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. 1864 ... 38hP + Tk2 + B2, where P, Q, 8, T, stand for the values of,, & when x = a and y = b, and;.. x being made = a, and y = b, after the differentiations. Hence, that f (a, b) may be a maximum or minimum, it is necessary that P, Q, 8, T, should all vanish. Also, B2 must be of invariable sign; but the conditions to ensure this are too complicated to find investigation here. 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 = f (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 (a;), then u would be a function of one variable x. We should then have In order that u may be a maximum or minimum, we must have, by Art. 211, du Hence, since 3/ is really independent of x, this equation must hold whatever be the function yjr' (x); du In order that u may be a maximum, the values of x and y du derived from the last equations must make negative, whatever (x) may be; hence, denoting by A, B, G, the values which (J, an (p)' resPectiTety assume for the values of x and y under consideration, we require that A + ZBtf (x) + CW (x)2 should be always negative, whatever i/r' (x) may be. Hence as in Art. 228, A must be negative, and generally AC--B must be positive. Similarly, that u may be a minimum we must have A positive, and generally AC--B positive. The preceding method may be rendered more symmetrical by supposing both x and y functions of a third variable t. Putting for shortness Dx for, and By for--, we have Also for values of x and y found from these equations, (dlu must preserve an invariable sign, whatever be the signs and values of Dx and Dy. From thi...

56 pages, Paperback

Published January 1, 2012

About the author

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