MEASUREMENT SCIENCE REVIEW, 19, (2019), No. 2, 70–78
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The Algebraic Structure of Quantity Calculus II: Dimensional Analysis and Differential and Integral Calculus
Álvaro P. Raposo1
1Department of Applied Mathematics, Universidad Politécnica de Madrid, Av. Juan de Herrera, 6, 28040, Madrid, Spain, [email protected]
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. Keywords: Quantity calculus, algebraic structure, dimensional analysis, pi-theorem.
1. INTRODUCTION two drawbacks which show it not to be an accurate descrip- In a previous paper [1] the author has produced a new alge- tion of the actual algebraic properties of quantity calculus. braic structure, axiomatically defined, called algebraic space First, it is based on the concept of unit, as all the structure is of quantities, which resembles accurately the algebra struc- built upon a starting set of base units. In fact, quantities are ture of quantity calculus. In that paper there is a review of introduced as objects of the form a number times a unit (just past attemps of giving a mathematical framework to quan- as Maxwell did over 100 years ago [7]), thus identifying the tity calculus and of the motivation to introduce yet another, concepts of quantity and quantity value, which are well dif- different, algebraic structure which is now summarized here. ferentiated by the VIM. This is quite unsatisfactory, for this Despite in the standard approach to quantity calculus (e.g. the approach blurs the distinction between the system of quan- International Vocabulary of Metrology (VIM) [2]) the calcu- tities (such as the ISQ) and the system of units (such as the lus focuses on quantities and dimensions as central concepts, SI). This situation has been recognized by several authors in the mathematical approach to the algebraic structure underly- previous years (see de Boer [8] for an excellent review and ing quantity calculus is based on units as the main concept. discussion, [9] for a recent one) who have claimed, instead, In the recent analysis of Kitano [3], for instance, or the first for an algebraic structure defined axiomatically and based on papers on this model, [4], [5], [6], the model is set up starting the concept of quantity and of dimension rather than on the from a set of base units (u1,...,un) and expresses any quan- concept of unit, which should be secondary. tity q in terms of them as The second drawback is the use of real exponents in (1). They are not necessary. It is a remarkable fact that the laws of α r1 rn q = u1 ···un , (1) Physics involve dimensionful quantities (in contrast with di- mensionless quantities) only through three operations: prod- where α,r1,...,rn are real numbers. In this equation α is the numerical value of q with respect to this system of units, uct of quantities, product by a scalar, and addition of quanti- r1 rn ties of the same kind. Therefore, only integer exponents are while u1 ···un is the unit in which q is measured. The set r1 rn found in physical equations. Some simple examples seem to of all units, that is {u1 ···un : ri ∈ R, i = 1,...,n} has the structure of a vector space over R, written in multiplicative prove the previous statement wrong, for instance, the veloc- form. The set of quantities, thus, adopts the form of a family ity v of a body of mass m for which the kinetic energy T is of rays, each identified and spanned by a unit as in (1) by known is given by letting α run through the reals. The rays coincide in a point, 2T the zero of the algebraic structure. The numbers (r1,...,rn) v = , (2) are referred to as the dimensions of the quantity q. r m This algebraic structure has served its main purpose: giv- so the dimensionful quantity T/m has an exponent 0.5. How- ing a proof of Buckingham’s Pi Theorem. However, it has ever, the dimension of T/m is already a squared one: [T/m]= DOI: 10.2478/msr-2019-0012
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−1 2 −1 (LT ) , so the use of the exponent 0.5 is only, say, acciden- fiber has its own zero element. The fiber dim 1D is called tal. The period t of a pendulum given its length l and the the set of dimensionless quantities or, more appropriately, acceleration of free fall, g, provides a similar example, for it quantities of dimension one. This fiber is, in addition to is given by the structure of vector space, an algebra over F and, in fact, l is naturally isomorphic with F by means of the map F → t = 2π , (3) −1 α α g dim 1D : 7→ 1Q, so we identify both whenever needed. s Notice that the structure of the fibers as one dimensional but, again, the quantity l/g has a squared dimension, [l/g]= vector spaces cannot be the ultimate model of quantity calcu- T2 , so the exponent 0.5 does not provide a dimension with lus, for it does not fit properly to quantized quantities. If a fractional exponent. A more critical example is a system of quantity is quantized, the corresponding fiber should not be units in which volume is taken as a base quantity and, e.g. the the whole vector space, but a discrete subset of it. However, liter as the corresponding base unit. In that case we should this issue will not be dealt with in this paper. admit fractional exponents for the derived units for length, A system of units is a section of the fiber bundle, that is, 1/3 2/3 (liter) , or surface area, (liter) . Nevertheless, experience a map σ : D → Q such that dim ◦ σ = idD . The system is in measurement science has shown, so far, that a suitable coherent if σ is a group homomorphism, and it is nonzero if choice of base quantities is possible which avoids such frac- σ(A) is not a zero for any dimension A. In[1] the conditions tional exponents to define units, and this is the case this paper for its existence are established. wants to support from the mathematical point of view. With In this paper we only consider spaces of quantities free of this goal in mind, the allowance of real exponents is unnec- zero divisors. Two advantages from such a choice are: (i) essarily oversized. It works, of course, since real numbers every nonzero quantity has an inverse and (ii) there exists a contain integer numbers, but it is not the algebraic structure nonzero coherent system of units. So we also assume without which best fits the quantity calculus. further notice that such a system is always available and all The paper [1] addresses these two issues and gives a way systems of units considered are nonzero and coherent. out by defining, in axiomatic form, a new algebraic structure When a coherent nonzero system of units σ is defined in a which resembles, more accurately, the calculus with quanti- space of quantities then we have the map ν : Q → F (where ties (i.e., only