Different definitions of Curl?

Questions
1

In Mathematica, the Curl of a second rank tensor in three dimensions yields a scalar.
This can be seen from the description in the documentation page:

In

Curl[f,{Subscript[x, 1],\[Ellipsis],Subscript[x, n]}]

if f is an array with depth k<n, it must have dimensions {n,[Ellipsis],n}, and the resulting curl is an array with depth n-k-1 of dimensions {n,[Ellipsis],n}.

It can also be seeing from an example in the same documentation page:

In[1]:= Curl[{{x y, x y^2, x y^3}, {x^2 y, x^2 y^2, x^2 y^3}, {x^3 y, 
   x^3 y^2, x^3 y^3}}, {x, y, z}]

Out[1]= 1/2 (x^3 - 3 x y^2 - 3 x^2 y^2 + 2 x y^3)

In Wikipedia, Tensor derivative (continuum mechanics) - Wikipedia, the curl of a second rank tensor in 3D is given as a second rank tensor. The k, m component of the curl of the tensor S is given as (translating the formula in Wikipedia to a Wolfram-like notation and implying index summation):

LeviCivitaTensor[3][[ i, j, k ]]*Derivative[S[[ j, m ]], x[[ i ]] ]

One way to try to reconcile this contradiction could be to assume that Wikipedia is interpreting the Curl of a second rank tensor as the Curl of each one of its rows (mapping the curl operation), but that seems contrary to the point of the Wikipedia article, which seems to be to present definitions that apply in different coordinate systems (for example, the divergence of a second rank tensor can be calculated as the divergence of each row in Cartesian coordinates, but not in cylindrical).

So, we seem to have two very different definitions for the Curl of a second rank tensor. Can anybody shed light on this?
Thanks in advance,
Otto Linsuain.

2

There is a related discussion in

https://mathematica.stackexchange.com/questions/191373/what-is-the-definition-of-curl-in-mathematica

The key point is that in Wolfram Language we use a definition based on Hodge duality.

3

Hello Jose,

Thanks for your reply and for the link. I have read it.
It is still somewhat uncomfortable in the that these operators (grad, div, curl) all have standard mathematical definitions. Occasionally one encounters some variability on how people treat, say, differentiating with respect to one index or the other (some people may talk about the pre-gradient and the post-gradient), but the difference between the result being a tensor and it being a scalar sounds too big.

Is there a way these two results could be reconciled or somehow connected?
Is the scalar answer Mathematica outputs related to the tensor that one would otherwise calculate?
In terms of relating integrals of a differential form over a boundary of a region to and integral of its external derivative over the region (as in Gauss’ or Stoke’s theorems) what integrals would be related when using this curl?

I would appreciate any references to material that could clarify these questions.
Thanks in advance,
Otto Linsuain