📝 题目
例 1 设 $x = x\left( {y,z}\right) ,y = y\left( {z,x}\right) ,z = z\left( {x,y}\right)$ 为由方程 $F(x,y$ , $z) = 0$ 所确定的隐函数. 证明:
$$ \frac{\partial x}{\partial y} \cdot \frac{\partial y}{\partial z} \cdot \frac{\partial z}{\partial x} = - 1 $$
💡 答案与解析
证 由隐函数定理知
$$ \frac{\partial x}{\partial y} = \frac{\frac{\partial F}{\partial y}}{\frac{\partial F}{\partial x}},\;\frac{\partial y}{\partial z} = - \frac{\frac{\partial F}{\partial z}}{\frac{\partial F}{\partial y}},\;\frac{\partial z}{\partial x} = - \frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial z}}, $$
所以得
$$ \frac{\partial x}{\partial y} \cdot \frac{\partial y}{\partial z} \cdot \frac{\partial z}{\partial x} = - 1 $$
评注 多元偏导数记号与一元微分记号不同,它不能理解成两个量之比.