问题及解答

设 $w=f(x+y+z,xyz)$, $f$ 具有二阶连续偏导数, 求 $\frac{\partial^2 w}{\partial x\partial z}$.

\[
\frac{\partial w}{\partial x}=f'_1\cdot 1+f'_2\cdot yz
\]

\[
\begin{split}
\frac{\partial^2 w}{\partial x\partial z}&=\frac{\partial}{\partial z}\Bigl(\frac{\partial w}{\partial x}\Bigr)\\
&=\frac{\partial}{\partial z}(f'_1+f'_2\cdot yz)\\
&=(f''_{11}\cdot 1+f''_{12}\cdot xy)+(f''_{21}\cdot 1+f''_{22}\cdot xy)yz+f'_2\cdot y\\
&=f''_{11}+xyf''_{12}+yzf''_{21}+xy^2 zf''_{22}+yf'_2
\end{split}
\]

又由题设, $f$ 的二阶导数连续, 故 $f''_{12}=f''_{21}$. 于是上式可化简为

\[
f''_{11}+(x+z)yf''_{12}+xy^2 zf''_{22}+yf'_2
\]