A History of the Progress of the Calculus of Variations During the Nineteenth Century |
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Page 49
... curvature at the point of the surface under consideration . These quantities are considered positive when the convexity of the surface is turned outwards . 80. A careful examination of the above investigation will shew that throughout ...
... curvature at the point of the surface under consideration . These quantities are considered positive when the convexity of the surface is turned outwards . 80. A careful examination of the above investigation will shew that throughout ...
Page 70
... curvature , then x , y , z may denote the co - ordinates of any point of the surface , and the limits of the inte- gration will depend upon the projections on the plane of ( x , y ) of these curves . In order to indicate what a quantity ...
... curvature , then x , y , z may denote the co - ordinates of any point of the surface , and the limits of the inte- gration will depend upon the projections on the plane of ( x , y ) of these curves . In order to indicate what a quantity ...
Page 84
... curvature . In order then to avoid . useless complication , we will suppose that the highest differential coefficient contained in Vis of the second order . In this case the equation H0 involves partial differential coefficients of the ...
... curvature . In order then to avoid . useless complication , we will suppose that the highest differential coefficient contained in Vis of the second order . In this case the equation H0 involves partial differential coefficients of the ...
Page 97
... curvature at any point of this surface , or more generally the radii of curvature of two normal sections at right angles , and by do the differential element of the surface , we shall suppose that among all surfaces of the same area the ...
... curvature at any point of this surface , or more generally the radii of curvature of two normal sections at right angles , and by do the differential element of the surface , we shall suppose that among all surfaces of the same area the ...
Page 107
... curvature at any point of a sur- face of revolution are the radius of curvature of the generating curve and the length of the normal at the point between the point and the axis of revolution . Denote these by p1 and Pai then 1 d2z dr.2 ...
... curvature at any point of a sur- face of revolution are the radius of curvature of the generating curve and the length of the normal at the point between the point and the axis of revolution . Denote these by p1 and Pai then 1 d2z dr.2 ...
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1+p² arbitrary constants axis Brunacci Calculus of Variations catenary chapter co-ordinates condition considered contains curvature curve Delaunay denote determined Differential Calculus differential equation double integral ds ds 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 Hence 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 minima values notation obtain occupies pages occur ordinary 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 - In fact this integral resembles the integral fVdydz ... and may be treated in the same way. We have merely indicated the transformations which must be applied to the portion JDUdxdydz ... of the variation SF; because since these transformations reduce to integration by parts they belong to the Integral Calculus rather than to the method of variations. 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...
Page 37 - 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.