In the literature I have encountered two different definitions of jets of smooth functions, and I was wondering how one could identify these definitions.


One definition is the often encountered definition of smooth functions up to equivalence. The other definition I have encountered is the more intrinsic definition which assigns to a manifold $M$
$$
\mathscr{J}(M) = \text{Hom}\_{C^\infty(M)}(D(M), C^\infty(M)),
$$
where $D(M)$ are the differential operators on smooth functions on $M$.
Is there any reference where these two approaches are identified? 


Thanks in advance. 



Equivalence of two definitions of jets of smooth functions
The definition of $k$-th order jet as an equivalence class $[f]\_x^k$ of a function $f\in C^\infty M$ at point $x\in M$, gives you a natural map  

\begin{align}
\mathcal{j}^k\colon C^\infty M &\to \mathscr{J}^k M\\
f &\mapsto (x\mapsto [f]\_x^k),
\end{align}
which may be shown to be the *universal* differential operator of order $\leq k$ *out* of $C^\infty M$. This means that any differential operator of order $\leq k$ out of $C^\infty M$, say $\Delta\colon C^\infty M \to P$, with $P$ a $C^\infty M$-module, factors uniquely through $\mathcal{j}^k$ via a $C^\infty M$-linear map $\phi\_\Delta\colon \mathscr{J}^k M \to P$. This implies that $D^k M$ (i.e. the module differential operators from $C^\infty M$ to $C^\infty M$ of order $\leq k$, with its left $C^\infty M$-module structure) is naturally isomorphic to $\text{Hom}\_{C^\infty M}(\mathscr{J}^k M,C^\infty M)=(\mathscr{J}^k M)^\*$, where $P^\*$ denotes the $C^\infty M$ dual. Using this we get
$$
\text{Hom}\_{C^\infty M}(D^k M,C^\infty M)= \text{Hom}\_{C^\infty M}((\mathscr{J}^k M)^\*,C^\infty M) = ((\mathscr{J}^k M)^\*)^\*=\mathscr{J}^k M
$$
where the last equality follows from the fact that $\mathscr{J}^k M$ is a finitely generated projective $C^\infty M$-module when $M$ is a smooth manifold. Passing to the limit with $k\to \infty$ gives you the isomorphism in your question.
A reference for this might be Krasil'shchik, Verbovetsky, *[Homological Methods in Equations of Mathematical Physics](https://arxiv.org/abs/math/9808130)*.


