第6章 6.7

(1)

$z = \sin(u^2+v^2) \qquad u = x+y \qquad v = xy$

$x_u = 2u\cos(u^2+v^2) \qquad z_v = 2v\cos(u^2+v^2)$

$u_x = 1 \qquad u_y = 1 \qquad v_x = y \qquad v_y = x$

$z_x = 2(x+y)\cos(x^2+x^2y^2+2xy+y^2) + 2xy^2(\cos(x^2+x^2y^2+2xy+y^2))$

$= 2(x+y+xy^2)\cos(x^2+x^2y^2+2xy+y^2)$

$z_y = 2(x+y+x^2y)\cos(x^2+x^2y^2+2xy+y^2)$

(2)

$z = uv \qquad u = \log\sqrt{x^2+y^2} \qquad v = \tan^{-1}\frac{y}{x}$

$z_u = v \qquad z_v = u$

$u_x = \frac{\dfrac{x}{\sqrt{x^2+y^2}}}{\sqrt{x^2+y^2}} = \frac{x}{x^2+y^2} \qquad u_y = \frac{\dfrac{y}{\sqrt{x^2+y^2}}}{\sqrt{x^2+y^2}} = \frac{y}{x^2+y^2}$

$v_x = \frac{-y}{x^2+y^2} \qquad v_y = \frac{x}{x^2+y^2}$

$z_x = \tan^{-1}\frac{x}{y}\cdot\frac{x}{x^2+y^2} + \log\sqrt{x^2+y^2}\cdot\frac{-y}{x^2+y^2} = \frac{1}{x^2+y^2}\left(x\tan^{-1}\frac{y}{x} - y\log\sqrt{x^2+y^2}\right)$

$z_y = \frac{1}{x^2+y^2}\left(y\tan^{-1}\frac{y}{x} + x\log\sqrt{x^2+y^2}\right)$