📝 题目
5.3.7 设 $u = u\left( {x,y,z}\right)$ 由方程 $F\left( {{u}^{2} - {x}^{2},{u}^{2} - {y}^{2},{u}^{2} - {z}^{2}}\right) = 0$ 所确定,证明:
$$ \frac{{u}_{x}^{\prime }}{x} + \frac{{u}_{y}^{\prime }}{y} + \frac{{u}_{z}^{\prime }}{z} = \frac{1}{u}. $$
5.3.7 设 $u = u\left( {x,y,z}\right)$ 由方程 $F\left( {{u}^{2} - {x}^{2},{u}^{2} - {y}^{2},{u}^{2} - {z}^{2}}\right) = 0$ 所确定,证明:
$$ \frac{{u}_{x}^{\prime }}{x} + \frac{{u}_{y}^{\prime }}{y} + \frac{{u}_{z}^{\prime }}{z} = \frac{1}{u}. $$