Multivariate Taylor’s Theorem

Motivation This short post derives Taylor’s Theorem for multivariate functions by employing Taylor’s Theorem in a single variable. Derivation Let $f : \mathbb{R}^d \rightarrow \mathbb{R}$. For vectors $x$ and $v$ in $\mathbb{R}^d$, define $g : \mathbb{R} \rightarrow \mathbb{R}$ by $g(t) = f(x + tv)$. If $g$ is $K$ times differentiable at zero, Taylor’s theorem in 1d tells us \begin{equation}\label{eq:1d}\tag{1} f(x + tv) = g(t) = \sum_{k = 0}^K \frac{t^k}{k!} g^{(k)}(0) + o(t^K) ...

Submit