I am looking to derive product rules for the curl of a 2nd-order tensor field contracted with a vector field (matrix vector multiplication), $$\nabla \times (\mathbf{A}(\mathbf{x}) \cdot \mathbf{u}(\mathbf{x}))$$ $$\nabla \times (\mathbf{u}(\mathbf{x}) \cdot \mathbf{A}(\mathbf{x}))$$
A general product rule would be ideal, but I am especially thinking of it for the case of $\mathbf{A} = \nabla\mathbf{v}$. I expect that there will be terms like $\nabla \times \nabla \mathbf{v}$, which is zero. Using index notation, for the first case, so far I have
$$\nabla \times (\mathbf{A} \cdot \mathbf{u})$$ $$= \epsilon_{ijk}\partial_i (\mathbf{A} \cdot \mathbf{u})_j $$ $$= \epsilon_{ijk}\partial_i (A_{jm}u_m) $$ $$= \epsilon_{ijk} (\partial_i A_{jm}) u_m + \epsilon_{ijk} A_{jm} (\partial_i u_m)$$ $$= (\nabla \times \mathbf{A}^T) \cdot \mathbf{u} + \epsilon_{ijk} A_{jm} (\partial_i u_m)$$
where this first term feels straight forward, but I'm not sure what to do with the second term. I have used the definition for the curl of a second order tensor field found here. It seems likely that this needs to be done component-wise and that there doesn't exist a nice expression. It would be great if there were, especially if terms could be expressed with $\nabla \times \mathbf{u}$ and $\nabla \times \mathbf{v}$.