Differential

“ To Lagrange, perhaps more than to any other, the theory of differential equations is indebted for its position as a science, rather than a collection of ingenious artifices for the solution of particular problems. ”

“ Analysis, which treats plane curves as equations in two variables, gives the direction of the tangent in terms of the differential coefficient, and declares accordingly that every continuous function has a differential coefficient, that is, may be differentiated, at every point. ”

“ 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. ”

“ Assuming the condition in its general form, take in the given differential equation a plane section of the surface φ = C parallel to the plane z, viz. put z constant, and consider the resulting differential equation in the two variables x, y, namely Xdx + Ydy = 0; let ψ (x, y, z) = constant, be its integral, the constant z entering, as a rule, in ψ because it enters in X and Y. Now differentiate the relation ψ (x, y, z) = ƒ (z) , where ƒ is a function to be determined, so obtaining
there exists a function σ of x, y, z such that
because ψ = constant, is the integral of Xdx + Ydy = 0 ”

“ The new ideal entities are called Derived Functions, or Differential Coefficients, and for a long time it was assumed as almost an axiom that every function continuous within given intervals must have, within those intervals, a derived function, or differential coefficient. ”

“ It will be observed that since the 's are linear functions of the 's, and the 's linear functions of the 's, the 's will be linear functions of the 's, so that a differential coefficient of the form will be independent of the 's, and function of the 's alone. ”

“ The integral, therefore, is the appropriate mathematical expression for a theory of action between particles at a distance, whereas the differential equation is the appropriate expression for a theory of action exerted between contiguous parts of a medium.
We have seen that the result of the integration satisfies the differential equation. We have now to shew that it is the only solution of that equation fulfilling certain conditions. ”

“ The researches above referred to are closely connected with the theory of functions and of groups. Endeavours have thus been made to determine the nature of the function defined by a differential equation from the differential equation itself, and not from any analytical expression of the function, obtained first by solving the differential equation. Instead of studying the properties of the integrals of a differential equation for all the values of the variable, investigators at first contented themselves with the study of the properties in the vicinity of a given point. ”

“ 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. ”

“ It may be said that the conceptions of differential quotient and integral, which in their origin certainly go back to Archimedes, were introduced into science by the investigations of Kepler, Descartes, Cavalieri, Fermat and Wallis.... The capital discovery that differentiation and integration are inverse operations belongs to Newton and Leibnitz.—Lie, Sophus. ”

“ Or we can say, and it is just the same thing in other words:
(speed) equals the derivative of (distance) with respect to (time)
Now we can tell what a differential equation is. It is simply an equation in which a derivative occurs, such as the last example. Perhaps the commonest kind of equation in physical problems is the differential equation.
SOLVING PHYSICAL PROBLEMS
Now we were able to change the equation about floor area into other forms, if we wanted to find length or width instead of floor area. When we did this, we ran into the inverse or opposite of multiplication: division. ”

“ 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 ”

“ Thus by our general theorem, if the differential equation allow a group of two parameters (and such a group is always integrable) , it can be solved by quadratures, our explanation sufficing, however, only provided the form Πƒ and the two infinitesimal transformations are not linearly connected. ”

“ And in all these relations we may interchange in any way if we interchange in the same way It follows that, within the limits of stability, when we choose for any one of the differential coefficients the quantities following the sign in the numerators of the others together with as those which are to remain constant in differentiation, the value of the differential coefficient as thus determined will be at least as small as when one or more of the constants in differentiation are taken from the denominators, one being still taken from each fraction, and v as before being constant. ”

“ 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. ”

“ The fundamental division of the Integral Calculus is founded on the same principle as that of the Differential Calculus, in distinguishing the integration of explicit differential formulas, and the integration of implicit differentials or of differential equations. The separation of these two cases is even much more profound in relation to integration than to differentiation. In the differential calculus, in fact, this distinction rests, as we have seen, only on the extreme imperfection of ordinary analysis. ”

“ 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. ”

“ When in a given set of differential equations the number of equations is greater than the number of dependent variables, the equations cannot be expected to have common solutions unless certain conditions of compatibility, obtainable by equating different forms of the same differential coefficients deducible from the equations, are satisfied. ”