integer exponents allowed) and is completely we identify the fiber of dimensionless quantities with F) de- focused on quantities and dimensions. It starts with the def- fined by D inition of the group of dimensions as a finitely generated ν(q)= qσ(dim(q))−1, (4) free Abelian group. Its elements are denoted A,B... and are referred to as dimensions. The identity of the group is denoted which assigns to a quantity q its numerical value with respect σ 1D . to the system of units ; it is a homomorphism. Notice that σ In the quantity calculus real numbers play a prominent role the composition ◦ dim : Q → Q gives the unit in the fiber as the measurement of a quantity is nothing but its compari- of each quantity. Then, a quantity q is written in terms of a son with the unit of its kind, and the result is a real number. system of units as This number and the unit make the quantity value. In the def- q ν q σ dim q (5) inition of the algebraic structure, the role of real numbers is = ( ) ◦ ( ). F taken by an abstract field . Nevertheless, this abstract field With the units at hand paper [1] explores the way in which is going to get more specific along the sections of this paper, quantities and the operations among them are expressed in first as an ordered field in section 3, then as the real numbers terms of the units. Ways to create new spaces from old ones in sections 4 and 5. With these elements, a summary of the (subspace, tensor product, quotient space) are also explored Q definition given in [1] is as follows. A space of quantities and, finally, the tool for comparison is defined, the homomor- D F Q with group of dimensions over the field is a set , whose phism of spaces of quantities, and utilized to give a character- elements are called quantities, together with a surjective map ization theorem of these structures. Finally, it is demonstrated Q D A dim : → and three operations. For each dimension , that any such structure, given a system of units, is isomorphic the set dim−1A is called a fiber, and the first two operations D dim D are defined within each fiber: addition of quantities and prod- with the projection fiber bundle F × −→ . uct by scalars in the field F. With them, the fiber aquires the As can be seen, paper [1] merely defines and develops the structure of a one-dimensional vector space over F. In addi- algebraic tools. Now it is the time to apply them to give a tion, the whole set Q has a product, the third operation, with sound justification to the two main uses of quantity calculus respect to which it is an Abelian monoid; the neutral element beyond the algebraic operations: dimensional analysis and differential and integral calculus. However, before address- is 1Q. As the three operations and the map dim are related by, first, dim is a homomorphism with respect to the product ing these two questions, a bit more of structure is needed over in Q, second, distribution of the product over the addition in the algebraic base: an order structure. And also a tool that each fiber and, third, association of the product of quantities appears once and again along the paper: changes of units. and product by a scalar. The structure is referred to as an Therefore, the layout is the following: changes of units are treated in section 2, and the order structure in section 3. Sec- Q dim D algebraic fiber bundle and denoted by the symbol −→ . tion 4 deals with dimensional analysis and states and proves A key and distinctive feature of this structure is that each
71 MEASUREMENT SCIENCE REVIEW, 19, (2019), No. 2, 70–78 the Pi-theorem in this context. Finally, in section 5, the basis iii. Let A and B be dimensions. Then of differential calculus and of integral calculus with quantities is established. σ˜ (AB)= χ(AB)σ(AB)= χ(A)χ(B)σ(A)σ(B)= σ˜ (A)σ˜ (B), 2. CHANGE OF UNITS χ σ In this section we study the way to change from a sys- because both, and are group homomorphisms and tem of units to another, that is, a change of scale. Con- the commutativity of the products. σ σ sider two systems of units 1 and 2. The quantity q can Let us go a step further by considering the set G of all scale be expressed in terms of both of them according to (5) factor maps χ : D → F∗. Let χ,ξ be two such maps; we can ν σ ν σ as q = 1(q) 1(dim(q)) = 2(q) 2(dim(q)). The quantity define a product in G componentwise, that is, for a dimen- σ 2(dim(q)) is not zero, so it has an inverse and we can solve sion A, (χξ)(A)= χ(A)ξ(A). With this product the set G ν for 2(q) as is an Abelian group, the identity element is the map A 7→ 1, i.e. with all scale factors 1, and the inverse element of χ is ν (q)= σ (dim(q))σ (dim(q))−1 ν (q), (6) 2 1 2 1 χ−1(A)= χ(A)−1. We refer to this group later in section 3. which is the conversion formula between the numerical val- ues with respect to both systems of units. The number 3. ORDER STRUCTURE IN A SPACE OF QUANTITIES −1 σ1(dim(q))σ2(dim(q)) is the conversion factor, or scale We all agree that 3 kg is a greater mass than 2 kg, and that factor, for the fiber of q. Notice that this equation seems dif- 3 kg and 5 m/s are not comparable. These examples show that ferent from the usual conversion factor formula, which would the addition of an order relation to the algebraic structure of read σ1 =(ν2/ν1)σ2; however, this form is not useful for a space of quantities must be studied in order to complete the mathematical purposes because ν1 may be zero, and thus di- mathematical model of quantity calculus. And it also warns vision is not allowed. On the other hand, σ2 is always nonzero us that the order needs not be complete, but a partial order. by definition, so the division is safe. The order we are about to define is borrowed from an order The association of the scale factor with each fiber (equiva- in the base field F so, in this section, F is assumed to be an lently, with each dimension in D) is a group homomorphism, ordered field, such as Q or R, and the order relation in the and the converse is also true: field is denoted by <. The first step is to define what we understand by an order Proposition 2.1. Given two systems of units σ and σ , 1 2 relation compatible with the algebraic structure of a space of the scale factor map χ : D → F∗ defined by χ(A) = −1 quantities. σ1(A)σ2(A) is a group homomorphism. Conversely, given a group homomorphism χ : D F∗ and a system of units σ, dim → Definition 3.1. An ordered space of quantities Q−→D over σ˜ D σ˜ A χ A σ A the map : → Q defined by ( )= ( ) ( ) is a system the ordered field F is a space of quantities with an order re- of units. lation ≺ in Q such that: A B PROOF. For the direct statement, let and be dimensions i. In each fiber there is a quantity greater than the zero of D in . Then the fiber.
