Curl of tensor * non-constant vector

[refind]

I know by definition that if T is a 2nd order tensor and v is a vector,
curl(Tv)=curl(T)v

but what if instead of constant vector v, I have w=grad(u), not constant but obviously an irrotational vector field. Is this still true:
curl(Tw)=curl(T)w ?

My guess is yes since curl(w)=0 but have no verification. Please advise. Thank you.

 

[Orodruin]

Are you familiar with index notation? If not I suggest learning it because it will make your tensor equations much more straightforward to interpret.

To answer your question, the answer will not depend on curl(w). The vector structure of the curl operator is not directly affecting w, since the w is contracted with one of the tensor indices.