The Algebraic Structure of Quantity Calculus II: Dimensional Analysis and Differential and Integral Calculus

Álvaro P. Raposo

doi:10.2478/msr-2019-0012

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The Algebraic Structure of Quantity Calculus II: Dimensional Analysis and Differential and Integral Calculus

Measurement Science Review

Volume 19 (2019): Issue 2 (April 2019)

By: Álvaro P. Raposo

Open Access

|May 2019

Abstract

In a previous paper, the author has introduced and studied a new algebraic structure which accurately describes the algebra underlying quantity calculus. The present paper is a continuation of that one, which extends the purely algebraic study by adding two more ingredients: an order structure and a topology. The goal is to give a solid justification of dimensional analysis and differential and integral calculus with quantities.

References

[1] Raposo, A. P. (2018). The algebraic structure of quantity calculus. Measurement Science Review 18(4), 147–157.10.1515/msr-2017-0021
Search in Google Scholar Back to article
[2] JCGM (2012). International vocabulary of metrology– basic and general concepts and associated terms (vim). 3rd edition.
Search in Google Scholar Back to article
[3] Kitano, M. (2013). Mathematical structure of unit sys- tems. Journal of Mathematical Physics 54, 052901–1–17.10.1063/1.4802876
Search in Google Scholar Back to article
[4] Drobot, S. (1953). On the foundations of dimensional analysis. Studia Mathematica 14 (1), 84–99.10.4064/sm-14-1-84-99
Search in Google Scholar Back to article
[5] Whitney, H. (1968). The mathematics of physical quan- tities: Part II: Quantity structures and dimensional anal- ysis. The American Mathematical Monthly 75, 227–256.10.1080/00029890.1968.11970972
Search in Google Scholar Back to article
[6] Carlson, D. (1979). A mathematical theory of physical units, dimensions and measures. Archive for Rational Mechanics and Analysis 70 (4), 289–304.10.1007/BF00281156
Search in Google Scholar Back to article
[7] Maxwell, J. (1873). Treatise on Electricity and Mag- netism. Oxford University Press.
Search in Google Scholar Back to article
[8] de Boer, J. (1994). On the history of quantity calculus and the international system. Metrologia 31, 405–429.10.1088/0026-1394/31/6/001
Search in Google Scholar Back to article
[9] Krystek, M. (2015). The term ‘dimension’ in the inter- national system of units. Metrologia 52, 297–300.10.1088/0026-1394/52/2/297
Search in Google Scholar Back to article
[10] Bridgman, P. Dimensional Analysis. Yale University Press.
Search in Google Scholar Back to article
[11] Buckingham, E. (1914). On physically similar systems: Illustrations of the use of dimensional equations. Phys- ical Review 4, 345–376.10.1103/PhysRev.4.345
Search in Google Scholar Back to article
[12] Fourier, J. (1822). Théorie analytique de la chaleur. Chez Firmin Didot, père et fils.
Search in Google Scholar Back to article
[13] Reynolds, O. (1883). An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of re- sistance in parallel channels. Philosophical transactions of the Royal Society of London 174, 935–982.10.1098/rstl.1883.0029
Search in Google Scholar Back to article
[14] Rayleigh, L. (1892). On the question of the stability of the flow of fluids. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 34(206), 59–70.10.1080/14786449208620167
Search in Google Scholar Back to article
[15] Spivak, M. (1980). Calculus. Publish or Perish, 2nd edition.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/msr-2019-0012 | Journal eISSN: 1335-8871

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Language: English

Page range: 70 - 78

Submitted on: Nov 15, 2018

Accepted on: Apr 10, 2019

Published on: May 2, 2019

Published by: Slovak Academy of Sciences, Institute of Measurement Science

In partnership with: Paradigm Publishing Services

Publication frequency: Volume open

Keywords:

Quantity calculus,

algebraic structure,

dimensional analysis,

pi-theorem

Related subjects:

Engineering,

Electrical engineering,

Control engineering, metrology and testing

© 2019 Álvaro P. Raposo, published by Slovak Academy of Sciences, Institute of Measurement Science
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 19 (2019): Issue 2 (April 2019)