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 $\displaystyle {f''(\mathbf {x} ,\mathbf {t} )=\sum _{i=1}^{n}\sum _{j=1}^{n}D_{ij}f(\mathbf {x} )t_{j}t_{i}}$. 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

$f(\mathbf {b} )-f(\mathbf {a} )={\frac {1}{1!}}f'(\mathbf {a} ,\mathbf {b} -\mathbf {a} )+{\frac {1}{2!}}f''(\mathbf {a} ,\mathbf {b} -\mathbf {a} )+...+{\frac {1}{(m-1)!}}f^{(m-1)}(\mathbf {a} ,\mathbf {b} -\mathbf {a} )+{\frac {1}{m!}}f^{(m)}(\mathbf {z} ,\mathbf {b} -\mathbf {a} )$

${\begin{aligned}\quad \quad \quad \quad \quad =\sum _{k=1}^{m-1}{\frac {1}{k!}}f^{(k)}(\mathbf {a} ,\mathbf {b} -\mathbf {a} )+{\frac {1}{m!}}f^{(m)}(\mathbf {z} ,\mathbf {b} -\mathbf {a} )\end{aligned}}$

Note that if $m=1$ 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