−1 χ(AB)= σ1(AB)σ2(AB) = ii. Let a1, a2, a3 be quantities in the same fiber. If a1 ≺ a2 −1 then a1 + a3 ≺ a2 + a3. σ1(A)σ1(B)(σ2(A)σ2(B)) = −1 −1 γ σ1(A)σ2(A) σ1(B)σ2(B) = χ(A)χ(B), iii. Let a and b be quantities and a scalar number in F. If a ≺ b and γ > 0, then γa ≺ γb and, if a ≺ b and γ < 0, because both, σ1 and σ2 are group homomorphisms and all then γb ≺ γa. the products are commutative. C For the converse we have to prove σ˜ is (i) a section of Q, iv. Let a, b and c be quantities, dim(c)= . Ifa ≺ b and (ii) nonzero, and (iii) coherent. 0C ≺ c then ac ≺ bc.
i. Let A be a dimension. Then Remarks: The first item of the definition is needed because, as noticed above, the order relation needs not be a total order, so we want at least to be able to make a comparison inside each fiber. A quantity that is greater than the zero of its fiber dim ◦ σ˜ (A)= dim(χ(A)σ(A)) = is called positive. The remaining three items are the compat- χ A σ A A A dim( ( ))dim( ( )) = 1D = , ibility conditions with each of the three operations in Q. The second part of item (iii) is not needed in a fiber, for it can where we have identified the number χ(A) with a dimen- be deduced from (ii), but that is not the case when it applies sionless quantity. Since A is arbitrary, dim ◦ σ˜ = idD . to the comparison of quantities of different dimensions, since ii. Since χ(D) ⊂ F∗ and σ is nonzero, clearly σ˜ is nonzero addition among them is not defined. as well. A number of immediate consequences follows from the previous definition.
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dim Proposition 3.2. Let Q−→D be an ordered space of quanti- In the following considerations there is no need to distin- ties over the field F. Then guish the symbols ≺ (order in Q) from < (order in F) for we have seen they share the same properties, so we only use the i. The fiber of dimensionless quantities is naturally order more familiar < for both. Nevertheless, the context always isomorphic with F. makes it clear which is the case. ii. Each fiber is order isomorphic with F. In the definition of an ordered space of quantities there is nothing to prevent a comparison of quantities of different di- iii. Zeros of different fibers are not comparable. mensions. We now turn to study this issue to see that, al- iv. The rule of signs applies to the product of quantities though the algebraic and order structures can hold such a comparison, the consequences are far from what it is desir- v. If a quantity is positive, its inverse is also positive (in its able for a quantity calculus. As we can see in the following corresponding fiber). paragraphs, all the discussion can be reduced to the fact that zeros of different fibers are not comparable (third item of the PROOF. i. Remember that the natural map between the previous proposition). field F and the fiber of dimensionless quantities F → −1 Let us assume there are quantities a and b, with dim(a)= dim (1D ) : α 7→ α1Q is an isomorphism of algebras, A 6= B = dim(b) such that a < b. We can assume 0B < b, oth- so we only need to prove it preserves the order. By (i) ′ erwise choose a quantity b greater than 0B and, by transitivity in definition 3.1, there is α ∈ F such that 0 ≺ α1 , 1D Q of the order, a < b < b′, so a < b′ as desired. so α 6= 0. Assume α < 0 in F, then −α−1 > 0 and so, α−1α Now, regarding a, consider the two possibilities either 0A < by (iii) in the same definition, 01D ≺− 1Q = −1Q, a or a < 0A. In the first case, again by the transitivity of the therefore 1 ≺ 0 . Multiply the later by the positive Q 1D order relation, we get quantity −1Q and get −1Q ≺ 01D , a contradiction with 0A b (7) the antisymmetry of the order relation. If, on the con- < . −1 ± −1 trary, α > 0 then so is α and, by the same argument, Define the sets B = {x ∈ dim (B) : 0A ≶ x}. It is easy to ′ ′ + ′ we get 01D ≺ 1Q. see that any b such that 0B < b is in B , because b = βb β A D for some number > 0 so, after multiplication in (7), we ii. Let be a dimension in and a a positive quantity in β ′ A −1 A α get 0A < b = b . Also, multiplication of (7) by −1 gives the fiber of dimension . The map F → dim ( ) : 7→ ′ ′ α −b < 0A and then, by a similar argument, if b < 0B then b is a is an order isomorphism (in fact, it is also a linear B− ′ ′ α α α α in . In other words, for any b such that 0B < b we have isomorphism) for, if 1 < 2 in F, then 0 < 2 − 1 ′ ′ −b < 0A < b . so we can multiply in 0A ≺ a and get 0A ≺ (α − α )a. 2 1 But this does not necessarily mean that there are quantities Now, by (ii) in the definition 3.1, we conclude α1a ≺ of dimension B as close to 0A as desired, for it may be the case α a, so the linear isomorphism preserves the order. 