I was trying to work out a integration by parts formula $S^2$ of the form $$\int_{S^2}f(x)\frac{\partial g}{\partial x_1}dx \tag{1}$$ where $f,g:\mathbb{R}^3\rightarrow\mathbb{R}$. Given $g$ and $\tilde{g}$ that agree on $S^2$, I am not sure if (1) independent from which of the two we choose.
Having realized this I tried two work purely on the manifolds. Does anyone know a good reference that explains how to use Stokes Theorem and an the volume form associated with the Riemanian metric to derive a integration by parts formula.
Thank you,
warsaga