Examples of the Processes of the Differential and Integral Calculus |
From inside the book
Results 1-5 of 64
Page 15
... factors of the first degree , as into ( x + a ) ( x + B ) , and then differentiating the product ( x + a ) " ( x + B ) " by the Theorem of Leibnitz ; but instead of doing so we shall make use of two formulæ given by Lagrange * . Let u ...
... factors of the first degree , as into ( x + a ) ( x + B ) , and then differentiating the product ( x + a ) " ( x + B ) " by the Theorem of Leibnitz ; but instead of doing so we shall make use of two formulæ given by Lagrange * . Let u ...
Page 30
... " 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 .
... " 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 .
Page 31
... factors , we find جرح d " u dy d d - [ { d - ( n - 1 ) } { / d / - ( n − 2 ) ... ( ́ - · ) 4 ] · = dx - ( 7 ) Change the independent variable in Ꮖ . и . d2 u du - ( 1 − y2 ) -y + n2 u = 0 dy2 dy from y to a , having given y = cos x ...
... factors , we find جرح d " u dy d d - [ { d - ( n - 1 ) } { / d / - ( n − 2 ) ... ( ́ - · ) 4 ] · = dx - ( 7 ) Change the independent variable in Ꮖ . и . d2 u du - ( 1 − y2 ) -y + n2 u = 0 dy2 dy from y to a , having given y = cos x ...
Page 74
... factor of one or other series to vanish . find ( 7 ) To expand sin næ in ascending powers of sin . Proceeding in the same manner as in the last example , we 712 n2 ( n2 - 2 " ) sin nx = sinnrπ { 1 - ( sin x ) 2 + ( sin x ) - & c . 1.2.3 ...
... factor of one or other series to vanish . find ( 7 ) To expand sin næ in ascending powers of sin . Proceeding in the same manner as in the last example , we 712 n2 ( n2 - 2 " ) sin nx = sinnrπ { 1 - ( sin x ) 2 + ( sin x ) - & c . 1.2.3 ...
Page 80
... factor of all the terms of the numerator or denominator of any of the series of fractions , the value which it has when a is put equal to a . These considerations frequently lead to simplifications of the process of evaluation . Ex ...
... factor of all the terms of the numerator or denominator of any of the series of fractions , the value which it has when a is put equal to a . These considerations frequently lead to simplifications of the process of evaluation . Ex ...
Other editions - View all
Common terms and phrases
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