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On Differentials
Published online by Cambridge University Press: 03 November 2016
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The article on “Differentials”, in the May Gazette—based on a discussion at the last Annual Meeting of the Association, and referring back to articles of July 1931 and February 1932 —has so clarified certain issues for the writer as to confirm a long-standing conviction that the standard definition of “differentials” is quite unsound, and that the view set out below (which he had developed in his own teaching ) is much truer to all the facts of the situation.
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- Copyright © Mathematical Association 1935
References
page no 79 note * Gazette, No. 214, pp. 401–3; 217, pp. 5–10; 228, pp. 68–79.
page no 79 note † See Note 686 on “The Notation of the Calculus”, Gazette, No. 166, p. 387 (Oct. 1923)—where some of the points dealt with below are summarized.
page no 79 note ‡ See, e.g. Professor Wilton's point, at the foot of p. 5, No. 217.
page no 79 note § No. 228, p. 77 (J. T. Combridge); also, p. 78.
page no 79 note ‖ No. 228, pp. 68–70.
page no 79 note ¶ No. 214, p. 401.
page no 79 note ** Compare, the standard notation for the Natural Numbers—as another perfect notation, which, however, took centuries to evolve; and, as an example of the opposite kind, the very imperfect notation for “roots” and “logarithms” (and terminology of the mathematical number-system).
page no 80 note * This is exactly the opposite view to that which is expressed at the top of p. 402 of No. 214: the difference being characteristic of the two different ways of approaching the subject of “differentials”. (See also Professor Wilton, No. 217, p. 5.)
page no 80 note † An “infinitesimal” is a variable used in a “limit” proposition, which is characterized by the fact that that variable → 0, “in the limit”
page no 80 note ‡ It is instructive to draw three diagrams: (1) for two given points P 1, P 2; (2) for a given point, P 1, and the variable point, P, of the graph of y; (3) for two variable points P, Q of that graph.
page no 81 note * It will be noted wherein, on the one hand, I agree—and, on the other hand, I do not agree—with Professor Wilton (No. 217, p. 7) and others, on these points.
page no 81 note † The usage has a remote resemblance to that of such “names” as “cm/sec” for the “ derived unit ” of speed: that unit being not itself actually a quotient, but based on a relationship (of proportion) which is expressible by means of an actual quotient.
page no 81 note ‡ Certain conditions (commonly satisfied) are necessary, in order that this limit-expression should give the second derived function, y".
page no 82 note * For example, dx 2 and d 2 y in § 3-and all the detail of what follows. Compare, the operator “D”—which, though not itself a quantity, is subject to operational processes such as are primarily characteristic of quantities (and because of its relationship—of a different kind—to quantities). It must have been something like this idea of “differentials” that inspired Lewis Carroll's whimsical notion of the Cheshire Cat's disembodied smile!
page no 82 note † y=u. v gives Δy=u. Δv+v. Δu + Δu. Δv, and, therefore, dy=u. dv + v. du; etc.
page no 83 note * The writer gave a presentation of the detail here involved, in an article on “The Integral Calculus Theorem” in No. 61, pp. 5–7 (January 1907).
page no 83 note † For the immediate purpose of this discussion, there is no significant loss of generality in assuming that x varies monotonically from a to b as u varies monotonically from a to β: the general case being reducible to that case.
page no 83 note ‡ The references to this application, in the Gazette articles, are rather slight, in view of its importance. In No. 214, p. 403, the approach is from the other point of view-to which reference has been made above in a footnote to § 2.
page no 84 note * It is to be noted that in this part of the Infinitesimal Calculus the emphasis is on “ expressions” rather than upon “functions”. Partial derivative, f x' (x, y, …), has a perfectly definite meaning—in terms of a given expression f(x, y, …) ; partial derived function, ∂u/∂x, may have many different meanings—according to the expression used for the function u. (See No. 166, p. 388.)
page no 84 note † It is important to note that f(x, y, …)=constant gives the same result. See § 8, below.
page no 85 note * An elementary argument, in graphical terms, for the “existence” of the primitive in this form is obtainable from dy=f(x, y) .dx, by constructing a polygonal line from the approximate relation Δy=f(x, y). Δx—using an arbitrary initial point and an arbitrary Δx. The “limit” of this line, when Δx→ 0, is a curve which “satisfies” the differential equation; and there is one such curve through each point of the xy-plane.
page no 85 note † Such an article as that of No. 223, pp. 105–11 (Underwood), may be regarded as typical of the justifiable manipulation of the “differentials” in such cases (or, see any book on Differential Equations).
page no 86 note * Meaning that f(x, y, λ + Δλ) gives, for such values of x, y, λ, an “ infinitesimal of higher order ” than Δλ.
page no 86 note † “ Consecutive ”, in terms of the λ-increment, Δλ—the tending to 0 of which gives the “ ultimate intersection ”.