Differential calculus

“ And when such a differential equation has been found, one can get no further until it has been integrated. Generally it is much easier to state the appropriate differential equation than to solve it:–the real trouble begins then only when one wants to integrate, unless indeed the equation is seen to possess some standard form of which the integral is known, and then the triumph is easy. ”

“ Leibnitz’s Differential Calculus. In this memoir the differential dx of a variable x, considered as the abscissa of a point of a curve, is said to be an arbitrary quantity, and the differential dy of a related variable y, considered as the ordinate of the point, is defined as a quantity which has to dx the ratio of the ordinate to the subtangent, and rules are given for operating with differentials. These are the rules for forming the differential of a constant, a sum (or difference) , a product, a quotient, a power (or root) . They are equivalent to our rules (i.) ”

“ Infinitesimals. In the Differential Calculus we are usually concerned with the derivative, that is, with the ratio of the differentials dy and dx. In some applications it is also useful to consider dx as an infinitesimal (see §15) , that is, as a variable whose values remain numerically small, and which, at some stage of the investigation, approaches the limit zero. ”

“ 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. ”

“ The definition of a differential coefficient, and the rules of differentiation are quite independent of any geometrical interpretation, such as that concerning tangents to a curve, and the tangent to a curve is properly defined by means of the differential coefficient of a function, not the differential coefficient by means of the tangent.
It may happen that the limit employed in defining the differential coefficient has one value when h approaches zero through positive values, and a different value when h approaches zero through negative values. ”

“ By triumphs must be understood the successes with which the calculus has been applied to the solution of problems otherwise intractable. Often in the consideration of physical relations one is able to build up an expression for the law governing the interaction of the parts or of the forces that govern them, such expression being naturally in the form of a differential equation, that is an equation containing differential coefficients with or without other algebraic quantities. ”

“ 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. ”

“ The terms at the limits and the integrand of the transformed integral must vanish separately, if the variation of the original integral vanishes. The process of freeing the original integral from the differentials of variations results in a differential equation, or a system of differential equations, for the determination of the form of the required curve, and in special terminal conditions, which serve to determine the constants that enter into the solution of the differential equations. Lagrange's method lent itself readily to applications. ”

“ When the second differential coefficient exists, or the first is differentiable, we have the relation
The limit expressed by the right-hand member of this equation may exist in cases in which ƒ′ (x) does not exist or is not differentiable. The result that, when the limit here expressed can be shown to vanish at all points of an interval, then ƒ (x) must be a linear function of x in the interval, is important. ”

“ Such men seem to forget the great lesson that the differential calculus has taught us. It represents one of the greatest developments of modern mathematics. It does not solve problems by absolute solutions but by such approximations as make the answer which cannot be reached very clear. ”

“ An untold amount of intellectual energy has been expended on the quadrature of the circle, yet no conquest has been made by direct assault. The circle-squarers have existed in crowds ever since the period of Archimedes. After innumerable failures to solve the problem at a time, even, when investigator possessed that most powerful tool, the differential calculus, persons versed in mathematics dropped the subject, while those who still persisted were completely ignorant of its history and generally misunderstood the conditions of the problem. "Our problem," ”

“ The introduction of partial differential coefficients enables us to solve at once for a surface a problem analogous to the problem of tangents for a curve; and it also enables us to take the first step in the solution of the problem of maxima and minima for a function of several variables. ”

“ As for subtraction, it follows at once; for if the function had itself had a negative sign, its differential coefficient would also be negative; so that by differentiating
,
we should get
.
But when we come to do with Products, the thing is not quite so simple.
Suppose we were asked to differentiate the expression
,
what are we to do? The result will certainly not be ; for it is easy to see that neither , nor , would have been taken into that product. ”

“ Take as an illustration the equation
.
Is this an exact differential or not? Apply the test.
which do not agree. Therefore, it is not an exact differential, and the two functions and have not come from a common original function.
It is possible in such cases to discover, however, an integrating factor, that is to say, a factor such that if both are multiplied by this factor, the expression will become an exact differential. There is no one rule for discovering such an integrating factor; but experience will usually suggest one. In the present instance will act as such. ”

