Taylor Series for Functions of Several Variables

In the previous lessons, the concept of differential of a function \(f: \mathbb {R}^n \rightarrow \mathbb {R}\) in a point \(\boldsymbol{x}_0^{~}\) of its existence field has been introduced and discussed. Considered an arbitrary point \(\boldsymbol{x}\) lying inside a (sufficiently small) neighbourhood of \(\boldsymbol{x}_0^{~}\) , the differential of f in \(\boldsymbol{x}_0^{~}\) is the part of the corresponding increment of this function ( \(f(\boldsymbol{x})-f(\boldsymbol{x}_0^{~})\) ) which is linear in the increment of the independent variable, i.e. in \(\boldsymbol{x}-\boldsymbol{x}_0^{~}\) (named as \(\Delta \boldsymbol{x}\) before). Another way to reread the differential lies in considering the approximation of a differentiable f in a (sufficiently small) neighbourhood of \(\boldsymbol{x}_0^{~}\) .