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

\begin{defn}
{\it $ \varphi : M^m \longrightarrow N^n$ is $\ccn$. Then the {\bf
differential} of $\varphi$  ,\\ $\varphi_* = d \varphi : TM
\longrightarrow TN$ \; is defined by \; $(d\varphi (X_p) \in
T_{\varphi(p)}N$ and )\\ \; $d\varphi(X_p)g = X_p (g \circ
\varphi)$\; ,$\forall g \in \ccn(\varphi(p))$ }

\end{defn}

\vspace{1em}

{\bf check} {\bf 1. $d\varphi(X_p)$ �� $\ccn(\varphi(p))$ ��������
linear derivation�� �ȴ�.} \\ i.e., \; $d \varphi(X_p) \in
T_{\varphi(p)}N $\\

\begin{proof}
\begin{eqnarray*}
d \varphi (X_p )(f  +  cg ) &=& X_p
( f + cg ) \circ \varphi \\
&=& X_p( f \circ \varphi + c g \circ \varphi )\\
&=& X_p(f \circ \varphi) + c X_p ( g \circ \varphi) \hspace{5em}
(\because X_p : \mbox{linear derivation})\\
&=&d\varphi(X_p)f + c d\varphi(X_p)g
\end{eqnarray*}

\begin{eqnarray*}
d \varphi (X_p ) (f g ) &=& X_p
(( fg) \circ \varphi )\\
&=& X_p ( (f \circ \varphi)(g \circ \varphi ))\\
&=& (g \circ \varphi)(p)X_p(f \circ \varphi) + (f \circ
\varphi)(p)X_p ( g \circ \varphi) \hspace{1em}
(\because X_p : \mbox{linear derivation})\\
&=&g(\varphi(p))d\varphi(X_p)f + f(\varphi(p))d\varphi(X_p)g
\end{eqnarray*}
\end{proof}

{\bf 2. $d\varphi_p := d\varphi\mid_{T_p M}:T_p M \longrightarrow
T_{\varphi(p)}N$ �� �����̴�.}\\

\begin{proof}
\begin{eqnarray*}
d \varphi (X_p +  cY_p )g &=& (X_p + c Y_p )(g \circ \varphi) \\
&=& X_p( g \circ \varphi) + c Y_p(g \circ \varphi )\\
&=& d \varphi(X_p)g + c d \varphi (Y_p) g\\
&=&(d\varphi(X_p) + c d\varphi(Y_p))g
\end{eqnarray*}
\end{proof}

{\bf 3. $d\varphi$ �� $\ccn$ �̴�.}\\

\begin{proof}
���� ���������� �����ϴ� ���� ���̰ڴ�. �ٽø��ϸ� $p$��
$\varphi(p)$�ٹ��� ������ chart��, $(U, x)$�� $(V, y)$�� ���Ͽ�
�����ϴ� ���� ���δٴ� ���̴�.\\

$\widehat{y}\circ
d\varphi\circ\widehat{x}^{-1}:(x(p),a_1,\cdot\cdot\cdot,a_m)\mapsto
(y(\varphi(p)),b_1,\cdot\cdot\cdot,b_n)$�̶� ����.


���������� $d\varphi(\frac{\pa}{\pa x_i}) = \disj c_j
\frac{\pa}{\pa y_j}$�� �θ�

$\frac{\pa}{\pa x_i}(y_k \circ \varphi) = d\varphi(\frac{\pa}{\pa
x_i})y_k = \disj c_j \frac{\pa y_k}{\pa y_j} = \disj c_j
\del_{jk}= c_k$�� �Ǿ�

$d\varphi(\frac{\pa}{\pa x_i}) = \disj \frac{\pa}{\pa x_i}(y_j
\circ \varphi)
\frac{\pa}{\pa y_j}$ �̴�.\\

$\therefore d\varphi(\underbrace{X_p}) \; = \; d\varphi(\disi a_i
\frac{\pa}{\pa x_i})\; =\; \disj (\underbrace{\disi a_i
\frac{\pa(y_j \circ \varphi)}{\pa x_i} }) \frac{\pa}{\pa y_j}\;
=\; \disj b_j \frac{\pa}{\pa y_j} $

$\hspace{2em} \sum a_i \frac{\pa}{\pa x_i} \hspace{14em}b_j$\\

���⿡�� $b_j = \disi a_i \frac{\pa(y_j \circ \varphi)}{\pa x_i}
$�� ���캸��

$\frac{\pa(y_j \circ \varphi)}{\pa x_i}\;=\; \frac{\pa}{\pa
u_i}(u_j \circ y \circ \varphi \circ x^{-1})$ �̹Ƿ� $\varphi$��
local chart������ Jacobian�� �Ǿ� $x$�� $\ccn$ function��  �Ǵ�
���� �� ��
�ִ�.\\

���� $\widehat{y}\circ d\varphi \circ \widehat{x}^{-1} : (x(p),
a_1, \cdots , a_m ) \mapsto (y(\varphi(p)), b_1, \cdots, b_n )$��
$\ccn$�̴�.\\

�ٽ� ó������ ���ư��� $\ccn$�̶�� ���� �������� �����ε� ������
$p \in M$ �� ���ؼ� ���� ���� �����ϹǷ� ��ǻ� $d\varphi$��
�뿪������ $\ccn$�̶�� ���� ���Ѵ�. �׷��Ƿ� ������ ������.
\end{proof}\\

{\bf 4. Chain rule : $d(\psi\circ\varphi)=d\psi\circ d\varphi$}.\\

\begin{proof} ������ $X_p$�� ���� $d(\psi\circ\varphi)(X_p)=d\psi\circ
d\varphi(X_p)$���� ������.


