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
$$f(x_{0} + h,y_{0} + k) = f(x_{0},y_{0}) + frac{hcdot frac{partial }{partial x} + kcdot frac{partial }{partial y}}{1!}f(x_{0},y_{0}) + frac{left ( hcdot frac{partial }{partial x} + kcdot frac{partial }{partial y}
ight )^{2}}{2!}f(x_{0},y_{0}) +...+ frac{left ( hcdot frac{partial }{partial x} + kcdot frac{partial }{partial y}
ight )^{n}}{n!}f(x_{0},y_{0}) + R_{n} \
R_{n} = frac{left ( hcdot frac{partial }{partial x} + kcdot frac{partial }{partial y}
ight )^{n+1}}{(n+1)!}f(x_{0} + heta h,y_{0} + heta k),0 < heta < 1$$
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：
$$Phi(t) = f(x_{0} + ht,y_{0}+kt),0leq tleq 1$$
当 $t = 1$ 时，就得到 $Phi(1) = f(x_{0} + h,y_{0}+k)$。
对 $Phi(t)$ 求导，有
$$Phi^{'}(t) = hf_{1}^{'} + kf_{2}^{'} = hfrac{partial f}{partial x} + kfrac{partial f}{partial y} = left ( hfrac{partial }{partial x} + kfrac{partial }{partial y} ight )f$$
$$Phi^{''}(t) = h^{2}f_{11}^{''} + 2hkf_{12}^{''} + k^{2}f_{22}^{''} = h^{2}frac{partial^{2} f}{(partial x)^{2}} + 2hkfrac{partial^{2} f}{partial x partial y} + k^{2}frac{partial^{2} f}{(partial y)^{2}} = left ( hfrac{partial }{partial x} + kfrac{partial }{partial y} ight )^{2}f$$
$$Phi^{'''}(t) = h^{3}f_{111}^{'''} + 3h^{2}kf_{112}^{'''} + 3hk^{2}f_{122}^{'''} + k^{3}f_{222}^{'''} = left ( hfrac{partial }{partial x} + kfrac{partial }{partial y} ight )^{3}f$$
当 $t = 0$ 时，得到
$$Phi (0) = f(x_{0},y_{0})$$
$$Phi^{'}(0) = left ( hfrac{partial }{partial x} + kfrac{partial }{partial y} ight )f(x_{0},y_{0})$$
$$Phi^{''}(0) = left ( hfrac{partial }{partial x} + kfrac{partial }{partial y} ight )^{2}f(x_{0},y_{0})$$
代入
$$Phi (t) = Phi (0) + Phi ^{'}(0)t + frac{Phi ^{''}(0)}{2}t^{2} + frac{Phi ^{'''}( heta)}{6}t^{3}$$
得
$$Phi (t) = f(x_{0},y_{0}) + left ( hfrac{partial }{partial x} + kfrac{partial }{partial y} ight )f(x_{0},y_{0})cdot t + frac{left ( hfrac{partial }{partial x} + kfrac{partial }{partial y} ight )^{2}}{2}f(x_{0},y_{0})cdot t^{2} + frac{left ( hfrac{partial }{partial x} + kfrac{partial }{partial y} ight )^{3}}{6}f(x_{0} + h heta,y_{0} + k heta)cdot t^{3}$$
所以有：
$$Phi (1) = f(x_{0},y_{0}) + left ( hfrac{partial }{partial x} + kfrac{partial }{partial y} ight )f(x_{0},y_{0})cdot + frac{left ( hfrac{partial }{partial x} + kfrac{partial }{partial y} ight )^{2}}{2}f(x_{0},y_{0})cdot + frac{left ( hfrac{partial }{partial x} + kfrac{partial }{partial y} ight )^{3}}{6}f(x_{0} + h heta,y_{0} + k heta)cdot$$