Editing Taylor's theorem (section)

== Generalizations of Taylor's theorem ==

=== Higher-order differentiability ===

A function {{math|''f'': '''R'''<sup>''n''</sup> → '''R'''}} is [[derivative|differentiable]] at {{math|'''''a''''' ∈ '''R'''<sup>''n''</sup>}} [[if and only if]] there exists a [[linear functional]] {{math|''L'' : '''R'''<sup>''n''</sup> → '''R'''}} and a function {{math|''h'' : '''R'''<sup>''n''</sup> → '''R'''}} such that

<math display="block"> f(\boldsymbol{x}) = f(\boldsymbol{a}) + L(\boldsymbol{x}-\boldsymbol{a}) + h(\boldsymbol{x})\lVert\boldsymbol{x}-\boldsymbol{a}\rVert,
\qquad \lim_{\boldsymbol{x}\to\boldsymbol{a}} h(\boldsymbol{x})=0. </math>

If this is the case, then <math display="inline">L = df(\boldsymbol{a})</math> is the (uniquely defined) [[differential of a function|differential]] of {{math|''f''}} at the point {{math|'''''a'''''}}. Furthermore, then the [[partial derivatives]] of {{math|''f''}} exist at {{math|'''''a'''''}} and the differential of {{math|''f''}} at {{math|'''''a'''''}} is given by

<math display="block"> df( \boldsymbol{a} )( \boldsymbol{v} ) = \frac{\partial f}{\partial x_1}(\boldsymbol{a}) v_1 + \cdots + \frac{\partial f}{\partial x_n}(\boldsymbol{a}) v_n. </math>

Introduce the [[multi-index notation]]

<math display="block"> |\alpha| = \alpha_1+\cdots+\alpha_n, \quad \alpha!=\alpha_1!\cdots\alpha_n!, \quad \boldsymbol{x}^\alpha=x_1^{\alpha_1}\cdots x_n^{\alpha_n} </math>

for {{math|''α'' ∈ '''N'''<sup>''n''</sup>}} and {{math|'''''x''''' ∈ '''R'''<sup>''n''</sup>}}.  If all the <math display="inline">k</math>-th order [[partial derivatives]] of {{math|''f'' : '''R'''<sup>''n''</sup> → '''R'''}} are continuous at {{math|'''''a''''' ∈ '''R'''<sup>''n''</sup>}}, then by [[symmetry of second derivatives|Clairaut's theorem]], one can change the order of mixed derivatives at {{math|'''''a'''''}}, so the short-hand notation

<math display="block"> D^\alpha f = \frac{\partial^{|\alpha|}f}{\partial\boldsymbol x^\alpha} = \frac{\partial^{\alpha_1 + \ldots + \alpha_n}f}{\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}}</math>

for the higher order [[partial derivatives]] is justified in this situation. The same is true if all the ({{math|''k'' − 1}})-th order partial derivatives of {{math|''f''}} exist in some neighborhood of {{math|'''''a'''''}} and are differentiable at {{math|'''''a'''''}}.<ref>This follows from iterated application of the theorem that if the partial derivatives of a function {{math|''f''}} exist in a neighborhood of {{math|'''''a'''''}} and are continuous at {{math|'''''a'''''}}, then the function is differentiable at {{math|'''''a'''''}}.  See, for instance, {{harvnb|Apostol|1974|loc=Theorem 12.11}}.</ref> Then we say that {{math|''f''}} is {{math|''k''}} '''times differentiable at the point&nbsp;{{math|''a''}}'''.

=== 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>

=== Example in two dimensions ===

For example, the third-order Taylor polynomial of a smooth function <math>f:\mathbb R^2\to\mathbb R</math> is, denoting <math>\boldsymbol{x}-\boldsymbol{a}=\boldsymbol{v}</math>,

<math display="block"> \begin{align}
P_3(\boldsymbol{x}) = f ( \boldsymbol{a} ) + {} &\frac{\partial f}{\partial x_1}( \boldsymbol{a} ) v_1 + \frac{\partial f}{\partial x_2}( \boldsymbol{a} ) v_2 + \frac{\partial^2 f}{\partial x_1^2}( \boldsymbol{a} ) \frac {v_1^2}{2!} +  \frac{\partial^2 f}{\partial x_1 \partial x_2}( \boldsymbol{a} ) v_1 v_2 + \frac{\partial^2 f}{\partial x_2^2}( \boldsymbol{a} ) \frac{v_2^2}{2!}  \\
& + \frac{\partial^3 f}{\partial x_1^3}( \boldsymbol{a} ) \frac{v_1^3}{3!} + \frac{\partial^3 f}{\partial x_1^2 \partial x_2}( \boldsymbol{a} ) \frac{v_1^2 v_2}{2!} + \frac{\partial^3 f}{\partial x_1 \partial x_2^2}( \boldsymbol{a} ) \frac{v_1 v_2^2}{2!} + \frac{\partial^3 f}{\partial x_2^3}( \boldsymbol{a} ) \frac{v_2^3}{3!}
\end{align}</math>