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\begin{document}
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{\large ���wI���K�@��V��@2007�N7��3���z�z} 

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�S�� ���n����CTA �O�p �~
\end{center}

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{\small

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{\tt http://www.ms.u-tokyo.ac.jp/$\sim$hirachi/courses/sugaku-I-2007/}
����_�E�����[�h�ł��܂��D�u�`�������ڂ��Ă��܂��D
}


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{\small
�ȉ��̖����ł���͈͂ŉ���, �V���P�O���P�R���܂łɃA�h�~�j�X�g���[�V�������̃��|�[�g��o�{�b�N�X�ɒ�o���邱��. 
}

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\prob [���ϐ��̒��Ԓl�̒藝] $D\subset\R^2$�Ƃ���B$D$�̔C�ӂ̓_$p,q$�ɑ΂��ĘA���ʑ�$\gamma\colon [0,1]\to D$��
$\gamma(0)=p,\gamma(1)=q$���݂������̂����݂���Ƃ�$D$�͘A���ł���Ƃ����B$D$���A���A$f\colon D\to\R$���A���ł����$f(D)$�͋�Ԃł��邱�Ƃ������B

\prob �Q�ϐ��̊֐�
$$
f(x,y)=\begin{cases}
(x^2+y^2)\sin\frac{1}{\sqrt{x^2+y^2}}&(x,y)\ne(0,0)\\
\qquad0 & (x,y)=(0,0)
\end{cases}
$$
�͕��ʂ̊e�_�őS�����\�ł��邪$C^1$-�֐��ł͂Ȃ����Ƃ������B


\prob [�s�񎮂̕Δ���]
$f_i,g_i,h_i$ $(i=1,2,3)$������\�ȂP�ϐ��֐��Ƃ���B
�������߂�
$$
\frac{\partial^3}{\partial x\partial y\partial z }
\left|
\begin{matrix}
f_1(x) &f_2(x) &f_3(x) \\
g_1(y) &g_2(y) &g_3(y) \\
h_1(z) &h_2(z) &h_3(z) 
\end{matrix}
\right|
$$
�i�������s�񎮂ŏ����܂��B�j

\prob [�ڕ��ʂ̊􉽓I�ȈӖ��Â�]
$\R^2$�̊J�W��$U$��Œ�`���ꂽ�S�����\�Ȋ֐�$f(x,y)$
�ɂ������Ȗ� $S=\{(x,y,f(x,y)):(x,y)\in U\}$��$S$��̓_
$P_0=(x_0,y_0,z_0)$��ʂ镽��$L$:
$$
 z-z_0=f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)
$$
���l����΁A$L$�͎��̈Ӗ���$S$�ɐڂ��Ă��邱�Ƃ������i������$L$��ڕ��ʂƂ�ԁj�F

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$P\in S$����$L$�ɂ��낵�������̑���$Q$�Ƃ����
$$
\lim_{P\to P_0}\frac{|P-Q|}{|P-P_0|}=0
$$



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