MEASUREMENT OF SCALAR QUANTITIES BY LIMITING PROCEDURE Текст научной статьи по специальности «Математика»

СУЧАСН1 ТЕНДЕНЦП РОЗВИТКУ МЕТОДИКИ НАВЧАННЯ МАТЕМАТИКИ У ВИЩ1Й ШКОЛ1

MEASUREMENT OF SCALAR QUANTITIES BY LIMITING PROCEDURE

(Метод границь для вимiрювання скалярних виразiв)

N. Gorchev, D. Petrov, «St Cyril and St Methodius» University of Veliko Turnovo,

BULGARIA, e-mail: n.g. kolev@uni. vt. bg, e-mail: d.petrov@uni-vt.bg

].......{

У po6omi представлений узагальнений nidxid до формулювання i обгрунтування методу меж для вимiру ненегативних скалярних величин. Для щег мети зроблет eidmeidm (в основному методичт) уточнення в гх акаоматичтй теорп. Знайдений споаб, що дозволяе простежити взаемозв 'язок при впорядкувант множини однорiдних величин.

К:иочо<и слова: скалярт величина, мгра величины.

I. Axiomatic Theory of the Positive Scalar Quantities

The basic notions in the axiomatic theory of positive scalar quantities are: A nonempty set S + of all positive scalar quantities which elements are unions of sets [J S+, I e N .

iel

Elements of S+ can be added to one another (" +i"), they are ordered (" <") (not necessary linearly ordered). Elements of S+ we will call positive scalar quantities and the elements of each St + - its states. [J S+ will be

iel

a set of homogeneous scalar quantities.

The axioms in the theory of positive scalar quantities are [3]:

A+ . (Trichotomy axiom). У a, b e S+ only one of the relation holds:

a =i b; a < b; b <i a (We will consider

states with a =i b to be identical.)

A2+ . У a, b e S+ 3!c e S+ such that c =ia +ib called sum of a and b with the properties:

©

a) a +i b =i b +i a, Va, b e S+ (commutativi-

ty);

6) (a +ib)+,.c =a +,.(b +ic), Va, b, c e s+ (associativity);

b) (a <i a +i b) a (b < b +i a), Va, b e S+ (positivity).

A+. Let a, b e S+ with. Then 3!c e S+ such that a = b +ic . We denote c =i a -ib . ^ S+ and Vn e N 3 b e S+ : n.b = a,

A+ . У a e S+ and Уп e N

def

where n.b = ib +. b + ... +. b.

г i "' г

A5+ . У a, b e S+ 3n e N

such that a < n.b .

A6. (Continuity axiom). Let

A+

al,a2,... e S+ and b,,b2,... e S+ are two sequences of states. If we assume that

a, <l a2 <l ... < an <i ... <,bn < ... < b,

1 i 2 i i n i i n i i 1

and that Vc e S+ 3n e N : bn - an <i c, then 3 !s e S+ such that

n-times

an < s < bm, Vn, m e N (states s1 and s2, for which s1 =. s2 we will consider equal.)

Definition 1. Axiomatic theory based on the above definitions and axioms A+ + A6+ is called axiomatic theory of positive scalar-additive continuous quantities.

Examples of positive scalar quantities:

(1) Length of a piecewise convex plane curve (a plane curve that can be broken in finite pieces such that if we take a piece with endpoints A and B, and connect the points with a line segment, we get a convex plane figure). For the corresponding subsets of homogeneous quantity (length) we can take length of a line segment (with set of states, say, S+) and length of an arc of a circle (with set of states, say, S+).

(2) Volume of a convex body as a subset of the homogeneous quantity volume.

(3) Surface area of a convex body is a subset of the homogeneous quantity surface area. Such subsets are surface areas of a prism, pyramid, or a pyramidal body ([2]).

(4) Mass of a body. Mass of a system of two bodies (that are motionless with respect to each other) is the sum of the masses.

(5) Relativistic energy of a particle. If m is its rest mass and p its momentum with respect to a fixed coordinate system. The total energy is E = p2c2 + m2c4 .

Now we consider measurement of positive scalar quantities.

Definition 2. Quantity S+ if called measurable if there exists a map ¡i: S+ — R + (R+

is the set of all real nonnegative numbers) with the properties:

a) if a = b, then ¡(a) = ¡(b), a, b e S+;

b) if a = b +ic, then

¡(a) = /~i(b) + ¡(c), a, b, c e S+ ;

c) 3e e Si + , such thate) = 1.

Definition 3. The number ¡¡a) is called

measure of quantity a e S+ and the map ¡ : S+ — R + is called measure on the set S+ .

