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\begin{document}
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{\large ���wI���K(�⑫)�@2007�N7��19���z�z} 

\smallskip

�S�� ���n����CTA �O�p �~
\end{center}

\smallskip
{\small

���K����
{\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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\prob
(a)
$f(x,y)$��$\R^2$���$C^1$�֐��Ƃ���B$f$�̃O���t$G=\{(x,y,f(x,y)): (x,y)\in\R^2\}$�̓_$P\in G$�ł̐ڕ��ʂɒ�������$P$��ʂ钼�������߂�(�����$P$�ł̖@���Ƃ��).

(b) $g:\R\to\R^3$��$C^1$�֐��ŔC�ӂ�$t$�ɑ΂���$g(t)\in G$����
$g(0)=P\in G$���݂����Ƃ���B���̂Ƃ�$\dfrac{dg}{dt}(0)\in\R^3$
��$G$��$P$�ł̖@���ɒ������邱�Ƃ������B


\prob
�R�������$\mathbf R^3$�ɂ����Č��_$O$����$(x,y,z)$�܂ł̋���$r$�Ɗp$\theta\in[0,\pi]$, $\varphi\in[0,\pi]$���������Ď��̂悤�ɋɍ��W���`����F
$$
x=r\sin\theta\cos \varphi,\quad y=r\sin\theta\sin \varphi,\quad z=r\cos \theta
$$

(a) ���v���V�A�� $\Delta=\dfrac{\pa^2}{\pa x^2}+\dfrac{\pa^2}{\pa y^2}+\dfrac{\pa^2}{\pa z^2}$�͋ɍ��W$(r,\theta,\varphi)$��
$$
\Delta=\frac{\pa^2}{\pa r^2}+\frac{2}{r}\frac{\pa}{\pa r}
+\frac{1}{r^2}\frac{\pa^2}{\pa \theta^2}
+\frac{\cos\theta}{r^2\sin\theta}\frac{\pa}{\pa\theta}
+\frac{1}{r^2\sin^2 \theta}\frac{\pa^2}{\pa\varphi^2}
$$
�Ə����邱�Ƃ������D

(b) $r$�ɂ݈̂ˑ�����֐�
$f(x,y,z)=g(r)=g(\sqrt{x^2+y^2+z^2})$��$r>0$�ɂ�����$\Delta f=0$���݂����̂�
$$g(r)=a+\frac{b}{r} \quad\text{(������$a,b$�͒萔)}$$
�̏ꍇ�Ɍ��邱�Ƃ������D[�q���g:$g(r)=h(r)r^{-1}$�Ƃ���$h$�Ɋւ�����������߂�D]

\prob
$\cos(\sqrt{x^2+y^2})$��$(x,y)=(0,0)$�ł̃e�C���[�W�J��
$x,y$�ɂ���4�����܂ŋ��߂�(���Ȃ킿��]����$(\sqrt{x^2+y^2})^4R(x,y)$������$\lim_{(x,y)\to(0,0)}R(x,y)=0$�̌`)�B

\prob ���̊֐��̋ɒl�����߂�D�܂�����炪�ő�C�ŏ��l��^���邩�𒲂ׂ�D

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(a) $f(x,y)=(x+y)e^{-x^2-y^2}$ �@�@�@(b) $f(x,y)=x^3+y^3-3xy$




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