I am getting really confused about the Green's formulae, the Divergence theorem and all those related equalities.
For example, How is this formula exactly called?
$$\int_\Omega \frac{\partial u}{\partial x_j} v \, dx = - \int_\Omega u \frac{\partial v}{\partial x_j} \, dx + \int_{\partial \Omega} uv \nu_j dS \quad \quad \boldsymbol{(1)}$$
Is this Divergence theorem? I have seen it called Green-Gauss formula, but searching on Google does not yield relevant results.
On the other hand,
$$\int_{\Omega}v \Delta u \ (d\Omega) =\int_{\partial \Omega}v (\nabla u )\vec{n} \ d (\partial \Omega) - \int_{\Omega} \nabla u \nabla v \ (d \Omega) \quad \quad \quad \boldsymbol{(2)}$$
What about this formula? This one seems to be called Green's first identity.
Is there any good reference where all these results are preciselly stated and named?
I could provide more equalities but I think you get the point. For example:
$$\int_{\Omega}(\nabla\cdot u)d(\Omega)=\int_{\partial \Omega} u \cdot \vec{n}\ d (\partial \Omega) \quad \quad \quad \boldsymbol{(3)}$$
Is this generalized Divergence Theorem?