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 Area Mathematics - General concepts and linear algebra / Scalar and vector fields IEV ref 102-05-29     en Laplacian,   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 U − rot rot UNote 1 to entry: In orthonormal Cartesian coordinates, the three components of the Laplacian of a vector field are: $\frac{{\partial }^{2}{U}_{x}}{\partial \text{\hspace{0.17em}}{x}^{2}}+\frac{{\partial }^{2}{U}_{x}}{\partial \text{\hspace{0.17em}}{y}^{2}}+\frac{{\partial }^{2}{U}_{x}}{\partial \text{\hspace{0.17em}}{z}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}},\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\partial }^{2}{U}_{y}}{\partial \text{\hspace{0.17em}}{x}^{2}}+\frac{{\partial }^{2}{U}_{y}}{\partial \text{\hspace{0.17em}}{y}^{2}}+\frac{{\partial }^{2}{U}_{y}}{\partial \text{\hspace{0.17em}}{z}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\frac{{\partial }^{2}{U}_{z}}{\partial \text{\hspace{0.17em}}{x}^{2}}+\frac{{\partial }^{2}{U}_{z}}{\partial \text{\hspace{0.17em}}{y}^{2}}+\frac{{\partial }^{2}{U}_{z}}{\partial \text{\hspace{0.17em}}{z}^{2}}$. 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 champΔU = grad div U − rot rot U Note 1 à l'article: En coordonnées cartésiennes orthonormées, les trois coordonnées du laplacien vectoriel sont: $\frac{{\partial }^{2}{U}_{x}}{\partial \text{\hspace{0.17em}}{x}^{2}}+\frac{{\partial }^{2}{U}_{x}}{\partial \text{\hspace{0.17em}}{y}^{2}}+\frac{{\partial }^{2}{U}_{x}}{\partial \text{\hspace{0.17em}}{z}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}},\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\partial }^{2}{U}_{y}}{\partial \text{\hspace{0.17em}}{x}^{2}}+\frac{{\partial }^{2}{U}_{y}}{\partial \text{\hspace{0.17em}}{y}^{2}}+\frac{{\partial }^{2}{U}_{y}}{\partial \text{\hspace{0.17em}}{z}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\frac{{\partial }^{2}{U}_{z}}{\partial \text{\hspace{0.17em}}{x}^{2}}+\frac{{\partial }^{2}{U}_{z}}{\partial \text{\hspace{0.17em}}{y}^{2}}+\frac{{\partial }^{2}{U}_{z}}{\partial \text{\hspace{0.17em}}{z}^{2}}$. 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), mvektorieller Laplace-Operator, m         es laplaciana vectoriallaplaciana de un campo vectorial         ko 라플라시안, <벡터장> 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> 