多元泰勒

通式

f(x,y)=limni=0ndnf(x0,y0)n! 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)=i=0ndnf(x0,y0)n!+o(ρn) 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)=i=0ndnf(x0,y0)n!+dn+1f(x0+θΔx,y0+θΔy)(n+1)! 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(x0,y0)=fx(x0+θΔx,y0+θΔy)Δx+fy(x0+θΔx,y0+θΔy)Δyf(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