Variable (mathematics)

“ The first lemma of the first book has been made the foundation of the method of limits:—
"Quantities and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the other than by any given difference, become ultimately equal."
In this, as well as in the lemmas following this, there are obscurities and difficulties. Newton appears to teach that a variable quantity and its limit will ultimately coincide and be equal. ”

“ The simplest problem in the calculus of variations proper is known to consist in finding a function of a variable such that the definite integral assumes a minimum value as compared with the values it takes when is replaced by other functions of with the same initial and final values. ”

“ In either case the problem is to determine, when possible, a single analytical expression which shall have the same value as the function at all points in the domain of the argument. For the representation of most functions for which the problem can be solved recourse must be had to limiting processes. Thus we may utilize infinite series, or infinite products, or definite integrals; or again we may represent a function of one variable as the limit of an expression containing two variables in a domain in which one variable remains constant and another varies. ”

“ We see that an explicit function in , , , etc., is simply something the value of which changes when , , , etc., are changing, either one at the time or several together. Because of this, the value of the explicit function is called the dependent variable, as it depends on the value of the other variable quantities in the function; these other variables are called the independent variables because their value is not determined from the value assumed by the function. For example, if , and are the independent variables, and is the dependent variable. ”

“ This question is now resolved in the most complete and the most simple manner, as are all those of which the differential calculus is composed. It is easy to conceive the general importance which it must have in any of the applications of the transcendental analysis, the fundamental resources of which it may be considered as augmenting, by permitting us to choose (in order to form the differential equations, in the first place, with more ease) that system of independent variables which may appear to be the most advantageous, although it is not to be finally retained. ”

“ From the Theory of Limits it is clear that if the derivative of a function exists for a certain value of the independent variable, the function itself must be continuous for that value of the variable. ”

“ We may say that the most marvelous of his creations rise from the general field of differential equations. We might cite further his researches in analysis situs, the realm of the invariants of a battered continuity. His double residues and studies in functions of many real variables are creations from which will spring a noble progeny. Even the lectures in which he presented the results of others scintillate with original thoughts.
To generalize in mathematics and science it is not enough simply to get together facts or ideas and to put them into new combinations. ”

“ The mathematical questions which have given birth to the Calculus of Variations consist generally in the investigation of the maxima and minima of certain indeterminate integral formulas, which express the analytical law of such or such a phenomenon of geometry or mechanics, considered independently of any particular subject. ”

“ In seeking to determine with precision in what determinations of values properly consist, we easily recognize that they are nothing else but veritable transformations of the functions to be valued; transformations which, in spite of their special end, are none the less essentially of the same nature as all those taught by analysis. In this point of view, the calculus of values might be simply conceived as an appendix, and a particular application of the calculus of functions, so that arithmetic would disappear, so to say, as a distinct section in the whole body of abstract mathematics. ”

“ In the most simple questions, and when we have really to resolve only a single equation with a single unknown quantity, this numerical resolution is none the less a very imperfect method, even when it is strictly sufficient. ”

“ In the differential and integral calculus it is often advantageous to regard as multiple quantities various sets of variables, especially the independent variables, or those which may be taken as such. ”

“ Use of the Differential Calculus as preparatory to that of the Integral. In forming the differential equations of any phenomenon whatever, it is very seldom that we limit ourselves to introduce differentially only those magnitudes whose relations are sought. To impose that condition would be to uselessly diminish the resources presented by the transcendental analysis for the expression of the mathematical laws of phenomena. ”

“ In this point of view the numerical solution of questions which we cannot resolve algebraically, and even the calculation of "Definite Integrals," whose general integrals we do not know, really make a part, in spite of all appearances, of the domain of arithmetic, in which we must necessarily comprise all that which has for its object the determination of the values of functions. ”

“ It is composed of what is called the Calculus, [2] taking this word in its greatest extent, which reaches from the most simple numerical operations to the most sublime combinations of transcendental analysis. The Calculus has the solution of all questions [Pg 34] relating to numbers for its peculiar object. Its starting point is, constantly and necessarily, the knowledge of the precise relations, i.e., of the equations, between the different magnitudes which are simultaneously considered; that which is, on the contrary, the stopping point of concrete mathematics. ”

“ If, indeed, the differential calculus was not rigorously complete, that is, if we did not know how to differentiate at will any function whatever, the differentiation to the second or higher order of each determinate function might engender special difficulties. ”