In practice measurement of a quantity consists of finding a functional ¡ : S+ — R+ that satisfies properties a) and b) from Definition 2 and also

c') ¡ : S+ — R + is normed up to a constant factor. If for the quantity is chosen a unit state e we must have ¡(e)= 1.

One can show that a functional ¡ : S+ — R + , satisfying a), b) and c') have

the properties:

1. monotonicity -

a b ^ ¡(a) < ¡(b), a, b e S;

2. continuity -

Vve (k,X) c R + 3!a e S+ : ¡(a) = a. The functional ¡ : S+ — R + is called measure of the quantity S+ .

II. Limiting procedure for measurement of positive scalar quantities

We apply limiting procedure (from [6]) for a system of homogeneous positive scalar continuous quantities.

The process of measurement is based on approximations:

Theorem 1. Let li. : S+ — R + is a

' ¡0 ¡0

measure on S + c S + and Va e S + \ S+

¡0 1 1 ¡0

neN

neN ^ S¡o such thdt.

j. Vn e N, pn <,a <i qn;

2. Vs > 0,3 v(s) > 0: Vn > v(s) we

have ¡¡o(qn)-Ulo(pn)<s.

Then the functional : S+ — R+ : ¡i (a) = lim h (pn ) = lim h (qn ) is wel1 de-

n—>w 0 n—w 0

fined and is a measure of the quantity S + .

Proof. We need to show that limits lim (pn) and lim ¡ii (qn) exist and they

n—w 0 n—w 0

are the same. From monotonicity we have ¡i0 (pn ) < ¡i0 (qm X Vm, n e N. Therefore there exist limsup (pn) = l1 and

n—w

liminf ! (qn) = l2. Obviously l1 < l2. Let

n—w 0

s > 0. Since 3 v(s) > 0: Vn > v(s) we

have ¡¡0 (qn)-!i0 (p«)<s it follows l2 -11 < s and thus l1 = l2 = l. It remains to show that the sequences {¡^ (pn)}neN and {M0 (qn )}neN both converge to l. Again from

condition ¡¡o (qn)-!i0 (p«)<s Vn >v(s) and from ! (qn) >l >! (pn) we get

G>

M0(4n ) - l <S and I -Mi0(Pn) <£. Therefore lim j (qn) = I = lim j (pn). We den —» 0 n—» 0

fine j (a) = l.

Now we show that the functional j: S+ — R + satisfies conditions a),b) and c'):

a) Let a = b, a, b e S+ and

{p'n)neN C ^M'n) neN C Si+ ,

{p"n\ neN C SC,{qn'} neN C Si0 are the co^e

sponding sequences for the states a, b e S+ respectively. Then the sequences

P2k+1 = pk-1, p2k = pk, q2k+1 = qk-1, q2k = qk satisfy the conditions of the theorem and so (as subsequences of a convergent sequence) they give one limit j (b) = j (a).

b) Let{pn}neN c SIM^ c S+ ,

ne N C Si+,{q: } :eN C Si0 are the co^e

sponding sequences for the states b, c e S+ respectively. Then

{p'n +,p:}neN C Si+ , {q'n +tql}neN C S+

are the sequences for the state a e S+

(since p'n +i p"n <i b +i c <i q'n +i q"n ). Vs > 0,3 v(s) > 0: V: > v(s) we have

u0( q'n +,q"n) -u0( p'n +i p: ) = = j (q'n) -j (p'n) + j (q:) -j ( p: ) < 2s.

Therefore

j (a) = lim^0(pn +Ip"n) =

n—» 0

=lim U0 (p'n) +lim U0 (p"n) = U (b) + U (c).

n—» 0 n—» 0

c') Let e0 be a unit state for S+ the

' i0

measure u : S+ — R+. But then e0 e S+

' i0 i0 i

and if we define the sequences

pn = e° = qn Vn e N. Then pn < e0 < qn

and so we can take e = e0 to be the unit state for S+ .i.e. j (e) = Vi0 (e°) = 1. QED

We will show monotonicity of j : S+ — R+. From a < b, a, b e S+ and

from property b)

we

get

u(a1 +i ••• +iak) =

= u(a1) + ... + u(ak), a13...,ak eS+,k >2. Therefore j(k.a ) = k.j(a) for any a e S+, k e N. From axiom A+ follows the existence of the states '1

a

n

n

a | e

S+, Vn e N. Thus

ju(a) = juj n. = n.U a | ^ juja | = —.j(a)

and therefore

u

m

.a | = j

a

m— | = m.j

v n

f a

V n

= m .j(a). n

b =i a +i (b - a) ^ j (b) = j (a) + j (b - a), a

and therefore j (b) > j (a).

