(Clairaut 定理)设 $E$ 是 $\mathbf{R}^n$ 的开子集合,并设 $f:\mathbf{E}\to \mathbf{R}^{m}$ 是 $E$ 上的二次连续可微函数.那么对于一切$x_0\in E$ 和 $1\leq i,j\leq n$,
\begin{align*}
\frac{\partial }{\partial x_j}\frac{\partial f}{\partial
x_i}(x_0)= \frac{\partial }{\partial x_i}\frac{\partial
f}{\partial x_j}(x_0)
\end{align*}

为$(a_1,a_2,\cdots,a_n)$.则

\label{eq:8.00}
\frac{\partial f}{\partial x_i}(x_0)=\lim_{\Delta x_{i}\to 0;\Delta
x_{i}\neq 0}\frac{f(a_1,\cdots,a_i+\Delta
x_{i},\cdots,a_n)-f(a_1,\cdots,a_i,\cdots,a_n)}{\Delta x_{i}}.

\begin{align*}
&\frac{\partial }{\partial x_j}\frac{\partial f}{\partial
x_i}(x_0)\\&=\lim_{\Delta x_j\to 0;\Delta x_j\neq
0}\lim_{\Delta x_i\to 0;\Delta x_i\neq
0}\frac{\frac{f(a_1,\cdots,a_j+\Delta x_j,\cdots,a_i+\Delta
x_i,\cdots,a_n)-f(a_1,\cdots,a_j+\Delta
x_j,\cdots,a_i,\cdots,a_n)}{\Delta x_i}-\frac{f(a_1,\cdots,a_j,\cdots,a_i+\Delta
x_i,\cdots,a_n)-f(a_1,\cdots,a_i,\cdots,a_n)}{\Delta x_i}}{\Delta x_j}.
\end{align*}

\begin{align*}
&\frac{\partial }{\partial x_i}\frac{\partial f}{\partial
x_j}(x_0)\\&=\lim_{\Delta x_i\to 0;\Delta x_i\neq
0}\lim_{\Delta x_j\to 0;\Delta x_j\neq
0}\frac{\frac{f(a_1,\cdots,a_j+\Delta x_j,\cdots,a_i+\Delta
x_i,\cdots,a_n)-f(a_1,\cdots,a_j+\Delta
x_j,\cdots,a_i,\cdots,a_n)}{\Delta x_i}-\frac{f(a_1,\cdots,a_j,\cdots,a_i+\Delta
x_i,\cdots,a_n)-f(a_1,\cdots,a_i,\cdots,a_n)}{\Delta x_i}}{\Delta x_j}.
\end{align*}