Green's first identity

It appears that I misread the question the first time. In any case, this seems to follow from the generalisation of the divergence theorem for rank $n$ tensors:

$$\int_{U}{\frac{\partial T_{i_1i_2 \ldots i_q \ldots i_n}}{\partial x_{i_q}} dU}=\int_{\partial U}{T_{i_1i_2 \ldots i_q \ldots i_n}n_{i_q} dS}$$

applied to $T \cdot \omega$ instead of $T$.

\begin{align} \int_{\partial U}{T_{i_1i_2 \ldots i_q \ldots i_r \ldots i_n}n_{i_q} \omega_{i_r} dS}& =\int_{U}{\frac{\partial (T_{i_1i_2 \ldots i_q \ldots i_r \ldots i_n}\omega_{i_r})}{\partial x_{i_q}} dU} \\ &=\int_{U}{\frac{\partial T_{i_1i_2 \ldots i_q \ldots i_r \ldots i_n}}{\partial x_{i_q}}\omega_{i_r} dU}+\int_{U}{T_{i_1i_2 \ldots i_q \ldots i_r \ldots i_n}\frac{\partial \omega_{i_r}}{\partial x_{i_q}} dU} \end{align}

Hopefully I haven't messed up again.