For any functional satisfying a), b) and c) we can show additivity [5]:

Let a e S+ be arbitrary state and let

u : S+ — R+ be as in the Theorem 1. The

' i0 i0

sequences {p'n}neN C S+ {q'n}neN C S!0 from

the theorem for a e S+ guarantees the existence the two sequences of rational numbers {an }neN (monotonically increasing) and

{Pn }neN (monotonically decreasing) that satisfy

lim an = limi j (pn) = limi j (qn) = limi Pn.

n—>» n 0 n—>» 0 n—>»

Moreover an ■ e <i a <i f3n • e, Vn e N so U(an "e) <U(a) <U(pn "e) ^ anU(e) < U(a) < PnU(e) «

an <u(a) <Pn, Vn eN. Thus u(a) = lim an = lim Pn = (a) and

n—» n—»

therefore ju = j.

To show continuity let V a e (k, A) e R +. There are two sequences of rational numbers {an }neN (monotonically increasing) and {Pn}neN (monotonically decreasing) for which an <a < Pn, Vn e N and lim an = lim Pn = a . But then

n—» n—»

a1 •e <i a2 •e <i ... <i an ■e <i ... <

<i Pn • e... <i P2 • e <i P1 • e. From axiom Ag follows 3!a e S+ for which b e Sa„ • e <i a < B„ • e, Vn e N. From Theorem

* n i i ! n '

1 we get

o

n

/ (a) = lim^0 (an e) =

= lim(an/0 (e))= 1lim

n^œ \ 0 N ' / n^œ

a,„ = a.

Now we consider the following example.

Let AB be an arc of a circle [4]. Divide it into n parts using points A = M0,M1,.Mn = B. They define n line segments Mt-1Mi, i = 1,..., n that form a partially linear curve p . On the other hand taking tangent lines to the points M., i = 0,..., n we get a second partially linear curve q . For their lengths we have l(p) < l(q). In this case we

n

define the relations p < AB < q.

n

In a more general situation when AB is a partially convex curve and p , q polygonal curves with endpoints A and B for which q

nn

encompasses AB and AB encompasses p

n

we have the inequalities l(p) < l(AB) < l(q). All this and the measurability of the length of

n

the curve AB follow from [1]. We define the

n

relation p < AB < q in this situation also.

n

If C is a point on the curve AB and we define polygonal curves p ,q for the curve

n n

AC and p , q for the curve CB we

have AnB = AC + CB ,

p' < AC < q', p" < CB < q" and

n

p' + p'' < AB < q' + q''.

Definition 4. Let (j S+, I e N be a set of

1eI

homogeneous scalar positive quantities. We say that a relation " < " is defined on Q if for some indices k, l e I the following properties are satisfied: (p < a a a < q ) o p < a < q ^

^¡k (p) < ¡k (q), a eS1, a q eSk

with measure /uk in S+ ; 2.

(p < a < q) a (p' < b < q') a (a = b) ^

p'<k q)a(p <k q'),

1.

where a, b e S+ and p, q, p', q' e S+. 3.

( p ' ^ a ' ^ q ' a p " < a ' ^ q ' ^ p ' p " < a' +i a" < q' +k q",

where p ' , p " , q ' , q " e S+ and a ' , a " e S+.

Theorem 2. Suppose we have a relation '' in Q be defined and that for a state a e S+ there are sequences

{pn)neN C S+k Mn }neN C SÏ for which:

1. Vn e N, pn < a < qn ;

2. Vs> 0,3 v(s) > 0 : Vn > v(s) we

have /(qn)-uk(p„)<s.

Then

I. there exists a measure /: S+ ^ R + ;

II. if a = b, then /(a ) = /(b), a, b e S+ ;

III. if a = i b + ic , then

/(a) = /(b) + /(c), a, b, c e S+ ; Proof:

I. From Definition 4 and condition 1 of the theorem we can conclude

/k(p„ ) ^ /kq ), Vm, n e N. Similarly as in the proof of Theorem 1 we take the sequenceS {-^n }neN ^ Sk+ ,{qn }neN ^ S+k and construct

the desired functional

def

/lia) = lim /k (qn) = lim /k ( p„ ).

n^œ n^œ

II. Let a =tb, a, b e S+,

{p„ }neN ^ Sk ,{qnlneN ^ S+k be the corresponding sequences for a e S+ and

{p'„ }neN ^ Sk ,{q'n}neN ^ S+k be the corresponding sequences for b e S+. From condition 2. of Definition 4 we easily get /(b)</(a)A/(a)< /(b) and so /i{a) = /(b).