“ The artifice consists in rejecting ab initio all terms of the equation which belong to R. We do not form R at all, but only φ.Δx, or φ.dx, which is the differential dy. In the same way, in all applications of the integral calculus to geometry or mechanics we form the element of an integral in the same way as the element of area y.dx is formed. ”

“ The fundamental artifice of the calculus is the artifice of forming differentials without first forming differential coefficients. From an equation containing x and y we can deduce a new equation, containing also Δx and Δy, by substituting x+Δx for x and y+Δy for y. If there is a differential coefficient Fundamental Artifice.of y with respect to x, then Δy can be expressed in the form φ.Δx + R, where lim.Δx = 0 (R/Δx) = 0, as in § 7 above. ”

“ This theorem (see Infinitesimal Calculus) gives the formula
(2) ,
where, c1, c2, c3, . . . are the values for x = x0 of the first, second, third, . . . differential coefficients of u with regard to x. The values of c1, c2, . . . can occasionally be calculated from the analytical expressions for the differential coefficients of u; but more generally they have to be calculated from the tabulated differences. For this purpose central-difference formulae are the best. If we write ”

“ Here we have obviously to deal with a function which is such that its second differential coefficient is proportional to itself. The only function we know that has this property is the exponential function (see p. 143) , and we may be certain therefore that the solution of the equation will be of that form.
Proceeding as before, by multiplying through by , and integrating, we get , and, as
where is a constant, and . ”

“ The total increment of the right-hand member of (A) found by the Theorem of Mean Value, (46) , § 106, is
(C) and since varies while and remain constant, we get the partial derivative with respect to . Since the increments (B) and (C) must be equal,
(D) In the same manner, if we take the increments in the reverse order,
(E) and also lying between zero and unity.
The left-hand members of (D) and (E) being identical, we have
(F) Taking the limit of both sides as and approach zero as limits, we have
(G) since these functions are assumed continuous. ”

“ It is a fundamental theorem that this equation cannot hold for all admissible functions u, unless the differential equation Li.. 'f§ f. Q§ =0
dx éy' 6y
is satisfied at every point of the curve along which the integral is taken. This is the principal equation for this problem. The curves that are determined by it are called the stationary curves, or the extremals, of the integral. We learn that curves the integral cannot be an extremum unless it is taken along a stationary curve. ”

“ It can be shown, from the fact that η1 is a quadratic polynomial in y1, that no differential equation of the second order can allow more than 8 really independent infinitesimal transformations, and that every homogeneous linear differential equation of the second order allows just 8, being in fact reducible to d2y/dx2 = 0. Since every group of more than two parameters has subgroups of two parameters, a differential equation of the second order allowing a group of more than two parameters can, as a rule, be solved by quadratures. ”

“ It appears natural enough now to inquire whether, under proper conditions for the forms of the rational functions a1, ... an, it may be possible to ensure that in each of the series φ1, ψ1, [χ] 1, ... the number of negative powers shall be finite. Herein Regular equations. lies, in fact, the limitation which experience has shown to be justified by the completeness of the results obtained. Assuming n integrals in which in each of φ1, ψ1, χ1 ... the number of negative powers is finite, there is a definite homogeneous linear differential equation having these integrals ”

“ When we pass to ordinary differential equations of the second order, that is, those expressing a relation between x, y, dy/dx and d2y/dx2, the number of types for which the solution can be found by a known procedure is very considerably reduced. Consider the general linear equation
where P, Q, R are functions of x only. There is no method always effective ”

“ Putting each of the fractions in the characteristic equations equal to dt, the equations enable us, starting from an arbitrary element x′0, y′0, z′0, p′0, q′0, about which all the quantities F′, dF′/dp′, &c., occurring in the denominators, are developable, to define, from the differential equation F′=0 alone, a connectivity of ∞1 elements, which we call a characteristic chain ”