Taylor's theorem in several variables

Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $k \ge 1$ be an integer, and let $f \colon U \to \mathbb{R}$ be of class C^k on $U$. Let $a \in U$. For a multi-index $\alpha = (\alpha_1, \dots, \alpha_n)$, write $|\alpha| = \alpha_1 + \dots + \alpha_n$, $\alpha! = \alpha_1! \cdots \alpha_n!$, $h^\alpha = h_1^{\alpha_1} \cdots h_n^{\alpha_n}$, and $D^\alpha f = \partial^{|\alpha|} f / \partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n}$. Then $$f(a + h) = \sum_{|\alpha| \le k} \frac{D^\alpha f(a)}{\alpha!} \, h^\alpha + R_k(h), \qquad \lim_{h \to 0} \frac{R_k(h)}{\|h\|^k} = 0.$$ If $f$ is of class $C^{k+1}$ and the segment from $a$ to $a + h$ lies in $U$, then $$R_k(h) = \sum_{|\beta| = k + 1} \frac{k + 1}{\beta!} \, h^\beta \int_0^1 (1 - t)^k \, D^\beta f(a + t h) \, dt.$$

Remarks

The sum over $|\alpha| \le k$ is the Taylor polynomial of order $k$ of $f$ at $a$. For $k = 1$ the statement says that $f$ is differentiable at $a$, with $f(a + h) = f(a) + \nabla f(a) \cdot h + o(\|h\|)$. One proof applies the one-variable Taylor theorem to $g(t) = f(a + t h)$ on $[0, 1]$, together with the chain rule. The case $k = 2$ is the second-order Taylor expansion.

Explanations

L1 human

Near a point, a smooth function of several variables looks like a polynomial in the displacement $h$. The coefficients of that polynomial are the partial derivatives at the point, divided by factorials. The error is smaller than $\|h\|^k$, so the approximation gets better quickly as $h$ shrinks.