8.2 Rozvoj funkce více proměnných

$$ \begin{align} f(x,y)&=f(x_0,y_0)+\left.\frac{\partial f}{\partial x}\right|_{x_0,y_0}(x-x_0)+\left.\frac{\partial f}{\partial y}\right|_{x_0,y_0}(y-y_0)\\ &+\frac{1}{2!}\left[\left.\frac{\partial^2f}{\partial x^2}\right|_{x_0,y_0}(x- x_0)^2+\left.\frac{2\,\partial^2f} {\partial x\,\partial y}\right|_{x_0,y_0}(x-x_0)(y-y_0)+ \left.\frac{\partial^2f}{\partial y^2}\right|_{x_0,y_0}(y-y_0)^2\right]\\ &+\frac{1}{3!}\left[\left.\frac{\partial^3f}{\partial x^3}\right|_{x_0,y_0}(x-x_0)^3+ \left.\frac{3\,\partial^3f}{\partial x^2\,\partial y}\right|_{x_0,y_0}(x-x_0)^2(y-y_0)\right.\\ &\left.\qquad\qquad\qquad\quad +\left.\frac{3\,\partial^3f}{\partial x\,\partial y^2}\right|_{x_0,y_0}(x-x_0)(y-y_0)^2+ \left.\frac{\partial^3f}{\partial y^3}\right|_{x_0,y_0}(y-y_0)^3\right]+\,\ldots\,, \end{align} $$

$$ f(x_1,\dots,x_k)=\sum_{n_1=0}^\infty\sum_{n_2=0}^\infty\cdots\sum_{n_k=0}^\infty \left.\left(\frac{\partial^{n_1+\cdots+n_k}f}{\partial x_1^{n_1}\cdots\partial x_k^{n_k}}\right)\right|_{(x_{01},\dots,x_{0k})}\frac{(x_1-x_{01})^{n_1}\cdots (x_k-x_{0k})^{n_k}}{n_1!\cdots n_k!}. $$