第6章 6.15

(1)

$z = f_{(ax+by)} \qquad b\frac{\partial z}{\partial x} = a\frac{\partial z}{\partial y}$

$z_x = af'_{(ax+by)} \qquad z_{(y)} = bf'_{(ax+by)}$

これから

$bz_x = az_y \qquad b\frac{\partial z}{\partial x} = a\frac{\partial z}{\partial y}$

(2)

$z = f_{(x^2+y^2)} \qquad y\frac{\partial z}{\partial x} = x\frac{\partial z}{\partial y}$

$z_x = 2xf'_{(x^2+y^2)} \qquad z_y = 2yf'_{(x^2+y^2)}$

これから

$yz_x = xz_y \qquad y\frac{\partial z}{\partial x} = x\frac{\partial z}{\partial y}$

(3)

$z = xy+xf_{(\frac{y}{x})} \qquad x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y} = z+xy$

$z_x = y+f_{(\frac{y}{x})}+x\left(-\frac{y}{x^2}\right)f'_{(\frac{y}{x})} = y+f_{(\frac{y}{x})}-\frac{y}{x}f'_{(\frac{y}{x})}$

$z_y = x+f'_{(\frac{y}{x})}$

これから

$x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y} = xy+xf_{(\frac{y}{x})}-yf'_{(\frac{y}{x})}+xy+yf'_{(\frac{y}{x})} = xy+xf_{(\frac{y}{x})}+xy = z+xy$