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

IEV ref102-05-29

en
Laplacian, <of a vector field>
vector ΔU associated at each point of a given space region with a vector U, equal to the gradient of the divergence of the vector field minus the rotation of the rotation of this vector field

ΔU = grad div Urot rot U

Note 1 to entry: In orthonormal Cartesian coordinates, the three components of the Laplacian of a vector field are:

2 U x x 2 + 2 U x y 2 + 2 U x z 2 , 2 U y x 2 + 2 U y y 2 + 2 U y z 2 , 2 U z x 2 + 2 U z y 2 + 2 U z z 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaalaaabaGaeyOaIy 7aaWbaaSqabeaacaaIYaaaaOGaamyvamaaBaaaleaacaWG4baabeaa aOqaaiabgkGi2kaaykW7caWG4bWaaWbaaSqabeaacaaIYaaaaaaaki abgUcaRmaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaamyv amaaBaaaleaacaWG4baabeaaaOqaaiabgkGi2kaaykW7caWG5bWaaW baaSqabeaacaaIYaaaaaaakiabgUcaRmaalaaabaGaeyOaIy7aaWba aSqabeaacaaIYaaaaOGaamyvamaaBaaaleaacaWG4baabeaaaOqaai abgkGi2kaaykW7caWG6bWaaWbaaSqabeaacaaIYaaaaaaakiaaysW7 caaMe8UaaiilaiaaywW7caaMe8UaaGjbVpaalaaabaGaeyOaIy7aaW baaSqabeaacaaIYaaaaOGaamyvamaaBaaaleaacaWG5baabeaaaOqa aiabgkGi2kaaykW7caWG4bWaaWbaaSqabeaacaaIYaaaaaaakiabgU caRmaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaamyvamaa BaaaleaacaWG5baabeaaaOqaaiabgkGi2kaaykW7caWG5bWaaWbaaS qabeaacaaIYaaaaaaakiabgUcaRmaalaaabaGaeyOaIy7aaWbaaSqa beaacaaIYaaaaOGaamyvamaaBaaaleaacaWG5baabeaaaOqaaiabgk Gi2kaaykW7caWG6bWaaWbaaSqabeaacaaIYaaaaaaakiaaysW7caaM e8UaaiilaiaaysW7caaMe8UaaGzbVpaalaaabaGaeyOaIy7aaWbaaS qabeaacaaIYaaaaOGaamyvamaaBaaaleaacaWG6baabeaaaOqaaiab gkGi2kaaykW7caWG4bWaaWbaaSqabeaacaaIYaaaaaaakiabgUcaRm aalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaamyvamaaBaaa leaacaWG6baabeaaaOqaaiabgkGi2kaaykW7caWG5bWaaWbaaSqabe aacaaIYaaaaaaakiabgUcaRmaalaaabaGaeyOaIy7aaWbaaSqabeaa caaIYaaaaOGaamyvamaaBaaaleaacaWG6baabeaaaOqaaiabgkGi2k aaykW7caWG6bWaaWbaaSqabeaacaaIYaaaaaaaaaa@A314@ .

Note 2 to entry: The Laplacian of the vector field U is denoted by ΔU or ∇2U, where Δ is the Laplacian operator.


fr
laplacien vectoriel, m
vecteur ΔU associé en chaque point d'un domaine déterminé de l'espace à un vecteur U, égal à la différence entre le gradient de la divergence du champ vectoriel et le rotationnel du rotationnel de ce champ

ΔU = grad div Urot rot U

Note 1 à l'article: En coordonnées cartésiennes orthonormées, les trois coordonnées du laplacien vectoriel sont:

2 U x x 2 + 2 U x y 2 + 2 U x z 2 , 2 U y x 2 + 2 U y y 2 + 2 U y z 2 , 2 U z x 2 + 2 U z y 2 + 2 U z z 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaalaaabaGaeyOaIy 7aaWbaaSqabeaacaaIYaaaaOGaamyvamaaBaaaleaacaWG4baabeaa aOqaaiabgkGi2kaaykW7caWG4bWaaWbaaSqabeaacaaIYaaaaaaaki abgUcaRmaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaamyv amaaBaaaleaacaWG4baabeaaaOqaaiabgkGi2kaaykW7caWG5bWaaW baaSqabeaacaaIYaaaaaaakiabgUcaRmaalaaabaGaeyOaIy7aaWba aSqabeaacaaIYaaaaOGaamyvamaaBaaaleaacaWG4baabeaaaOqaai abgkGi2kaaykW7caWG6bWaaWbaaSqabeaacaaIYaaaaaaakiaaysW7 caaMe8UaaiilaiaaywW7caaMe8UaaGjbVpaalaaabaGaeyOaIy7aaW baaSqabeaacaaIYaaaaOGaamyvamaaBaaaleaacaWG5baabeaaaOqa aiabgkGi2kaaykW7caWG4bWaaWbaaSqabeaacaaIYaaaaaaakiabgU caRmaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaamyvamaa BaaaleaacaWG5baabeaaaOqaaiabgkGi2kaaykW7caWG5bWaaWbaaS qabeaacaaIYaaaaaaakiabgUcaRmaalaaabaGaeyOaIy7aaWbaaSqa beaacaaIYaaaaOGaamyvamaaBaaaleaacaWG5baabeaaaOqaaiabgk Gi2kaaykW7caWG6bWaaWbaaSqabeaacaaIYaaaaaaakiaaysW7caaM e8UaaiilaiaaysW7caaMe8UaaGzbVpaalaaabaGaeyOaIy7aaWbaaS qabeaacaaIYaaaaOGaamyvamaaBaaaleaacaWG6baabeaaaOqaaiab gkGi2kaaykW7caWG4bWaaWbaaSqabeaacaaIYaaaaaaakiabgUcaRm aalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaamyvamaaBaaa leaacaWG6baabeaaaOqaaiabgkGi2kaaykW7caWG5bWaaWbaaSqabe aacaaIYaaaaaaakiabgUcaRmaalaaabaGaeyOaIy7aaWbaaSqabeaa caaIYaaaaOGaamyvamaaBaaaleaacaWG6baabeaaaOqaaiabgkGi2k aaykW7caWG6bWaaWbaaSqabeaacaaIYaaaaaaaaaa@A314@ .

Note 2 à l'article: Le laplacien vectoriel du champ vectoriel U est noté ΔU ou ∇2U, où Δ est l'opérateur laplacien.


de
Laplace-Operator (angewandt auf eine vektorielle Feldgröße), m
vektorieller Laplace-Operator, m

es
laplaciana vectorial
laplaciana de un campo vectorial

ja
ラプラシアン, <ベクトル場の>

pl
laplasjan wektorowy

pt
laplaciano vectorial

sr
лапласијан, <векторског поља> м јд

sv
Laplaceoperator (på ett vektorfält)

zh
拉普拉斯算子, <向量场的>

Publication date: 2008-08
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