Gauss-Green Theorem
Green's Formulas
What I'm trying to do is to demonstrate that all Green's formulas follow from the Gauss-Green theorem, which are all given above. I am aware of that the process would be similar to integration by parts in 1-dimension, but the format of these formulas still seem different. So I'm still stucked at where to start the integration, what should be the u and what should be the dv of the process of integration by parts, or I'm totally at the wrong track.
Could anyone please give me a hint? Thanks in advance!
In the Gauss-Green formula, replace $f(x)$ by $f(x)g(x)$. Then since $$ \partial_{x_i}(f(x)g(x)) = \partial_{x_i}(f(x))g(x) + \partial_{x_i}(g(x))f(x)$$ by product rule, we easily get the first of the Green's formulas you posted.
The other formulas just follow if you're familiar with the definitions of $\Delta f$ and $\frac{\partial f}{\partial \nu}$.