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

IEV ref 102-05-29

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:

$\frac{{\partial }^{2}{U}_{x}}{\partial \text{\hspace{0.17em}}{x}^{2}}+\frac{{\partial }^{2}{U}_{x}}{\partial \text{\hspace{0.17em}}{y}^{2}}+\frac{{\partial }^{2}{U}_{x}}{\partial \text{\hspace{0.17em}}{z}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}},\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\partial }^{2}{U}_{y}}{\partial \text{\hspace{0.17em}}{x}^{2}}+\frac{{\partial }^{2}{U}_{y}}{\partial \text{\hspace{0.17em}}{y}^{2}}+\frac{{\partial }^{2}{U}_{y}}{\partial \text{\hspace{0.17em}}{z}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\frac{{\partial }^{2}{U}_{z}}{\partial \text{\hspace{0.17em}}{x}^{2}}+\frac{{\partial }^{2}{U}_{z}}{\partial \text{\hspace{0.17em}}{y}^{2}}+\frac{{\partial }^{2}{U}_{z}}{\partial \text{\hspace{0.17em}}{z}^{2}}$.

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 champ

ΔU = grad div Urot 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), m
vektorieller Laplace-Operator, m

es
laplaciana vectorial
laplaciana de un campo vectorial

ko
라플라시안, <벡터장>

ja
ラプラシアン, <ベクトル場の>

 nl BE laplaciaan, m

pl
laplasjan wektorowy

pt
laplaciano vectorial

sr
лапласијан, <векторског поља> м јд

sv
Laplaceoperator (på ett vektorfält)

zh