2 that a′ < b′ for any quantity a′ with dimension A and any pos- Notice this item is different from the first one because itive quantity b′. In this later case, the whole fiber dim−1(A) the isomorphism is not canonical; it depends on the arbi- lies above negative elements of dim−1(B) and below posi- −1 trarily chosen quantity a. Now we can denote as positive tive elements of dim (B). However, while 0A is comparable all quantities greater than zero in a fiber, and as negative with all elements of the fiber of B (except its zero), 0B is com- their negatives, which are less than zero. And we can parable with no element at all of the fiber of A. also define the absolute value of a quantity q as |q| = q To explore another scenario, assume in addition that there if q is positive or zero, or |q| = −q in case it is negative. is a′ in dim−1(A) which is greater than b, so we have a < ′ ′ A B b < a . By transitivity we get 0B < b < a so the same argu- iii. Let and be different dimensions. We already know ± A B ment can be utilized here and define the sets A of elements 0A 6= 0B because dim(0A)= 6= = dim(0B). Assume −1 in dim (B) greater or less, respectively, than 0B. Moreover, A B 0 ≺ 0 . Then, after multiplication by −1, by (iii) in ′ ′′ since we also have 0A < b < a , let a be any positive quantity the definition, we get 0B ≺ 0A, a contradiction with the in dim−1(A), then a′′ = αa′ for some number α > 0 so, after antisymmetry of the order relation.. ′ ′′ multiplication by α, 0A < αb < αa = a , i.e. we can find iv. Let a and b be positive quantities of dimensions A and an element in the fiber of B between 0A and any positive ele- B respectively. Then, by (iv) in the definition, 0AB ≺ ab. ment in the fiber of A. The same applies to negative elements. Now, if both quantities were negative, then by (iii) in Therefore, in this case we can say that there are elements in −1 the same definition, −a and −b are positive, so 0AB ≺ dim (B) as close to 0A as we desire. Even more, the closer −1 (−a)(−b)= ab. Similarly, in the cases of different signs an element in the fiber dim (B) is to 0B, the closer it is to we get ab ≺ 0AB. 0A. However, 0A and 0B remain uncomparable. A similar discussion can be made in the case a < 0A, 0B < b v. Let a be a positive quantity with dim(a)= A. Since it and a < b, leading to similar conclusions. is not zero, it is invertible and a−1 is not zero as well Since the scenarios depicted by such ordinations of quan- so, by the second item of this proposition, it is greater tities of different dimensions are not in accordance with the or less than 0A−1 . In the later case, multiplication by the
positive quantity a yields 1Q ≺ 01D , a contradiction.
73 MEASUREMENT SCIENCE REVIEW, 19, (2019), No. 2, 70–78 actual use of quantity calculus in Physics, in the remaining we σ˜ (A)= χ(A)σ(A) for A ∈ D. The orders defined by σ and σ˜ only consider order structures in which quantities of different in each fiber, denoted < and <˜ , respectively, are easily related dimensions are not comparable. Notice that, by item (ii) of by the following proposition. the proposition 3.2, in this case the order is characterized by −1 A defining an element in each fiber as positive, i.e. giving an Proposition 3.5. In the fiber dim ( ) the orders < and <˜ χ A χ A orientation to each fiber. But this choice is bound with alge- are the same if ( ) > 0 or reversed if ( ) < 0. braic restrictions. For instance, either if we choose a quantity −1 PROOF. Let a1,a2 be quantities in the fiber dim (A) such a, with dimension A, to be positive or we choose −a to be that a1 < a2 with respect to a system of units σ, that is a1 = positive, in both cases a2 is a positive quantity in the fiber of α1σ(A) and a2 = α2σ(A) with α1,α2 ∈ F and α1 < α2 in A2. Thus, we are free to choose an orientation only for the F. With respect to another system of units σ˜ we have a1 = fibers of a basis in the group of dimensions. This fact gives −1 −1 α1χ(A) σ˜ (A) and a2 = α2χ(A) σ˜ (A). Now, if χ(A) > 0 full meaning to the following definition. −1 −1 −1 so is χ (A) and, thus, α1χ(A) < α2χ(A) , so a1<˜ a2, dim χ A ˜ Definition 3.3. An ordered space of quantities Q −→ D is while if ( ) < 0 the opposite is the case and a2
74 MEASUREMENT SCIENCE REVIEW, 19, (2019), No. 2, 70–78 current, as in the SI). It is completely arbitrary that the charge the quantities in the same fiber by the factor χ(A)−1 and re- of the proton has been chosen to be positive and the charge of fer them to the same original unit, so we take advantage of the electron to be negative, and the physics involved do not the later to define an action of the group G on the space of change if we switch the signs of the charges. quantities Q. For a quantity q, denote by χ[q] the action of χ ∈ G on q, defined as χ[q]= χ(dim(q))−1q. It is an action G 4. DIMENSIONAL ANALYSIS for the identity in given by 1G [q]= q and, given two maps χ ξ G we have In this section we assume the base field F to be R, the set , ∈ of real numbers. 