二阶偏导数求导对比：两个变量的三元函数和三个变量的二元函数

题目一：两个变量的三元函数

已知函数 $u$ $=$ $f(x+y,xy,\frac{x}{y})$, 求 $\frac{{\partial }^{2}u}{\partial x^2}$, $\frac{{\partial }^{2}u}{\partial x\partial y}$, $\frac{{\partial }^{2}u}{\partial y^2}$.

其中，$f$ 具有二阶连续偏导数。

难度评级：

解析一

为了方便计算，我们首先进行如下标注：

$$\begin{array}{}\text{(1)}& \{\begin{array}{ll}x+y& \leftrightarrow 1\\ xy& \leftrightarrow 2\\ \frac{x}{y}& \leftrightarrow 3\end{array}\end{array}$$

同时，在上面的 $(1)$ 式中，分别对 $x$ 和 $y$ 求偏导数，有：

$$\begin{array}{}\text{(2)}& \{\begin{array}{ll}x+y& \leftrightarrow 1\{\begin{array}{l}\partial x\to 1\\ \partial y\to 1\end{array}\\ \\ xy& \leftrightarrow 2\{\begin{array}{l}\partial x\to y\\ \partial y\to x\end{array}\\ \\ \frac{x}{y}& \leftrightarrow 3\{\begin{array}{l}\partial x\to \frac{1}{y}\\ \partial y\to \frac{-x}{y^2}\end{array}\end{array}\end{array}$$

接着，先求解一阶偏导数：

$$\begin{array}{rl}\frac{\partial u}{\partial x}& =f_1^{\prime }+y{f}_2^{\prime }+\frac{1}{y}{f}_3^{\prime }\\ \\ \frac{\partial u}{\partial y}& =f_1^{\prime }+x{f}_2^{\prime }–\frac{x}{y^2}{f}_3^{\prime }\end{array}$$

于是：

$$\begin{array}{r}\frac{{\partial }^{2}u}{\partial x^2}\\ \\ =& \frac{\partial u}{\partial x}(f_1^{\prime }+y{f}_2^{\prime }+\frac{1}{y}{f}_3^{\prime })\\ \\ =& f_{11}^{\prime \prime }+y{f}_{12}^{\prime \prime }+\frac{1}{y}{f}_{13}^{\prime \prime }\\ –& y(f_{21}^{\prime \prime }+y{f}_{22}^{\prime \prime }+\frac{1}{y}{f}_{23}^{\prime \prime })\\ –& \frac{1}{y}(f_{31}^{\prime \prime }+y{f}_{32}^{\prime \prime }+\frac{1}{y}{f}_{33}^{\prime \prime })\\ \\ \Rightarrow & \text{ 二阶偏导数连续 }\Rightarrow \{\begin{array}{l}{f}_{12}^{\prime \prime }=f_{21}^{\prime \prime }\\ f_{13}^{\prime \prime }=f_{31}^{\prime \prime }\\ f_{23}^{\prime \prime }=f_{32}^{\prime \prime }\end{array}\\ \\ =& {\mathit{f}}_{\mathbf{11}}^{\mathit{\prime }\mathit{\prime }}\mathbf{+}\mathbf{2}\mathit{y}{\mathit{f}}_{\mathbf{12}}^{\mathit{\prime }\mathit{\prime }}\mathbf{+}\frac{\mathbf{2}}{\mathit{y}}{\mathit{f}}_{\mathbf{13}}^{\mathit{\prime }\mathit{\prime }}\mathbf{+}{\mathit{y}}^{\mathbf{2}}{\mathit{f}}_{\mathbf{22}}^{\mathit{\prime }\mathit{\prime }}\mathbf{+}\mathbf{2}{\mathit{f}}_{\mathbf{23}}^{\mathit{\prime }\mathit{\prime }}\mathbf{+}\frac{\mathbf{1}}{{\mathit{y}}^{\mathbf{2}}}{\mathit{f}}_{\mathbf{33}}^{\mathit{\prime }\mathit{\prime }}\end{array}$$

