here's the doubt I have: Is it possible to express Taylor's polynomial in terms of the differential of the function (at least in the two dimentional case)? For instance:
Let's call $f: D\subset\mathbb{R}^n \to \mathbb{R}$ such that $f\in C^2(D)$ and $x_o \in D$ and $\vec{h} = (x-x_0, y-y_0)$.
Then my books reports this ($Hf(x_0)$ is the Hessian Matrix of $f$ in $x_0$):
$f(x) = f(x_0) + \vec{\nabla} f|_{x_0} \cdot \vec{h} + \frac{1}{2} \vec{h}^t Hf(x_o)\vec{h} + o(|h^2|)$
To my unexpert eye this looks like
$f(x) = f(x_0) +df(x_o,h) + \frac{1}{2}d^2f(x_0,h) + o(|h^2|)$
Am I wrong? I didn't find anything on internet cause I don't actually know what to type to look for it...