1 −1 1 Dimensional analysis is traditionally rooted in two re- χ[ξ[q]] = χ dim ξ(dim(q))− q ξ(dim(q))− q = sults known as Bridgman’s theorem [10] and Buckingham’s −1 −1 χ (dim(q)) ξ(dim (q)) q =(χξ)[q], Pi-theorem [11]. However, the result of Bridgman is un- necessary in the present context, let us explain why. In because dim(ξ(dim(q))−1q) = dim(q). Now assume the his book, Bridgman distinguishes between primary and sec- quantities a1,...al,b, with dimensions dim(ai) = Ai, i = ondary quantities. The former are those which can be directly 1,...,l, dim(b) = B, are related by a formula such as measured, such as length or time. The later are those com- f (a1,...,al)= b. We define the concept of homogeneity in puted from primary quantities, for instance velocity. The re- terms of the map f and the action of G just introduced.. sult of Bridgman states that, if q1,...,qn are primary quanti- ties, any secondary quantity obtained from them has the form Definition 4.1. Let A1,...,Al,B be dimensions in D. A map −1 −1 −1 α r1 rn α f : dim (A)×···×dim (A ) → dim (B) is dimensionally q1 ···qn , where ,r1,...,rn are real numbers (as in (1)). l This is another reason for the general acceptance of the so homogeneous if it commutes with the action of the group G in far established model of quantity calculus discussed in the the space of quantities. That is, for any χ ∈ G and any ai ∈ Ai, introduction. Notwithstanding, in the context of a space of i = 1,...,l, quantities as understood in this paper this is not necessary be- χ χ χ cause it is included in the axioms of the algebraic structure. f ( [a1],..., [al]) = [ f (a1,...,al)]. Moreover, in that definition the only operations considered Lemma 4.2. Let f : dim−1(A) × ··· × dim−1(A ) → between quantities are product, product by a scalar, and ad- l dim−1(B) be a homogeneous nonzero map and let dition of quantities of the same kind. Therefore, an expres- {A ,...,A }, after reordering if necessary, be a maxi- sion like that of Bridgman is not permitted unless the expo- 1 k mal independent subset of {A ,...,A }. Then the set nents of the quantities are integer numbers. It is a matter of 1 l {A ,...,A ,B} is not independent in D, that is, there are in- fact that equations of Physics involve only expressions of the 1 k teger numbers n n n such that α m1 mn α , 1,..., k form q1 ···qn , where ∈ R but m1,...,mn ∈ Z. Indeed, the description of a theory of quantity calculus with integer Bn = An1 ···Ank . exponents instead of real ones is one of the leitmotifs of the 1 k present paper and its predecessor [1]. PROOF. Assume the contrary, i.e. {A1,...,Ak,B} is an in- Therefore, we are left only with the task of stating and dependent set in D. Let a1,...,al,b be quantities of the ap- proving the Pi-theorem in the context of the present alge- propriate dimensions: dim(ai)= Ai, i = 1,...,l, dim(b)= B, bra structure of quantity calculus. The first step towards such that b = f (a1,...,al) 6= 0. By the assumption we can ∗ such a theorem is the concept of dimensionally homoge- choose arbitrary scalars in F , α1,...,αk,β and find a group ∗ neous equation, which seems to have been conceived firstly homomorphism χ : D → F such that χ(Ai)= αi, i = 1,...,k by Fourier [12] and succesfully employed by Reynolds [13] and χ(B)= β. Take, for instance, α1 = ··· = αk = 1, β = 2. or Rayleigh [14]. The naive idea is that all the terms in a valid Then the action of χ on the quantities involved is χ[ai]= ai, equation of Physics must have the same dimensions. Another i = 1,...,l (for the cases i > k, there is a relationship be- approach says that such an equation, which describes a law tween Ai and A1,...,Ak, so we can choose χ(Ai)= 1 as well), of nature, must be invariant under changes in the system of χ 1 [b]= 2 b. Now, since the map f is homogeneous we find units, which is arbitrary. We take the later path to define the concept of dimensional homogeneity, that is, we consider ho- b = f (a1,...,al)= f (χ[a1],..., χ[al]) mogeneity as a kind of symmetry for it states an invariance 1 property under suitable transformations. = χ[ f (a1,...,al)] = b, The mathematical tool for dealing with symmetries is the 2 group theory. In sections 2 and 3 a change of units has been a contradiction. characterized by a group homomorphism χ : D → F∗, called a scale factor map, and the set of all those maps are gathered We also need a lemma on the extraction of nth roots of together in the group G . In a fiber dim−1(A), the new unit suitable quantities. is obtained from the previous by multiplication by the fac- dim D tor χ(A), while the numerical values of quantities get multi- Lemma 4.3. Let Q −→ be an ordered space of quantities σ A D plied by χ(A)−1; the quantities themselves are not modified. over R oriented by a system of units , let ∈ be a dimen- The same numerical values are obtained if we multiply all sion and q a quantity with dim(q)= An, n an integer number.
