Editing Taylor's theorem (section)

=== Taylor's theorem for multivariate functions ===
Using notations of the preceding section, one has the following theorem.
{{math theorem|name=Multivariate version of Taylor's theorem<ref>Königsberger Analysis 2, p. 64 ff.</ref>|math_statement= Let {{math|''f'' : '''R'''<sup>''n''</sup> → '''R'''}} be a {{math|''k''}}-times [[continuously differentiable]] function at the point {{math|'''''a''''' ∈ '''R'''<sup>''n''</sup>}}. Then there exist functions {{math|''h''<sub>''α''</sub> : '''R'''<sup>''n''</sup> → '''R'''}}, where <math>|\alpha|=k,</math> such that

<math display="block">\begin{align}
& f(\boldsymbol{x}) = \sum_{|\alpha|\leq k} \frac{D^\alpha f(\boldsymbol{a})}{\alpha!} (\boldsymbol{x}-\boldsymbol{a})^\alpha  + \sum_{|\alpha|=k} h_\alpha(\boldsymbol{x})(\boldsymbol{x}-\boldsymbol{a})^\alpha, \\
& \mbox{and}\quad \lim_{\boldsymbol{x}\to \boldsymbol{a}}h_\alpha(\boldsymbol{x})=0.
\end{align}</math>}}

If the function {{math|''f'' : '''R'''<sup>''n''</sup> → '''R'''}} is {{math|''k'' + 1}} times [[continuously differentiable]] in a [[closed ball]] <math>B = \{ \mathbf{y} \in \R^n : \left\|\mathbf{a}-\mathbf{y}\right\| \leq  r\}</math> for some <math>r > 0</math>, then one can derive an exact formula for the remainder in terms of {{nowrap|({{math|''k''+1}})-th}} order [[partial derivatives]] of ''f'' in this neighborhood.<ref>{{cite web | title = Higher-Order Derivatives and Taylor's Formula in Several Variables | last = Folland | first = G. B. | url = https://sites.math.washington.edu/~folland/Math425/taylor2.pdf | website = Department of Mathematics {{!}} University of Washington | access-date = 2024-02-21 }}</ref> Namely,

<math display="block"> \begin{align}
& f( \boldsymbol{x} ) = \sum_{|\alpha|\leq k} \frac{D^\alpha f(\boldsymbol{a})}{\alpha!} (\boldsymbol{x}-\boldsymbol{a})^\alpha  + \sum_{|\beta|=k+1} R_\beta(\boldsymbol{x})(\boldsymbol{x}-\boldsymbol{a})^\beta, \\
& R_\beta( \boldsymbol{x} ) = \frac{|\beta|}{\beta!} \int_0^1 (1-t)^{|\beta|-1}D^\beta f \big(\boldsymbol{a}+t( \boldsymbol{x}-\boldsymbol{a} )\big) \, dt.
\end{align}
</math>

In this case, due to the [[continuous function|continuity]] of ({{math|''k''+1}})-th order [[partial derivative]]s in the [[compact set]] {{math|''B''}}, one immediately obtains the uniform estimates

<math display="block"> \left|R_\beta(\boldsymbol{x})\right| \leq \frac{1}{\beta!} \max_{|\alpha|=|\beta|} \max_{\boldsymbol{y}\in B} |D^\alpha f(\boldsymbol{y})|, \qquad \boldsymbol{x}\in B. </math>