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Since weknow how to find ṙ(t) from r(t) and then how to find r̈(t), we are already in aposition to answer the question whether a pair of functions (x(t), y(t)) can describethe motion of the body m about the body M. To answer this question, one must findẍ(t) and ÿ(t) and check whether Eqs. (5.8) hold. The system (5.8) is an example ofa system of so-called differential equations.
At this point we can only check whethera set of functions is a solution of the system. How to find the solution or, better expressed, how to investigate the properties of solutions of differential equations, isstudied in a special and, as one can now appreciate, critical area of analysis – thetheory of differential equations.The operation of finding the rate of change of a vector quantity, as has beenshown, reduces to finding the rates of change of several numerical-valued functions– the coordinates of the vector.
Thus we must first of all learn how to carry outthis operation easily in the simplest case of real-valued functions of a real-valuedargument, which we now take up.5.1.2 Functions Differentiable at a PointWe begin with two preliminary definitions that we shall shortly make precise.1765Differential CalculusDefinition 01 A function f : E → R defined on a set E ⊂ R is differentiable at apoint a ∈ E that is a limit point of E if there exists a linear function A · (x − a) ofthe increment x − a of the argument such that f (x) − f (a) can be represented asf (x) − f (a) = A · (x − a) + o(x − a) as x → a, x ∈ E.(5.9)In other words, a function is differentiable at a point a if the change in its valuesin a neighborhood of the point in question is linear up to a correction that is infinitesimal compared with the magnitude of the displacement x − a from the point a.Remark As a rule we have to deal with functions defined in an entire neighborhoodof the point in question, not merely on a subset of the neighborhood.Definition 02 The linear function A · (x − a) in Eq.
(5.9) is called the differentialof the function f at a.The differential of a function at a point is uniquely determined; for it followsfrom (5.9) thato(x − a)f (x) − f (a)= lim A += A,limEx→aEx→ax−ax−aso that the number A is unambiguously determined due to the uniqueness of thelimit.Definition 1 The numberf (a) = limEx→af (x) − f (a)x−a(5.10)is called the derivative of the function f at a.Relation (5.10) can be rewritten in the equivalent formf (x) − f (a)= f (a) + α(x),x −awhere α(x) → 0 as x → a, x ∈ E, which in turn is equivalent tof (x) − f (a) = f (a)(x − a) + o(x − a) as x → a, x ∈ E.(5.11)Thus, differentiability of a function at a point is equivalent to the existence of itsderivative at the same point.If we compare these definitions with what was said in Sect.
5.1.1, we can conclude that the derivative characterizes the rate of change of a function at the pointunder consideration, while the differential provides the best linear approximation tothe increment of the function in a neighborhood of the same point.If a function f : E → R is differentiable at different points of the set E, then inpassing from one point to another both the quantity A and the function o(x − a) in5.1 Differentiable Functions177Eq.
