A treatise on infinitesimal calculus Volume 2; containing differential and integral calculus, calculus of variations, applications to algebra and geometry, and analytical mechanics

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. 1854 Excerpt: ...a surface be z = /(a?, y), dz = £ dx + zt dy; (84) and as z and zt are variables, from (83) and (84) we infer that dxdydz' (85) that is, the variations and the differentials of the coordinates of the point under inquiry are proportional to each other; the new position therefore of the displaced point is on the original surface, and therefore the displacement has been wholly in the tangent plane. 205. It is good to consider a difficult subject, such as that under discussion, from another point of view. We have conceived the quantity involving the unknown function to be resolved into its elements, and the definite integral of these elements to be the finite quantity which is the subject of inquiry; and the limits have been taken to be values whose symbols have subscripts 1 and 0. Now imagine the definite integral to re Price, Vol. n. M m present some property of a plane curve, and between the values xx and x0; this restriction is convenient to fix our thoughts; and let the quantity xx--x0 be resolved into ra elements, and fi £2 £3£«-! be the values of x corresponding to the points of division, and the corresponding values of y be yx y2.. yn-i ' then, as the definite integral is the sum of a series of quantities, of each of which the element-function is a type; so if we replace the definite integral by its equivalent series given in equation (12), Art. 6, it will be a function of #0 fi £2 fn-i #TM that is of n +1 variables; and when the elements are infinitesimal, of an infinite number of variables: this then is a distinguishing mark of the Calculus of Variations; its immediate subjects of inquiry are functions of an infinite number of variables generally independent of each other; but as these functions consist of a series of ter...