I want to define an identity for the expansion of the gradient of the cross product between two vector fields:
$\nabla(\mathbf{f}\times\mathbf{g})=[\nabla\otimes(\mathbf{f}\times\mathbf{g})]^T= \begin{bmatrix} \frac{\partial}{\partial x}(f_yg_z-f_zg_y) & \frac{\partial}{\partial y}(f_yg_z-f_zg_y) & \frac{\partial}{\partial z}(f_yg_z-f_zg_y)\\ \frac{\partial}{\partial x}(f_zg_x-f_xg_z) & \frac{\partial}{\partial y}(f_zg_x-f_xg_z) & \frac{\partial}{\partial z}(f_zg_x-f_xg_z)\\ \frac{\partial}{\partial x}(f_xg_y-f_yg_x) & \frac{\partial}{\partial y}(f_xg_y-f_yg_x) & \frac{\partial}{\partial z}(f_xg_y-f_yg_x) \end{bmatrix}$
Wolfram Alpha gave me this:
$\nabla(\mathbf{f}\times\mathbf{g})=\mathbf{g}\cdot(\nabla\times\mathbf{f})-\mathbf{f}\cdot(\nabla\times\mathbf{g})$
I think this is wrong because the cross product between two vector fields yields another vector field and the gradient of a vector field is a second-order tensor field. However, the right-hand side from Wolfram Alpha is a scalar field (curl of a vector field yields another vector field and the dot product between two vector fields yields a scalar field). FWIW I entered del(cross(f, g)) in Wolfram Alpha.
I am looking for an identity in vectorial or matrix-vector notation (not Einstein summation) where the outer product is expanded over $\mathbf{f}$ and $\mathbf{g}$.