Htykuut

I read in a book the following unproven statement:

$int_{s} utimes A n , ds = int_v ( utimes nablacdot A + mathcal{E}: A^T ) , dv$

with a: 1st order tensor, n: normal vector of s, A: 2nd order tensor, and $mathcal{E}$ the permutation 3rd order tensor.

I don't know much about properties to modify the left hand side cross product so I am trying to develop both sides using Einstein notation but I can't obtain the equality..

Any hint ?

EDIT: I am able to demonstrate the equation if I assume $frac{partial u_j}{partial x_j}$ is one, which makes sense if u is the position vector. Since this assumption is not mentioned in my book, I would like to know if there is another way ..

$ int_{partial v} left( utimes A n right)_i , ds = ... = int_{v} mathcal{E}_{ijk} left[ frac{partial u_j}{partial x_j }A_{kj} + u_jfrac{partial A_{kK}}{partial x_K} right] , dv $

and

$int_v left( utimes nablacdot A + mathcal{E}: A^T right)_i , dv = int_v mathcal{E}_{ijk} left[ u_jfrac{partial A_{kK}}{partial x_K} + A_{jk}^T right] , dv = int_v mathcal{E}_{ijk} left[ A_{kj} + u_j frac{partial A_{kK}}{partial x_K} right] , dv $