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IEVref: | 102-05-29 | ID: | |

Language: | en | Status: Standard | |

Term: | Laplacian, <of a vector field> | ||

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Definition: | vector Δ 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 fieldU Δ grad div − Urot rot UNote 1 to entry: In orthonormal Cartesian coordinates, the three components of the Laplacian of a vector field are: $\frac{{\partial}^{2}{U}_{x}}{\partial \text{\hspace{0.17em}}{x}^{2}}+\frac{{\partial}^{2}{U}_{x}}{\partial \text{\hspace{0.17em}}{y}^{2}}+\frac{{\partial}^{2}{U}_{x}}{\partial \text{\hspace{0.17em}}{z}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}},\text{\hspace{1em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\partial}^{2}{U}_{y}}{\partial \text{\hspace{0.17em}}{x}^{2}}+\frac{{\partial}^{2}{U}_{y}}{\partial \text{\hspace{0.17em}}{y}^{2}}+\frac{{\partial}^{2}{U}_{y}}{\partial \text{\hspace{0.17em}}{z}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{1em}}\frac{{\partial}^{2}{U}_{z}}{\partial \text{\hspace{0.17em}}{x}^{2}}+\frac{{\partial}^{2}{U}_{z}}{\partial \text{\hspace{0.17em}}{y}^{2}}+\frac{{\partial}^{2}{U}_{z}}{\partial \text{\hspace{0.17em}}{z}^{2}}$. Note 2 to entry: The Laplacian of the vector field or ∇U^{2}, where Δ is the Laplacian operator. U | ||

Publication date: | 2008-08 | ||

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Internal notes: | 2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO | ||

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Δ** U** =

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

$\frac{{\partial}^{2}{U}_{x}}{\partial \text{\hspace{0.17em}}{x}^{2}}+\frac{{\partial}^{2}{U}_{x}}{\partial \text{\hspace{0.17em}}{y}^{2}}+\frac{{\partial}^{2}{U}_{x}}{\partial \text{\hspace{0.17em}}{z}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}},\text{\hspace{1em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\partial}^{2}{U}_{y}}{\partial \text{\hspace{0.17em}}{x}^{2}}+\frac{{\partial}^{2}{U}_{y}}{\partial \text{\hspace{0.17em}}{y}^{2}}+\frac{{\partial}^{2}{U}_{y}}{\partial \text{\hspace{0.17em}}{z}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{1em}}\frac{{\partial}^{2}{U}_{z}}{\partial \text{\hspace{0.17em}}{x}^{2}}+\frac{{\partial}^{2}{U}_{z}}{\partial \text{\hspace{0.17em}}{y}^{2}}+\frac{{\partial}^{2}{U}_{z}}{\partial \text{\hspace{0.17em}}{z}^{2}}$.

Note 2 to entry: The Laplacian of the vector field ** U** is denoted by Δ