Formulate Gauss-Bonnet

In Geometry, the formula of Gauss-Bonnet is a property connecting the geometry and the Topologie of the Surface S. It bears the name of the mathematicians Carl Friedrich Gauss, which was aware of a version of the theorem, but never published it, and Pierre Ossian Bonnet, which published a particular case in 1848 of it.

Statement

That is to say M a compact surface (without edge); then the integral of the Courbure of Gauss makes it possible to find the Caractéristique of Euler surface

$\backslash \; int\_M\; K\; \backslash ;\; dA=2\; \backslash \; pi\; \backslash \; chi\; (M)$

For a compact variety on board, the formula becomes

$\backslash \; int\_M\; K\; \backslash ;\; dA+\; \backslash \; int\_\; \{\backslash \; partial\; M\}\; k\_g\; \backslash ;\; ds=2\; \backslash \; pi\; \backslash \; chi\; (M)$

by noting $k\_g$ the geodetic curve at the points of the edge $\backslash \; partial\; M$.

If the edge $\backslash \; partial\; M$ is only regular per pieces, the formula still holds by taking instead of the integral $\backslash \; int\_\; \{\backslash \; partial\; M\}\; k\_g\; \backslash ;\; ds$ the sum of the corresponding integrals on the regular portions of the edge, plus the sum of the Angle S formed with the angular points.

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