JoannaEMC

The Laplace operator page has a questionable equation on it. The page indicates the vector Laplace operator returns a vector quantity. But the equation in Cartesian coordinates does not show a vector quantity result. It does show the scaler Laplace operator operating on each of the three cartesian components. The result of these three scaler Laplace operations will be three scaler quantities. But there are no unit vectors or vector notation on the right side of the equation indicating the quantity is a vector.

Are the unit vectors missing from the following equation?

$$\nabla ^{2}\mathbf {A} =(\nabla ^{2}A_{x},\nabla ^{2}A_{y},\nabla ^{2}A_{z}),$$

Copied Laplace operator page text:

The vector Laplace operator, also denoted by $\nabla ^{2}$, 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 $\mathbf {A}$ is defined as $$\nabla ^{2}\mathbf {A} =\nabla (\nabla \cdot \mathbf {A} )-\nabla \times (\nabla \times \mathbf {A} ).$$

In Cartesian coordinates, this reduces to the much simpler form as $$\nabla ^{2}\mathbf {A} =(\nabla ^{2}A_{x},\nabla ^{2}A_{y},\nabla ^{2}A_{z}),$$ where $A_{x}$, $A_{y}$, and $A_{z}$ are the components of the vector field $\mathbf {A}$, and $\nabla ^{2}$ just on the left of each vector field component is the (scalar) Laplace operator. This can be seen to be a special case of Lagrange's formula; see Vector triple product.