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i. If n is an odd integer there exists a quantity a, with so it is dimensionless. dim(a)= A, such that an = q. Now, in order to take advantage of the homogeneity of f we define a suitable scale factor map χ : D → R∗. Since ii. If n is an even integer and q is positive there exists a {A1,...,Ak} is independent in D we can certainly choose quantity a, with dim(a)= A, such that an = q. arbitrarily k nonzero real numbers as image of χ(Ai), i = χ A ν ν PROOF. The hypothesis on q allows us to write q = βσ(An), 1,...,k. Let us take ( i) = (ai), i = 1,...,k, where χ χ where β ∈ R and, in case (ii), β > 0. In both cases, there is the map defined in (4). Then the action of is [ai]= 1 ν(a )−1a = σ(A ), the unit in the fiber, for i = 1,...,k. How- exists in R the nth root of this number, say β n , and we just i i i 1 ever, for j = k + 1,...,l the result is different, for define a = β n σ(A). m χ A m j χ A j The point in the previous lemma is to understand to what ( j) = ( j )= m m extent the condition on q being positive in case (ii) is not an χ A j1 A jk χ A m j1 χ A m jk ( 1 ··· k )= ( 1) ··· ( k) = empty one, since the orientation of fibers is dependent on m m ν m j1 ν m jk ν j1 jk the system of units, which is arbitrary. However, the orien- (a1) ··· (ak) = (a1 ···ak ) tation of the fiber with dimension An is not arbitrary when χ ν n is even. Consider two systems of units, σ and σ˜ which since both, and , are homomorphisms. m j πm j m j1 m jk give different orientations to the fiber with dimension A, say From (10) we have a j = j a1 ···ak . Therefore −m j mj −m j mj σ A σ A σ An σ A n σ A n σ An χ A m j ν π π ν ˜ ( )= − ( ). Then ˜ ( )= ˜ ( ) =(− ( )) = ( ) ( j) = ( j a j ) = j (a j ) which is a positive so the fiber dim−1(An) has only one possible orientation. real number, so we conclude We are now ready to state and prove the Pi-theorem. χ A π−1ν ( j)= j (a j), j = k + 1,...,l (11) dim Theorem 4.4 (Pi-theorem with integer exponents). Let Q−→ D be an ordered space of quantities over the field R oriented and the action is A A B D by a system of units. Let 1,..., l, ∈ be dimensions −1 χ[a j]= π jν(a j) a j = π jσ(A j), j = k + 1,...,l. (12) and {A1,...,Ak}, after reordering if necessary, be a maximal independent subset of {A1,...,Al}. −1 −1 −1 With a parallel argument we arrive at If f : dim (A1) ×···× dim (Al) → dim (B) is a ho- 1 mogeneous map and a1,...,al are positive quantities in their n1 nk χ(B)=(ν(a1) ···ν(ak) ) n , (13) respective fibers, dim(ai)= Ai,i = 1,...,l, then there are in- l−k teger numbers n,n1,...,nk and a function g : R → R such so the action on b is that −1 n1 nk 1 χ[b]=(ν(a1) ···ν(ak) ) n b (14) n1 nk n π π f (a1,...,al)= a1 ···ak g( k+1,..., l), which is equal, by the homogeneity of f , to for suitable real numbers πk+1,..., πl. χ χ χ χ The real numbers πk+1,...,πl are the l − k independent di- f ( [a1],..., [ak], [ak+1],..., [al]) mensionless quantities that can be constructed with the orig- = f (σ(A1),...,σ(Ak),πk+1σ(Ak+1),...,πlσ(Al)). (15) inal quantities a1,...,al (the so called pi-groups which give the name to the theorem). Define the real function g : Rl−k → R by
PROOF. By the hypothesis for each A j not in the subset g(πk+1,...,πl) {A1,...,Ak} there are integer numbers m j,m j1,...,m jk such σ −1 σ A σ A π σ A π σ A that = (B) f ( ( 1),..., ( k), k+1 ( k+1),..., l ( l)). Am j Am j1 Am jk (16) j = 1 ··· k , k ≤ j ≤ l. (8)
By lemma 4.2 there are integer numbers n,n1,...,nk such that The right hand side of (16) lies in the fiber of dimensionless quantities which, once again, is identified with the base field Bn An1 Ank = 1 ··· k . (9) R. Now, by (14) and (15) and bringing together (9) and (16) For each j = k + 1,...,l define we find
1 −m −m m f (a1,...,al)= b = π = a a j1 ···a jk j , (10) j j 1 k 1 ν n1 ν nk n ( (a1) ··· (ak) ) × which is well defined because, by (8), the quantity f (σ(A1),...,σ(Ak),πk+1σ(Ak+1),...,πlσ(Al)) = −m j1 −m jk A−m j a1 ···a has dimension j and so the m jth root does 1 k π n1 nk exist. The dimension of j can be thus computed as (ν(a1) ···ν(ak) ) n σ(B)g(πk+1,...,πl)= n n 1 1 1 k n π π a1 ···a g( k+1,..., l). π −m j1 −m jk m j k dim( j)= dim(a j)dim a1 ···ak A A−1 = j j = 1D , 76 MEASUREMENT SCIENCE REVIEW, 19, (2019), No. 2, 70–78
5. DIFFERENTIAL AND INTEGRAL CALCULUS S = {z ∈ R : dim( f (z)) = dim( f (x)) and z < y}, which is a Differential and integral calculus are, possibly, the most nonempty set (x ∈ S) and with upper bound y. Denote y0 the important mathematical tools in Physics. Therefore, their use least upper bound of S, which is an accumulation point of S, with quantities has to be justified in the context of a space of and consider the following two possible choices. quantities. The first step, however, is to provide the space of First, if y0 ∈ S, then y0 < y and dim( f (y0)) = dim( f (x)) = quantities with a topology so that the concept of limit can be A. We now show that the inverse image of the set {A}, which properly handled. Since all these concepts make sense in a is open in D, by the map dim ◦ f is not open in Q, for y0 is complete field and we want also to preserve the order struc- in that inverse image but it is not an interior point. In fact, ture, we need a complete ordered