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

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.

Challenge Problems for a Theory of Degree Multiplication (with partial answer key)

“…I hope to have convinced the reader that it is useful to appeal to a notion of quotient among quantities across different dimensions, that it is not something that is provided for under current foundations for degree semantics, and that we can get it by importing a system of quantity calculus. I've illustrated how to do this, using the dimension-centric approach of Raposo (2018Raposo ( , 2019. At the minimum, we got a lexical entry for per out of it.…”

“…Yet the algebraic foundations for quantity calculus remain a topic of discussion to this day. I will be drawing on a recent proposal by Raposo (2018Raposo ( , 2019, who distinguishes between two different approaches to the algebraic foundations for quantity calculus: a unit-centric approach, where a quantity is defined as a combination of a number and a unit, and a dimension-centric approach, under which "the dimension is an intrinsic property of a quantity, in contrast to its numerical value, which depends on the unit chosen, or the unit itself, which can be changed arbitrarily." So, for example, the quantity named by 'five kilometers' bears no intrinsic relation to the number five, but it does bear an intrinsic relation to the dimension 'length'.…”

An Algebraic Foundation of Amended Dimensional Analysis

“…where r is the rank of D. Each such equation corresponds to a maximal independent subtuple A of D that does not contain [y 0 ]. If (18) is not already a system of simultaneous equations, it can be made into one by setting Λ (ℓ) = lcm W (1)0 , . .…”

“…are not real numbers, but it restricts the scope of dimensional analysis. • The exponents W j and W kj are usually assumed to be rational or real numbers [6, p. 293], but Quade [16] and more recently Raposo [18] use only integer exponents (see also [1]). • It is usually implicitly assumed that for any φ there is just one ψ such that (2) holds, or at least that it suffices to consider one ψ or, at the very least, deal with one ψ at a time.…”

Technical Foundations of CORSIKA 8: New Concepts for Scientific Computing

“…The implementation is based on template metaprogramming and keeps track of the dimensions of quantities via a set of integers (since fractional powers of dimensions do not occur in physical equations [14]) representing the powers of each base dimension, i.e., mass, length, time, etc., of the International System of Units (SI). While in principle sufficient to express any calculation, the strict enforcement of correct dimensions by the compiler prevents volitional "sloppy" handling of units as we often want to do it in particle physics with natural units, e.g.…”