Examples of the Processes of the Differential and Integral CalculusJ. and J.J. Deighton, 1846 - 529 стор. |
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Сторінка iii
... given in Elementary Treatises on the subject , but I have also introduced demonstrations of propositions which , although important and interesting , do not usually find a place in works devoted to the exposition of the principles of ...
... given in Elementary Treatises on the subject , but I have also introduced demonstrations of propositions which , although important and interesting , do not usually find a place in works devoted to the exposition of the principles of ...
Сторінка iv
... given in works which are likely to be in the hands of students . I have done so because I think it a matter of some importance that the use of this method should be extended as much as possible , since it shortens and simplifies many of ...
... given in works which are likely to be in the hands of students . I have done so because I think it a matter of some importance that the use of this method should be extended as much as possible , since it shortens and simplifies many of ...
Сторінка 30
... , d " 1 d n y " = € dy dx d d d nx T = E to n factors . dx This may be put under the form d { , ( ༦-1 ) , € dx 2 ) a d dx 6 d ( n - 2 ) . -1 dv d dx Now by the theorem given in Ex . 18 , 30 CHANGE OF THE INDEPENDENT VARIABLE .
... , d " 1 d n y " = € dy dx d d d nx T = E to n factors . dx This may be put under the form d { , ( ༦-1 ) , € dx 2 ) a d dx 6 d ( n - 2 ) . -1 dv d dx Now by the theorem given in Ex . 18 , 30 CHANGE OF THE INDEPENDENT VARIABLE .
Сторінка 31
... given y = cos x . The result is d'u + n2 u = 0 . dx2 ( 8 ) Change the independent variable in ( 1 - y2 ) 2 d2 u dy2 du 2 a - · 2y ( 1 y3 ) - from y to a , having given y = € 2x 2x dy - 1 - y u = 0 The result is d2 u + a ( c2 * + 1 ) u ...
... given y = cos x . The result is d'u + n2 u = 0 . dx2 ( 8 ) Change the independent variable in ( 1 - y2 ) 2 d2 u dy2 du 2 a - · 2y ( 1 y3 ) - from y to a , having given y = € 2x 2x dy - 1 - y u = 0 The result is d2 u + a ( c2 * + 1 ) u ...
Сторінка 32
... given a2 = 40 . Since we have d Ꮎ = 40 , x = 2 d r du dx du dx2 = = = du do = do dx 1 du 2 do du de = + x du 2 do x d'u do 2 do dx a2 d2 u 4 d02 du + 0 a do d2 u d 02 , whence 1 du d2 u du d2 u u + 32 VARIABLE . CHANGE OF THE INDEPENDENT.
... given a2 = 40 . Since we have d Ꮎ = 40 , x = 2 d r du dx du dx2 = = = du do = do dx 1 du 2 do du de = + x du 2 do x d'u do 2 do dx a2 d2 u 4 d02 du + 0 a do d2 u d 02 , whence 1 du d2 u du d2 u u + 32 VARIABLE . CHANGE OF THE INDEPENDENT.
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a² b2 a²x² angle arbitrary constant assume asymptote becomes branches C₁ Cambridge circle co-ordinates condition Crelle's Journal curvature curve cycloid determine differential coefficients differential equation dx dx dx dy dx dx² dy dx dy dy dy dy dz dz dz eliminate ellipse equal Euler factor formula fraction function Geometry gives Hence hypocycloid infinite intersection John Bernoulli Let the equation lines of curvature locus logarithmic logarithmic spiral Multiply negative origin parabola perpendicular radius SECT singular points singular solution spiral Substituting subtangent surface tangent plane theorem triangle University of Cambridge vanish whence x²)³