Taylor's Formula for Functions from Rn To R

# Taylor's Formula for Functions from Rn To R

Let $S \subseteq \mathbb{R}^n$ be open, $\mathbf{x} \in S$, and $f : S \to \mathbb{R}$. If $\mathbf{t} \in \mathbb{R}^n$ then we know that the directional derivative of $f$ at $\mathbf{x}$ in the direction of $\mathbf{t}$ is given by the formula:

(1)
\begin{align} \quad f'(\mathbf{x}, \mathbf{t}) = \nabla f(\mathbf{x}) \cdot \mathbf{t} = D_1f(\mathbf{x})t_1 + D_2f(\mathbf{x})t_2 + ... + D_nf(\mathbf{x})t_n = \sum_{i=1}^{n} D_if(\mathbf{x})t_i \end{align}

We will generalize this definition to define higher order directional derivatives.

 Definition: Let $S \subseteq \mathbb{R}^n$ be open, $\mathbf{x} \in S$, and $f : S \to \mathbb{R}$. Let $\mathbf{t} \in \mathbb{R}^n$. If all of the second order partial derivatives of $f$ at $\mathbf{x}$ exist, i.e., $D_{i, j} f(\mathbf{x})$ exist where $i, j \in \{ 1, 2, ..., n \}$ then the Second Order Directional Derivative of $f$ at $\mathbf{x}$ in the Direction of $\mathbf{t}$ is defined as $\displaystyle{f''(\mathbf{x}, \mathbf{t}) = \sum_{i=1}^{n} \sum_{j=1}^{n} D_{ij} f(\mathbf{x}) t_j t_i}$. If all of the third order partial derivatives of $f$ at $\mathbf{x}$ exist, i.e., $D_{i,j,k} f(\mathbf{x})$ exist where $i, j, k \in \{ 1, 2, ..., n \}$ then the Third Order Directional Derivative of $f$ at $\mathbf{x}$ in the Direction of $\mathbf{t}$ is defined as $\displaystyle{f'''(\mathbf{x}, \mathbf{t}) = \sum_{i=1}^{n} \sum_{j=1}^{n} \sum_{k=1}^{n} D_{i,j,k} f(\mathbf{x}) t_k t_j t_i}$. In general, if all of the $m^{\mathrm{th}}$ order partial derivatives of $f$ at $\mathbf{x}$ exist, i.e, $D_{i_1, i_2, ..., i_{m}}$ exist where $i_1, i_2, ..., i_m \in \{ 1, 2, ..., n \}$ then the $m^{\mathrm{th}}$ Order Directional Derivative of $f$ at $\mathbf{x}$ in the Direction of $\mathbf{t}$ is defined as $\displaystyle{f^{(m)}(\mathbf{x}, \mathbf{t}) = \sum_{i_1=1}^{n} \sum_{i_2=1}^{n} ... \sum_{i_m=1}^{n} D_{i_1, i_2, ..., i_m} f(\mathbf{x}) t_{i_m} t_{i_{m-1}} ..., t_{i_1}}$.

We can now state a very important result known as Taylor's formula which is somewhat of a generalization to the Mean Value theorem for differentiable functions.

 Theorem (Taylor's Formula): Let $S \subseteq \mathbb{R}^n$ be open and let $f : S \to \mathbb{R}$. If $f$ and all of its partial derivatives of order less than $m$ are differentiable on $S$, and $\mathbf{a}, \mathbf{b} \in S$ are such that $L(\mathbf{a}, \mathbf{b}) \subset S$, then there exists a $\mathbf{z} \in L(\mathbf{a}, \mathbf{b})$ such that $\displaystyle{f(\mathbf{b}) - f(\mathbf{a}) = \frac{1}{1!} f'(\mathbf{a}, \mathbf{b} - \mathbf{a}) + \frac{1}{2!} f''(\mathbf{a}, \mathbf{b} - \mathbf{a}) + ... + \frac{1}{(m-1)!} f^{(m-1)} (\mathbf{a}, \mathbf{b} - \mathbf{a}) + \frac{1}{m!} f^{(m)} (\mathbf{z}, \mathbf{b} - \mathbf{a}) \\ \quad \quad\quad \quad = \sum_{k=1}^{m-1} \frac{1}{k!} f^{(k)} (\mathbf{a}, \mathbf{b} - \mathbf{a}) + \frac{1}{m!} f^{(m)}(\mathbf{z}, \mathbf{b} - \mathbf{a})}$.

Note that if $m = 1$ and satisfies the hypotheses of the theorem above, then the formula above reduces to $f(\mathbf{b}) - f(\mathbf{a}) = f'(\mathbf{z})(\mathbf{b} - \mathbf{a})$ for some $\mathbf{z} \in L(\mathbf{a}, \mathbf{b})$. But this is simply The Mean Value Theorem for Differentiable Functions from Rn to Rm for the case when $f$ is a differentiable multivariable real-valued function.