A Treatise on Infinitesimal Calculus: Differential calculus. 1857University Press, 1857 - Calculus |
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Page 79
... substituting these in ( 50 ) , we have Р ( dr ) dr dr + Q + R = 0 ; dx dy dz ( 53 ) and this is the equivalent expression . The converse operation is performed by a similar process . It will be observed that we have hereby passed from ...
... substituting these in ( 50 ) , we have Р ( dr ) dr dr + Q + R = 0 ; dx dy dz ( 53 ) and this is the equivalent expression . The converse operation is performed by a similar process . It will be observed that we have hereby passed from ...
Page 80
... substituting x2 + y2 2 and y = { a2 — x2 } ; x dx dy - { a2 — x2 } + ' dx Du = { a2 — x2 } + - a result which is manifestly as it ought to be , if we consider that the relation between u and x , after the elimination of y , is u = sin ...
... substituting x2 + y2 2 and y = { a2 — x2 } ; x dx dy - { a2 — x2 } + ' dx Du = { a2 — x2 } + - a result which is manifestly as it ought to be , if we consider that the relation between u and x , after the elimination of y , is u = sin ...
Page 98
... substituting in equation ( 7 ) , we have x x2 ƒ ( x ) = ƒ ( 0 ) + ƒ ′ ( 0 ) + ƒ ” ( 0 ) 12 + + f " ( 0 ) ... xn 1.2.3 ... n + ... ( 8 ) ƒ ( 0 ) , ƒ ' ( 0 , ƒ ” ( 0 ) ... being the values of f ( x ) , ƒ ' ( x ) , ƒ ” ( x ) , ...
... substituting in equation ( 7 ) , we have x x2 ƒ ( x ) = ƒ ( 0 ) + ƒ ′ ( 0 ) + ƒ ” ( 0 ) 12 + + f " ( 0 ) ... xn 1.2.3 ... n + ... ( 8 ) ƒ ( 0 ) , ƒ ' ( 0 , ƒ ” ( 0 ) ... being the values of f ( x ) , ƒ ' ( x ) , ƒ ” ( x ) , ...
Page 126
... substituting , we -day sin 0 do + cos 0 ( d0 ) 2 dy - ( sin 0 ) 3 d03 cos 0 dy У + + = 0 ; ( sin 0 ) 2 sin 0 do ( sin 0 ) 2 d2y + y = 0 . d02 Ex . 2. Eliminate a between d2y 1 dy + dx2 x dx + y = 0 , and x2 = 40 ; and express the result ...
... substituting , we -day sin 0 do + cos 0 ( d0 ) 2 dy - ( sin 0 ) 3 d03 cos 0 dy У + + = 0 ; ( sin 0 ) 2 sin 0 do ( sin 0 ) 2 d2y + y = 0 . d02 Ex . 2. Eliminate a between d2y 1 dy + dx2 x dx + y = 0 , and x2 = 40 ; and express the result ...
Page 138
... substituting from ( 109 ) , we have D2u = dt2 d2u ' dx'2 +2 ( d2u dx'dy ' xy + :) d2u y2 + . ... ( 113 ) dy ' 2 also the t - differential of ( 111 ) is d2u dt2 = n ( n - 1 ) tn - 2 F ( x , y , z , ... ) ; ( 114 ) whence , equating ( 113 ) ...
... substituting from ( 109 ) , we have D2u = dt2 d2u ' dx'2 +2 ( d2u dx'dy ' xy + :) d2u y2 + . ... ( 113 ) dy ' 2 also the t - differential of ( 111 ) is d2u dt2 = n ( n - 1 ) tn - 2 F ( x , y , z , ... ) ; ( 114 ) whence , equating ( 113 ) ...
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Common terms and phrases
a₁ algebraical ascending powers b₁ becomes calculated change of sign changes sign coefficients constant curve d²u d²x d²y d2y dx2 d³u d³y denominator determine differential equation dr dr dr dx dx dr dx dx dx dy dx dz dx² dx2 dy2 dy d2u dy dx dy dy dy dz dy² dy³ dz dx dz dz equal equicrescent Evaluate explicit function expression F(xo F(xo+h factor finite quantity fraction given Hence homogeneous function increases increments indeterminate form infinite infinitesimal Infinitesimal Calculus infinity logarithm loge maxima and minima minimum value negative partial derived-functions particular values positive primitive equation proper fraction replaced result roots Similarly sin x singular value substituting suppose symbols Taylor's Series tion total differentials vanish variation whence x+▲x zero
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