通式
$$ f(x,y)=\lim_{ n \to \infty } \sum_{i=0}^{n} \frac{ \mathrm{d}^{n}f(x_{0},y_{0})}{n!} $$
皮亚诺余项
$$ f(x,y)=\sum_{i=0}^{n} \frac{ \mathrm{d}^{n}f(x_{0},y_{0})}{n!}+o(\rho^{n}) $$
拉格朗日余项
$$ f(x,y)=\sum_{i=0}^{n} \frac{ \mathrm{d}^{n}f(x_{0},y_{0})}{n!}+\frac{ \mathrm{d}^{n+1}f(x_{0}+\theta\Delta x,y_{0}+\theta\Delta y)}{(n+1)!} $$
特例：拉格朗日中值定理
$$f(x,y)-f(x_{0},y_{0})=\frac{ \partial f }{ \partial x } (x_{0}+\theta\Delta x,y_{0}+\theta\Delta y)\Delta x+\frac{ \partial f }{ \partial y }(x_{0}+\theta\Delta x,y_{0}+\theta\Delta y)\Delta y$$