9.7.1 泰勒定理

首先我们考虑一个二元函数. Suppose $f(x, y)$ 是一个函数，其所有的 $n$ 阶偏导数在某个圆域上存在且连续 $D$ 以…为圆心 $(a, b)$ . Then, if $(a + h, b + k)$ 位于 $D$ , 我们有

$$\begin{aligned} f(a + h, b + k) &= f(a, b) + \frac{1}{1!} \left( h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y} \right) f(a, b) + \frac{1}{2!} \left( h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y} \right)^2 f(a, b) \ &\quad + \dots + \frac{1}{(n-1)!} \left( h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y} \right)^{n-1} f(a, b) \ &\quad + \frac{1}{n!} \left( h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y} \right)^n f(a + \theta h, b + \theta k) \end{aligned} \quad (9.35)$$

where $0 < \theta < 1$ . 这里我们引入了记号

$$\left( h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y} \right)^r f(a, b)$$

表示该表达式的值

$$\begin{aligned} &h^r \frac{\partial^r f}{\partial x^r} + \binom{r}{1} h^{r-1} k \frac{\partial^r f}{\partial x^{r-1} \partial y} + \binom{r}{2} h^{r-2} k^2 \frac{\partial^r f}{\partial x^{r-2} \partial y^2} + \dots \ &\quad + \binom{r}{r-1} h k^{r-1} \frac{\partial^r f}{\partial x \partial y^{r-1}} + k^r \frac{\partial^r f}{\partial y^r} \end{aligned}$$

在该点处 $(a, b)$ .

这个结果是通过反复使用链式法则得到的。. Setting $x = a + ht$ and $y = b + kt$ , where $0 \leq t \leq 1$ , 我们得到

$$g(t) = f(a + ht, b + kt)$$

这是一个单变量函数, 以便, 由定理 9.4, 它有一个泰勒展开式

$$g(t) = g(0) + \frac{t}{1!} g'(0) + \frac{t^2}{2!} g''(0) + \dots + \frac{t^{n-1}}{(n-1)!} g^{(n-1)}(0) + \frac{t^n}{n!} g^{(n)}(\theta t)$$

where $0 \leq \theta \leq 1$ . 的导数 $g$ 可用链式法则求得:

$$\begin{aligned} g' &= \frac{dg}{dt} = \frac{dx}{dt} \frac{\partial f}{\partial x} + \frac{dy}{dt} \frac{\partial f}{\partial y} = h \frac{\partial f}{\partial x} + k \frac{\partial f}{\partial y} = \left( h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y} \right) f \ g'' &= \frac{d^2 g}{dt^2} = \frac{d}{dt} \left( h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y} \right) f = \left( h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y} \right) \left( h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y} \right) f \ &= \left( h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y} \right)^2 f \end{aligned}$$