Vector Laplacian

In mathematics and physics, the vector Laplace operator, denoted by $scriptstyle\; nabla^2$, 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.

Definition

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

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

In Cartesian coordinates, this reduces to the much simpler form (see proof)

$nabla^2\; mathbf\{A\}\; =\; (nabla^2\; A\_x,\; nabla^2\; A\_y,\; nabla^2\; A\_z)$

where $A\_x$, $A\_y$, and $A\_z$ are the components of $mathbf\{A\}$.

For expressions of the vector Laplacian in other coordinate systems see Nabla in cylindrical and spherical coordinates.

Generalization

The Laplacian of any tensor field $T$ ("tensor" includes scalar and vector) is defined as the divergence of the gradient of the tensor:

$nabla^2\; T\; =\; nabla\; cdot\; (nabla\; T)$

For the special case where $T$ is a scalar (a tensor of rank zero), the Laplacian takes on the familiar form.

If $T$ is a vector, the gradient is a covariant derivative which results in a tensor of second rank, and the divergence of this is again a vector (a tensor of first rank). The formula for the vector Laplacian above may be used to avoid tensor math and may be shown to be equivalent to the divergence of the gradient of the vector.

Use in physics

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

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

where the term with the vector Laplacian of the velocity field $muleft(nabla\; ^2\; mathbf\{v\}right)$ represents the viscous stresses in the fluid.

Another example is the wave equation for the electric field that can be derived from the Maxwell equations in the absence of charges and currents:

$nabla^2\; mathbf\{E\}\; -\; mu\_0\; epsilon\_0\; frac\{partial^2\; mathbf\{E\}\}\{partial\; t^2\}\; =\; 0.$

Previous equation can be written also as:

$Box,\; mathbf\{E\}\; =\; 0,$

where

$Box=frac\{1\}\{c^2\}\; frac\{partial^2\}\{partial\; t^2\}-nabla^2,$

is the D'Alembertian

See also

• Vector Laplacian/Proofs

References

• MathWorld Vector Laplacian. .