How this equation with Laplace operator simplifies

Question

I am trying to understand how this equation is simplified.

$$ - \int_{\Omega} \nabla^2 ( uv) \hspace{3pt}dx = \int_{\Omega} \nabla u \nabla v \hspace{3pt}dx - \oint_{\Gamma} \frac{\partial u}{\partial n}v \hspace{3pt}ds$$

Where, $\Omega \in \mathbb{R^3}$ and $\nabla u$ is gradient of $u$.

My question is how does LHD simplify to RHS - what theorem or property is used?

Answer 1

One of Green's identities states that $$ - \int_\Omega (\nabla^2 u) v \, dx = \int_\Omega \nabla u\cdot\nabla v \, dx - \oint_\Gamma \frac{\partial u}{\partial n} v \, ds.$$ Is that what you meant to write?