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I would like to calculate the following integration:

\oint_{\Sigma}f(x,y)g(x,y)\mathbf{n}\cdot d\mathbf{s}

where g(x,y)=0 on \Sigma, and \mathbf{n} is the outward pointing unit normal field of the boundary \Sigma.

In this case does the integral equals to 0?

Thanks!