2変数関数 \( z = f(x,y) \) の2階以上の偏微分について。
1階偏導関数
$$ \frac{\partial f}{\partial x}, \quad \frac{\partial f}{\partial y}$$
2階偏導関数
\begin{eqnarray}
\frac{\partial}{\partial x} \left(\frac{\partial f}{\partial x}\right) &=& \frac{\partial^2 f}{\partial x^2}\\
\frac{\partial}{\partial x} \left(\frac{\partial f}{\partial y} \right) &=& \frac{\partial^2 f}{\partial x\partial y}\\
\frac{\partial}{\partial y} \left(\frac{\partial f}{\partial x} \right) &=& \frac{\partial^2 f}{\partial y\partial x} \\
\frac{\partial}{\partial y} \left(\frac{\partial f}{\partial y} \right) &=& \frac{\partial^2 f}{\partial y^2}
\end{eqnarray}
3階偏導関数
$$ \frac{\partial}{\partial x} \left(\frac{\partial^2 f}{\partial x\partial y}\right) = \frac{\partial^3 f}{\partial x^2 \partial y} , \ \ \mbox{etc.}$$
偏微分の順序に関する定理
2階偏導関数について，偏微分の順序を変えてもよい。すなわち，
$$\frac{\partial^2 f}{\partial x\partial y} = \frac{\partial^2 f}{\partial y\partial x}, $$
3階以上の高次偏導関数についても同様。