Higher order differentials meaning? If we consider a map $f : \mathbb{R}^n\to \mathbb{R}^m $. Then I understand the differential $f'(p) :\mathbb{R}^n\to \mathbb{R}^m$ as the map that given $v$ gives me the change to first order of the map $f$ along $v$. Is this right? But what about $f''(p) : \mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R}^m$ and $f'''(p) : \mathbb{R}^n\times \mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R}^m$ what is the interpretation of $(v_1,v_2), (v_1,v_2,v_3)$ and of  $f''(p)(v_1,v_2)$ and $f'''(p)(v_1,v_2,v_3)$?
 A: Denote by $e_i$ the standard basis vectors of $\mathbb{R}^n$. I'll use the notation $D^nf|_{p} \colon \mathbb{R}^n \times \dots \times \mathbb{R}^n \rightarrow \mathbb{R}^m$ for the $n$-th derivative of $f$ at $p$. Then we have
$$ Df|_{p}(e_i) = \frac{\partial f}{\partial x^i}(p) $$
and more generally, for an arbitrary $v \in \mathbb{R}^n$, we have
$$ Df|_{p}(v) = \left. \left( \frac{d}{dt} f(p + tv)\right)\right\rvert_{t = 0} $$
so $Df|_{p}(v)$ gives you the directional derivative of $f$ at $p$ in the direction of $v$.
If you unravel the definition of $D^2f|_{p}$, you'll find out that
$$ D^2f|_{p}(e_i, e_j) = \frac{\partial f}{\partial x^i x^j}(p). $$
Thus, the components of the tensor $D^2f|_{p}$ with respect to the standard basis give you all the second partial derivatives of $f$ (with respect to the associated coordinates). More generally, for $v,w \in \mathbb{R}^n$, we have
$$ D^2f|_{p}(v,w) = \frac{\partial}{\partial s} \left. \left( \left.\left(  \frac{\partial}{\partial t} f(p + tv + sw)\right)\right\rvert_{t=0} \right) \right\rvert_{s=0} = \left. \left( \frac{\partial^2}{\partial s \partial t} f(p + tv + sw) \right) \right\rvert_{t=s=0}$$
which is a "second order directional derivative". You first differentiate $f$ around $p$ in the direction of $v$ and get a function around $p$. Then you differentiate the resulting function in the direction of $w$ at $p$ and the result is $D^2 f|_{p}(v,w)$.
This generalizes naturally to $Df^{n}$. The second description works for arbitrary maps $f \colon V \rightarrow W$ between finite dimensional real vector spaces (where $V,W$ don't come with a natural basis so there is no meaning to $\frac{\partial f}{\partial x^i}$, etc).
