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 $h^\alpha=h_1^{\alpha_1}\cdots h_n^{\alpha_n}$.

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

Proof sketch: Fix $h$ and consider the one-variable function $\phi(t)=f(a+th)$ for $t$ near $0$. Apply the one-dimensional Taylor theorem to $\phi$ at $t=0$ and translate the derivatives $\phi^{(j)}(0)$ into directional derivatives expressed in terms of partial derivatives of $f$ at $a$. The remainder estimate follows from the one-dimensional remainder estimate and smoothness of $f$.