Vector Laplacian


In mathematics and physics, the vector Laplace operator, denoted by, 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 a scalar field and returns a scalar quantity, the vector Laplacian applies to a vector field, returning a vector quantity. When computed in orthonormal Cartesian coordinates, the returned vector field is equal to the vector field of the scalar Laplacian applied to each vector component.

Definition

The vector Laplacian of a vector field is defined as
In Cartesian coordinates, this reduces to the much simpler form:
where,, and are the components of. This can be seen to be a special case of Lagrange's formula; see Vector triple product.
For expressions of the vector Laplacian in other coordinate systems see Del in cylindrical and spherical coordinates.

Generalization

The Laplacian of any tensor field is defined as the divergence of the gradient of the tensor:
For the special case where is a scalar, the Laplacian takes on the familiar form.
If is a vector, the gradient is a covariant derivative which results in a tensor of second degree, and the divergence of this is again a vector. 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 Jacobian matrix shown below for the gradient of a vector:
And, in the same manner, a dot product, which evaluates to a vector, of a vector by the gradient of another vector can be seen as a product of matrices:
This identity is a coordinate dependent result, and is not general.

Use in physics

An example of the usage of the vector Laplacian is the Navier-Stokes equations for a Newtonian incompressible flow:
where the term with the vector Laplacian of the velocity field represents the viscous stresses in the fluid.
Another example is the wave equation for the electric field that can be derived from
the Maxwell's equations in the absence of charges and currents:
The previous equation can also be written as:
where
is the D'Alembertian, used in the Klein–Gordon equation.