다변수 함수에 대한 테일러 정리

다변수 함수에 대한 테일러 정리

Taylor's Theorem with Remainder for Multivariable Function

정리1

$F : \mathbb{R}^{n} \to \mathbb{R}$를 $C^{k}$ 함수, $\mathbf{a} = (a_{1}, \dots, a_{n}) \in \mathbb{R}^{n}$라고 하자. 그러면 다음을 만족하는 $C^{k-2}$ 함수 $h_{ij}$가 존재한다.

$$ F(x_{1}, \dots, x_{n}) = F(a_{1}, \dots, a_{n}) + \sum_{i} \dfrac{\partial F}{\partial x_{i}}(\mathbf{a})(x_{i} - a_{i}) + \sum_{i,j}h_{ij}(\mathbf{x})(x_{i} - a_{i}) (x_{j} - a_{j}) $$

증명

$$ \begin{align*} F(\mathbf{x}) - F(\mathbf{a}) =&\ \int_{0}^{1} \dfrac{d}{dt} \left[ F(t(\mathbf{x} - \mathbf{a}) + \mathbf{a}) \right]dt \\ =&\ \int_{0}^{1} \left( \sum_{i} \dfrac{\partial F}{\partial x_{i}}\left( t(\mathbf{x} - \mathbf{a}) + \mathbf{a} \right)(x_{i}-a_{i}) \right) dt & \text{by } \href{https://freshrimpsushi.github.io/posts/3134}{\text{chain rule}} \\ =&\ \sum_{i}(x_{i} - a_{i}) \int_{0}^{1} \left( \dfrac{\partial F}{\partial x_{i}}\left( t(\mathbf{x} - \mathbf{a}) + \mathbf{a} \right) \right) dt \end{align*} $$

적분부분을 $g_{i}(\mathbf{x})$라고 표기하자. $g_{i}(\mathbf{x}) = \displaystyle \int_{0}^{1} \left( \dfrac{\partial F}{\partial x_{i}}\left( t(\mathbf{x} - \mathbf{a}) + \mathbf{a} \right) \right) dt$라고 하면,

$$ \begin{equation} F(\mathbf{x}) - F(\mathbf{a}) = \sum_{i}(x_{i} - a_{i}) \int_{0}^{1} \left( \dfrac{\partial F}{\partial x_{i}}\left( t(\mathbf{x} - \mathbf{a}) + \mathbf{a} \right) \right) dt = \sum_{i} g_{i}(\mathbf{x}) (x_{i} - a_{i}) \end{equation} $$

$g_{i}(\mathbf{a})$의 값은 다음과 같다.

$$ g_{i}(\mathbf{a}) = \int_{0}^{1} \dfrac{\partial F}{\partial x_{i}} \left(t(\mathbf{a} - \mathbf{a}) + \mathbf{a} \right) dt = \int_{0}^{1} \dfrac{\partial F}{\partial x_{i}}\left( \mathbf{a} \right) dt = \dfrac{\partial F}{\partial x_{i}}\left( \mathbf{a} \right) $$

그러면 $(1)$을 이끌어냈던 것과 같은 방법으로 다음의 식을 얻을 수 있다.

$$ g_{i}(\mathbf{x}) - g_{i}(\mathbf{a}) = \sum_{j} h_{ij}(\mathbf{x}) (x_{j}-a_{j}) $$

이제 정리하면

$$ \begin{align*} F(\mathbf{x}) =&\ F(\mathbf{a}) + \sum_{i}g_{i}(\mathbf{x})(x_{i}-a_{i}) \\ =&\ F(\mathbf{a}) + \sum_{i}\left( g_{i}(\mathbf{a}) + \sum_{j} h_{ij}(\mathbf{x}) (x_{j}-a_{j}) \right)(x_{i}-a_{i}) \\ =&\ F(\mathbf{a}) + \sum_{i} g_{i}(\mathbf{a})(x_{i}-a_{i}) + \sum_{i,j} h_{ij}(\mathbf{x})(x_{i}-a_{i})(x_{j}-a_{j}) \\ =&\ F(\mathbf{a}) + \sum_{i} \dfrac{\partial F}{\partial x_{i}}\left( \mathbf{a} \right)(x_{i}-a_{i}) + \sum_{i,j} h_{ij}(\mathbf{x})(x_{i}-a_{i})(x_{j}-a_{j}) \end{align*} $$

같이보기


  1. Richard S. Millman and George D. Parker, Elements of Differential Geometry (1977), p213-214 ↩︎

댓글