Vector Laplacian

In mathematics and physics, the vector Laplace operator, denoted by ${\displaystyle \scriptstyle \nabla ^{2}}$, 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 a scalar field and returns a scalar quantity, the vector Laplacian applies to a vector field, returning a vector quantity. When computed in rectangular cartesian coordinates, the returned vector field is equal to the vector field of the scalar Laplacian applied on the individual elements.

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

In Cartesian coordinates, this reduces to the much simpler form:

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