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Each of these formulas provides a particular functionf : R+ → R+ defined on the set R+ of positive real numbers with values in thesame set.Example 2 Let X be the set of inertial coordinate systems and c : X → R the function that assigns to each coordinate system x ∈ X the value c(x) of the speed of lightin vacuo measured using those coordinates. The function c : X → R is constant, thatis, for any x ∈ X it has the same value c. (This is a fundamental experimental fact.)Example 3 The mapping G : R2 → R2 (the direct product R2 = R × R = Rt × Rxof the time axis Rt and the spatial axis Rx ) into itself defined by the formulasx = x − vt,t = t,is the classical Galilean transformation for transition from one inertial coordinatesystem (x, t) to another system (x , t ) that is in motion relative to the first atspeed v.The same purpose is served by the mapping L : R2 → R2 defined by the relationsx = x − vt1 − ( vc )2,141 Some General Mathematical Concepts and Notationt − ( v2 )xt = c.1 − ( vc )2This is the well-known (one-dimensional) Lorentz12 transformation, which playsa fundamental role in the special theory of relativity.
The speed c is the speed oflight.Example 4 The projection pr1 : X1 × X2 → X1 defined by the correspondencepr1X1 × X2 (x1 , x2 ) −→ x1 ∈ X1 is obviously a function. The second projectionpr2 : X1 × X2 → X2 is defined similarly.Example 5 Let P(M) be the set of subsets of the set M. To each set A ∈ P(M) weassign the set CM A ∈ P(M), that is, the complement to A in M. We then obtain amapping CM : P(M) → P(M) of the set P(M) into itself.Example 6 Let E ⊂ M. The real-valued function χE : M → R defined on the set Mby the conditions (χE (x) = 1 if x ∈ E) ∧ (χE (x) = 0 if x ∈ CM E) is called thecharacteristic function of the set E.Example 7 Let M(X; Y ) be the set of mappings of the set X into the set Y and x0a fixed element of X. To any function f ∈ M(X; Y ) we assign its value f (x0 ) ∈ Yat the element x0 .
This relation defines a function F : M(X; Y ) → Y . In particular,if Y = R, that is, Y is the set of real numbers, then to each function f : X → R thefunction F : M(X; R) → R assigns the number F (f ) = f (x0 ). Thus F is a functiondefined on functions. For convenience, such functions are called functionals.Example 8 Let Γ be the set of curves lying on a surface (for example, the surfaceof the earth) and joining two given points of the surface.
To each curve γ ∈ Γ onecan assign its length. We then obtain a function F : Γ → R that often needs to bestudied in order to find the shortest curve, or as it is called, the geodesic between thetwo given points on the surface.Example 9 Consider the set M(R; R) of real-valued functions defined on the entirereal line R. After fixing a number a ∈ R, we assign to each function f ∈ M(R; R)the function fa ∈ M(R; R) connected with it by the relation fa (x) = f (x + a). Thefunction fa (x) is usually called the translate or shift of the function f by a. Themapping A : M(R; R) → M(R; R) that arises in this way is called the translationof shift operator. Thus the operator A is defined on functions and its values are alsofunctions fa = A(f ).12 H.A.
Lorentz (1853–1928) – outstanding Dutch theoretical physicist. Poincaré called these transformations Lorentz transformations in honor of Lorentz, who stimulated the research of symmetries in Maxwell’s equations. They were used by Einstein in 1905 in the formulation of his theoryof special relativity.1.3 Functions15This last example might seem artificial if not for the fact that we encounter realFoperators at every turn. Thus, any radio receiver is an operator f −→ fˆ that transforms electromagnetic signals f into acoustic signals fˆ; any of our sensory organsis an operator (transformer) with its own domain of definition and range of values.Example 10 The position of a particle in space is determined by an ordered tripleof numbers (x, y, z) called its spatial coordinates.
The set of all such ordered triplescan be thought of as the direct product R × R × R = R3 of three real lines R.A particle in motion is located at some point of the space R3 having coordinates(x(t), y(t), z(t)) at each instant t of time. Thus the motion of a particle can beinterpreted as a mapping γ : R → R3 , where R is the time axis and R3 is threedimensional space.If a system consists of n particles, its configuration is defined by the position ofeach of the particles, that is, it is defined by an ordered set (x1 , y1 , z1 ; x2 , y2 , z2 ; .
. . ;xn , yn , zn ) consisting of 3n numbers. The set of all such ordered sets is called theconfiguration space of the system of n particles. Consequently, the configurationspace of a system of n particles can be interpreted as the direct product R3 × R3 ×· · · × R3 = R3n of n copies of R3 .To the motion of a system of n particles there corresponds a mapping γ : R →R3n of the time axis into the configuration space of the system.Example 11 The potential energy U of a mechanical system is connected with themutual positions of the particles of the system, that is, it is determined by the configuration that the system has.
