I am not a specialist in differential geometry, so I have some difficulties in finding the right words for the following natural things:

First of all it seems that there is a lot of nonequivalent definitions of the tangent vectors of higher order. I will define the one that I need. Suppose we have a $C^{k}$ manifold $M$ of dimension $n$. Let $p\in M$, and let $\varphi:U\to\mathbb{R}^{n}$ be a chart. Then for any $l\le k$ the span of $$\frac{\partial} {\partial x_1}, ...,\frac{\partial} {\partial x_n}, \frac{\partial^2} {\partial x_1\partial x_1},\frac{\partial^2} {\partial x_1\partial x_2},...,\frac{\partial^2} {\partial x_n\partial x_n},...,\frac{\partial^l} {\partial x_n...\partial x_n}$$ does not actually depend on $\varphi$. The elements of this span is what I mean by tangent vectors of order $l$.

Q1: How do you call this guys in general (without the specification of $l$, and in the way that it is clear what it is)?

Q2: How to say that I am acting with such thing on a vector function (formally this does not make any sense, since they are defined as functional on germs of scalar functions)?

I only understand how to write it formally correct just for usual tangent vectors, using the differential of our vector function and the identification between tangent vectors and elements on my codomain. However I don't want to write down such formula without explanation, neither I want to write this whole story in my paper.

Q3: What's the story in the infinite-dimensional case?

In fact I need to state a simple fact (at least in good cases), but I don't know how to write it, and I don't know to which extent it actually ramains correct. The fact is following: I have a $C^{k}$ manifold $M$ (possibly infinite-dimensional), a $C^{l}$ map $\psi:M\to X$, where $X$ is a topological vector space, a functional $v\in X^{*}$ and a tangent vector (sort of), that I described above, say $d$. Then, $d\left(\psi\right)\in X$ and $\left<d\left(\psi\right), v\right>=d\left(v\circ\psi\right)$, in other words, operation of differentiation is implemented coordinatewise.

UPD Q4: If I have a field of such vectors, is the obtained object called a "differential operator"?

Thank you.

jets, where an $\ell$-jet is the collection of derivatives up to order $\ell$ of a given $C^\ell$-function on $M$, taken at one point $p$. The set of $\ell$-jets at a point $p\in M$ forms a vector space that is dual to the space you are describing. However, the global structure of the jet bundle is more complicated. So you may try to find something on jets in the literature. $\endgroup$ – Sebastian Goette Oct 28 '15 at 16:25