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Area Mathematics - General concepts and linear algebra / Scalar and vector fields

IEV ref102-05-33

en
second Green formula
second Green theorem
second Green identity
identity resulting from the divergence theorem applied to the vector field f 1 grad f 2 f 2 grad f 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadAgadaWgaaWcba GaaGymaaqabaGccaaMc8ocbeGaa83zaiaa=jhacaWFHbGaa8hzaiaa ykW7caWGMbWaaSbaaSqaaiaaikdaaeqaaOGaeyOeI0IaamOzamaaBa aaleaacaaIYaaabeaakiaaykW7caWFNbGaa8NCaiaa=fgacaWFKbGa aGPaVlaadAgadaWgaaWcbaGaaGymaaqabaaaaa@4B1E@ , where f1 and f2 are two scalar fields given at each point of a three-dimensional domain V limited by a closed surface S

V ( f 1 Δ f 2 f 2 Δ f 1 ) dV= S ( f 1 grad f 2 f 2 grad f 1 ) e n dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaapmfabaWaaeWaae aacaWGMbWaaSbaaSqaaKqzaeGaaGymaaWcbeaakiaayIW7imaacqWF uoarcaaMi8UaamOzamaaBaaaleaajugabiaaikdaaSqabaGccqGHsi slcaWGMbWaaSbaaSqaaKqzaeGaaGOmaaWcbeaakiaayIW7cqWFuoar caaMi8UaamOzamaaBaaaleaajugabiaaigdaaSqabaaakiaawIcaca GLPaaaaSqaaiaabAfaaeqaniabgUIiYlabgUIiYlabgUIiYdqcLbua caGGKbGccaWGwbGaeyypa0Zaa8GvaeaadaqadaqaaiaadAgadaWgaa WcbaqcLbqacaaIXaaaleqaaOGaaGPaVJqabKqzafGaa43zaiaa+jha caGFHbGaa4hzaOGaaGPaVlaadAgadaWgaaWcbaqcLbqacaaIYaaale qaaOGaeyOeI0IaamOzamaaBaaaleaajugabiaaikdaaSqabaGccaaM c8EcLbuacaGFNbGaa4NCaiaa+fgacaGFKbGccaaMc8UaamOzamaaBa aaleaajugabiaaigdaaSqabaaakiaawIcacaGLPaaaaSqaaKqzaeGa ae4uaaWcbeqdcqWIs4U0cqGHRiI8cqGHRiI8aOGaeyyXICTaaCyzam aaBaaaleaajugabiaab6gaaSqabaqcLbuacaGGKbGccaWGbbaaaa@81AA@

where dV is the volume element, endA is the vector surface element and Δ is the Laplacian operator

Note 1 to entry: The second Green formula is symmetric with respect to f1 and f2.


fr
formule de Green, f
deuxième formule de Green, f
identité résultant de l'application du théorème de la divergence au champ vectoriel f 1 grad f 2 f 2 grad f 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadAgadaWgaaWcba GaaGymaaqabaGccaaMc8ocbeGaa83zaiaa=jhacaWFHbGaa8hzaiaa ykW7caWGMbWaaSbaaSqaaiaaikdaaeqaaOGaeyOeI0IaamOzamaaBa aaleaacaaIYaaabeaakiaaykW7caWFNbGaa8NCaiaa=fgacaWFKbGa aGPaVlaadAgadaWgaaWcbaGaaGymaaqabaaaaa@4B1E@ , où f1 et f2 sont deux champs scalaires donnés en tout point d'un domaine tridimensionnel V délimité par une surface fermée S

V ( f 1 Δ f 2 f 2 Δ f 1 ) dV= S ( f 1 grad f 2 f 2 grad f 1 ) e n dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaapmfabaWaaeWaae aacaWGMbWaaSbaaSqaaKqzaeGaaGymaaWcbeaakiaayIW7imaacqWF uoarcaaMi8UaamOzamaaBaaaleaajugabiaaikdaaSqabaGccqGHsi slcaWGMbWaaSbaaSqaaKqzaeGaaGOmaaWcbeaakiaayIW7cqWFuoar caaMi8UaamOzamaaBaaaleaajugabiaaigdaaSqabaaakiaawIcaca GLPaaaaSqaaiaabAfaaeqaniabgUIiYlabgUIiYlabgUIiYdqcLbua caGGKbGccaWGwbGaeyypa0Zaa8GvaeaadaqadaqaaiaadAgadaWgaa WcbaqcLbqacaaIXaaaleqaaOGaaGPaVJqabKqzafGaa43zaiaa+jha caGFHbGaa4hzaOGaaGPaVlaadAgadaWgaaWcbaqcLbqacaaIYaaale qaaOGaeyOeI0IaamOzamaaBaaaleaajugabiaaikdaaSqabaGccaaM c8EcLbuacaGFNbGaa4NCaiaa+fgacaGFKbGccaaMc8UaamOzamaaBa aaleaajugabiaaigdaaSqabaaakiaawIcacaGLPaaaaSqaaKqzaeGa ae4uaaWcbeqdcqWIs4U0cqGHRiI8cqGHRiI8aOGaeyyXICTaaCyzam aaBaaaleaajugabiaab6gaaSqabaqcLbuacaGGKbGccaWGbbaaaa@81AA@

où dV est l’élément de volume, endA est l’élément vectoriel de surface et Δ est l'opérateur laplacien

Note 1 à l'article: La formule de Green est symétrique par rapport à f1 et f2.


de
zweite Greensche Formel, f

es
segunda fórmula de Green
fórmula de Green

ja
グリーンの第二定理

pl
druga tożsamość Greena

pt
segunda fórmula de Green

sr
друга Гринова формула, ж јд

sv
Greens andra formel

zh
格林第二公式

Publication date: 2017-07
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