Let Q be the set of possible configurations of a system.This is a certain subset of the configuration space of the system. To each positionq ∈ Q there corresponds a certain value U (q) of the potential energy of the system.Thus the potential energy is a function U : Q → R defined on a subset Q of theconfiguration space with values in the domain R of real numbers.Example 12 The kinetic energy K of a system of n material particles depends ontheir velocities.
The total mechanical energy of the system E, defined as E = K +U ,that is, the sum of the kinetic and potential energies, thus depends on both the configuration q of the system and the set of velocities v of its particles. Like the configuration q of the particles in space, the set of velocities v, which consists of nthree-dimensional vectors, can be defined as an ordered set of 3n numbers. The ordered pairs (q, v) corresponding to the states of the system form a subset Φ in thedirect product R3n × R3n = R6n , called the phase space of the system of n particles(to be distinguished from the configuration space R3n ).The total mechanical energy of the system is therefore a function E : Φ → Rdefined on the subset Φ of the phase space R6n and assuming values in the domainR of real numbers.In particular, if the system is isolated, that is, no external forces are acting on it,then by the law of conservation of energy, at each point of the set Φ of states of thesystem the function E will have the same value E0 ∈ R.161 Some General Mathematical Concepts and NotationFig.
1.61.3.2 Elementary Classification of MappingsWhen a function f : X → Y is called a mapping, the value f (x) ∈ Y that it assumesat the element x ∈ Y is usually called the image of x.The image of a set A ⊂ X under the mapping f : X → Y is defined as the setf (A) := y ∈ Y | ∃x (x ∈ A) ∧ y = f (x)consisting of the elements of Y that are images of elements of A.The setf −1 (B) := x ∈ X | f (x) ∈ Bconsisting of the elements of X whose images belong to B is called the pre-image(or complete pre-image) of the set B ⊂ Y (Fig. 1.6).A mapping f : X → Y is said to besurjective (a mapping of X onto Y ) if f (X) = Y ;injective (or an imbedding or injection) if for any elements x1 , x2 of Xf (x1 ) = f (x2 ) ⇒ (x1 = x2 ),that is, distinct elements have distinct images;bijective (or a one-to-one correspondence) if it is both surjective and injective.If the mapping f : X → Y is bijective, that is, it is a one-to-one correspondencebetween the elements of the sets X and Y , there naturally arises a mappingf −1 : Y → X,defined as follows: if f (x) = y, then f −1 (y) = x that is, to each element y ∈ Yone assigns the element x ∈ X whose image under the mapping f is y.
By thesurjectivity of f there exists such an element, and by the injectivity of f , it is unique.Hence the mapping f −1 is well-defined. This mapping is called the inverse of theoriginal mapping f .1.3 Functions17Fig. 1.7It is clear from the construction of the inverse mapping that f −1 : Y → X is itselfbijective and that its inverse (f −1 )−1 : X → Y is the same as the original mappingf : X → Y.Thus the property of two mappings of being inverses is reciprocal: if f −1 isinverse for f , then f is inverse for f −1 .We remark that the symbol f −1 (B) for the pre-image of a set B ⊂ Y involvesthe symbol f −1 for the inverse function; but it should be kept in mind that the preimage of a set is defined for any mapping f : X → Y , even if it is not bijective andhence has no inverse.1.3.3 Composition of Functions and Mutually Inverse MappingsThe operation of composition of functions is on the one hand a rich source of newfunctions and on the other hand a way of resolving complex functions into simplerones.If the mappings f : X → Y and g : Y → Z are such that one of them (in ourcase g) is defined on the range of the other (f ), one can construct a new mappingg ◦ f : X → Z,whose values on elements of the set X are defined by the formula(g ◦ f )(x) := g f (x) .The compound mapping g ◦ f so constructed is called the composition of themapping f and the mapping g (in that order!).Figure 1.7 illustrates the construction of the composition of the mappings fand g.You have already encountered the composition of mappings many times, both ingeometry, when studying the composition of rigid motions of the plane or space,and in algebra in the study of “complicated” functions obtained by composing thesimplest elementary functions.The operation of composition sometimes has to be carried out several times insuccession, and in this connection it is useful to note that it is associative, that is,h ◦ (g ◦ f ) = (h ◦ g) ◦ f.181 Some General Mathematical Concepts and NotationProof Indeed, h ◦ (g ◦ f )(x) = h (g ◦ f )(x) = h g f (x) = = (h ◦ g) f (x) = (h ◦ g) ◦ f (x).This circumstance, as in the case of addition and multiplication of several numbers, makes it possible to omit the parentheses that prescribe the order of the pairings.If all the terms of a composition fn ◦ · · · ◦ f1 are equal to the same function f ,we abbreviate it to f n .It is well known, for example, that the square root of a positive number a can becomputed by successive approximations using the formula1a,xn +xn+1 =2xnstarting from any initial approximation x0 > 0.
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