By George W. Hart
This publication bargains with the mathematical houses of dimensioned amounts, resembling size, mass, voltage, and viscosity.
Beginning with a cautious exam of ways one expresses the numerical result of a size and makes use of those leads to next manipulations, the writer carefully constructs the idea of dimensioned numbers and discusses their algebraic constitution. the result's a unification of linear algebra and standard dimensional research that may be prolonged from the scalars to which the normal research is perforce limited to multidimensional vectors of the kind usually encountered in engineering, structures conception, economics, and different applications.
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Additional info for Multidimensional Analysis: Algebras and Systems for Science and Engineering
R. ) CRC Handbook of Chemistry and Physics, 1993. 8 for a discussion of units of probability. 38 1. The Mathematical Foundations of Science and Engineering separated spatially into XY Z, or radial and tangential components, which are treated as independent dimensions; (b) there can be two independent dimensions of time; (c) volume may be treated independently of length. See Isaacson and Isaacson  for examples. However, these are issues of physical modeling in each problem domain, which have to do with the class of transformations of interest, and not something to be rigidized into a single basis.
The first is that rather than making a dimensioned algebra unnecessary, nondimensionalization actually assumes a dimensioned algebra is already in place. The fact that one chooses y / R as a dimensionless variable, as opposed to perhaps y + R or yR, is a consequence of the algebra of dimensioned scalars. ;gR, where 9 is the acceleration of gravity. Only an algebra that deals in dimensioned quantities and operations upon them could justify such a choice. The second point to observe is that nondimensionalization does not generalize to dimensioned vectors and matrices.
8,) although '1 meterV2' is, to my knowledge, not needed. Note that traditional dimensional analysis does not argue for this condition, since a putative irrational law such as 'area = heightV2 . width 2- V2 , would be dimensionally consistent. (D) In exponentiations of the form x Y , if x is not dimensionless, then the result may be undefined for noninteger y, because v of (r, v) is a vector of integers. While this may appear severe in prohibiting '1 y'meter,' if one only considers equations, this is not a major restriction over (C).
Multidimensional Analysis: Algebras and Systems for Science and Engineering by George W. Hart