Vector laplacian

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In mathematics and physics, the vector Laplace operator or vector Laplacian, denoted by $\nabla^2$ , named after Pierre-Simon Laplace, is a differential operator defined over a vector field. The vector Laplacian is similar to the scalar laplacian. Whereas the scalar Laplacian applies to scalar field and returns a scalar quantity, the vector laplacian applies to the vector fields and returns a vector quantity.

[edit] Definition

The vector laplacian of a vector field $\mathbf{A}$ is defined as

$\nabla^2 \mathbf{A} = \nabla(\nabla \cdot \mathbf{A}) - \nabla \times (\nabla \times \mathbf{A})$

A simpler method for evaluating the vector Laplacian is:

$\nabla^2 \mathbf{A} = (\nabla^2 A_x, \nabla^2 A_y, \nabla^2 A_z)$

Where $A x$ , $A y$ , and $A z$ are the components of $\mathbf{A}$ .

[edit] Usage in Physics

An example of the usage of the vector Laplacian is the Navier-Stokes equations for a Newtonian Incompressible flow:

$\rho \left(\frac{\partial \mathbf{v}}{\partial t}+ ( \mathbf{v} \cdot \nabla ) \mathbf{v}\right)=\rho \mathbf{f}-\nabla p +\mu\left(\nabla ^2 \mathbf{v}\right)$