By George W. Hart
ISBN-10: 1461242088
ISBN-13: 9781461242086
ISBN-10: 1461286972
ISBN-13: 9781461286974
This publication bargains with the mathematical houses of dimensioned amounts, comparable to 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 ends up in next manipulations, the writer conscientiously constructs the thought 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 type usually encountered in engineering, platforms conception, economics, and different applications.
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Additional info for Multidimensional Analysis: Algebras and Systems for Science and Engineering
Example text
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). Equations that are initially written as equating products of dimensioned quantities with rational exponents can be reexpressed using only integer exponents, simply by raising both sides of the equation to the power of the greatest common denominator of the exponents.
Quantities such as mass, length, and time are usually considered primitive or fundamental because of the direct way in which they can be measured and physically concatenated. 3. Physically, there is something peculiar about basis elements with dimensions of [area) or [mass5 ). If we wish to have [length) or [mass) be mathematically primitive, this can only be accomplished by choosing the integer exponent constraint, option (D); otherwise the primitive dimensions could just as well be derived from their higher powers.
All quantities may be treated differently depending on the context. , when considered to be the ratio of an arc to a radius. , (a) directional quantities such as length or force can be 21 22 It is an interesting property of the physical world that a small finite number of basic dimensions apparently suffices for any application. This is especially intriguing in light of the fact that it is not clear what the exact number is and there is apparently no simple criterion for deciding. From this perspective, a fundamental contribution of nineteenth century science was to enlarge the dimension of the known space by one, to include [charge], which had not previously been a quantified separate dimension.
Multidimensional Analysis: Algebras and Systems for Science and Engineering by George W. Hart
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