(5.9) may change (a result at which we have already arrived explicitly in (5.11)).This circumstance should be noted in the very definition of a differentiable function,and we now write out this fundamental definition in full.Definition 2 A function f : E → R defined on a set E ⊂ R is differentiable at apoint x ∈ E that is a limit point of E iff (x + h) − f (x) = A(x)h + α(x; h),(5.12)where h → A(x)h is a linear function in h and α(x; h) = o(h) as h → 0, x + h ∈ E.The quantitiesΔx(h) := (x + h) − x = handΔf (x; h) := f (x + h) − f (x)are called respectively the increment of the argument and the increment of the function (corresponding to this increment in the argument).They are often denoted (not quite legitimately, to be sure) by the symbols Δxand Δf (x) representing functions of h.Thus, a function is differentiable at a point if its increment at that point, regardedas a function of the increment h in its argument, is linear up to a correction that isinfinitesimal compared to h as h → 0.Definition 3 The function h → A(x)h of Definition 2, which is linear in h, is calledthe differential of the function f : E → R at the point x ∈ E and is denoted df (x)or Df (x).Thus, df (x)(h) = A(x)h.From Definitions 2 and 3 we haveΔf (x; h) − df (x)(h) = α(x; h),and α(x; h) = o(h) as h → 0, x + h ∈ E; that is, the difference between the increment of the function due to the increment h in its argument and the value of thefunction df (x), which is linear in h, at the same h, is an infinitesimal of higherorder than the first in h.For that reason, we say that the differential is the (principal) linear part of theincrement of the function.As follows from relation (5.12) and Definition 1,A(x) = f (x) =limh→0x+h,x∈Ef (x + h) − f (x),h1785Differential Calculusand so the differential can be written asdf (x)(h) = f (x)h.(5.13)In particular, if f (x) ≡ x, we obviously have f (x) ≡ 1 anddx(h) = 1 · h = h,so that it is sometimes said that “the differential of an independent variable equalsits increment”.Taking this equality into account, we deduce from (5.13) thatdf (x)(h) = f (x) dx(h),(5.14)df (x) = f (x) dx.(5.15)that is,The equality (5.15) should be understood as the equality of two functions of h.From (5.14) we obtaindf (x)(h)= f (x),dx(h)(5.16)that is, the function dfdx(x) (the ratio of the functions df (x) and dx) is constant andequals f (x).
For this reason, following Leibniz, we frequently denote the derivativeby the symbol dfdx(x) , alongside the notation f (x) proposed by Lagrange.5In mechanics, in addition to these symbols, the symbol ϕ̇(t) (read “phi-dot of t”)is also used to denote the derivative of the function ϕ(t) with respect to time t.5.1.3 The Tangent Line; Geometric Meaning of the Derivativeand DifferentialLet f : E → R be a function defined on a set E ⊂ R and x0 a given limit point of E.We wish to choose the constant c0 so as to give the best possible description of thebehavior of the function in a neighborhood of the point x0 among constant functions.More precisely, we want the difference f (x) − c0 to be infinitesimal compared withany nonzero constant as x → x0 , x ∈ E, that isf (x) = c0 + o(1) as x → x0 , x ∈ E.(5.17)This last relation is equivalent to saying limEx→x0 f (x) = c0 .
If, in particular,the function is continuous at x0 , then limEx→x0 f (x) = f (x0 ), and naturally c0 =f (x0 ).5 J.L. Lagrange (1736–1831) – famous French mathematician and specialist in theoretical mechanics.5.1 Differentiable Functions179Now let us try to choose the function c0 + c1 (x − x0 ) so as to havef (x) = c0 + c1 (x − x0 ) + o(x − x0 )as x → x0 , x ∈ E.(5.18)This is obviously a generalization of the preceding problem, since the formula (5.17)can be rewritten asf (x) = c0 + o (x − x0 )0 as x → x0 , x ∈ E.It follows immediately from (5.18) that c0 = limEx→x0 f (x), and if the functionis continuous at this point, then c0 = f (x0 ).If c0 has been found, it then follows from (5.18) thatc1 =limEx→x0f (x) − c0.x − x0And, in general, if we were seeking a polynomial Pn (x0 ; x) = c0 + c1 (x − x0 ) +· · · + cn (x − x0 )n such thatf (x) = c0 + c1 (x − x0 ) + · · · + cn (x − x0 )n + o (x − x0 )n as x → x0 , x ∈ E,(5.19)we would find successively, with no ambiguity, thatc0 =c1 =lim f (x),Ex→x0limf (x) − c0,x − x0limf (x) − [c0 + · · · + cn−1 (x − x0 )n−1 ],(x − x0 )nEx→x0...cn =Ex→x0assuming that all these limits exist.
