Suppose that $\Omega$ is a compact region with a smooth boundary in a Riemann surface and that $\Phi$ is a real-valued function which is positive on $\Omega$ and vanishes on the boundary of $\Omega$. Show that

$\int_{\partial\Omega} i \space\partial\Phi \ge0$.

The problem makes a suggestion: Consider $\Omega ⊂ C$ and see that the integral is, in traditional notation,the flux of the gradient of $\Phi$ through the boundary.

But it is not enough to direct me, therefore any help is welcome.


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