��ϳ� IA No.9��

�� 9.1

$\mbox{${\mathbb{R}}$}$ �Υ٥��ȥ� $v=(v_1,v_2,\dots,v_m), w=(w_1,w_2,\dots,w_m)$ ��Ф�� Ĥ�����

$\displaystyle \langle v, w \rangle=\sum_{j=1}^m v_j w_j$

��롣
$\mbox{${\mathbb{R}}$}$ �Υ٥��ȥ� �ΥΥ��

$% latex2html id marker 1085 $\displaystyle \vert\vert v\vert\vert=\sqrt{\langle v,v\rangle } $$

��롣
( �Τ��ιֵ��Ѥ��Υ(�桼��åɵ�Υ)�� $d(v,w)=\vert\vert v-w\vert\vert$ ��ΤȰ��פ��롣)
�� $P\in M_{l,m}($ $\mbox{${\mathbb{R}}$}$ ��Ф����ǥΥ�� ��

$% latex2html id marker 1095 $\displaystyle \vert\vert P\vert\vert=\sup_{\substa... ...{\mathbb{R}}$}^l\\ Vert\vert v\vert\vert\leq 1}} \vert\vert P\cdot v\vert\vert $$

��롣

�� 9.1

��Ѥ��Ǥ��롣
�٥��ȥ��Ф��ƤΥΥ�� Υ��θ�� ʤ��
1. $\forall v \in$ $\mbox{${\mathbb{R}}$}$ $% latex2html id marker 1108 $ \vert\vert v\vert\vert\geq 0$$ .
2. $\vert\vert v\vert\vert=0 {\Leftrightarrow} v=0$ .
3. $\forall v,w \in$ $\mbox{${\mathbb{R}}$}$ $% latex2html id marker 1114 $ ^m \quad \vert\vert v+w\vert\vert\leq \vert\vert v\vert\vert +\vert\vert w\vert\vert$$ .
4. $\forall c\in$ $\mbox{${\mathbb{R}}$}$ $\forall v \in$ $\mbox{${\mathbb{R}}$}$ $% latex2html id marker 1120 $ ^m \quad \vert\vert c v\vert\vert=\vert c\vert\cdot \vert\vert v\vert\vert$$ .
�� Ф��ơ��κ��ǥΥ��Ͼ��ͭ�¤Ǥ��ꡢ $P\mapsto \vert\vert P\vert\vert$ �ϥΥ��θ��
Ǥ�դι�� (��Ⱦ軻��ǽ�ʥ��)Ǥ�դΥ٥��ȥ��Ф�� $% latex2html id marker 1128 $ \vert\vert P\cdot v\vert\vert \leq \vert\vert P\vert\vert\cdot \vert\vert v\vert\vert$$ ��Ω�ġ�

�� 9.1 �� $\mbox{${\mathbb{R}}$}$ �γ��Ǥ��Ȥ�� $f: U \to$ $\mbox{${\mathbb{R}}$}$ �� Ǥ��Ȥ��롣 (��Ͱ�μ��Ʊ��Ǥ��뤳�Ȥ��ա�) $x_0\in U$ �ˤ��ơ� $Df\vert _{x_0}$ ��Ȥ��ƲĵդǤ��Ȳ��ꤷ�� εչ�� Ȥ�� μ¿� �򼡤Τ褦�ʾ��褦�ˤȤ롣

$\begin{displaymath} x \in B_0\overset{\operatorname{def}}{=}B_{r_0}(x_0) \implie... ...f(x_0)\vert\vert<\frac{1}{2 \vert\vert L\vert\vert} \end{cases}\end{displaymath}$

�� Ȥ��ˤ�ꤳ�Τ褦�� ¸�ߤ��롣 ��ΤȤ��

�� ñ�ͤǤ��롣
$r_1= \frac{r_0}{2 \vert\vert L\vert\vert}$ , $B_1=B_{r_1}(f(x_0))$ �Ȥ��ȡ� ��줿 ��ؿ� ��¸�ߤ��ơ�

$\displaystyle f \circ g\vert _{B_1}=id_{B_1}$

��ʤꤿ�ġ�

��ξ��Υ��ϡ��ʲ��(Newton ˡ)�Ǥ��롣 ��

�εչ��ΤȤ��

��֤��ʬ��ʪ�� Newton ˡ�Ȥϰۤʤ롣

�� 9.2 ��β��β��ǡ� ��Τ褦�� $B_1\times B_0$ �� $\mbox{${\mathbb{R}}$}$ �ʹؿ��ͤ��褦��

$% latex2html id marker 1196 $\displaystyle \varphi(y,x)=x+ L (y-f(x)) \qquad (x\in B_0, y\in B_1) $$

��ȡ� $\varphi$ �� 롣

�� 9.3 ��β��Τ�Ȥǡ�� $\varphi$ ��ͤ��ơ� ��Τ褦�� δؿ�� $\{g_j\}_{j=1}^\infty$ ��롣

$\begin{displaymath} \begin{cases} g_1(y)=x_0 & \text{\rm {(}��ؿ�\rm {)}} \\ g_{j+1}(y)=\varphi(y,g_j(y)) \end{cases}\end{displaymath}$

�� $\{g_j\}$ �Ϥ��ؿ� �˳��«��롣

�� 9.2

$\displaystyle f\circ g={\operatorname{id}}$

�ΤȤ��

$\displaystyle Dg\vert _y= (Df\vert _{g(y)})^{-1}$

�Ȥ��ˡ� �� ʤ顢 �� Ǥ��롣

�ռ��Ǥϡ��Ͱ�� ʤ��

�Ǥ�

��ʬ $Df\vert _{x_0}$ �� ĵդǤ��뤳�Ȥ�Ŭ�ѤΥݥ��ȤǤ��롣 ��Τ褦�ʹͤ��Ѥ��ơ� ��Ͱ�μ��㤦��ˤ⡢�մؿ��Ѥ��뤳�Ȥ��Ǥ��롣 ��Ǥ��ޤ��ʹͤ��Τ߽񤤤Ƥ��(�ܺ٤ϸ�渦��)

̿�� 9.3 (��ؿ��) $\mbox{${\mathbb{R}}$}$ $^{n+s} \to$ $\mbox{${\mathbb{R}}$}$ �� $C^\infty$ ��ǡ��ʤ��

$\displaystyle \hat f: (x,y)\to (f(x,y),y)\in$ $\displaystyle \mbox{${\mathbb{R}}$}$ $% latex2html id marker 1257 $\displaystyle ^{n+s} \quad (x\in$$ $\displaystyle \mbox{${\mathbb{R}}$}$ $\displaystyle ^n, y\in$ $\displaystyle \mbox{${\mathbb{R}}$}$ $\displaystyle ^s)$

�� $% latex2html id marker 1263 $ \operatorname{det}(D \hat f\vert _{(x_0,y_0)})\neq 0$$ ��Ȥ��롣 �Ȥ�� ΤȤ� $\hat f$ �ζɽ�Ū�ʵռ��

$% latex2html id marker 1269 $\displaystyle \hat g:(x,y)\to (g(x,y),y) \quad (x\in$$ $\displaystyle \mbox{${\mathbb{R}}$}$ $\displaystyle ^n, y\in$ $\displaystyle \mbox{${\mathbb{R}}$}$ $\displaystyle ^s)$