二元泰勒公式

用多个变量的一个多项式来近似表达一个给定的多元函数，并能具体的估算出误差的大小。

定义：函数 $f(x,y)$ 在含 $(x_{0},y_{0})$ 的某一邻域内连续且有直到 $n+1$ 阶的连续偏导数，$(x_{0} + h, y_{0} + k)$ 为此邻域内一点，则有

$$f(x_{0} + h,y_{0} + k) = f(x_{0},y_{0}) + \frac{h\cdot \frac{\partial }{\partial x} + k\cdot \frac{\partial }{\partial y}}{1!}f(x_{0},y_{0}) + \frac{\left ( h\cdot \frac{\partial }{\partial x} + k\cdot \frac{\partial }{\partial y} \right )^{2}}{2!}f(x_{0},y_{0}) +...+ \frac{\left ( h\cdot \frac{\partial }{\partial x} + k\cdot \frac{\partial }{\partial y} \right )^{n}}{n!}f(x_{0},y_{0}) + R_{n} \\
R_{n} = \frac{\left ( h\cdot \frac{\partial }{\partial x} + k\cdot \frac{\partial }{\partial y} \right )^{n+1}}{(n+1)!}f(x_{0} + \theta h,y_{0} + \theta k),0 < \theta < 1$$

类比于一元泰勒公式，每个多项式有两部分构成，一部分是包含偏导数的系数部分，另一部分是 $x-x_{0},y-y_{0}$ 的幂次项。

上面的定义式不太直观，在这个公式中多了很多交叉的项，如果只写到二阶，则形式如下：

$$f(x,y) = f(x_{0},y_{0}) + f_{x}^{'}(x_{0},y_{0})(x-x_{0}) + f_{y}^{'}(x_{0},y_{0})(y-y_{0}) + \frac{f_{xx}^{''}(x_{0},y_{0})}{2!}(x-x_{0})^{2} + \frac{f_{xy}^{''}(x_{0},y_{0})}{2!}2(x-x_{0})(y-y_{0}) + \frac{f_{yy}^{''}(x_{0},y_{0})}{2!}(y-y_{0})^{2} + R_{n}$$

或者是写成下面的形式

$$f(x_{0} + h,y_{0}+k) = f(x_{0},y_{0}) + f_{x}^{'}(x_{0},y_{0})h + f_{y}^{'}(x_{0},y_{0})k + \frac{f_{xx}^{''}(x_{0},y_{0})}{2!}h^{2} + \frac{f_{xy}^{''}(x_{0},y_{0})}{2!}2hk + \frac{f_{yy}^{''}(x_{0},y_{0})}{2!}k^{2} + R_{n}$$

下面来看下这个一长串的定义式是怎么推导出来的：

我们是利用一元泰勒公式来推导的，引入一元函数：

$$\Phi(t) = f(x_{0} + ht,y_{0}+kt),0\leq t\leq 1$$

当 $t = 1$ 时，就得到 $\Phi(1) = f(x_{0} + h,y_{0}+k)$。

对 $\Phi(t)$ 求导，有

$$\Phi^{'}(t) = hf_{1}^{'} + kf_{2}^{'} = h\frac{\partial f}{\partial x} + k\frac{\partial f}{\partial y} = \left ( h\frac{\partial }{\partial x} + k\frac{\partial }{\partial y}\right )f$$

$$\Phi^{''}(t) = h^{2}f_{11}^{''} + 2hkf_{12}^{''} + k^{2}f_{22}^{''} = h^{2}\frac{\partial^{2} f}{(\partial x)^{2}} + 2hk\frac{\partial^{2} f}{\partial x \partial y} + k^{2}\frac{\partial^{2} f}{(\partial y)^{2}} = \left ( h\frac{\partial }{\partial x} + k\frac{\partial }{\partial y}\right )^{2}f$$

$$\Phi^{'''}(t) = h^{3}f_{111}^{'''} + 3h^{2}kf_{112}^{'''} + 3hk^{2}f_{122}^{'''} + k^{3}f_{222}^{'''} = \left ( h\frac{\partial }{\partial x} + k\frac{\partial }{\partial y}\right )^{3}f$$

当 $t = 0$ 时，得到

$$\Phi (0) = f(x_{0},y_{0})$$

$$\Phi^{'}(0) =  \left ( h\frac{\partial }{\partial x} + k\frac{\partial }{\partial y}\right )f(x_{0},y_{0})$$

$$\Phi^{''}(0) = \left ( h\frac{\partial }{\partial x} + k\frac{\partial }{\partial y}\right )^{2}f(x_{0},y_{0})$$

代入

$$\Phi (t) = \Phi (0) + \Phi ^{'}(0)t + \frac{\Phi ^{''}(0)}{2}t^{2} + \frac{\Phi ^{'''}(\theta)}{6}t^{3}$$

得

$$\Phi (t) = f(x_{0},y_{0}) + \left ( h\frac{\partial }{\partial x} + k\frac{\partial }{\partial y}\right )f(x_{0},y_{0})\cdot t + \frac{\left ( h\frac{\partial }{\partial x} + k\frac{\partial }{\partial y}\right )^{2}}{2}f(x_{0},y_{0})\cdot t^{2} + \frac{\left ( h\frac{\partial }{\partial x} + k\frac{\partial }{\partial y}\right )^{3}}{6}f(x_{0} + h\theta,y_{0} + k\theta)\cdot t^{3}$$

所以有：

$$\Phi (1) = f(x_{0},y_{0}) + \left ( h\frac{\partial }{\partial x} + k\frac{\partial }{\partial y}\right )f(x_{0},y_{0})\cdot + \frac{\left ( h\frac{\partial }{\partial x} + k\frac{\partial }{\partial y}\right )^{2}}{2}f(x_{0},y_{0})\cdot + \frac{\left ( h\frac{\partial }{\partial x} + k\frac{\partial }{\partial y}\right )^{3}}{6}f(x_{0} + h\theta,y_{0} + k\theta)\cdot$$