For a problem in the textbook I am reading, I need to prove that $\int_Vw_{i,j}v_jdV = \int_Sw_iv_jn_jdS$, where $S$ is the boundary of the volume $V$, $v_i$ is the velocity vector field of a continuum in $V$, and $w_i$ is one half of the curl of the velocity vector field; that is, $w_i=(1/2)\epsilon_{ijk}v_{k,j}$.
I know that $w_{i,j}v_j=(w_iv_j)_{,j}-w_iv_{j,j}$.
Integrating over $V$ and using the Divergence Theorem, we get: $\int_Vw_{i,j}v_jdV=\int_V(w_iv_j)_{,j}dV-\int_Vw_iv_{j,j}dV=\int_Sw_iv_jn_jdS+\int_Vw_iv_{j,j}dV$
Hence we need to show that $0=\int_Vw_iv_{j,j}dV$. To my knowledge, this is not true, in general. The vector field $v(x_1,x_2,x_3)=(x_1+x_2,-x_1+x_2,x_3)$ has constant nonzero curl and divergence, so the integral will be nonzero. Am I missing something? Thank you in advance.