Otherwise condition (5.19) cannot be fulfilled,and the problem has no solution.If the function f is continuous at x0 , it follows from (5.18), as already pointedout, that c0 = f (x0 ), and we then arrive at the relationf (x) − f (x0 ) = c1 (x − x0 ) + o(x − x0 )as x → x0 , x ∈ E,which is equivalent to the condition that f (x) be differentiable at x0 .From this we findc1 =limEx→x0f (x) − f (x0 )= f (x0 ).x − x0We have thus proved the following proposition.Proposition 1 A function f : E → R that is continuous at a point x0 ∈ E that isa limit point of E ⊂ R admits a linear approximation (5.18) if and only if it isdifferentiable at the point.1805Differential CalculusFig. 5.3The functionϕ(x) = c0 + c1 (x − x0 )(5.20)with c0 = f (x0 ) and c1 = f (x0 ) is the only function of the form (5.20) that satisfies(5.18).Thus the functionϕ(x) = f (x0 ) + f (x0 )(x − x0 )(5.21)provides the best linear approximation to the function f in a neighborhood of x0 inthe sense that for any other function ϕ(x) of the form (5.20) we have f (x) − ϕ(x) =o(x − x0 ) as x → x0 , x ∈ E.The graph of the function (5.21) is the straight liney − f (x0 ) = f (x0 )(x − x0 ),(5.22)passing through the point (x0 , f (x0 )) and having slope f (x0 ).Since the line (5.22) provides the optimal linear approximation of the graph ofthe function y = f (x) in a neighborhood of the point (x0 , f (x0 )), it is natural tomake the following definition.Definition 4 If a function f : E → R is defined on a set E ⊂ R and differentiableat a point x0 ∈ E, the line defined by Eq.
(5.22) is called the tangent to the graph ofthis function at the point (x0 , f (x0 )).Figure 5.3 illustrates all the basic concepts we have so far introduced in connection with differentiability of a function at a point: the increment of the argument,the increment of the function corresponding to it, and the value of the differential.The figure shows the graph of the function, the tangent to the graph at the pointP0 = (x0 , f (x0 )), and for comparison, an arbitrary line (usually called a secant)passing through P0 and some point P = P0 of the graph of the function.The following definition extends Definition 4.5.1 Differentiable Functions181Definition 5 If the mappings f : E → R and g : E → R are continuous at a pointx0 ∈ E that is a limit point of E and f (x) − g(x) = o((x − x0 )n ) as x → x0 , x ∈ E,we say that f and g have nth order contact at x0 (more precisely, contact of orderat least n).For n = 1 we say that the mappings f and g are tangent to each other at x0 .According to Definition 5 the mapping (5.21) is tangent at x0 to a mapping f :E → R that is differentiable at that point.We can now also say that the polynomial Pn (x0 ; x) = c0 + c1 (x − x0 ) + · · · +cn (x − x0 )n of relation (5.19) has contact of order at least n with the function f .The number h = x − x0 , that is, the increment of the argument, can be regardedas a vector attached to the point x0 and defining the transition from x0 to x = x0 + h.We denote the set of all such vectors by T R(x0 ) or T Rx0 .6 Similarly, we denote byT R(y0 ) or T Ry0 the set of all displacement vectors from the point y0 along the yaxis (see Fig.
5.3). It can then be seen from the definition of the differential that themapping(5.23)df (x0 ) : T R(x0 ) → T R f (x0 ) ,defined by the differential h → f (x0 )h = df (x0 )(h) is tangent to the mappingh → f (x0 + h) − f (x0 ) = Δf (x0 ; h),(5.24)defined by the increment of a differentiable function.We remark (see Fig. 5.3) that if the mapping (5.24) is the increment of the ordinate of the graph of the function y = f (x) as the argument passes from x0 to x0 + h,then the differential (5.23) gives the increment in the ordinate of the tangent to thegraph of the function for the same increment h in the argument.5.1.4 The Role of the Coordinate SystemThe analytic definition of a tangent (Definition 4) may be the cause of some vagueuneasiness.
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