Both Wikipedia and Mathworld define the vector Laplacian of a vector field ${\bf A}$ to be $$ \nabla^2 {\bf A} := {\bf \nabla} (\nabla \cdot {\bf A}) - \nabla \times (\nabla \times {\bf A}). $$ This definition only makes sense for vector fields on $\mathbb{R}^3$, because of the presence of the curls in the second term.
But it seems to me that the alternate, more general definition $$ (\nabla^2 {\bf A})^j := \partial_i \partial_i A^j $$ using abstract index notation (which is equivalent for vector fields on $\mathbb{R}^3$) is much more natural because:
- It makes the connection to the scalar Laplacian much clearer, thereby motivating the name of the operator.
- It applies to any vector field on Euclidean space of any dimension - and even more generally, on any Riemannian manifold (if the partial derivatives are promoted to covariant ones). Moreover, this more general operator remains $\mathrm{SO}(n)$ invariant for any dimension of vector space.
Is there any motivation for restricting to the less general (and arguably less intuitive) standard definition?