Recently I was studying to minimize a functional containing multiple integrals and found the following article in Wikipedia :
Clearly the functional above has been divided into two parts, one corresponds to the plane region $D$, and another corresponds to the contour $C$ around the plane $D$. Now to minimize the functional we need to find the first variation of $\displaystyle{V[\phi]}$. We know that the first variation is defined by $$\delta V=\bigg[\frac{\partial}{\partial \epsilon}V[\phi(x,y,z,\epsilon)]\bigg]_{\epsilon=0}$$ where $\phi(x,y,z,\epsilon)=\phi(x,y,z)+\epsilon\delta \phi(x,y,z)$.
Now equating $\delta V$ to zero, we could get two separate equation, one extremal as Poisson equation and another as a boundary condition corresponding to the extremal. But I cannot figure out how the second expression above has been derived in terms of $u$ and $v$. Also at the third expression it has been mentioned about an application of Divergence theorem, but this is also not clear to me as Divergence theorem relates between surface integral and volume integral, whereas the above expression only contains line and surface integrals. Can someone explain in detail about the hidden calculations in the above discussion? Any help is appreciated.