What is missing in Physics[Vectors] is the algebra of dyadic products
(also known as the tensor product) of vectors. Let's write for the space
of vectors in the three-dimensional Euclidean space, and by that I don't
mean ; I mean vectors as Euclid would have understood them,
that is, pointed arrows in 3D. That's what Physics[Vectors]
expresses as , , etc.
Linear operators on are called second order tensors (in the
Euclidean space setting). Let's write for the the set of those
operators. itself is a linear space in the usual sense of addition
and multiplication by scalars.
The gradient of a vector field is a linear operator on , and therefore
a second order tensor. So what I have asked for in the original post,
amounts to a wish for having support for the algebra and calculus of
second order tensors in the Physics[Vectors] package.
Physics[Vectors] introduces the , , vectors for a basis of the
(3-dimensional) vector space associated with the point in
cylindrical coordinates. What is missing is a basis for the (9-dimensional)
space of second order tensors at . These may be constructed from
, , through the dyadic products ,
(see below.) A good naming scheme for these in Physics[Vectors] would be , , etc.
Then the gradient of a vector field would be a linear combination of those
nine basis elements. One can show that if
In that setting, the derivative of in the direction of an arbitrary vector
would be the vecor (grad
Those familiar with the Navier-Stokes equations of fluid mechanics
will recognize (grad as the expression for the
fluid's convective acceleration. Similarly, applications in nonlinear elastiicity
invariably call for calculating gradients of vector fileds.
The major hurdle toward adding this feature would be the
implementation of the algebra of dyadic products in Physics[Vectors].
I can't tell how much effort that will take.
For the record, the dyadic product of two vectors is
a second order tensor, that is, an object in , defined through the
) , for all
where is the usual dot product of vectors.