ENG 103 - Total differentials

ENG 103 - Total differentials - UNIVERSITY OF CALIFORNIA...

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U NIVERSITY OF C ALIFORNIA D AVIS College of Engineering ENG 103: Fluid Mechanics J.-P. Delplanque Total (Exact) Differentials Definition If ) , ( y x F is a function of two variables, then the total differential of F is: dy y F dx x F dF ! ! + ! ! = Inverse problem Consider two functions ) , ( y x P and ) , ( y x Q which satisfy the following equation: dy y x Q dx y x P D ) , ( ) , ( + = Under which conditions is there a function ) , ( y x F for which D is the total differential (i.e., dF D = )? dF D = can be re-written: dy y F dx x F dy y x Q dx y x P ! ! + ! ! = + ) , ( ) , ( which leads to ) , ( y x P x F = ! ! and ) , ( y x Q y F = ! ! Since we know that for functions with continuous second order derivatives x y F y x F ! ! ! = ! ! ! 2 2 , a necessary condition for such a function ) , ( y x F to exist is: x Q y P ! ! = ! ! (it turns out that this condition is both necessary and sufficient).
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U NIVERSITY OF C ALIFORNIA D AVIS College of Engineering ENG 103: Fluid Mechanics J.-P. Delplanque
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ENG 103 - Total differentials - UNIVERSITY OF CALIFORNIA...

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