ΔU = grad div U − rot rot U

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

${\displaystyle \frac{{\partial }^2{U}_x}{\partial x^2}+\frac{{\partial }^2{U}_x}{\partial y^2}+\frac{{\partial }^2{U}_x}{\partial z^2},\frac{{\partial }^2{U}_y}{\partial x^2}+\frac{{\partial }^2{U}_y}{\partial y^2}+\frac{{\partial }^2{U}_y}{\partial z^2},\frac{{\partial }^2{U}_z}{\partial x^2}+\frac{{\partial }^2{U}_z}{\partial y^2}+\frac{{\partial }^2{U}_z}{\partial z^2}}$.

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

ΔU = grad div U − rot rot U

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

${\displaystyle \frac{{\partial }^2{U}_x}{\partial x^2}+\frac{{\partial }^2{U}_x}{\partial y^2}+\frac{{\partial }^2{U}_x}{\partial z^2},\frac{{\partial }^2{U}_y}{\partial x^2}+\frac{{\partial }^2{U}_y}{\partial y^2}+\frac{{\partial }^2{U}_y}{\partial z^2},\frac{{\partial }^2{U}_z}{\partial x^2}+\frac{{\partial }^2{U}_z}{\partial y^2}+\frac{{\partial }^2{U}_z}{\partial z^2}}$.

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