Wikipedia states
$ \int_\Omega \varphi\, \operatorname{div}\, \vec v \; \mathrm d V = \int_{\partial \Omega} \varphi\, \vec v \cdot \mathrm d \vec S - \int_\Omega \vec v\cdot \nabla\, \varphi \; \mathrm dV$
though without source (if anyone got one, please link it).
https://en.wikipedia.org/wiki/Integration_by_parts
Applying this to $\vec{v}$=$\nabla{u}$ we get
$ \int_\Omega \varphi\, \Delta\, u \; \mathrm d V = \int_{\partial \Omega} \varphi\, \nabla u \cdot \mathrm d \vec S - \int_\Omega \nabla u\cdot \nabla\, \varphi \; \mathrm dV$
However, in the Appendix of Evans Partial Differential Equations, the following is stated:
$ \int_\Omega \varphi\, \Delta\, u \; \mathrm d V = \int_{\partial \Omega} \varphi\, \frac{du}{dn} \mathrm dS - \int_\Omega \nabla u\cdot \nabla\, \varphi \; \mathrm dV$
with outward poinitng unit normal n.
Is $d\vec{S}$ defined as $n\cdot dS$, or how to make Sense of this?
Also if someone can recommend a short, mostly informal text to refresh knowledge about Integration on surfaces, it would be appreciated.
Yes, $d\vec S$ is an alternative notation for $\vec n\,dS$, and $\dfrac{\partial u}{\partial n} = \nabla u\cdot\vec n$ is the directional derivative in the normal direction (at the boundary).