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Homogeneous functions and the dimension method.10 . The dimension of a physical quantity and the properties of functional relationsbetween physical quantities.Physical laws establish interconnections between physical quantities, so that ifcertain units of measurement are adopted for some of these quantities, then the unitsof measurement of the quantities connected with them can be expressed in a certainway in terms of the units of measurement of the fixed quantities. That is how thebasic and derived units of different systems of measurement arise.In the International System, the basic mechanical units of measurement are takento be the unit of length (the meter, denoted m), mass (the kilogram, denoted kg), andtime (the second, denoted s).The expression of a derived unit of measurement in terms of the basic mechanicalunits is called its dimension.
This definition will be made more precise below.The dimension of any mechanical quantity is written symbolically as a formulaexpressing it in terms of the symbols L, M, and T proposed by Maxwell6 as thedimensions of the basic units mentioned above. For example, the dimensions ofvelocity, acceleration, and force have respectively the forms[v] = LT −1 ,[a] = LT −2 ,[F ] = MLT −2 .If physical laws are to be independent of the choice of units of measurement, oneexpression of that invariance should be certain properties of the functional relationx0 = f (x1 , . .
. , xk , xk+1 , . . . , xn )(*)between the numerical characteristics of the physical√quantities.Consider, for example, the relation c = f (a, b) = a 2 + b2 between the lengthsof the legs and the length of the hypotenuse of a right triangle. Any change of scaleshould affect all the lengths equally, so that for all admissible values of a and b therelation f (αa, αb) = ϕ(α)f (a, b) should hold, and in the present case ϕ(α) = α.A basic (and, at first sight, obvious) presupposition of dimension theory is that arelation (*) claiming physical significance must be such that when the scales of thebasic units of measurement are changed, the numerical values of all terms of thesame type occurring in the formula must be multiplied by the same factor.In particular, if x1 , x2 , x3 are basic independent physical quantities and the relation (x1 , x2 , x3 ) → f (x1 , x2 , x3 ) expresses the way a fourth physical quantitydepends on them, then, by the principle just stated, for any admissible values ofx1 , x2 , x3 the equalityf (α1 x1 , α2 x2 , α3 x3 ) = ϕ(α1 , α2) α3 )f (x2 , x2 , x3 ),(**)must hold with some particular function ϕ.6 J.C.
Maxwell (1831–1879) – outstanding British physicist. He created the mathematical theory ofthe electromagnetic field, and is also famous for his research in the kinetic theory of gases, optics,and mechanics.8.3 The Basic Laws of Differentiation453The function ϕ in (**) characterizes completely the dependence of the numericalvalue of the physical quantity in question on a change in the scale of the basic fixedphysical quantities. Thus, this function should be regarded as the dimension of thatphysical quantity relative to the fixed basic units of measurement.We now make the form of the dimension function more precise.a) Let x → f (x) be a function of one variable satisfying the condition f (αx) =ϕ(α)f (x), where f and ϕ are differentiable functions.Show that ϕ(α) = α d .b) Show that the dimension function ϕ in Eq.
(**) always has the form α1d1 · α2d2 .dα3 3 , where the exponents d1 , d2 , d3 are certain real numbers. Thus if, for example, the basic units of L, M, and T are fixed, then the set (d1 , d2 , d3 ) of exponentsexpressed in the power representation Ld1 M d2 T d3 can also be regarded as the dimension of the given physical quantity.c) In part b) it was found that the dimension function is always a power function,that is, it is a homogeneous function of a certain degree with respect to each of thebasic units of measurement. What does it mean if the degree of homogeneity of thedimension function of a certain physical quantity relative to one of the basic unitsof measurement is zero?20 The Π -theorem and the dimension method.Let [xi ] = Xi (i = 0, 1, .
. . , n) be the dimensions of the physical quantities occurring in the law (*).Assume that the dimensions of x0 , xk+1 , . . . , xn can be expressed in terms of thedimensions of x1 , . . . , xk , that is,p1pk[x0 ] = X0 = X1 0 · · · Xk 0 ,p1pk[xk+i ] = Xk+i = X1 i · · · Xk i(i = 1, . . . , n − k).d) Show that the following relation must then hold, along with (*):p1pkp1pkp1pkα1 0 · · · αk 0 x0 = f α1 x1 , . .
. , αk xk , α1 1 · · · αk 1 xk+1 , . . . , α1 n−k · · · αk n−k xn .(***)e) If x1 , . . . , xk are independent, we set α1 = x1−1 , . . . , αk = xk−1 in (***). Verifythat when this is done, (***) yields the equalityx0p1x1 0pk· · · xk 0= f 1, . . . , 1,xk+1p1x1 1,...,kp· · · xk 1xnp1x1 n−kpk· · · xk n−k,which is a relationΠ = f (1, .
