I am integrating by parts $\int\limits_\Omega (\nabla \cdot \vec{u}) (\nabla \cdot \vec{v}) dx $ I found a general integration by parts formula in a book Mécanique des milieux continus et discrets Nicolas Moës, but I am not sure if I am using it properly, my first approach was $\int\limits_\Omega (\nabla \cdot \vec{u}) (\nabla \cdot \vec{v}) dx = \int\limits_{\partial \Omega} (\vec{v} \cdot \vec{n}) (\nabla \cdot \vec{u}) dS - \int\limits_\Omega (\Delta \vec{u})\cdot \vec{v} dx$ but then I decided to expand by components and found another answer
$\int\limits_\Omega (\nabla \cdot \vec{u}) (\nabla \cdot \vec{v}) dx = \int\limits_{\partial \Omega} (\vec{v} \cdot \vec{n}) (\nabla \cdot \vec{u}) dS - \int\limits_\Omega (\nabla \cdot {}^t\nabla\vec{u})\cdot \vec{v} dx$
Could someone help me find the correct answer please