j}=0$ �̶�� $\,\,\disj \dis{c_j\frac{\pa
x_i}{\pa x_j}=0}$ �̴�. �׷���\\

$\hspace{4em}\dis{\frac{\pa x_i}{\pa
x_j}=\frac{\pa\{(u_i\circ\phi)\circ\phi^{-1}\}}{\pa u_j}=\frac{\pa
u_i}{\pa u_j}=\del_{ij}}$ �̹Ƿ�

$\forall c_i=0$ �� ���� �� �ְ� ���� ������
�ϼ��Ǿ���.

\end{proof}


{\bf Note.} $p$�ٹ濡 �ΰ��� chart $(U,x),(V,y)$�� ���� �� ��
chart�� ���� �����Ǵ� $T_p M$�� basis $\dis{\frac{\pa}{\pa
x_i}}$�� $\dis{\frac{\pa}{\pa y_i}}$��

$\dis{\frac{\pa}{\pa x}=\frac{\pa}{\pa y}\frac{\pa y}{\pa x}}$ ��
���谡 �ִ�. �̸� ���̱� ���� ������ �� tangent vector $X$��
$\dis{\frac{\pa}{\pa y_i}}$ �� ǥ���ϸ�
$X=\displaystyle{\sum^n_{i=1}}(Xy_j)\frac{\partial }{\partial
y_j}$ �ε� Ư�� $X=\dis{\frac{\pa}{\pa x_i}}$�� ��쿡��
$\dis{\frac{\pa}{\pa x_i}}=\disj\dis{(\frac{\pa y_j}{\pa x_i
})\frac{\pa}{\pa y_j}}$ �� �ȴ�. ������ ���� ���� ���� �����Ϸ���
��İ��� �����Ͱ� �ƴ� �຤�ͷ� �������ٴ� ����̴�. ����
$\dis{\frac{\pa}{\pa x}=\frac{\pa y}{\pa x}\frac{\pa}{\pa y}}$ ��
�ƴ� $\dis{\frac{\pa}{\pa x}=\frac{\pa}{\pa y}\frac{\pa y}{\pa
x}}$ �� �����ϰ� �ȴ�.

���⿡�� $\frac{\pa y}{\pa x}$�� coordinate transition map��
Jacobi ����̰� ($\because \frac{\pa y_i}{\pa x_j} = \frac{\pa
(y_i \circ x^{-1}) }{\pa u_j} = \frac{\pa(u_i \circ y \circ
x^{-1})}{\pa u_j}$)

$\frac{\pa}{\pa x} = (\frac{\pa}{\pa x_1}, \cdots, \frac{\pa}{\pa
x_n})$, $\frac{\pa}{\pa y} = (\frac{\pa}{\pa y_1}, \cdots,
\frac{\pa}{\pa y_n})$���� �־����� �຤���̴�.

(�̰Ͱ� $\frac{d y}{d t} = \frac{\pa y}{\pa x} \frac{d x}{d t}$��
���غ���, �� ��쿡 $\frac{d x}{d t}$, $\frac{d y}{d t}$��
�຤�Ͱ� �ƴ϶� �������̴�.)


\end{document}
