二元函数Taylor公式

castelu

定理（Taylor定理）　若函数$f$在点$P_0(x_0,y_0)$的某邻域$U^*(P_0)$内有直到$n+1$阶的连续偏导数，则对$U(P_0)$内任一点$(x_0+h,y_0+k)$，存在相应的$\theta \in (0,1)$，使得
$$f(x_0+h,y_0+k)=f(x_0,y_0)+(h \frac{\partial}{\partial x}+k \frac{\partial}{\partial y})f(x_0,y_0)+$$
$$\frac{1}{2!}(h \frac{\partial}{\partial x}+k \frac{\partial}{\partial y})^2f(x_0,y_0)+\cdots+$$
$$\frac{1}{n!}(h \frac{\partial}{\partial x}+k \frac{\partial}{\partial y})^nf(x_0,y_0)+$$
$$\frac{1}{(n+1)!}(h \frac{\partial}{\partial x}+k \frac{\partial}{\partial y})^{n+1}f(x_0+\theta h,y_0+\theta k)。$$

　　上式称为二元函数$f$在点$P_0$的$n$阶Taylor公式，其中
$$(h \frac{\partial}{\partial x}+k \frac{\partial}{\partial y})^m f(x_0,y_0)=\sum\limits_{i=0}^m C_m^i \frac{\partial^m}{\partial x^i \partial y^{m-i}}h^ik^{m-i}。$$