Linear applications notation details I have much trouble in clearly understanding what sounds like simple notation in linear applications in physics. Based on this image

authors make the following develoment

which I would like to decipher, or even know exactly how it is called.
 A: The notation is explained in the article mikuszefski cites in a comment, but can be understood in a fairly intuitive way via the $\phi$ with the curved arrow at the top of the picture. 
Without reference to the notation, just looking at the curved arrow, $\phi$ is a function that maps one big region to another, and in particular the sub-region $X$ to the sub-region $x.$
The authors formalize this idea with two lines:
$\phi: R_X \times [t_0,t_f]\to R_x\times [t_0,t_f] $
$(X_0,t)\mapsto \phi(X,t)=(x,t) $
These can be read, respectively, as:
"$\phi$ maps the region $R_X$ in the time interval $[t_0,t_f]$ to the region $R_x$ during the same time interval." 
"$(X,t)$ is mapped by the function $\phi$ to $(x,t).$"
I don't think the English is fixed in stone. While it may be standard in this context, I think the notation $\times$ is very confusing. It is meant to associate a time interval with the regions, but of course the symbol is loaded with other associations for someone embroiled in vector calculus.    
There may be conventions regarding $\mapsto$ and $\to.$ Readers of the Wiki article can decide whether the distinctions are meaningful or worth fussing about.
