Vector Laplacian

The vector Laplacian to Pierre Simon Laplace is a differential operator which acts on a vector-valued function. While the scalar Laplace operator acting on a scalar function and returns a scalar function as a result, affects the vectorial Laplace operator on a vector function and results in a vectorial function. In physics, the vectorial Laplacian immersed in the wave equations for electromagnetic fields (see the derivation of the electromagnetic wave equation). Thus, the vectorial Laplacian immersed in the D' Alembert operator of when it is applied to a vector-valued function.

Definition

Be with a vector field. Using the Grassmann identity applies

Where the nabla operator. With so is the divergence and designated by the rotation of the vector field. Therefore, the vector is defined by the Laplace operator

Coordinate representations

Cartesian coordinates

In three-dimensional Cartesian coordinates we obtain the form

Wherein the respective Cartesian components of the vector field and the scalar are Laplace operator.

Cylindrical coordinates

If the vector field in cylindrical coordinates given by the component representation

Said the coordinate system adapted, dependent on the orthogonal basis point is, then:

Spherical coordinates

Is the vector field in spherical coordinates (wherein the polar angle and the azimuth angle) with component representation

Said the coordinate system adapted, dependent on the orthogonal basis point is, then: