第6章 6.18

(1)

$f_{(x,y)} = \log(x^2+y^2)$

$f_x = \frac{2x}{x^2+y^2} \qquad f_y = \frac{2y}{x^2+y^2} \qquad f_{xx} = \frac{2(x^2+y^2)-4x^2}{(x^2+y^2)^2} \qquad f_{yy} = \frac{2(x^2+y^2)-4y^2}{(x^2+y^2)^2}$

$\Delta f = \frac{-2x^2+2y^2+2x^2-2y^2}{(x^2+y^2)^2} = 0$

(2)

$f_{(x,y)} = \tan^{-1}\dfrac{y}{x}$

$f_x = \frac{-y}{x^2+y^2} \qquad f_y = \frac{x}{x^2+y^2} \qquad f_{xx} = \frac{2xy}{(x^2+y^2)^2} \qquad f_{yy} = \frac{-2xy}{(x^2+y^2)^2}$

$\Delta f = \frac{2xy-2xy}{(x^2+y^2)^2} = 0$

(3)

$f_{(x,y)} = e^{ax}(\cos by + \sin by)$

$f_x = ae^{ax}(\cos by+\sin by) \qquad f_y = be^{ax}(-\sin by+\cos by)$

$f_{xx} = a^2 e^{ax}(\cos by+\sin by) \qquad f_{yy} = -b^2 e^{ax}(\cos by+\sin by)$

$\Delta f = (a^2-b^2)e^{ax}(\cos by+\sin by)$

(4)

$f_{(x,y,z)} = \dfrac{1}{\sqrt{x^2+y^2+z^2}}$

$f_x = -\frac{x}{\sqrt{(x^2+y^2+z^2)^3}} \qquad f_y = -\frac{y}{\sqrt{(x^2+y^2+z^2)^3}} \qquad f_z = -\frac{z}{\sqrt{(x^2+y^2+z^2)^3}}$

$f_{xx} = -\frac{\sqrt{(x^2+y^2+z^2)^3} - x\cdot\dfrac{3}{2}\cdot 2x\sqrt{x^2+y^2+z^2}}{(x^2+y^2+z^2)^3} = -\frac{\sqrt{x^2+y^2+z^2}\,(-2x^2+y^2+z^2)}{(x^2+y^2+z^2)^3}$

同様に考えて

$f_{yy} = -\frac{\sqrt{x^2+y^2+z^2}\,(x^2-2y^2+z^2)}{(x^2+y^2+z^2)^3}$

$f_{zz} = -\frac{\sqrt{x^2+y^2+z^2}\,(x^2+y^2-2z^2)}{(x^2+y^2+z^2)^3}$

$\Delta f = \frac{\sqrt{x^2+y^2+z^2}\,\{(-2x^2+y^2+z^2)+(x^2-2y^2+z^2)+(x^2+y^2-2z^2)\}}{(x^2+y^2+z^2)^3} = 0$