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

IEV ref102-05-32

en
first Green formula
identity resulting from the divergence theorem applied to the vector field f 1 grad f 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadAgadaWgaaWcba GaaGymaaqabaGccaaMc8ocbeGaa83zaiaa=jhacaWFHbGaa8hzaiaa ykW7caWGMbWaaSbaaSqaaiaaikdaaeqaaaaa@3FC0@ , 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 ( grad f 1 grad f 2 + f 1 Δ f 2 ) dV= S f 1 grad f 2 e n dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaapmfabaWaaeWaae aaieqajugqbiaa=DgacaWFYbGaa8xyaiaa=rgakiaaykW7caWGMbWa aSbaaSqaaKqzaeGaaGymaaWcbeaakiabgwSixNqzafGaa83zaiaa=j hacaWFHbGaa8hzaOGaaGPaVlaadAgadaWgaaWcbaqcLbqacaaIYaaa leqaaOGaey4kaSIaamOzamaaBaaaleaajugabiaaigdaaSqabaGcca aMc8ocdaGae4hLdqKaaGjcVlaadAgadaWgaaWcbaqcLbqacaaIYaaa leqaaaGccaGLOaGaayzkaaaaleaajugabiaabAfaaSqab0Gaey4kIi Vaey4kIiVaey4kIipajugqbiaacsgakiaadAfacqGH9aqpdaWdwbqa aiaadAgadaWgaaWcbaqcLbqacaaIXaaaleqaaaqaaKqzaeGaae4uaa WcbeqdcqWIs4U0cqGHRiI8cqGHRiI8aOGaaGPaVNqzafGaa83zaiaa =jhacaWFHbGaa8hzaOGaaGPaVlaadAgadaWgaaWcbaqcLbqacaaIYa aaleqaaOGaeyyXICTaaCyzamaaBaaaleaajugabiaab6gaaSqabaqc LbuacaGGKbGccaWGbbaaaa@7D05@

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

Note 1 to entry: In English the first Green formula is sometimes called “first Green theorem” or “first Green identity”.


fr
première formule de Green, f
identité résultant de l'application du théorème d'Ostrogradski au champ vectoriel f 1 grad f 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadAgadaWgaaWcba GaaGymaaqabaGccaaMc8ocbeGaa83zaiaa=jhacaWFHbGaa8hzaiaa ykW7caWGMbWaaSbaaSqaaiaaikdaaeqaaaaa@3FC0@ , où f1 et f2 sont deux champs scalaires donnés en tout point d'un domaine tridimensionnel V délimité par une surface fermée S

V ( grad f 1 grad f 2 + f 1 Δ f 2 ) dV= S f 1 grad f 2 e n dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaapmfabaWaaeWaae aaieqajugqbiaa=DgacaWFYbGaa8xyaiaa=rgakiaaykW7caWGMbWa aSbaaSqaaKqzaeGaaGymaaWcbeaakiabgwSixNqzafGaa83zaiaa=j hacaWFHbGaa8hzaOGaaGPaVlaadAgadaWgaaWcbaqcLbqacaaIYaaa leqaaOGaey4kaSIaamOzamaaBaaaleaajugabiaaigdaaSqabaGcca aMc8ocdaGae4hLdqKaaGjcVlaadAgadaWgaaWcbaqcLbqacaaIYaaa leqaaaGccaGLOaGaayzkaaaaleaajugabiaabAfaaSqab0Gaey4kIi Vaey4kIiVaey4kIipajugqbiaacsgakiaadAfacqGH9aqpdaWdwbqa aiaadAgadaWgaaWcbaqcLbqacaaIXaaaleqaaaqaaKqzaeGaae4uaa WcbeqdcqWIs4U0cqGHRiI8cqGHRiI8aOGaaGPaVNqzafGaa83zaiaa =jhacaWFHbGaa8hzaOGaaGPaVlaadAgadaWgaaWcbaqcLbqacaaIYa aaleqaaOGaeyyXICTaaCyzamaaBaaaleaajugabiaab6gaaSqabaqc LbuacaGGKbGccaWGbbaaaa@7D05@

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: En anglais, la première formule de Green est parfois appelée «first Green theorem» ou «first Green identity».


de
erste Greensche Formel, f

es
primera fórmula de Green

ja
グリーンの第一定理

pl
pierwsza tożsamość Greena

pt
primeira fórmula de Green

sr
прва Гринова формула, ж јд

sv
Greens första formel

zh
格林第一公式

Publication date: 2008-08
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