I am currently studying electrodynamic and came across the following vectoridentity, but I am unable to prove it:
$$ \vec{f} \times ( \nabla \times \vec{f} ) -\vec{f}(\nabla\cdot\vec{f}) = \nabla \cdot (1/2 \cdot f^2 \delta - \vec{f} \otimes \vec{f} ) $$
$$ \vec{f} \times (rot(\vec{f})) -\vec{f}(\nabla\cdot\vec{f})= div (1/2 \cdot f^2 E - \vec{f} \otimes \vec{f} ) $$
The second equation is the same as the first.
Edit: I totally forgot the term $ -\vec{f}(\nabla\cdot\vec{f}). $
$ \delta $ , E is the unit tensor (I first wrote it as Kronecker symbol $ \delta_{ij} $).
Switched + to - on the right side between the tensors.