UNDER CONSTRUCTION!!
Let $M$, $N$ be real smooth manifolds, $U,V$ open subsets of $N$, $a\in U\cap V\subset M$ and $f\colon U\to N$, $g\colon V\to N$ smooth maps. We say that $f$ is tangent of order $k\in\mathbb{N}$ to $g$ and we right $f\sim_k g$ if $f(a) = g(a)$ and for some (therefore every) pair of smooth charts $\phi\colon W\to\mathbb{R}^m$ around $a$, such that $\phi(a)=0$, $\psi\colon W'\to\mathbb{R}^n$, $\psi(f(a)) = 0$, we have
where $h(x)$ is a function defined and continuous on $W\cap f^{-1}(W')\cap g^{-1}(W')$ and such that $h(x)\to 0$ whenever $x\to a$. Equivalently, the Taylor series of all componenets of $f$ and $g$ agree up to order $k$. Tangency of order $k$ at point $x$ is a relation of equivalence on the set of smooth functions locally defined around $x$. The corresponding class of equivalence $f$ is the $k$-jet_ $j^k(f)$. $k$-jets of smooth maps $M\to N$ at point $x\in M$ form a set of $k$-jets $J^k(x)=J^k(M,N,x)$, with canonical structure of a real vector space. Note the obvious projections $J^{k+1}(x)\to J^k(x)$; these define an inverse system of vector spaces. Its limit is the space $J^\infty(x) = J^\infty(M,N,x)\colon = \lim_k J^k(M,N,x)$ of $\infty$-jets at $x$ of smooth maps from $M$ to $N$. A formal function is a an $\infty$-jet at $x$ of smooth functions from $M$ to $\mathbb{R}$.
(Emile Borel’s theorem on formal power series.) Let $\mathcal{C}^\infty_x$ be the vector space of germs of smooth real functions around a point $x\in\mathbb{R}^n$. Then a linear map $\mathrm{can}\colon\mathcal{C}^\infty_x\to J^\infty(\mathbb{R}^n,\mathbb{R},x)$ of taking the Taylor series at $x$ is a surjection.
Taylor series of a smooth function does not need to converge and if it does it may converge to a different function. For example, for the Cauchy function (equal $e^{-1/x^2}$ for $x\neq 0$ and equal $1$ for $x=0$) all Taylor coefficients at $0$ vanish. Of course, the germs of real analytic functions around $x$ could be identified with the corresponding formal functions: the restriction of $\mathrm{can}$ to the subspace of all germs of analytic functions is injective.
Let $R$ be a commutative unital $k$-algebra. For every $r\in R$ denote by $l(r)$ the $k$-linear map $l(r)(r') = r\cdot r'$ of muliplication by $r$ from the left. The function $r\mapsto l(r)$ defines a morphism of rings $l\colon R\to\operatorname{End}_k(R)$. For $f\in End_k(R)$ let us define $[f,r] := f\circ l(r) - l(r)\circ f\in\operatorname{End}_k(R)$. We say that (not necessarily cocomplete!) filtration $F_0\subset F_1\subset F_2\subset \ldots \subset End_k R$ is diferencial filtration if $[f,r] \in F_k$ for each $f\in F_{k+1}$ and for all $k\in\mathbb{N}_0$. Ring $End_k(R)$ has a maximal differential filtration in the sense of inclusion at each filtered level; its colimit is by definition the subring of regular differential operators $Diff(R)\subset End_k(R)$.
Regular diferential operator of order $s\in\mathbb{N}_0$ is a $k$-linear map $D\in\operatorname{End}_k(R)$ such that for all $r_0, r_1,\dots,r_s\in R$ we have $[\ldots[D,r_0],r_1],\ldots,r_s]=0$. Regular diferential operators of the order $s$ form a subspace $Diff_s(R)\subset End_k R$; regular diferential operator is an element of the space $Diff(R) = \cup_{s=0}^\infty Diff_s(R)$. The left multiplication $l(r)$ by the element $r\in R$ is a regular diferencial operator of order $0$; thus, if $R$ is an integral domain, then $R\subset\operatorname{Diff}_0(R)$. Diferential filtration $Diff_0(R)\subset Diff_1
\subset\ldots\subset Diff_k(R)
\subset\ldots\subset End_k(R)$ is typically not cocomplete, i.e. its union
$Diff(R)\neq End_k(R)$. $C^\infty$-diferential operator on $C^\infty$-manifold $M$ is a regular diferencial operator over $C^\infty(M)$. They have the tensoriality property, i.e. if $f|_U=0$ on open subspace $U$, then $(D f)|_U = 0$.
Émile Borel, Sur quelques points de la théorie des fonctions, Annales scientifiques de l’École Normale Supérieure, Sér. 3, 12 (1895), p. 9-55.
A. Cannas da Silva, A. Weinstein, Geometric models for noncommutative algebras}, Berkeley Math. Lec. Notes Series, AMS 1999 http://math.berkeley.edu/~alanw/Models.pdf
J. Dieudonné, Introduction to the theory of formal groups, Marcel Dekker, New York 1973.
J. Dieudonné, Linearly compact spaces and double vector spaces over sfields, Amer. J. Math. 1951
N. Durov, S. Meljanac, A. Samsarov, Z. \v{S}koda, A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra, Journal of Algebra 309:1, str. 318-359 (2007) math.RT/0604096.
S. Helgason, Differential geometry, Lie groups and symmetric spaces, Acad. Press 1978; Amer. Math. Soc. 2001.
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