. . , 1, Π1 , . . . , Πn−k )involving the dimensionless quantities Π, Π1 , . . . , Πn−k .(****)4548 Differential Calculus in Several VariablesThus we obtain the followingΠ-theorem of dimension theory If the quantities x1 , . . . , xk in relation (*) areindependent, this relation can be reduced to the function (****) of n − k dimensionless parameters.f) Verify that if k = n, the function f in relation (*) can be determined up to anumerical√ multiple by using the Π -theorem. Use this method to find the expressionc(ϕ0 ) l/g for the period of oscillation of a pendulum (that is, a mass m suspendedby a thread of length l and oscillating near the surface of the earth, where ϕ0 is theinitial displacement angle). √g) Find a formula P = c mr/F for the period of revolution of a body of massm held in a circular orbit by a central force of magnitude F .h) Use Kepler’s law (P1 /P2 )2 = (r1 /r2 )3 , which establishes for circular orbitsa connection between the ratio of the periods of revolution of planets (or satellites)and the ratio of the radii of their orbits, to find, as Newton did, the exponent α inthe law of universal gravitation F = G mr1αm2 .8.4 The Basic Facts of Differential Calculus of Real-ValuedFunctions of Several Variables8.4.1 The Mean-Value TheoremTheorem 1 Let f : G → R be a real-valued function defined in a region G ⊂ Rm ,and let the closed line segment [x, x + h] with endpoints x and x + h be contained in G.
If the function f is continuous at the points of the closed line segment[x, x + h] and differentiable at points of the open interval ]x, x + h[, then thereexists a point ξ ∈ ]x, x + h[ such that the following equality holds:f (x + h) − f (x) = f (ξ )h.(8.53)Proof Consider the auxiliary functionF (t) = f (x + th)defined on the closed interval 0 ≤ t ≤ 1. This function satisfies all the hypotheses ofLagrange’s theorem: it is continuous on [0, 1], being the composition of continuousmappings, and differentiable on the open interval ]0, 1[, being the composition ofdifferentiable mappings.
Consequently, there exists a point θ ∈ ]0, 1[ such thatF (1) − F (0) = F (θ ) · 1.But F (1) = f (x + h), F (0) = f (x), F (θ ) = f (x + θ h)h, and hence the equality just written is the same as the assertion of the theorem.8.4 Real-valued Functions of Several Variables455We now give the coordinate form of relation (8.53).If x = (x 1 , . . . , x m ), h = (h1 , . . . , hm ), and ξ = (x 1 + θ h1 , . . . , x m + θ hm ),Eq. (8.53) means thatf (x + h) − f (x) = f x 1 + h1 , . . . , x m + hm − f x 1 , . . .
, x m =⎛ ⎞ h1∂f∂f(ξ ), . . . , m (ξ ) ⎝ · · · ⎠ == f (ξ )h =∂x∂x 1hm= ∂1 f (ξ )h1 + · · · + ∂m f (ξ )hm ==m∂i f x 1 + θ h1 , . . . , x m + θ hm hi .i=1Using the convention of summation on an index that appears as both superscriptand subscript, we can finally writef x 1 + h1 , . . . , x m + hm − f x 1 , . . . , x m == ∂i f x 1 + θ h1 , . .
. , x m + θ hm hi ,(8.54)where 0 < θ < 1 and θ depends on both x and h.Remark Theorem 1 is called the mean-value theorem because there exists a certain“average” point ξ ∈ ]x, x + h[ at which Eq. (8.53) holds. We have already noted inour discussion of Lagrange’s theorem (Sect.
5.3.1) that the mean-value theorem isspecific to real-valued functions. A general finite-increment theorem for mappingswill be proved in Chap. 10 (Part 2).The following proposition is a useful corollary of Theorem 1.Corollary If the function f : G → R is differentiable in the domain G ⊂ Rm andits differential equals zero at every point x ∈ G, then f is constant in the domain G.Proof The vanishing of a linear transformation is equivalent to the vanishing of allthe elements of the matrix corresponding to it. In the present casedf (x)h = (∂1 f, .
. . , ∂m f )(x)h,and therefore ∂1 f (x) = · · · = ∂m f (x) = 0 at every point x ∈ G.By definition, a domain is an open connected set. We shall make use of this fact.We first show that if x ∈ G, then the function f is constant in a ball B(x; r) ⊂ G.Indeed, if (x +h) ∈ B(x; r), then [x, x +h] ⊂ B(x; r) ⊂ G. Applying relation (8.53)or (8.54), we obtainf (x + h) − f (x) = f (ξ )h = 0 · h = 0,4568 Differential Calculus in Several Variablesthat is, f (x + h) = f (x), and the values of f in the ball B(x; r) are all equal to thevalue at the center of the ball.Now let x0 , x1 ∈ G be arbitrary points of the domain G.
By the connectednessof G, there exists a path t → x(t) ∈ G such that x(0) = x0 and x(1) = x1 . Weassume that the continuous mapping t → x(t) is defined on the closed interval 0 ≤t ≤ 1. Let B(x0 ; r) be a ball with center at x0 contained in G. Since x(0) = x0 andthe mapping t → x(t) is continuous, there is a positive number δ such that x(t) ∈B(x0 ; r) ⊂ G for 0 ≤ t ≤ δ. Then, by what has been proved, (f ◦ x)(t) ≡ f (x0 ) onthe interval [0, δ].Let l = sup δ, where the upper bound is taken over all numbers δ ∈ [0, 1] such that(f ◦ x)(t) ≡ f (x0 ) on the interval [0, δ]. By the continuity of the function f (x(t))we have f (x(l)) = f (x0 ).
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