I would like to ask about the relation between the higher order derivatives of a function $f\colon \mathbb{R}^n\to\mathbb{R}$. Let me first fix some notation.
By the derivative of order $0$ I mean the function $f$ itself. By the derivative of order $k$ of a function $f\colon \mathbb{R}^n \to \mathbb{R}$ at point $a$ I mean a $k$-linear mapping $L\colon (\mathbb{R}^n)^k \to \mathbb{R}$ satisfying: $$\lim_{h\to o}\frac{\|f^{(k-1)}(a + h) - f^{(k-1)}(a) - L(h, \cdot, \dots, \cdot)\|_{{\mathcal L_{k-1}}(\mathbb{R}^n, \mathbb{R})}}{\|h\|_{\mathbb{R}^n}} = 0,$$ where ${\mathcal L}_k(\mathbb{R}^n, \mathbb{R})$ is the set of all $k$-linear mappings $(\mathbb{R}^n)^k \to \mathbb{R}$ with norm defined by $$\|L\|_{\mathcal{L}_k(\mathbb{R}^n, \mathbb{R})} := \sup\{|L(u^1,\dots,u^k)| : \|u^1\|_{\mathbb{R}^n} \leq 1, \dots, \|u^k\|_{\mathbb{R}^n} \leq 1\}.$$ I denote $L = f^{(k)}(a)$. Thus, $k$-th derivative assigns to a point $a \in \mathbb{R}^n$ a $k$-linear mapping which is the best linear approximation of the $(k-1)$-th derivative at $a$.
A differential $k$-form assigns to each point of $\mathbb{R}^n$ an element of $\mathcal{A}_k(\mathbb{R}^n, \mathbb{R})$, which is the subspace of $\mathcal{L}(\mathbb{R}^n, \mathbb{R})$ of the alternating $k$-linear mapings.
What is the relation between the $k$-th derivative and differential $k$-forms? Firstly, by applying the exterior derivative to $f$, we get exactly the first derivative of $f$. Can something more be said about the higher order derivatives? Obviously, if $k > n$ this makes no sense since $\mathcal{A}_k({\mathbb{R}^n,\mathbb{R}})$ is trivial. But otherwise, can something be said? I guess the motivation for these two objects is different, the first is for approximating a function localy (e.g., in Taylor series), whereas the latter one is for integrating. Any anweres are appreciated.