$d(\psi\circ\varphi)(X_p)(f)=X_p(f\circ\psi\circ\varphi)$


$\hspace{7.5em}=d\varphi(X_p)(f\circ\psi)$

$\hspace{7.5em}=d\psi(d\varphi(X_p) )(f)$

$\hspace{7.5em}=(d\psi\circ d\varphi)(X_p)(f)$


\end{proof}\\

{\bf 5. $M$���� � $\sigma$�� tangent vector}.\\

tangent vector of $\sigma$ at t =
$\frac{d\sigma}{dt}:=d\sigma(\frac{d}{dt})=\sigma_*(\frac{d}{dt})\in
T_{\sigma(t)}M $

$\frac{d\sigma}{dt}\cdot
g=d\sigma(\frac{d}{dt})g=\frac{d}{dt}(g\circ\sigma)$=directional
derivatives of $g$ in the derivation of $\frac{d\sigma}{dt}$.

Ư�� $g=x_i$�� ��쿡��  $X=\disi(Xx_i)\frac{\pa}{\pa x_i}$ �̹Ƿ�

$\dis{\frac{d\sigma}{dt}}=\disi(\frac{d\sigma}{dt}\cdot
x_i)\frac{\pa}{\pa x_i}=\disi
\frac{d}{dt}(x_i\circ\sigma)\frac{\pa}{\pa
x_i}=\disi\frac{d\sigma_i}{dt}\frac{\pa}{\pa x_i}$.\\\\



{\bf Geometric interpretation of $d\varphi : d\varphi(X_p)= ?$}

chain rule�� ����
$d\varphi(\frac{d\sigma}{dt})=\frac{d}{dt}(\varphi\circ\sigma)$ ��
�ȴ�. ���� �Ŀ��� �̸� �� �� �ִ�.


$d\varphi(\frac{d\sigma}{dt})=d\varphi(d\sigma(\frac{d}{dt}))=d(\varphi\circ\sigma)(\frac{d}{dt})
=\frac{d}{dt}(\varphi\circ\sigma)$ .\\

���� $\varphi$�� ��ü������ �־��� ��쿡 $d\varphi(X_p)$��
����ϴ� �Ѱ��� ����� ���� $X_p$�� "fit"�ϴ� (i.e.,$\sigma(0)=p
\,\,\, and \,\,\,\frac{d\sigma}{dt}|_0=X_p$) ��� ������ �ƹ���
� $\sigma$�� ��´�. �׸��� $\varphi\circ\sigma$�� ����� ��
�װ��� tangent vector $\frac{d}{dt}(\varphi\circ\sigma)$��
����ϸ� �ȴ�. ���� ��쿡 �̷��� ������� $d\varphi$�� ��ü������
����� �� �ִ�.\\

{\bf Remark.} �տ��� coordinate function $x_i:U\subset
M\rightarrow
  \rb$�� ���� �� differential�ν��� $dx_i$ �� $\{\frac{\pa}{\pa x_i}\}$�� dual
  basis�ν���
  $\{dx_i\}$ �� ��� ��ġ�Ѵ�. �̸� ���̱� ���� ���� real valued function f�� ���Ͽ� $df$�� $f_*$��
  �����ؼ� ��������.\\

  $<$Definition of $df>$\\

  $f:M\rightarrow\rb,\ccn$�� ���� $f_*(X_p)=(f_*(X_p)\cdot t)\frac{d}{dt}=X_p(t\circ
  f)\frac{d}{dt}=X_p(f)\frac{d}{dt}$.\\

  Define   $df:TM\overset{f_*}{\rightarrow} T\rb\rightarrow
  \rb$  $\,\,\,\,i.e.,\,\,\,\,\,df_p:T_pM\overset{f_*}{\rightarrow}T_{f(p)}\rb\cong\rb:linear $

  $\hspace{8.4em}a\frac{d}{dt}\mapsto a\hspace{6.5em}X_p\mapsto(X_pf)\frac{d}{dt}\mapsto X_pf$\\

  $df_p\in (T_pM)^*$�̹Ƿ� $df_p=\sum a_idx_i$�� ǥ�õǰ� �� �� ���ǿ� ����

  $df_p(\frac{\pa}{\pa x_i})=\paxi
  f(p)=\frac{\pa}{\pa x_i}|_pf=\frac{\pa f}{\pa x_i}(p)$�̴�.



  $df_p(\paxj)=\sum a_idx_i(\paxj)=a_j\hspace{2em}$  $\therefore df_p=\sum(df_p(\paxi))dx_i=\sum(\frac{\pa f}{\pa x_i}(p))dx_i$

  $\therefore df=\sum\frac{\pa f}{\pa x_i} dx_i$.\\

  ���� coordinate function $x_i$�� ���� $x_i$ ���� $U$���� real
  valued function�̹Ƿ� �̰��� differential $"dx_i"$�� ������ �� �ְ�, �̸� �� ���뿡
  �����Ű��\\


  $\hspace{5em}"dx_i"=\sum\frac{\pa x_i}{\pa x_j} dx_j=dx_i$.

�� �� ���� �º��� coordinate function $x_i$�� differential�μ���
$dx_i$�̰� �캯�� $\paxi$�� dual basis�� �� ���ε� �� ����
��ġ���� �� �� �ִ�.









\end{document}
