Suppose is a compact surface with boundary and metric . Denote by the Gauss curvature of and by the geodesic curvature of . Define a functional

**Lemma:** The first variation of the functional

where , correspond to the curvature . Suppose there is another metric , similarly we can define replacing by and . Then

**Proposition:** ;

*Proof:* The first conclusion is easy to see from the lemma.

So we need to prove

By using , we get

which exactly means the former identity.

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