“ They are those in which the differential equations are found to be immediately ready for integration, because they contain, at their first formation, only the infinitesimals which relate to the functions sought, or to the really independent variables, without its being necessary to introduce, differentially, other functions as intermediaries. If in these new cases we introduce these last functions, since, by hypothesis, they will enter directly and not by their differentials, ordinary algebra will suffice to eliminate them, and to bring the question to depend on only the integral calculus. ”

“ The Differential Calculus, however, shows that the mind is feeling after even a more definite mode of contrast than that indicated above. We cannot truly know the Law which governs an organic group of phenomena, unless we allow ourselves to conceive the idea of a Singular Solution of that Law, i.e., an individual which is constant in some relations as to which the other individuals are variable, and variable in some relations as to which the others are constant. ”

“ We usually form much too vague an idea of what an equation is, when we give that name to every kind of relation of equality between any two functions of the magnitudes which we are considering. For, though every equation is evidently a relation of equality, it is far from being true that, reciprocally, every relation of equality is a veritable equation, of the kind of those to which, by their nature, the methods of analysis are applicable. ”

“ The complete solution of every mathematical question divides itself necessarily into two parts, of natures essentially distinct, and with relations invariably determinate. We have seen that every mathematical inquiry has for its object to determine unknown magnitudes, according to the relations between them and known magnitudes. Now for this object, it is evidently necessary, in the first place, to ascertain with precision the relations which exist between the quantities which we are considering. ”

“ A variable is said to vary continuously through an interval , when starts with the value and increases until it takes on the value in such a manner as to assume the value of every number between and in the order of their magnitudes. ”

“ The Calculus of Values, or Arithmetic, would appear, at first view, to present a field as vast as that of algebra, since it would seem to admit as many distinct questions as we can conceive different algebraic formulas whose values are to be determined. ”

“ Thus, for example, in the case of rectifications, it will be first necessary to calculate dy, or dy and dz, by differentiating the equation or the equations of each curve proposed; after eliminating these expressions, the general differential formula above enunciated will then contain only ds and dx; having arrived at this point, the elimination of the infinitesimals can be completed only by the integral calculus. ”

“ In fact, even if we could integrate every equation of the first order relating to a single function, that would not be sufficient for obtaining the final integral of an equation of any order whatever, inasmuch as every differential equation is not reducible to that of an immediately inferior order. ”

“ A great many practical problems occur where we have to deal with functions of such a nature that they have a greatest (maximum) value or a least (minimum) value, [1] and it is very important to know what particular value of the variable gives such a value of the function. For instance, suppose that it is required to find the dimensions of the rectangle of greatest area that can be inscribed in a circle of radius 5 inches. ”

“ The general notions of functionality, limits and continuity are explained in the article Function. Illustrations of the more immediate ways in which these notions present themselves in the development of the differential and integral calculus will be useful in what follows.
33. Let y be given as a function of x, or, more generally, let x and y be given as functions of a variable t. The first of these cases is included in the second by putting x = t. If certain conditions are satisfied the aggregate of the points determined by the functional relations form a curve. ”

“ What we know in Algebra. In the present condition of algebra, the complete resolution of the equations of the first four degrees, of any binomial equations, of certain particular equations of the higher degrees, and of a very small number of exponential, logarithmic, or circular equations, constitute the fundamental methods which are presented by the calculus of direct functions for the solution of mathematical problems. ”

“ This transformation is called changing the independent variable from x to y.
Illustrative Example 1. Change the independent variable from x to y in the equation
Solution. Substituting from (35) , (36) , (37) ,
Reducing, we get
a much simpler equation. ”

“ Such expansions are just as important in analysis, geometry and mechanics as expansions of functions of one variable. Among the problems which are solved by means of such expansions are the problem of maxima and minima for functions of more than one variable (see Maxima and Minima) . ”

“ It is even certain that the last special improvements in mathematical analysis have contributed to extend our resources in that respect, by introducing within the domain of the calculus certain definite integrals, which in some respects supply the place of new simple functions, although they are far from fulfilling all the necessary conditions, which has prevented me from inserting them in the table of true analytical elements. ”

“ There are nevertheless, in mathematics, some examples of so-called Induction, in which the conclusion does bear the appearance of a generalization grounded on some of the particular cases included in it. A mathematician, when he has calculated a sufficient number of the terms of an algebraical or arithmetical series to have ascertained what is called the law of the series, does not hesitate to fill up any number of the succeeding terms without repeating the calculations. ”

“ From the considerations indicated in this chapter, we see that, while the differential calculus constitutes by its nature a limited and perfect system, to which nothing essential remains to be added, the integral calculus, or the simple system of integration, presents necessarily an inexhaustible field for the activity of the human mind, independently of the indefinite applications of which the transcendental analysis is evidently susceptible. ”