So generally we define a differential relation to be $\mathcal{R} \subset X^{(r)}.$ In the case that $X=M\times N$ is it possible to have $\mathcal{R}=X^{(1)}$? So in this case the formal solutions would be (my guess) just fiberwise linear bundle maps that covers a continuous map and the genuine (holonomic) solutions would be bundle maps that are the derivative of a smooth map $f: M \to N$. In none of the books (Gromov, Eliashberg & Mishachev) i have seen they treat this case. Does this mean that the case $\mathcal{R}=X^{(1)}$ cannot be treated and what i said before is plainly wrong? On the other case, if something can be said, under what conditions the *h*-principle (lets say surjectiveness of $\pi\_0$ ) holds? 



Question about the h-principle
Yes, it is possible to have $\mathcal{R}=X^{(1)}$, i.e. to have a system of partial differential equations that consists of no equations at all. Then as you say the formal solutions are sections of $X^{(1)} \to M$, while the holonomic solutions are sections of $X \to M$. The surjectivity is obvious, since the fibers of $X^{(1)} \to X$ are affine spaces (which Gromov points out, but you can check as an exercise), so contractible, so $\pi\_0$ is onto, i.e. if you have no differential equations to solve then nothing interesting in PDE happens.


To explain the notation, $p \colon X \to M$ is a fiber bundle (or maybe something more general, but this is good enough) and $X^{(1)}$ is the set of pairs $(m,x,u)$ so that $m \in M$, $x \in X\_m$, $u \colon T\_m M \to T\_x X$ is a linear map so that $p'(x) \circ u \colon T\_m M \to T\_m M$ is the identity linear map. Every section $s \colon M \to X$ gives a section $s' \colon M \to X^{(1)}$ by differentiating. A section $M \to X^{(1)}$ is holonomic if it is $s'$ for some $s$.


