Difference between revisions of "Taylor's Formula in Several Variables"

Latest revision as of 15:40, 12 November 2021

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 $f(\mathbf {b} )-f(\mathbf {a} )=f'(\mathbf {z} )(\mathbf {b} -\mathbf {a} )$ 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.