field as base field, that is, any open ball in the fiber centered on y0 and radius less than −1 the field of real numbers so, in this section, it is also assumed |y − y0| contains points not in S and not in (dim ◦ f ) (A). F = R. Therefore, dim ◦ f is not a continuous map, and so is f be- dim Since the definition of a space of quantities Q −→ D is cause dim is continuous. purely algebraic, the only hint to impose a topology on it is Second, if y0 ∈/ S, apply the same argument to the com- that the operations in Q be continuous maps, as well as the plementary set of {A} in D, which is again an open set. Its projection map, dim, on the group of dimensions D. inverse image by dim ◦ f contains y0 which, again, is not an We also want to take advantage of the order structure de- interior point. fined in section 3. Remember that the standard topology of Therefore, a necessary condition for a map f : R → Q to be R can be thought of as induced by its order, so the same can continuous is that f (R) is contained in a single fiber in Q. be applied to each fiber. With this topology the operations of It is also usual to consider a variable quantity as a parame- addition and product by scalar numbers are both continuous ter for another, e.g. time as parameter for the length travelled as they are in R. by a moving particle. Since any single fiber is homeomorphic On the other hand, the fibers which make the space of quan- with R, the previous proposition applies to this case as well. tities are bound by the algebraic properties which are sum- It is stated in the following corollary. marized in the group structure of the group of dimensions D. This is, by definition, a finitely generated free Abelian Corollary 5.2. Let A be a dimension in D and dim−1(A) its group. There are several topologies that can be assumed in fiber in Q. If a map f : dim−1(A) → Q is continuous, then its the group D which make it into a topological group, but we image is contained in a single fiber of Q. choose to endow it with the discrete topology because, first, it is the choice coherent with our previous choice of discard- In the following, we only consider maps of the form f : −1 −1 −1 ing an order relation among quantities of different dimensions R → dim (B) or f : dim (A) → dim (B), which include and, second, it enables us to state and prove proposition 5.1 all continuous maps, so in fact we are considering maps of which is quite a natural result to be expected. the form R → R and, thus, differential and integral calculus dim can be readily established. But before we give a definition of Finally, in [1] it is shown that a space of quantities Q −→ derivative and of integral with quantities it proves useful to D over the field R is algebraically isomorphic with D × R point how limits of functions can be dealt with in the fashion with suitable operations defined therein. This is to say, Q of metric spaces. Once we have adopted a topology in Q the can be seen as a trivial fiber bundle with base D and fiber R. concept of limit of a map in Q is established. Nevertheless, Therefore, it is only natural to assume as topology for Q the the following lemma states that the definition of limit in Q product topology in D × R. Accordingly to the previous two can be reduced to that in R due to that topology. paragraphs, a base for this topology is that of open intervals D in each fiber. The map dim : Q → is trivially continuous. Lemma 5.3. Let A,B be dimensions in D, a,b quantities with Thus, the space of quantities can be described in this respect dim(a)= A, dim(b)= B, and f : dim−1(A) → dim−1(B) a as a bundle whose fibers are 1-dimensional positively oriented map. The limit of f at a is b if for any positive quantity ε with linearly ordered order-topological vector spaces. dim(ε)= B, there is a positive quantity δ with dim(δ)= A, Now we consider quantities as variables in order to define such that, given a′ ∈ dim−1(A), if |a′ − a| < δ then | f (a′) − maps and deal with them towards the definition of derivative. b| < ε. A variable quantity often depends on a real parameter, so we consider first this case and study the conditions for such a map PROOF. A basis for the topology in Q is obtained from the to be continuous. The first proposition shows that a continu- product of a basis in D (single elements) and a basis in R ous map in Q must lie on a single fiber. (open intervals or balls); we get, thus, open intervals in each fiber. Now apply the standard definition of limit in a given R R Proposition 5.1. Let f : → Q be continuous. Then f ( ) topology, but take into account that any open set containing a lies in a single fiber of Q. can be substituted by an open ball in the fiber dim−1(A) cen- PROOF. Let us prove the equivalent statement: if f (R) is tered in a, and the same applies for b. An open ball centered not contained in a single fiber, then f is not continuous. on a is described a |a′ − a| < δ, while the open ball centered Let x,y be real numbers such that dim( f (x)) 6= dim( f (y)) on b can be described as |b′ − b| < ε, so the lemma follows. and assume x < y. Define the subset of R of the numbers For the sake of completeness the definitions of the deriva- less than y and with image by f in the same fiber that x, tive and of the integral follow.
