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

题目二：三个变量的二元函数

设 $u$ $=$ $f(x+y+z,x^2+y^2+z^2)$, 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：

解析二

首先，求解一阶偏导数：

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

于是：

$$\begin{array}{r}\frac{{\partial }^{2}u}{\partial x^2}\\ \\ =& \frac{\partial u}{\partial x}(f_1^{\prime }+2x{f}_2^{\prime })\\ \\ =& f_{11}^{\prime \prime }+2x{f}_{12}^{\prime \prime }+2f_2^{\prime }+2x(f_{21}^{\prime \prime }+2x{f}_{22}^{\prime \prime })\\ \\ =& {\mathit{f}}_{\mathbf{11}}^{\mathit{\prime }\mathit{\prime }}\mathbf{+}\mathbf{4}\mathit{x}{\mathit{f}}_{\mathbf{12}}^{\mathit{\prime }\mathit{\prime }}\mathbf{+}\mathbf{2}{\mathit{f}}_{\mathbf{2}}^{\mathit{\prime }}\mathbf{+}\mathbf{4}{\mathit{x}}^{\mathbf{2}}{\mathit{f}}_{\mathbf{22}}^{\mathit{\prime }\mathit{\prime }}\end{array}$$

于是：

$$\begin{array}{r}\frac{{\partial }^{2}u}{\partial x\partial y}\\ \\ =& \frac{\partial u}{\partial y}(f_1^{\prime }+2y{f}_2^{\prime })\\ \\ =& f_{11}^{\prime \prime }+2y{f}_{12}^{\prime \prime }+2x(f_{21}^{\prime \prime }+2y{f}_{22}^{\prime \prime })\\ \\ =& {\mathit{f}}_{\mathbf{11}}^{\mathit{\prime }\mathit{\prime }}\mathbf{+}\mathbf{2}\mathbf{(}\mathit{x}\mathbf{+}\mathit{y}\mathbf{)}{\mathit{f}}_{\mathbf{12}}^{\mathit{\prime }\mathit{\prime }}\mathbf{+}\mathbf{4}\mathit{x}\mathit{y}{\mathit{f}}_{\mathbf{22}}^{\mathit{\prime }\mathit{\prime }}\end{array}$$

于是：

$$\begin{array}{r}\frac{{\partial }^{2}u}{\partial y^2}\\ \\ =& \frac{\partial u}{\partial y}(f_1^{\prime }+2y{f}_2^{\prime })\\ \\ =& f_{11}^{\prime \prime }+2y{f}_{12}^{\prime \prime }+2f_2^{\prime }+2y(f_{21}^{\prime \prime }+2y{f}_{22}^{\prime \prime })\\ \\ =& {\mathit{f}}_{\mathbf{11}}^{\mathit{\prime }\mathit{\prime }}\mathbf{+}\mathbf{4}\mathit{y}{\mathit{f}}_{\mathbf{12}}^{\mathit{\prime }\mathit{\prime }}\mathbf{+}\mathbf{2}{\mathit{f}}_{\mathbf{2}}^{\mathit{\prime }}\mathbf{+}\mathbf{4}{\mathit{y}}^{\mathbf{2}}{\mathit{f}}_{\mathbf{22}}^{\mathit{\prime }\mathit{\prime }}\end{array}$$

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