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 scalar field and returns a scalar quantity, the vector Laplacian applies to the vector fields and returns 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.
For the special case where is a scalar (a tensor of rank zero), the Laplacian takes on the familiar form.
If is a vector (a tensor of first rank), the gradient is a covariant derivative which results in a tensor of second rank, 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 (a tensor of 2nd rank) can be seen as a product of matrices:
This identity is a coordinate dependent result, and is not general.