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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

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    What is $\widetilde{g}$?2012-11-04
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    $\tilde{g}$ is another function $\tilde{g}:\mathbb{R}\rightarrow\mathbb{R}$. $g=\tilde{g}$ on $S^2$. Is (1) dependent on the particular choice of $g$?2012-11-04

2 Answers 2