SUO: Re: Basics of Process and Event, 3D and 4D
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A couple of off-list questions about the definition of a manifold
make me think that this additional comment might be useful here.
Pretty much the whole idea is contained in that "manifold picture"
that shows how two charts from an atlas -- think of aerial photos --
overlap in their mapping of the underlying "real world" space.
The rest is just analytic details that I kinds say blah-blah
over unless I have an exam problem to solve, which I don't.
The problem here is to stand back and think about how to
get the right linguistic-logical analogue, where the
different charts are different languages or codes
or ways of talking and thinking about a commonly
underlying world. I started an impressionist
interpretation here:
http://suo.ieee.org/ontology/msg04782.html
There is also a reading from a Diff Geom for Engineers book:
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DIF. Differential Geometry
Preface
01. http://suo.ieee.org/ontology/msg04056.html
1. Introduction
02. http://suo.ieee.org/ontology/msg04057.html
03. http://suo.ieee.org/ontology/msg04058.html
2. Manifolds And Their Maps
2.1. Differentiable Manifolds
04. http://suo.ieee.org/ontology/msg04059.html
05. http://suo.ieee.org/ontology/msg04060.html
06. http://suo.ieee.org/ontology/msg04061.html
2.2. Examples
07. http://suo.ieee.org/ontology/msg04062.html
2.3. Manifold Maps
08. http://suo.ieee.org/ontology/msg04063.html
3. Tangent Spaces
09. http://suo.ieee.org/ontology/msg04065.html
The above material is excerpted from:
| Brian F. Doolin & Clyde F. Martin,
|'Introduction to Differential Geometry for Engineers',
| Marcel Dekker, New York, NY, 1990.
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This is the math that made it possible for Einstein to dream up relativity,
which demands a way to relate the data collected in one reference frame to
the data collected in another reference frame and reconstruct what it says
about the objective world that generated these very diverse kinds of data.
We could get a big leg up on intercommunity problems by working out the
qualitative analogue of this concept, a thing I've been working on for
the last 10 or 15 years in my differential logic work.
http://suo.ieee.org/ontology/msg04799.html
Jon Awbrey
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