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.
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$\begingroup$ It means $\varphi$ is a map that takes the space $R_\mathbf{X} \times [t_0,t_\mathrm{final}[$ and maps it to the new space $R_\mathbf{x} \times [t_0,t_\mathrm{final}[$, i.e. the time remains unchanged, but the coordinates will be transformed in some given way. The second line just says any pair of $\mathbf{X}$ and $t$ transformes with $\varphi(\mathbf{X},t)$ to a new pair called $(\mathbf{x},t)$. $\endgroup$– noahCommented May 5, 2017 at 10:05
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$\begingroup$ Thanks ! What about the cross ? $\endgroup$– Blue_ElephantCommented May 5, 2017 at 10:09
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$\begingroup$ The symbol X between $R_X$ and $[t_0,t_{f}]$ is really confusing. It's just supposed to label R with the time interval. $\endgroup$– danielCommented May 5, 2017 at 10:09
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$\begingroup$ Any difference between the two arrows (one of them has a small vertical bar at its root) ? Also, $:$ means 'maps' then ? $\endgroup$– Blue_ElephantCommented May 5, 2017 at 10:13
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$\begingroup$ I'm going to say no difference. The first line is, "phi maps blah1 to blah2." The second line is (X,t) are taken/mapped by phi to (x,t). So one is talking about what $\phi$ does to the entire region, and the other about what $\phi$ does to the sub-region or state (X,x). And the : means "this is what it does;" or depending on context, "such that." $\endgroup$– danielCommented May 5, 2017 at 10:22
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1 Answer
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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.
