  In this note, we provide a important considerations of a familiar topic: the
gradient of a vector field. The gradient of a vector field is a common quantity
represented in continuum mechanics. However, even for Cartesian coordinate
systems, there are two different representations for this quantity in common
use, which leads to ambiguity in some results. We review the history leading to
the \emph{Gibbsian representation} for the gradient of a vector field, and
provide some suggestions to help clarify the meaning of such terms when
represented in conventional Gibbsian vector or tensor notation. In an appendix,
we briefly expand on the connection with the Gibbsian representation of the
deformation and rotation tensors in the framework of geometric algebra (GA).
