📝 题目
5.3.8 设 $u = u\left( {x,y,z}\right)$ 由方程 $\frac{{x}^{2}}{{a}^{2} + u} + \frac{{y}^{2}}{{b}^{2} + u} + \frac{{z}^{2}}{{c}^{2} + u} = 1$ 所确定,证明:
$$ {\left( \frac{\partial u}{\partial x}\right) }^{2} + {\left( \frac{\partial u}{\partial y}\right) }^{2} + {\left( \frac{\partial u}{\partial z}\right) }^{2} = {2x}\frac{\partial u}{\partial x} + {2y}\frac{\partial u}{\partial y} + {2z}\frac{\partial u}{\partial z}. $$