于是：

$$\begin{array}{r}\frac{{\partial }^{2}u}{\partial x\partial y}\\ \\ =& \frac{\partial u}{\partial y}(f_1^{\prime }+y{f}_2^{\prime }+\frac{1}{y}{f}_3^{\prime })\\ \\ =& f_{11}^{\prime \prime }+x{f}_{12}^{\prime \prime }–\frac{x}{y^2}{f}_{13}^{\prime \prime }+f_2^{\prime }\\ –& y(f_{21}^{\prime \prime }+x{f}_{22}^{\prime \prime }–\frac{x}{y^2}{f}_{23}^{\prime \prime })–\frac{1}{y^2}{f}_3^{\prime }\\ –& \frac{1}{y}(f_{31}^{\prime \prime }+x{f}_{32}^{\prime \prime }–\frac{x}{y^2}{f}_{33}^{\prime \prime })\\ \\ =& {\mathit{f}}_{\mathbf{11}}^{\mathit{\prime }\mathit{\prime }}\mathbf{+}\mathbf{(}\mathit{x}\mathbf{+}\mathit{y}\mathbf{)}{\mathit{f}}_{\mathbf{12}}^{\mathit{\prime }\mathit{\prime }}\mathbf{+}\left(\frac{\mathbf{1}}{\mathit{y}}–\frac{\mathit{x}}{{\mathit{y}}^{\mathbf{2}}}\right){\mathit{f}}_{\mathbf{13}}^{\mathit{\prime }\mathit{\prime }}\\ & \mathbf{+}{\mathit{f}}_{\mathbf{2}}^{\mathit{\prime }}\mathbf{+}\mathit{x}\mathit{y}{\mathit{f}}_{\mathbf{22}}^{\mathit{\prime }\mathit{\prime }}–\frac{\mathbf{1}}{{\mathit{y}}^{\mathbf{2}}}{\mathit{f}}_{\mathbf{3}}^{\mathit{\prime }}–\frac{\mathit{x}}{{\mathit{y}}^{\mathbf{3}}}{\mathit{f}}_{\mathbf{33}}^{\mathit{\prime }\mathit{\prime }}\end{array}$$

于是：

$$\begin{array}{r}\frac{{\partial }^{2}u}{\partial y^2}\\ \\ =& \frac{\partial u}{\partial y}(f_1^{\prime }+y{f}_2^{\prime }+\frac{1}{y}{f}_3^{\prime })\\ \\ =& f_{11}^{\prime \prime }+x{f}_{12}^{\prime \prime }–\frac{x}{y^2}{f}_{13}^{\prime \prime }\\ –& x(f_{21}^{\prime \prime }+x{f}_{22}^{\prime \prime }–\frac{x}{y^2}{f}_{23}^{\prime \prime })+\frac{2x}{y^3}{f}_3^{\prime }\\ –& \frac{x}{y^2}(f_{31}^{\prime \prime }+x{f}_{32}^{\prime \prime }–\frac{x}{y^2}{f}_{33}^{\prime \prime })\\ \\ =& {\mathit{f}}_{\mathbf{11}}^{\mathit{\prime }\mathit{\prime }}\mathbf{+}\mathbf{2}\mathit{x}{\mathit{f}}_{\mathbf{12}}^{\mathit{\prime }\mathit{\prime }}–\mathbf{2}\frac{\mathit{x}}{{\mathit{y}}^{\mathbf{2}}}{\mathit{f}}_{\mathbf{13}}^{\mathit{\prime }\mathit{\prime }}\mathbf{+}{\mathit{x}}^{\mathbf{2}}{\mathit{f}}_{\mathbf{22}}^{\mathit{\prime }\mathit{\prime }}\\ & \mathbf{-}\mathbf{2}\frac{{\mathit{x}}^{\mathbf{2}}}{{\mathit{y}}^{\mathbf{2}}}{\mathit{f}}_{\mathbf{23}}^{\mathit{\prime }\mathit{\prime }}\mathbf{+}\frac{{\mathit{x}}^{\mathbf{2}}}{{\mathit{y}}^{\mathbf{4}}}{\mathit{f}}_{\mathbf{33}}^{\mathit{\prime }\mathit{\prime }}\mathbf{+}\frac{\mathbf{2}\mathit{x}}{{\mathit{y}}^{\mathbf{3}}}{\mathit{f}}_{\mathbf{3}}^{\mathit{\prime }}\end{array}$$