Taylor's Formula in Several Variables

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:

${\begin{aligned}\quad f'(\mathbf {x} ,\mathbf {t} )=\nabla f(\mathbf {x} )\cdot \mathbf {t} =D_{1}f(\mathbf {x} )t_{1}+D_{2}f(\mathbf {x} )t_{2}+...+D_{n}f(\mathbf {x} )t_{n}=\sum _{i=1}^{n}D_{i}f(\mathbf {x} )t_{i}\end{aligned}}$

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x}} exist, i.e., Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_{i,j,k} f(\mathbf{x})} exist where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i, j, k \in \{ 1, 2, ..., n \}} then the Third Order Directional Derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x}} in the Direction of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{t}} is defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^{\mathrm{th}}} order partial derivatives of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x}} exist, i.e, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_{i_1, i_2, ..., i_{m}}} exist where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i_1, i_2, ..., i_m \in \{ 1, 2, ..., n \}} then the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^{\mathrm{th}}} Order Directional Derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x}} in the Direction of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{t}} is defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \subseteq \mathbb{R}^n} be open and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : S \to \mathbb{R}} . If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} and all of its partial derivatives of order less than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} are differentiable on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{a}, \mathbf{b} \in S} are such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(\mathbf{a}, \mathbf{b}) \subset S} , then there exists a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{z} \in L(\mathbf{a}, \mathbf{b})} such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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})}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \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}) \end{align}}

Note that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = 1} and satisfies the hypotheses of the theorem above, then the formula above reduces to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(\mathbf{b}) - f(\mathbf{a}) = f'(\mathbf{z})(\mathbf{b} - \mathbf{a})} for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is a differentiable multivariable real-valued function.

Licensing

Content obtained and/or adapted from:

• Taylor's Formula for Functions from Rn to R, mathonline.wikidot.com under a CC BY-SA license