Examples of the Processes of the Differential and Integral Calculus |
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Page 13
... expressions , it appears that d ( 2/4 ) d € 47 x7 = a " - " x2 - r da ( ~ 2 / 2 ) * ( 15 ) uv = x " log x , d ' ( uv ) da · = n ( n − 1 ) ... ( n − r + 1 ) x2 - r { log x + r . - r ( r1 ) - 1.2 r ( r− 1 ) ( r− 2 ) - 1 ( n − r + 1 ) ...
... expressions , it appears that d ( 2/4 ) d € 47 x7 = a " - " x2 - r da ( ~ 2 / 2 ) * ( 15 ) uv = x " log x , d ' ( uv ) da · = n ( n − 1 ) ... ( n − r + 1 ) x2 - r { log x + r . - r ( r1 ) - 1.2 r ( r− 1 ) ( r− 2 ) - 1 ( n − r + 1 ) ...
Page 29
... expression for the radius of curvature when a is the independent variable is dy + da - d'y dx2 When y is made the independent variable , it becomes { 1+ (税) dy dx dy ( 3 ) Transform d2 y dx2 ' dy 3 CHANGE OF THE INDEPENDENT VARIABLE . 29.
... expression for the radius of curvature when a is the independent variable is dy + da - d'y dx2 When y is made the independent variable , it becomes { 1+ (税) dy dx dy ( 3 ) Transform d2 y dx2 ' dy 3 CHANGE OF THE INDEPENDENT VARIABLE . 29.
Page 31
... expressing ď3 u and in terms of the differentials of u and y with dy regard to a , we may effect the required transformation more simply by differentiating successively and simplifying at each step . CHANGE OF THE INDEPENDENT VARIABLE . 31.
... expressing ď3 u and in terms of the differentials of u and y with dy regard to a , we may effect the required transformation more simply by differentiating successively and simplifying at each step . CHANGE OF THE INDEPENDENT VARIABLE . 31.
Page 33
... = dr cos d Ꮎ - sin 0 , dy dr = de ᏧᎾ sin ✪ + r cos 0 ; dr sin + cos dy ᏧᎾ and therefore da dr cos Ꮎ - r sin o de Substituting this expression for dy 9 we find dx + { + } ( dy 23 ( 13 ) 3 CHANGE OF THE INDEPENDENT VARIABLE . 33.
... = dr cos d Ꮎ - sin 0 , dy dr = de ᏧᎾ sin ✪ + r cos 0 ; dr sin + cos dy ᏧᎾ and therefore da dr cos Ꮎ - r sin o de Substituting this expression for dy 9 we find dx + { + } ( dy 23 ( 13 ) 3 CHANGE OF THE INDEPENDENT VARIABLE . 33.
Page 35
... expressions become more com- plicated . Such cases however seldom occur . If the independent variables enter into multiple integrals , we cannot substitute directly the values of the original diffe- rentials in terms of the new ...
... expressions become more com- plicated . Such cases however seldom occur . If the independent variables enter into multiple integrals , we cannot substitute directly the values of the original diffe- rentials in terms of the new ...
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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