Examples of the Processes of the Differential and Integral Calculus |
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Page 11
... € “ * cos ( n x + P ) . Hence as before , ď u dx = ( a2 + n2 ) 3ça * ax cos ( nx + rp ) . Similarly , if u = € " * sin n x , ď u dx = a'r ( a2 + n2 ) 2 ea * sin ( na + rQ ) . > 1 - ( − ) ' − 1 ( SUCCESSIVE DIFFERENTIATION . 11.
... € “ * cos ( n x + P ) . Hence as before , ď u dx = ( a2 + n2 ) 3ça * ax cos ( nx + rp ) . Similarly , if u = € " * sin n x , ď u dx = a'r ( a2 + n2 ) 2 ea * sin ( na + rQ ) . > 1 - ( − ) ' − 1 ( SUCCESSIVE DIFFERENTIATION . 11.
Page 17
... ( − ) } } ' + ' = ( a2 + x2 ) { x + a ( − ) } } ' + ' = ( a2 + x2 ) r + 1 { cos ( r + 1 ) 9 − ( − ) a sin ( r + 1 ) 0 } , - { cos ( r + 1 ) 0 + ( − ) a sin ( r + 1 ) 0 } . Hence we have d 9 ( a ) 1 a2 2 SUCCESSIVE DIFFERENTIATION . 17.
... ( − ) } } ' + ' = ( a2 + x2 ) { x + a ( − ) } } ' + ' = ( a2 + x2 ) r + 1 { cos ( r + 1 ) 9 − ( − ) a sin ( r + 1 ) 0 } , - { cos ( r + 1 ) 0 + ( − ) a sin ( r + 1 ) 0 } . Hence we have d 9 ( a ) 1 a2 2 SUCCESSIVE DIFFERENTIATION . 17.
Page 18
D. F. Gregory. Hence we have d 9 ( a ) 1 a2 + x2 ( - ) ' r ( 1 ) ... 2.1 sin ( +1 ) 8 a + ( a2 + x2 ) Liouville , Jour . de l'Ecole Polytechnique , Cah . 21 , p . 157 . ( 21 ) In the same way if we had the function we should find d u = x ...
D. F. Gregory. Hence we have d 9 ( a ) 1 a2 + x2 ( - ) ' r ( 1 ) ... 2.1 sin ( +1 ) 8 a + ( a2 + x2 ) Liouville , Jour . de l'Ecole Polytechnique , Cah . 21 , p . 157 . ( 21 ) In the same way if we had the function we should find d u = x ...
Page 21
... Hence multiplying by ( e2 + 1 ) ' + 1 we must have ɑ ‚ € TM a + ɑ , -16 ( r − 1 ) * + & c . + a1e * ( 1 ) . ( ε * + 1 ) ' + 1 d u dx ' = -2x -3- Now as u = € -E + € - & c . d'u dx 2x − ( ~ = ( − ) ' { l'e ̄ ′ — 2 ′ e ̃2 * + 3 ′ ...
... Hence multiplying by ( e2 + 1 ) ' + 1 we must have ɑ ‚ € TM a + ɑ , -16 ( r − 1 ) * + & c . + a1e * ( 1 ) . ( ε * + 1 ) ' + 1 d u dx ' = -2x -3- Now as u = € -E + € - & c . d'u dx 2x − ( ~ = ( − ) ' { l'e ̄ ′ — 2 ′ e ̃2 * + 3 ′ ...
Page 36
... Hence we have dx - dx d Ꮎ . de Substituting these values in the double integral it becomes SS V ( dx dy dx dy - do dr dr de dr de . If we had three variables x , y , z to be transformed into three others p , q , r , we should have ...
... Hence we have dx - dx d Ꮎ . de Substituting these values in the double integral it becomes SS V ( dx dy dx dy - do dr dr de dr de . If we had three variables x , y , z to be transformed into three others p , q , r , we should have ...
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