III. Let a, b, c e S+ ,

{p'n }neN ^ Sk+ ,{q'n)neN ^ S+k be the corresponding sequences for b e S+ and

{p"n }neN c S+, {qn}neN c S; be the corresponding sequences for c e S+. From condition 3 of Definition 4 follows that the sequences

{p'n + k P"n)n,N c S+, {q'n + k q"n}nGN c S+ correspond to state a e S+.

Since Vs> 0, 3v(s)> 0 we have that for every n >v the following takes place

vk (q'n +kq"n )-Vk (p'n +kPl ) =

= vk (q'„ )-Vk (p'n)+vk (q"n)- vk ( p"„ ) < 2s

and therefore

v(a) =lim vk (q'n + kq"n) =

n^ro

=lim vk (q'n)+lim vk (ql) = vW+v(c).

n ^ro n ^ro

QED

Returning to the previous example we take S+ to be the states of lengths of arcs of

circles and S+ to be the states of lengths of partially linear curves. We can apply Theorem 2 for these sets. It follows that the length of an arc of angle a of a circle with radius r is uniquely defined and it must agree with the more naive definition 2^r a . S+ and S+ are both subsets of states of lengths of partially convex curves (say S+) so the functional V(-) defined in Theorem 2 must agree with

v (•) on s+ u s;c s+ .

Definition 5. Let S+ and S+ are sets of homogeneous positive scalar quantities and suppose a relation '' is defined on S+ = S+ u S+. We define

a < kl P, a, P e S+1 as follows:

1. a, P e S+, a<kl P O a <r P, r = k, l;

2. ae S+ a P e S+, a <kl P Oa< P.

Corollary. Suppose under the assumptions of Theorem 2 the relation <ik from Definition 5 is defined. Then the measure ¡и: S+ ^ R + from Theorem 2 satisfies properties a), b) and c) of Definition 2 if Uk : S+ ^ R + does.

Ш. Conclusion. We present a general (for the methodological purposes) approach for formulation and justification the limiting procedure for measurement of positive scalar quantities. This note can introduce to the field students in pedagogy of mathematics and expose them to more general notions and ideas. This in turn can help them understand the subject better and appreciate the compromises that must be made in a course for school students.

References

1Адамар Ж Элементарная геометрия, ч. II. Стереометрия / ЖАдамар. - М. : Учпедгиз, 1961.

2. Gorchev N. On the notion of surface area for rotational surfaces / N. Gorchev //Дидактика математики: проблеми i дослiдження: мiжнар. зб. наук. робт /редкол.: О.1.Скафа (наук. ред.) та т.; Донецький нац. ун-т; 1нститут педагоаки. Акад. пед. наук Украгни; Нацюнальний пед. ун-т ш М.П.Драгоманова. - Донецьк, 2013. - Вип. 39. - С. 103-108.

3. Колмогоров А.Н. Математика - наука и професия / А.Н.Колмогоров. -М. : Наука, 1988.

4. Лебег А. Об измерение величин / АЛебег. -М.: ГУПИ, 1938.

5. Петканчин Б. Основы на математиката / Б.Петканчин. - София: Наука и искусство, 1968.

6. Фихтенгольц ГМ. Курс дифференциального и интегрального исчисления / Г.М.Фихтенгольц. -М.: Наука, 1969.

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Резюме. Горчев Н., Петров Д, МЕТОД ГРАНИЦ ДЛЯ ИЗМЕРЕНИЯ СКАЛЯРНЫХ ВЕЛИЧИН. В работе предложен обобщенный подход к разработке и обоснованию способа границ измерения положительных скаляров. Сделаны соответствующие (в основном методические) разъяснения в их аксиоматической теории, а также введена специальная реляция, позволяющая продолжить реляцию упорядочевания в множестве однородных величин.

Ключевые слова: скалярные величины, измерения величины.

Abstract. Gorchev N., Petrov D. MEASUREMENT OF SCALAR QUANTITIES BY LIMITING PROCEDURE. In this note we define and develop a general approach for determination of positive scalar quantities. We make appropriate generalization in their axiomatic theory (mainly from methodological point of view). In particular we introduce a relation that extends the ordering in the set of homogeneous positive scalar quantities.

Key words: scalar quantities, measuring of quantities.

Стаття представлена професором В.Б.Мтушевим.

Надшшла доредакци 28.02.2014р.

-dD