From: DGoncz@replikon.net   
      
   The Change of Variable Theorem is of interest as it generalizes the   
   transformations between coordinate systems.   
      
   I have found pictures of something like unit volumes for the Cartesian,   
   Cylindrical, and Spherical coordinate systems.   
      
   There are 16 orthogonal systems listed at Math World under Orthogonal   
   Coordinates and it is asserted on that page they are all degenerate   
   cases of elliptical coordinates, a mistake; Giankoplis gives the   
   derivation of the degenerate cases from the matrix equation.   
      
   I do not have a chart and may have to program in Mathcad. That's pretty   
   easy using the transformation equations to Cartesian coordinates tracing   
   along each edge of the coordinate system specific unit volume (not the   
   differential volume, but a substantial "chunk" of spaces, near the   
   origin, in each system). I think I can articulate some of those limits   
   here today before trying it:   
      
   For each coordinate axis with range from 0 to oo, apply limits of 1/2 to 1.   
   For each coordinate axis with range from 0 to 2pi, apply limits of pi/2 to   
   3pi/2.   
   For each coordinate axis with range from 0 to pi, apply limits of pi/4 to   
   3pi/4.   
   There are others, however, and there are inequalities.   
      
   Ideally the volume of each "chunk" would be 8 since the obvious chunk of   
   Cartesian space is:   
   x=[-1,1]; y=[=1,1]; z=[-1,1].   
      
   I am open to suggestions today and have not started programming yet.   
      
   Cheers,   
   Douglas Goncz   
   Replikon Research FCN 7837774974   
      
   On Sunday, June 6, 2021 at 10:22:08 AM UTC-4, Douglas Dana Edward^2   
   Parker-Goncz (fully) wrote  (I wrote):   
   ...   
   >   
   > I am interesting in seeing and counting (enumerating) the most basic   
   > unit volumes and their associated surfaces for inclusion in a computer   
   > software library of "atomic" features from which "everything" (to second   
   > order) may be designed, in a attempt to provide a reasonable and   
   > nontrivial basis for para-universal constructors.   
   >   
   ...   
   > It seems to me that including advanced math in CAD representations of   
   > manufacturable objects would reduce file sizes and eliminate   
   > digitization and tiling errors which are becoming a problem as the   
   > resolution of additive and subtractive manufacturing machinery   
   > increases, which is why I am writing about this here.   
   >   
   > This post would go to sci.math were it not for the ubiquitous use of   
   > change of coordinate system in solving the most advanced physics   
   > problems. Briefly, when an initial, constraining, or terminal condition   
   > of a physics problem is representable most effectively in a coordinate   
   > system other than Cartesian, translating the entire problem into that   
   > system can provide a solution where no other method will work. The   
   > solution to the Navier-Stokes equations with viscosity for the case of   
   > flow over a sphere is a famous example--after the change of system the   
   > problem is thereby reduced from 3 dimensions to only 1, and is readily   
   > solved.   
   >   
   >   
   > Douglas Goncz   
   > Replikon Research FCN 783774974   
      
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