kaoyan2advanced 高等数学 第61题
📝 题目
### 第61题
设 $u=f(x, y, z), z=z(x, y)$ 是由方程 $\varphi(x+y, z)=1$ 所确定的隐函数，其中 $f$ 和 $\varphi$ 有二阶连续偏导数且 $\varphi_{2}^{\prime} \neq 0$ ，则 $\displaystyle \frac{\partial^{2} u}{\partial x \partial y}=$ $\_\_\_\_$ ．
💡 答案解析
**答案**：$\displaystyle \frac{\partial^2 u}{\partial x\partial y}=f_{xz}\cdot\frac{\partial z}{\partial y}+f_{yz}\cdot\frac{\partial z}{\partial x}+f_{zz}\cdot\frac{\partial z}{\partial x}\frac{\partial z}{\partial y}+f_z\cdot\frac{\partial^2 z}{\partial x\partial y}$，其中$\displaystyle \frac{\partial z}{\partial x}=-\frac{\varphi_1'}{\varphi_2'}$，$\displaystyle \frac{\partial z}{\partial y}=-\frac{\varphi_1'}{\varphi_2'}$，$\displaystyle \frac{\partial^2 z}{\partial x\partial y}=-\frac{\varphi_{11}''\varphi_2'-\varphi_1'\varphi_{21}''}{(\varphi_2')^2}$（具体表达式略） **解析**： 步骤1：$u=f(x,y,z)$，则$\displaystyle \frac{\partial u}{\partial x}=f_x+f_z\frac{\partial z}{\partial x}$。 步骤2：$\displaystyle \frac{\partial^2 u}{\partial x\partial y}=f_{xy}+f_{xz}\frac{\partial z}{\partial y}+f_{zy}\frac{\partial z}{\partial x}+f_{zz}\frac{\partial z}{\partial x}\frac{\partial z}{\partial y}+f_z\frac{\partial^2 z}{\partial x\partial y}$。 步骤3：由$\varphi(x+y,z)=1$隐函数求导得$\displaystyle \frac{\partial z}{\partial x}=-\frac{\varphi_1'}{\varphi_2'}$，$\displaystyle \frac{\partial z}{\partial y}=-\frac{\varphi_1'}{\varphi_2'}$，再求二阶混合偏导。 **难度**：★★★★☆