Source: https://homepages.warwick.ac.uk/staff/C.M.Elliott/DziEll13a.pdf
Definition 2.11 of the weak derivative:
A function $f \in L^1(\Gamma)$ has the weak derivative $v_i=D_if \in L^1(\Gamma)$, $i \in \{1,...,n+1\}$, if for every function $\phi \in C^1_0(\Gamma)$ we have the relation $$\int_\Gamma f D_i\phi \,dA =- \int_\Gamma\phi v_i \,dA+ \int_\Gamma f\phi H\ n_id\,A$$ where $\Gamma$ is a hypersurface in $\mathbb R^{n+1}$ and $H$ is its mean curvature.
Question:
I am familiar with the "standard" definition of the weak derivative (https://en.wikipedia.org/wiki/Weak_derivative). How are these two different defintions to be reconciled and what role does the mean curvature get into the equation? Some explanation/interpretation would be much appreciated.