Khanh: Created page with "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 kn..."

2021-11-12T21:40:45Z

Created page with "Let <math>S \subseteq \mathbb{R}^n</math> be open, <math>\mathbf{x} \in S</math>, and <math>f : S \to \mathbb{R}</math>. If <math>\mathbf{t} \in \mathbb{R}^n</math> then we kn..."

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

<div style="text-align: center;"><math>\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}</math></div>

We will generalize this definition to define higher order directional derivatives.
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
:'''Definition:''' Let <math>S \subseteq \mathbb{R}^n</math> be open, <math>\mathbf{x} \in S</math>, and <math>f : S \to \mathbb{R}</math>. Let <math>\mathbf{t} \in \mathbb{R}^n</math>. If all of the second order partial derivatives of <math>f</math> at <math>\mathbf{x}</math> exist, i.e., <math>D_{i, j} f(\mathbf{x})</math> exist where <math>i, j \in \{ 1, 2, ..., n \}</math> then the <strong>Second Order Directional Derivative of <math>f</math> at <math>\mathbf{x}</math> in the Direction of <math>\mathbf{t}</math></strong> is defined as <math>\displaystyle{f''(\mathbf{x}, \mathbf{t}) = \sum_{i=1}^{n} \sum_{j=1}^{n} D_{ij} f(\mathbf{x}) t_j t_i}</math>. If all of the third order partial derivatives of <math>f</math> at <math>\mathbf{x}</math> exist, i.e., <math>D_{i,j,k} f(\mathbf{x})</math> exist where <math>i, j, k \in \{ 1, 2, ..., n \}</math> then the '''Third Order Directional Derivative of <math>f</math> at <math>\mathbf{x}</math> in the Direction of <math>\mathbf{t}</math>''' is defined as <math>\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}</math>. In general, if all of the <math>m^{\mathrm{th}}</math> order partial derivatives of <math>f</math> at <math>\mathbf{x}</math> exist, i.e, <math>D_{i_1, i_2, ..., i_{m}}</math> exist where <math>i_1, i_2, ..., i_m \in \{ 1, 2, ..., n \}</math> then the <strong><math>m^{\mathrm{th}}</math> Order Directional Derivative of <math>f</math> at <math>\mathbf{x}</math> in the Direction of <math>\mathbf{t}</math></strong> is defined as <math>\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}}</math>.
</blockquote>

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.

<blockquote style="background: white; border: 1px solid black; padding: 1em;">
:'''Theorem (Taylor's Formula):''' Let <math>S \subseteq \mathbb{R}^n</math> be open and let <math>f : S \to \mathbb{R}</math>. If <math>f</math> and all of its partial derivatives of order less than <math>m</math> are differentiable on <math>S</math>, and <math>\mathbf{a}, \mathbf{b} \in S</math> are such that <math>L(\mathbf{a}, \mathbf{b}) \subset S</math>, then there exists a <math>\mathbf{z} \in L(\mathbf{a}, \mathbf{b})</math> such that
:<math> 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})</math>

:<math>\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}</math>
</blockquote>

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

== Licensing ==
Content obtained and/or adapted from:
* [http://mathonline.wikidot.com/taylor-s-formula-for-functions-from-rn-to-r Taylor's Formula for Functions from Rn to R, mathonline.wikidot.com] under a CC BY-SA license

Taylor's Formula in Several Variables - Revision history

Khanh: Created page with "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 kn..."

Khanh: Created page with "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 kn..."