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dim Definition 5.4. Let Q−→D be a space of quantities over the 6. ACKNOWLEDGEMENT field R, let A,B be dimensions in D and let f : dim−1(A) → Thanks are due to the anonymous reviewers whose insightful dim−1(B) be a map. The map f is differentiable at a ∈ and accurate comments have improved this paper. dim−1(A) if the following limit exists f (a + h) − f (a) REFERENCES lim , h→0A h [1] Raposo, A. P. (2018). The algebraic structure of quantity calculus. Measurement Science Review 18(4), 147–157. where dim(h)= A. In such case, the value of the limit is denoted f ′(a) and referred to as the derivative of f at a. [2] JCGM (2012). International vocabulary of metrology– basic and general concepts and associated terms (vim). The definition shows, in addition, that the dimension of the 3rd edition. derivative is that of the image of the map times the inverse of [3] Kitano, M. (2013). Mathematical structure of unit sys- the independent variable quantity: tems. Journal of Mathematical Physics 54, 052901–1– dim( f ′(a)) = dim( f (a))dim(a)−1 = BA−1. 17. [4] Drobot, S. (1953). On the foundations of dimensional Starting from this definition the rest of the differential cal- analysis. Studia Mathematica 14 (1), 84–99. culus in R can be traslated step by step to Q straighforward, so we will not go further into it here. [5] Whitney, H. (1968). The mathematics of physical quan- Now we turn to the definition of the integral of a map. tities: Part II: Quantity structures and dimensional anal- Since the goal is to traslate the concepts defined in R to a ysis. The American Mathematical Monthly 75, 227– space of quantities, the simpler case of Riemannian definition 256. of integral suffices to our purposes. [6] Carlson, D. (1979). A mathematical theory of physical Let [a′, a′′] be an interval in the fiber dim−1(A) and let P = units, dimensions and measures. Archive for Rational ′ ′′ Mechanics and Analysis {a0,a1,...,an}, where a = a0 < a1 < ··· < an = a , be a 70 (4), 289–304. partition of that interval. Let f : dim−1(A) → dim−1(B) be a [7] Maxwell, J. (1873). Treatise on Electricity and Mag- map bounded in [a′, a′′]. For each i = 1,...,n define netism. Oxford University Press. [8] de Boer, J. (1994). On the history of quantity calculus mi = inf{ f (a) : ai−1 ≤ a ≤ ai}, (17) and the international system. Metrologia 31, 405–429. M = sup{ f (a) : a ≤ a ≤ a }. i i−1 i [9] Krystek, M. (2015). The term ‘dimension’ in the inter- The lower and upper sums of f for the partition P are, respec- national system of units. Metrologia 52, 297–300. tively, [10] Bridgman, P. Dimensional Analysis. Yale University n Press. L( f ,P)= ∑ mi(ai − ai−1), [11] Buckingham, E. (1914). On physically similar systems: i=1 Illustrations of the use of dimensional equations. Phys- n (18) ical Review 4, 345–376. U( f ,P)= ∑ Mi(ai − ai−1). i=1 [12] Fourier, J. (1822). Théorie analytique de la chaleur. Then the properties regarding lower and upper sums with dif- Chez Firmin Didot, père et fils. ferent partitions can be readily proven (see, e.g. [15]) and the [13] Reynolds, O. (1883). An experimental investigation of following definition stated the circumstances which determine whether the motion dim of water shall be direct or sinuous, and of the law of re- Definition 5.5. Let Q−→D be a space of quantities over the sistance in parallel channels. Philosophical transactions A B −1 A field R, let , be dimensions in D and let f : dim ( ) → of the Royal Society of London 174, 935–982. −1 B be a bounded map in the interval a′ a′′ dim ( ) [ , ] ⊂ [14] Rayleigh, L. (1892). On the question of the stabil- −1 A . The map f is integrable in that interval if dim ( ) ity of the flow of fluids. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science sup{L( f ,P) : P a partition of [a′, a′′]} = 34(206), 59–70. inf{U( f ,P) : P a partition of [a′, a′′]}. [15] Spivak, M. (1980). Calculus. Publish or Perish, 2nd a′′ edition. In such a case, this number is denoted a′ f (a)da and re- ferred to as the integral of f in the interval´ [a′, a′′]. Received November 15, 2018. The definition and (18) show that the dimension of the in- Accepted April 10, 2019. tegral is
a′′ dim f (a)da = dim ∑mi(ai − ai−1) = ˆa′ ! dim( f (a))dim(a)= BA.
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