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!



Differential between tangent sheaves
You might want to try Ramanan's [Global Calculus](http://www.google.com/url?sa=t&source=web&cd=1&ved=0CBIQFjAA&url=http%253A%252F%252Fwww.amazon.com%252FGlobal-Calculus-Graduate-Studies-Mathematics%252Fdp%252F0821837028&ei=-q0oTO-ZBePesAavhok9&usg=AFQjCNED1_dGrlHkytEUuyDiutnunakcCg&sig2=ZC8MnQVdRimCkJuGrGlVVg) which does a bit of introductory manifold theory, and uses the mechanism of sheaves throughout.


