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\begin{document}
 \parindent=0cm
  \section*{Vector field.}

  Let $\pi:TM\rightarrow M$ be a canonical projection.

  $\hspace{3.8em}X_p\mapsto p$

\begin{defn}{\it
A {\bf vector field} X on $A\subset M$ is a "section" of
$\pi:TM\rightarrow M$ over $A$, i.e., A vector field is a map
$X:A\rightarrow TM$ with $\pi\circ X=id.$}\\
\end{defn}

*�׸� 21*\\

�� vector field $X$�� �� �� $p$�� ���� $X_p$�� ������Ű��
map�̴�.\\

\begin{thm}
Let $X$ be a vector field. Then the followings are equivalent;

(1) $X$ is $\ccn$.

(2) $ (U,x)$is a coordinate chart on $M$ and $X(p)=\sum
a_i(p)\frac{\pa}{\pa x_i}(p)$ on $U$

$\Rightarrow a_i\in \ccn(U,\rb)$.

(3) $V$ is open in $M$, $f\in\ccn(V,\rb)\Rightarrow
Xf\in\ccn(V,\rb).$
\end{thm}

\begin{proof}
$(1)\Rightarrow(2)$

$a_i(p)=Xx_i(p)=dx_i(X_p)$ and $dx_i$ is a coordinate function on
$\pi\inv(U)=\widehat{U}$

$\therefore dx_i$ is $\ccn\Rightarrow a_i=dx_i\circ
X\in\ccn(U,\rb)$.

$(2)\Rightarrow(3)$

$Xf$�� locally $\ccn$���� ���̸� �ȴ�. $Xf=\sum a_i\frac{\pa
f}{\pa x_i }$ �ε� $a_i$�� $f$ �� $\frac{\pa f}{\pa x_i}$��
$\ccn$�̹Ƿ� $Xf$�� locally $\ccn$�̴�.

$(3)\Rightarrow(1)$
$\hspace{1em}X:U\hspace{1.5em}\rightarrow\hspace{1.5em}
\widehat{U}=\pi\inv(U)$

$\hspace{6.5em}\vphi\downarrow\hspace{6em}\downarrow\hat{\vphi}$

$\hspace{6em}\vphi(U)\overset{induced\,\,by\,\,X}{\dashrightarrow}\vphi(U)\times\rb^n$

$\hspace{6.5em}x\hspace{2.5em}\mapsto\hspace{1.5em}(x,a_1,\cdot\cdot\cdot,a_n)$

�� coordinate function��  $id,a_1,\cdot\cdot\cdot,a_n$�ε�
$a_i=Xx_i$�̰� $x_i$�� $\ccn$�̹Ƿ� (3)�� ���� $\forall a_i$��
$\ccn$�̴�. ���� �� coordinate function�� $\ccn$�̹Ƿ� $X$��
$\ccn$�� �ȴ�.
\end{proof}

\begin{defn}{\it Let $X$ be a $\ccn$ vector field on $M$. A $\ccn$
curve $\sigma$ is an {\bf integral curve} of $X$, if
$\frac{d\sigma}{dt}(t)=X(\sigma(t))$ for $\forall t\in
dom(\sigma)$.}\\
\end{defn}

$X$�� local�ϰԴ�  open subset of $\rb^n$������
 $\ccn$ vector field�� �� �� �ִ�.

$X=(f_1(x),\cdot\cdot\cdot,f_n(x))$(or $X=\sum f_i(x)\paxi$)��
�θ� integral curve�� ã�´ٴ� ����
$\sigma(t)=(x_1(t),\cdot\cdot\cdot,x_n(t))$ such that
$\frac{d\sigma}{dt}=X(\sigma(t))$ �� �����ϴ� $\sigma$�� ã��
�����̰� �� �̴� �Ʒ��� system of o.d.e. $(*)$�� Ǫ�� �Ͱ� ����.

\[ (*) \left\{
         \begin{array}{cc}
               \frac{dx_1}{dt}=& f_1(x) \\
                     \,\,\, \cdot         \\
                           \,\,\,\cdot     \\
                             \,\,\,\cdot \\
             \frac{dx_n}{dt}=& f_n(x)
          \end{array} \right. \]\\

{\bf �� 1.} $X=y\frac{\pa}{\pa x}+\frac{\pa}{\pa y}=(y,1)$����
�־����� �� integral curve $\sigma$��?

\[ (*) \left\{
         \begin{array}{cc}
               \frac{dx}{dt}=& y\\
             \frac{dy}{dt}=& 1
          \end{array} \right. \]\\

$\therefore y=t+c,x=\frac{1}{2}t^2+ct+d.$\\

{\bf �� 2.} $X=y\frac{\pa}{\pa x}+x\frac{\pa}{\pa y}=(y,x)$ with
initial condition $(x(0),y(0))=(1,0)$.

�̺й������� Ǯ�� $-\infty<t<\infty$�� ���� $\sigma(t)=(cosht,sinht)$�� ��´�.\\

�� 1,2�� �޸� ������ ��츦 ����. ������ ��� ���� ���缺�� ���ѵǰų� �ذ� �������� ���� ���� �ִ�.\\

{\bf �� 3.} $X(x)=x^2$ on $\rb$ with initial condition $x(0)=1$.

$\frac{dx}{dt}=x^2\Rightarrow \int \frac{dx}{x^2}=\int
dt\hspace{2em}\therefore x=\frac{1}{1-t}$.

�� �Լ��� $t=-\infty$���� $t=1$������ ���ǵȴ�. ����
$\rb$��ü���� ���ǵǴ� �ش� �������� �ʴ´�.\\

{\bf �� 4.}$\frac{dx}{dt}=3x^{\frac{2}{3}}$ on $\rb$ with initial
condition $x=0$ at $t=0$.

�̺й������� Ǯ�� $x(t)=t^3$ ������ $x(t)\equiv 0$ ���� �ذ� �ȴ�.
���� �ذ� �����ϰ� �������� �ʴ´�.



\end{document}
