Let $M$ be a smooth manifold. Its structure sheaf $\mathcal{O}_M$ is the sheaf of smooth real-valued functions. Together they form a ringed space $(M,\mathcal{O}_M)$. The tangent sheaf $\mathcal{T}_M$ is a sheaf of modules over the structure sheaf. It can be defined as the sheaf of derivations of the structure sheaf.

A smooth map of manifolds $f: M \rightarrow N$ induces a morphism $df: \mathcal{T}_M \rightarrow f^*(\mathcal{T}_N)$ of $\mathcal{O}_M$-modules, where $f^*(\mathcal{T}_N)$ is the inverse image of $\mathcal{T}_N$. It is called the differential or pushforward of the map $f$.

Does anyone have a reference for the definition of the differential? In which kind of textbook would this be explained? It seems to be somewhere between differential geometry and algebraic geometry but I could not find it in any textbook in neither of these areas.

(I am not looking for the differential between tangent *bundles* which is explained in detail in every basic book on differential geometry.)

Thanks in advance!

isthe usual map between tangent bundles, since a map of bundles $E' \rightarrow E$ over a map of spaces $f:X' \rightarrow X$ is "the same" as an $O_ {X'}$-linear map $\underline{E}' \rightarrow f^{\ast}(\underline{E})$, using the associated sheaves of modules $\underline{E}'$ and $\underline{E}$. Do you still need a reference? – Boyarsky Jun 28 '10 at 12:35