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, https://en.wikipedia.org/wiki/Tensor_derivative_(continuum_mechanics), 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.
