问 . 何谓高阶全微分?
答：一般来说, 二元函数 z=f(\mathrm{x}, \mathrm{y}) 的全微分 d z=\frac{\partial z}{\partial x} d x \frac{\partial z}{\partial y} d y, 仍是 \mathrm{x} 与 \mathrm{y} 的二元函数, d x 与 d y 看作与 \mathrm{x}, \mathrm{y} 无关的常数.
定义 如果二元函数 z=f(\mathrm{x}, \mathrm{y}) 的高阶偏导数连续, 全微分 d z 的全微分 d(d z), 称为函数 z=f （x,y）的二阶全微分, 表为 d^2 z.
一般情况, \mathrm{k}-1 阶全微分 d^{k-1} z 的全微分 d\left(d^{k-1} z\right) 称为函数 z=f(\mathrm{x}, \mathrm{y}) 的 \mathrm{k} 阶全微分, 表为 d^k z.二阶以及二阶以上的全微分, 统称为高阶全微分。
根据定义, 二元函数 z=f(x, y) 的高阶全微分是
\begin{aligned}
& d^2 z=d(d z)=d\left(\frac{\partial z}{\partial x} d x+\frac{\partial z}{\partial y} d y\right) \\
& =\frac{\partial}{\partial x}\left(\frac{\partial z}{\partial x} d x+\frac{\partial z}{\partial y} d y\right) d x+\frac{\partial}{\partial y}\left(\frac{\partial z}{\partial x} d x+\frac{\partial z}{\partial y} d y\right) d y \\
& =\frac{\partial^2}{\partial x} d x^2+2 \frac{\partial^2 z}{\partial x \partial y} d x d y+\frac{\partial^2 z}{\partial y^2} d y^2 \\
& =\left(d x \frac{\partial}{\partial x}+d y \frac{\partial}{\partial y}\right)^2 z
\end{aligned}
\begin{aligned} & d^3 z=d\left(d^2 z\right)=\frac{\partial^3 z}{\partial x^3} d x^3+3 \frac{\partial^3 z}{\partial x^2 \partial y} d x^2 d y+3 \frac{\partial^3 z}{\partial x \partial y^2} d x d y^2+\frac{\partial 3 z}{\partial y^3} d y^3 \\ & =\left(d x \frac{\partial}{\partial x}+d y \frac{\partial}{\partial y}\right)^3 \cdot z \end{aligned}
一般来说， d^n z=\left(d x \frac{\partial}{\partial x}+d y \frac{\partial}{\partial y}\right)^n \cdot z , 并把 d^n f(x, y)_{\text {在点 }}(a, b) 的值, 表为 d^n f(a, b)=\left(d x \frac{\partial}{\partial x}+d y \frac{\partial}{\partial y}\right)^n f(a, b)
由此不难给出 \mathrm{n} 元函数 u=f\left(x_1, x_2, \ldots, x_n\right) 的全微分是
d^k u=\left(d x_1 \frac{\partial}{\partial x_1}+d x_2 \frac{\partial}{\partial x_2}+\ldots+d x_n \frac{\partial}{\partial x_n}\right)^k \cdot u
有了二元函数的高阶全微分, 二元函数 f(x, y) 在点 (a, b) 的泰勒公式（设 d x=h, d y=k ), 即
\begin{aligned}
& f(a+h, b+k)=f(a, b)+\frac{1}{1!}\left(h \frac{\partial}{\partial x}+k \frac{\partial}{\partial y}\right) f(a, b) \\
& +\frac{1}{2!}\left(h \frac{\partial}{\partial x}+k \frac{\partial}{\partial y}\right)^2 f(a, b) \\
& +\cdots+\frac{1}{n!}\left(h \frac{\partial}{\partial x}+k \frac{\partial}{\partial y}\right)^n f(a, b) \\
& +\frac{1}{(n+1)!}\left(h \frac{\partial}{\partial x}+k \frac{\partial}{\partial y}\right)^{n+1} f(a+\theta h, b+\theta k),\quad 0<\theta<1
\end{aligned}
可用高阶全微分的简单的表为
\begin{aligned}
& f(a+h, b+k)=f(a, b)+\frac{1}{1!} d f(a, b)+\frac{1}{2!} d^2 f(a, b) \\
& +\cdots+\frac{1}{n!} d^n f(a, b) \\
& +\frac{1}{(n+1)!} d^{n+1} f(a+\theta h, b+\theta k), 0<\theta<1
\end{aligned}
欢迎大家进行探讨~