“ This transformation can always be effected. On the other hand, every differential equation of any order in relation to a single function might evidently always be reduced to the first order, by introducing a suitable number of auxiliary differential equations, containing at the same time the different anterior derivatives regarded as new functions to be determined. ”

“ Now here again, of all the paths that lead from one position to another, there is one for which the average action is a minimum, and there is only one. The principle of least action suffices, then, to determine the differential equations which define the variations of the parameters q. ”

“ The equations which express the condition of equilibrium, as also its statement in words, are to be interpreted in accordance with the general usage in respect to differential equations, that is, infinitesimals of higher orders than the first relatively to those which express the amount of change of the system are to be neglected. But to distinguish the different kinds of equilibrium in respect to stability, we must have regard to the absolute values of the variations. ”

“ But in justice to those who maintain the precedence of structure, it must be added that function, which, as we hold, is anterior to structure, nevertheless, regarded as an activity modified and different from what it was, assumes its differential, distinguishing characters only in proportion as the adaptation becomes perfect, and as equilibrium is established between that portion of internal reaction which it represents and the external action which it withstands. ”

“ In order to obtain invariable physical laws, we have to proceed to differential equations, showing the direction of change at each moment, not the integral change after a finite interval, however short. But for the purposes of daily life many sequences are to all in tents and purposes invariable. With the behaviour of human beings, however, this is by no means the case. ”

“ 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. ”

“ When may we say that we have a complete mechanical explanation of the phenomenon? It will be, on the one hand, when we know the differential equations which are satisfied by the co-ordinates of these hypothetical molecules m, equations which must, in addition, conform to the laws of dynamics; and, on the other hand, when we know the relations which define the co-ordinates of the molecules m as functions of the parameters q, attainable by experiment. ”

“ Since the formulas deduced in the Differential Calculus generally involve derivatives of y with respect to x, such formulas as (A) and (B) are especially useful when the parametric equations of a curve are given. ”

“ Solution. Let radius of base, = altitude; then = volume,
Substitute in (51) ,
But , , = 5, = - 10.
∴ cu. ft. per sec., increase. Ans.
126. Total differentials. Multiplying (51) and (52) through by dt, we get
(55) (56) and so on. [6] Equations (55) and (56) define the quantity , which is called a total differential of or a complete differential, and
are called partial differentials. ”

“ A rational ordinary differential equation, with x as independent and y as dependent variable, is then one which equates to zero a rational function of y, the order k of the differential equation being that of the highest differential coefficient y (k) which enters; only such equations are here discussed. Such an equation P=0 is called irreducible when, firstly, being arranged as an integral polynomial in y (k) , this polynomial Irreducibility of a rational equation.is not the product of other polynomials in y (k) also of rational form ”

“ 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. ”

“ They resemble each other more than they do any one irreversible process. This is evident from an examination of the differential equations which control them; the differential with respect to time is always of an even order, because the essential sign of time can be reversed. ”

“ Examples of analytical expressions which represent continuous functions that are not differentiable have been given by Riemann, Weierstrass, Darboux and Dini (see § 24) . The most important theorem in regard to differentiable functions is the “theorem of intermediate value.” (See Infinitesimal Calculus.)
13. Analytic Function.—If ƒ (x) and its first n differential coefficients, denoted by ƒ′ (x) , ƒ′′ (x) , ... ƒ (n) (x) , are continuous in the interval between a and a + h, then
where Rn may have various forms, some of which are given in the article Infinitesimal Calculus. ”

“ In such equations even a differential element of volume is to be supposed to be sufficiently large to average out the diverse agitations of the molecules, and to register only the general unbalanced residuum which to ordinary observation is the motion of the body. ”

“ A very much closer approximation may be obtained, if we regard at any given point of the medium and for light of a given period, as linear functions of and the nine differential coefficients We shall write and diff. ”

“ But what is very little for one may be very big for another, and what seems very complex to one may seem simple to another. In part I have already answered by saying precisely in what cases differential equations become too simple for the laws of chance to remain applicable. ”

“ Thus by differentiating and casting off as evanescent all differentials of a lower order than that which we are working with, we may arrive at forms of which we know the integrals, and by integrating get back to the results in ordinary numbers, which we were in search of but could not attain directly. ”