A Treatise on the Differential Calculus; And the Elements of the Integral Calculus with Numerous Examples

by Isaac Todhunter

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1855 ...p(a + A, b+k)-?a, b) = jPh2+3Qh2k + 3Shk2+ TF) + R2, du d2u where P, Q, S, T, stand for the values of, &c, 'TM ' dx3' da?dy when x = a and y = b, and x being made = a, and y = b, after the differentiations. Hence, that if (a, b) may be a maximum or minimum, it is necessary that P, Q, S, T, should all vanish. Also, i?2 must be of invariable sign; but the conditions to ensure this are too complicated to find investigation here. 230. If AC-B=0, then?(a + h, b + k)-pa, b) is always of the same sign as A when h and k are taken small enough, except when =--. In this case, in order to ensure a maximum or minimum, it is necessary that Ph2 + 3 Qhk + 3 SM? + Tk3 It s should vanish when j =---?; and also that i?2 should then have the same sign as A. If these conditions be satisfied, we have a maximum if A be negative, and a minimum if A be positive. 231. The following is another method of investigating the conditions that a function of two independent variables may admit of a maximum or minimum. Let u = cp (x, y), where x and y are required the maxima and minima values of u. If y, instead of being independent of x, were equal to some function of x, say (x), then u would be a function of one variable x. We should then have du fdu fdu In order that u may be a maximum or minimum, we must have, by Art. (211), = 0 dx' therefore () + () If () = 0. Hence, since y is really independent of x, this equation must hold whatever be the function ty' (x); therefore = 0, (Uo. In order that it may be a maximum, the values of x and y derived from the last equations must make T-j negative, whatever / ( e) may be; hence, denoting by A, B, C, the, ., fd2u / du, fd2u. values which J yyj an -J respectively assume for the values of x and y under co...

50 pages, Paperback

Published January 1, 2012

About the author

English mathematician known for his writings on the history of mathematics.