A History of the Progress of Calculus of Variations During the Nineteenth Century

A History of the Progress of Calculus of Variations During the Nineteenth Century

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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. 1861 edition. Excerpt: ...determining y the equation dy dx Lastly, suppose that we have d'V_n dV a dV ( dV y A dy'" ' dy" 2dydy" dy'dy")-dy dydy')+dydy")-' Spitzer shews in the same manner as in Art. 273, that the form of V must in this case be yf (x) + + (x, y)' + x (x, y, y'); thus in SI Vdx the unintegrated part is lf (x) hydx, and this will not vanish unless f (x) vanishes. Then I Vdx is exactly integrable, and its maximum or minimum can be sought by ordinary methods. 277. We may also obtain the results of the preceding article dV by another method. Suppose-3-775 = 0, then the left-hand member of equation (4) of Art. 275 takes the form (2E+ H-B) w" + (2E + D"-B') w' + (A-F' + E") to, where A, B, D, E, .Phave the same meaning as in Art. 276. The above expression may be written thus 2E+ D'-B) to' +A-F' + E") w, so that we have to determine the sign of f(2E+ H-B) w' + A-F' + E") w wdx. Suppose u such a quantity that (2E+ D'-B) u' + A-F' + E')u = 0. Then the expression which we have to examine may be written j 12E+ H-B)w'-(2E+ D'-B) u' 1 wdx. Integrate by parts; then the terms remaining under the integral sign will be f l(B-2E-U) w'-(B-2E-D')-' Ql dx, that is, fB-2E-U) (to'- wY dx. This agrees with the result at the bottom of page 303. 278. In his sixteenth section Spitzer examines some exceptional cases which occur in finding the maximum or minimum of Vdx, when V involves x, y, y', y," and y'." He does not here prove that V must have specific forms in certain cases, but he assumes specific forms for V and shews that certain exceptional cases do thence arise. The following four forms for V are examined. 1. f(x, y, y', y") + y'"ir(x, y, y;y"). 2. fx, y, y')+ y"+(x, y, y') + x x, y, y', ...show more

Product details

  • Paperback | 126 pages
  • 189 x 246 x 7mm | 240g
  • Rarebooksclub.com
  • Miami Fl, United States
  • English
  • black & white illustrations
  • 1236618998
  • 9781236618993