问题描述： 设z=f(xy,x^2+y^2),其中f具有连续的二阶偏导数,求α^2z/αxαy.

步骤 1：定义函数
给定函数为 \(z = f(u, v)\)，其中 \(u = xy\)，\(v = x^2 + y^2\)，且 \(f\) 具有连续的二阶偏导数。

步骤 2：计算一阶偏导数
首先，计算 \(z\) 对 \(x\) 的偏导数 \(\frac{\partial z}{\partial x}\)：
\[
\frac{\partial z}{\partial x} = \frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial x} + \frac{\partial f}{\partial v} \cdot \frac{\partial v}{\partial x} = y \frac{\partial f}{\partial u} + 2x \frac{\partial f}{\partial v}
\]
其中，\(\frac{\partial f}{\partial u}\) 和 \(\frac{\partial f}{\partial v}\) 分别表示 \(f\) 对 \(u\) 和 \(v\) 的偏导数。

步骤 3：计算二阶偏导数
接下来，计算 \(\frac{\partial z}{\partial x}\) 对 \(y\) 的偏导数 \(\frac{\partial^2 z}{\partial x \partial y}\)：
\[
\frac{\partial^2 z}{\partial x \partial y} = \frac{\partial}{\partial y} \left( y \frac{\partial f}{\partial u} + 2x \frac{\partial f}{\partial v} \right)
\]
\[
= \frac{\partial f}{\partial u} + y \left( x \frac{\partial^2 f}{\partial u^2} + 2y \frac{\partial^2 f}{\partial u \partial v} \right) + 2x \left( x \frac{\partial^2 f}{\partial v \partial u} + 2y \frac{\partial^2 f}{\partial v^2} \right)
\]
\[
= \frac{\partial f}{\partial u} + yx \frac{\partial^2 f}{\partial u^2} + 2y^2 \frac{\partial^2 f}{\partial u \partial v} + 2x^2 \frac{\partial^2 f}{\partial v \partial u} + 4xy \frac{\partial^2 f}{\partial v^2}
\]
由于 \(\frac{\partial^2 f}{\partial u \partial v} = \frac{\partial^2 f}{\partial v \partial u}\)，上式可以简化为：
\[
\frac{\partial^2 z}{\partial x \partial y} = \frac{\partial f}{\partial u} + (2x^2 + xy) \frac{\partial^2 f}{\partial u \partial v} + 4xy \frac{\partial^2 f}{\partial v^2}
\]