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"Region" seems to be a very important concept in Mathematica's latest version. Where can one find the more thorough introductory details about the concept of Region instead of just the shattering examples in the document?

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Do you mean something like the ElementMesh creation tutorial for Regions? – user21 Aug 7 '14 at 8:24
My impression is that tutorial writing for regions has not caught up with the new functions (except for FEM). For instance Integrals over Regions does not mention new region functionality at all. An introduction to basic region specification and use for regions other than mesh regions would be helpful, especially to those not satisfied by the guides linked in the overview guide Geometric Computation pointed out by Mr.Wizard below. – Michael E2 Aug 7 '14 at 11:44
@Michael On reflection I thoroughly agree. – Mr.Wizard Aug 7 '14 at 19:58
up vote 6 down vote accepted

I think this question is broad since Regions are used in various fields for various purposes. Think computational geometry (Voronoi diagrams, Delaunay triangulations, Convex hull etc.), finite element methods (PDEs), solid geometry etc. In addition to the links provided by Mr.Wizard, here are some additional links that might get you started:

Solving Partial Differential Equations with Finite Elements

Element Mesh Generation

Element Mesh Visualization

Finite Element Programming

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That's more like it. +1 – Mr.Wizard Aug 7 '14 at 19:58

I am not sure what you are looking for but in case any of these are new to you:

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A good start. OP presumably looked - like I did - for an introduction with examples, rather than a list of functions or gallery of images. – cormullion Aug 7 '14 at 7:46

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