Taylor's Theorem in several variables

Taylor's Theorem (several variables, one standard form): Let $U\subseteq\mathbb{R}^n$ be open and let $f:U\to\mathbb{R}$ be of class $C^{k+1}$ on a neighborhood of $a\in U$ . Using multi-index notation, there exists a remainder $R_k(h)$ such that for $h$ sufficiently small (with $a+h\in U$ ),

f(a+h)=\sum_{|\alpha|\le k}\frac{D^\alpha f(a)}{\alpha!}\,h^\alpha + R_k(h),

and

\frac{R_k(h)}{\|h\|^k}\to 0 \quad \text{as } h\to 0.

Here $\alpha=(\alpha_1,\dots,\alpha_n)$ , $|\alpha|=\alpha_1+\cdots+\alpha_n$ , $\alpha!=\alpha_1!\cdots \alpha_n!$ , and $h^\alpha=h_1^{\alpha_1}\cdots h_n^{\alpha_n}$ .

Remarks

Taylor's theorem is the basis for local approximation, classification of critical points, and higher-order error bounds in multivariable calculus.