Advanced calculus; . y,A = lim^ = CfdA = jfrdrd^, (12) where the double integrals are extended over the area desired. The elements of volume which are required for triple integration(§§ 133, 134) over a volume in space may readily be written down forthe three cases of rectangular, polar, and cylindrical coordinates. In thefirst case space is supposed to be divided up by planes x = a, y = b,z = c perpendicular to the axes and spaced at infinitesimal intervals; inthe second case the division is made by the spheres r = a concentricwith the pole, the planes cf> = b through the polar axis, an
Advanced calculus; . y,A = lim^ = CfdA = jfrdrd^, (12) where the double integrals are extended over the area desired. The elements of volume which are required for triple integration(§§ 133, 134) over a volume in space may readily be written down forthe three cases of rectangular, polar, and cylindrical coordinates. In thefirst case space is supposed to be divided up by planes x = a, y = b,z = c perpendicular to the axes and spaced at infinitesimal intervals; inthe second case the division is made by the spheres r = a concentricwith the pole, the planes cf> = b through the polar axis, and the cones6 — c of revolution about the polar axis ; in the third case by the cylin-ders r = a, the planes = b, and the planes z = c. The infinitesimal TAYLORS FORMULA; ALLIED TOPICS 81 volumes into which space is divided then differ from dv = dxdydzj dv = r3 sin ddrdd$, dv = rdrdfydz (13) respectively by infinitesimals of higher order, and fiidxdydz, J I {>*sin Odrd^dO, j j irdrdfr Iz (13). are the formulas for the volumes. 41. The direction of a line in space is represented by the three angleswhich the line makes with the positive directions of the axes or by thecosines of those angles, the direction cosines of the line. From the defi-nition and figure it appears that dx _ du dz l = cosa = —, m = cosB=-f-. ra = cosy = — (11) ds ds dx v 7 are the direction cosines of the tangent to the arc at the point; of thetangent and not of the chord for the reasonthat the increments are replaced by the differ-entials. Hence it is seen that for the direc-tion cosines of the tangent the proportion I: m : n = dx : dy : dz (H) holds. The equations of a space curve are in terms of a variable parameter t* At the point (xQ} y0, zQ) wheret = t0 the equations of the tangent lines would then be x — x0 _ y — y0 __ z — zQ x — x0 __ y — y0 ___ z — z0 (dx\ (dy\ (dz\ f(t0) g%) K%) As the cosine of the angle 9 between the two directions given by thedirection cosines I, ?n, n and V
Size: 1758px × 1422px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, bookpublisherbostonnewyorketcgi
