用多個變量的一個多項式來近似表達一個給定的多元函數，並能具體的估算出誤差的大小。
定義：函數 $f(x,y)$ 在含 $(x_{0},y_{0})$ 的某一鄰域內連續且有直到 $n+1$ 階的連續偏導數，$(x_{0} + h, y_{0} + k)$ 為此鄰域內一點，則有
$$f(x_{0} + h,y_{0} + k) = f(x_{0},y_{0}) + \frac{h\cdot \frac{\partial }{\partial x} + k\cdot \frac{\partial }{\partial y}}{1!}f(x_{0},y_{0}) + \frac{\left ( h\cdot \frac{\partial }{\partial x} + k\cdot \frac{\partial }{\partial y} \right )^{2}}{2!}f(x_{0},y_{0}) +...+ \frac{\left ( h\cdot \frac{\partial }{\partial x} + k\cdot \frac{\partial }{\partial y} \right )^{n}}{n!}f(x_{0},y_{0}) + R_{n} \\
R_{n} = \frac{\left ( h\cdot \frac{\partial }{\partial x} + k\cdot \frac{\partial }{\partial y} \right )^{n+1}}{(n+1)!}f(x_{0} + \theta h,y_{0} + \theta k),0 < \theta < 1$$
類比於一元泰勒公式，每個多項式有兩部分構成，一部分是包含偏導數的系數部分，另一部分是 $x-x_{0},y-y_{0}$ 的冪次項。
上面的定義式不太直觀，在這個公式中多了很多交叉的項，如果只寫到二階，則形式如下：
$$f(x,y) = f(x_{0},y_{0}) + f_{x}^{'}(x_{0},y_{0})(x-x_{0}) + f_{y}^{'}(x_{0},y_{0})(y-y_{0}) + \frac{f_{xx}^{''}(x_{0},y_{0})}{2!}(x-x_{0})^{2} + \frac{f_{xy}^{''}(x_{0},y_{0})}{2!}2(x-x_{0})(y-y_{0}) + \frac{f_{yy}^{''}(x_{0},y_{0})}{2!}(y-y_{0})^{2} + R_{n}$$
或者是寫成下面的形式
$$f(x_{0} + h,y_{0}+k) = f(x_{0},y_{0}) + f_{x}^{'}(x_{0},y_{0})h + f_{y}^{'}(x_{0},y_{0})k + \frac{f_{xx}^{''}(x_{0},y_{0})}{2!}h^{2} + \frac{f_{xy}^{''}(x_{0},y_{0})}{2!}2hk + \frac{f_{yy}^{''}(x_{0},y_{0})}{2!}k^{2} + R_{n}$$
下面來看下這個一長串的定義式是怎么推導出來的：
我們是利用一元泰勒公式來推導的，引入一元函數：
$$\Phi(t) = f(x_{0} + ht,y_{0}+kt),0\leq t\leq 1$$
當 $t = 1$ 時，就得到 $\Phi(1) = f(x_{0} + h,y_{0}+k)$。
對 $\Phi(t)$ 求導，有
$$\Phi^{'}(t) = hf_{1}^{'} + kf_{2}^{'} = h\frac{\partial f}{\partial x} + k\frac{\partial f}{\partial y} = \left ( h\frac{\partial }{\partial x} + k\frac{\partial }{\partial y}\right )f$$
$$\Phi^{''}(t) = h^{2}f_{11}^{''} + 2hkf_{12}^{''} + k^{2}f_{22}^{''} = h^{2}\frac{\partial^{2} f}{(\partial x)^{2}} + 2hk\frac{\partial^{2} f}{\partial x \partial y} + k^{2}\frac{\partial^{2} f}{(\partial y)^{2}} = \left ( h\frac{\partial }{\partial x} + k\frac{\partial }{\partial y}\right )^{2}f$$
$$\Phi^{'''}(t) = h^{3}f_{111}^{'''} + 3h^{2}kf_{112}^{'''} + 3hk^{2}f_{122}^{'''} + k^{3}f_{222}^{'''} = \left ( h\frac{\partial }{\partial x} + k\frac{\partial }{\partial y}\right )^{3}f$$
當 $t = 0$ 時，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：
$$\Phi (1) = f(x_{0},y_{0}) + \left ( h\frac{\partial }{\partial x} + k\frac{\partial }{\partial y}\right )f(x_{0},y_{0})\cdot + \frac{\left ( h\frac{\partial }{\partial x} + k\frac{\partial }{\partial y}\right )^{2}}{2}f(x_{0},y_{0})\cdot + \frac{\left ( h\frac{\partial }{\partial x} + k\frac{\partial }{\partial y}\right )^{3}}{6}f(x_{0} + h\theta,y_{0} + k\theta)\cdot$$