Examples of the Processes of the Differential and Integral CalculusJ and J. J. Deighton, 1846 - 529 стор. |
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Сторінка 10
... results of the labours of mathe- maticians in this field of research are referred to various Memoirs of Liouville in the Journal de l'Ecole Polytech- nique , Vol . XIII . , and in Crelle's Journal ; to two papers by Professor Kelland in ...
... results of the labours of mathe- maticians in this field of research are referred to various Memoirs of Liouville in the Journal de l'Ecole Polytech- nique , Vol . XIII . , and in Crelle's Journal ; to two papers by Professor Kelland in ...
Сторінка 15
... ) x = € + a dx a ) -ar ax ( " * X ) . dx This result , when generalized , is of great importance in the solution of Differential Equations . If the function to be differentiated be ( a + SUCCESSIVE DIFFERENTIATION . 15.
... ) x = € + a dx a ) -ar ax ( " * X ) . dx This result , when generalized , is of great importance in the solution of Differential Equations . If the function to be differentiated be ( a + SUCCESSIVE DIFFERENTIATION . 15.
Сторінка 19
... results are useful in the theory of definite in- tegrals . In the following examples the functions are reduced to the required forms by differentiation in the same way as in Ex . 11 . ( 22 ) Let u = ( 1 d ' Therefore x - x $ = du dx dr ...
... results are useful in the theory of definite in- tegrals . In the following examples the functions are reduced to the required forms by differentiation in the same way as in Ex . 11 . ( 22 ) Let u = ( 1 d ' Therefore x - x $ = du dx dr ...
Сторінка 30
... result is of such extreme complexity , that it happens for- tunately that we have seldom to employ these transformations for high orders of differentials ; and where this is necessary , that the nature of the case usually gives us the ...
... result is of such extreme complexity , that it happens for- tunately that we have seldom to employ these transformations for high orders of differentials ; and where this is necessary , that the nature of the case usually gives us the ...
Сторінка 31
... result is ď2 x + x − € 3 0 . dy2 ( 4 ) Change the variable in du dy + и ( 1+ y2 ) } a from y to w , when a = log { y + ( 1 + y2 ) } } . du a The result is + u = ( ε * + € ̃ * ) . dx 2 ( 5 ) Change the variable in y2 dy ? d'u du + Ay dy ...
... result is ď2 x + x − € 3 0 . dy2 ( 4 ) Change the variable in du dy + и ( 1+ y2 ) } a from y to w , when a = log { y + ( 1 + y2 ) } } . du a The result is + u = ( ε * + € ̃ * ) . dx 2 ( 5 ) Change the variable in y2 dy ? d'u du + Ay dy ...
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a² b2 a²x² angle arbitrary constant asymptote axis becomes C₁ c²x² Cambridge circle co-ordinates condition curvature curve cycloid cylinder determine differential coefficients differential equation dx dx dx dy dx dx² dy dx dy dy dy dy dz eliminate ellipse equal Euler find the value formula function Geometry gives Hence hypocycloid infinite Integrating with respect intersection John Bernoulli Let the equation lines of curvature locus logarithmic logarithmic spiral maximum minimum Multiply negative origin parabola perpendicular radius radius of curvature singular solution spiral Substituting subtangent surface tangent plane theorem tractory triangle vanish whence x²)³