# Herglotz formula

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An integral relation between two closed isometric oriented regular surfaces. Let local coordinates $u$ and $v$ be introduced on the surfaces $S_1$ and $S_2$ so that equality of the coordinates realizes an isometric mapping. Let

$$ds^2=Edu^2+2Fdudv+Gdv^2$$

be the first fundamental form, having the same coefficients for both surfaces in the given coordinates, let $K$ be the Gaussian curvature, let $H_\alpha$ be the mean curvatures, and let

$$\sqrt{EG-F^2}(\lambda_\alpha du^2+2\mu_\alpha dudv+\nu_\alpha dv^2)$$

be the second fundamental forms of the surfaces $S_\alpha$. Herglotz' formula then takes the following form:

$$\int\limits_{S_1}\begin{vmatrix}\lambda_2-\lambda_1&\mu_2-\mu_1\\\mu_2-\mu_1&\nu_2-\nu_1\end{vmatrix}(\mathbf n,\mathbf x)d\tau=\int\limits_{S_2}H_2d\tau-\int\limits_{S_1}H_1d\tau,$$

where $\mathbf x=\mathbf x(u,v)$ is the position vector of $S_1$, $\mathbf n$ is the unit vector of the normal to $S_1$ and $d\tau$ is the surface element. It was obtained by G. Herglotz .