I'm trying to show the following integral equality, but I really can't come up with a proof. The context here is the one of an introductive book to continuum mechanics, so everything is smooth and there are no differetiability problems.
Let $R$ be a control volume in the space, with $v$ a velocity field and $r(x)=x-0$ the position vector. Then $$\int_R r \times \operatorname{div}(v \otimes v)dV = \int_{\partial R}r \times (v\cdot n)vdA$$
Clearly, the divergence theorem must be used. The fact is that I really don't know how to move things so to get something like $\int_R \operatorname{div}(\ldots)dV$, because that cross product is tricky to solve.
EDIT:
I'm trying to work with components. As suggested, I try to take the divergence of the r.h.s.
$r \times (v \cdot n) v$ has i-th component $\varepsilon_{ijk} x_j v_ln_lv_k$ and hence I take the i-th derivative, since I want to compute its divergence
$$\varepsilon_{ijk} \frac{\partial}{\partial x_i} \bigl(x_j v_ln_lv_k \bigr) $$ but after this I really get lost in the computations. How should I move from here?