  In this article we construct three infinite families of endofunctors
$J_d^{(n)}$, $J_d^{[n]}$, and $J_d^n$ on the category of left $A$-modules,
where $A$ is a unital associative algebra over a commutative ring $\mathbb{k}$,
equipped with an exterior algebra $\Omega^\bullet_d$. We prove that these
functors generalize the corresponding classical notions of nonholonomic,
semiholonomic, and holonomic jet functors, respectively. Our functors come
equipped with natural transformations from the identity functor to the
corresponding jet functors, which play the r\^{o}les of the classical
prolongation maps. This allows us to define the notion of linear differential
operators with respect to $\Omega^{\bullet}_d$. We show that if $\Omega^1_d$ is
flat as a right $A$-module, the semiholonomic jet functor satisfies the
semiholonomic jet exact sequence $0 \rightarrow \bigotimes^n_A \Omega^1_d
\rightarrow J^{[n]}_d\rightarrow J^{[n-1]}_d \rightarrow 0$. Moreover, we
construct a functor of quantum symmetric forms $S^n_d$ associated to
$\Omega^\bullet_d$, and proceed to introduce the corresponding noncommutative
analogue of the Spencer $\delta$-complex. We give necessary and sufficient
conditions under which the holonomic jet functor $J_d^n$ satisfies the
(holonomic) jet exact sequence, $0\rightarrow S^n_d \rightarrow J_d^n
\rightarrow J_d^{n-1} \rightarrow 0$. In particular, for $n=1$ the sequence is
always exact, for $n=2$ it is exact for $\Omega^1_d$ flat as a right
$A$-module, and for $n\ge 3$, it is sufficient to have $\Omega^1_d$,
$\Omega^2_d$, and $\Omega^3_d$ flat as right $A$-modules and the vanishing of
the Spencer cohomology $H^{\bullet,2}$.
