Vector Laplacian

In mathematics and physics, the vector Laplace operator, denoted by ∇ 2 {displaystyle abla ^{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 orthonormal Cartesian coordinates, the returned vector field is equal to the vector field of the scalar Laplacian applied to each vector component. In mathematics and physics, the vector Laplace operator, denoted by ∇ 2 {displaystyle abla ^{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 orthonormal Cartesian coordinates, the returned vector field is equal to the vector field of the scalar Laplacian applied to each vector component. The vector Laplacian of a vector field A {displaystyle mathbf {A} } is defined as In Cartesian coordinates, this reduces to the much simpler form: where A x {displaystyle A_{x}} , A y {displaystyle A_{y}} , and A z {displaystyle A_{z}} are the components of A {displaystyle mathbf {A} } . 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. The Laplacian of any tensor field T {displaystyle mathbf {T} } ('tensor' includes scalar and vector) is defined as the divergence of the gradient of the tensor: For the special case where T {displaystyle mathbf {T} } is a scalar (a tensor of degree zero), the Laplacian takes on the familiar form. If T {displaystyle mathbf {T} } is a vector (a tensor of first degree), 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: