A History of the Progress of the Calculus of Variations During the Nineteenth Century |
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Page 3
... means of the relation between x and y given by df df = 0 dy dp ...... ( 1 ) , and the fact that the integration is taken between fixed limits , the integral denoted by K df бу + dy dp Sp } dx vanishes . We must then examine the integral ...
... means of the relation between x and y given by df df = 0 dy dp ...... ( 1 ) , and the fact that the integration is taken between fixed limits , the integral denoted by K df бу + dy dp Sp } dx vanishes . We must then examine the integral ...
Page 11
... means of its peculiar notation , and on the other hand to explain the connexion of the subject with the ordinary principles of the Differential Calculus . He adds that those readers who wish to confine themselves to the Calculus of ...
... means of its peculiar notation , and on the other hand to explain the connexion of the subject with the ordinary principles of the Differential Calculus . He adds that those readers who wish to confine themselves to the Calculus of ...
Page 16
... means by his assertion ; in the problem which he has discussed in the pages immediately preceding his assertion he has confined himself to the case in which the limits of the integrations are all constants , and his results agree with ...
... means by his assertion ; in the problem which he has discussed in the pages immediately preceding his assertion he has confined himself to the case in which the limits of the integrations are all constants , and his results agree with ...
Page 19
... means a change of x into X and p into and a corresponding subtraction . This however is too restricted a meaning of the operation denoted by 8 , for it is necessary to have the power of supposing that the curve ye in the figure is not ...
... means a change of x into X and p into and a corresponding subtraction . This however is too restricted a meaning of the operation denoted by 8 , for it is necessary to have the power of supposing that the curve ye in the figure is not ...
Page 23
... mean the indefinite integral V'de without any con- stant added ; but v may stand for something different , as for ex- V'dx . In this case in order to find what arises from ample for ( * If ( dxdV ' - dV'Sx ) , we must take the integral ...
... mean the indefinite integral V'de without any con- stant added ; but v may stand for something different , as for ex- V'dx . In this case in order to find what arises from ample for ( * If ( dxdV ' - dV'Sx ) , we must take the integral ...
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Common terms and phrases
1+p³ arbitrary constants axis Calculus of Variations catenary chapter co-ordinates condition considered curvature curve Delaunay denote determined Differential Calculus differential equation double integral ds ds dv dv dv dx dv dy dx dx dx dy dz dx dz dx² dy dx dy dy dy dz dx dz dy dz dz Euler exact differential coefficient example expression formula geodesic gives indefinitely small independent variable Integral Calculus integral sign investigation involves Jacobi's Jacobi's theorem Lacroix Lagrange maxima and minima maximum or minimum memoir method minimum area minimum value notation obtain occupies pages occur Ostrogradsky partial differential equation plane Poisson problem quantities remarks respect result Sarrus second order shew solution Stegmann Strauch suppose surface theorem tion treatise triple integral vanish volume Y₁ zero
Popular passages
Page 140 - It is true that one of the fundamental principles of this method consists in removing as much as possible the differential coefficients of the variations which occur under the integral sign; but the calculus of variations only indicates this operation and refers the execution of it to the Integral Calculus. CHAPTER VI. DELAUNAY. 133. THE Academy of Sciences at Paris proposed the following as the subject of competition for their great mathematical prize in 1842 ; To find the limiting equations which...
Page 229 - Jacobi in 1837, and the memoir which Jacobi then published has given rise to an extensive series of commentaries and developments. Before however we proceed to Jacobi's investigations, we will give an analysis of Legendre's memoir and of some others connected with it. • 197. Legendre's memoir is entitled Memoire sur la maniere de distinguer les maxima des minima dans le Calcul des Variations.
Page 142 - ... can be calculated. It will be necessary to effect successive integrations, and to take each integral between appropriate limits, and these can be determined in the following manner. The order of the successive integrations being arbitrary, we can suppose that we integrate first with respect to z, then with respect to y, and then with respect to x. In the first integration y and x are regarded as constants, and the integration with respect to z extends over all the values of z which render f(x,...
Page 37 - ... of Variations involving the variation of a certain double integral, the limits of the integration being also variable; it is the earliest example of the solution of such a problem. Gauss himself says on page 67, " Sed quum calculus variationum integralium duplicium pro casu ubi etiam limites tanquam variabiles spectari debent, hactenus parum excultus sit, hanc disquisitionem subtilem paullo profundius petere oportet.
Page 345 - Bjorling discusses another particular example before considering the general equation, namely, among all surfaces which can be formed by the motion of a straight line which always remains parallel to a fixed plane, to determine that of minimum area.
Page 286 - Let there be any linear differential expression which involves x, y, and the differential coefficients of y with respect to x...
Page 409 - ... that y is then a minimum and so is F. These results do not agree with those in the book. The case in which y = k seems there overlooked. If y = k we have h = k = a. And it may be seen that the relation on the 14th line of page 165 of the book may be satisfied by supposing a = y and the angle CPY zero. 352. On page 365 the following problem is suggested ; to construct upon a given base a curve such that the superficial area of the surface generated by its revolution round AB may be given, and...