**5 Geometry and analysis**

*Surfaces with boundary are introduced. The divergence theorem of Gauss is derived and used to show that the total Gauss curvature of a compact regular surface does not depend on the Riemannian metric.*

**5.1 The divergence theorem**

In this section we want to derive a two-dimensional analogue of the fundamental theorem of calculus. In this theorem the integral of a derivative over a one-dimensional interval is identified with the difference of the values at the end-points. This term in the values at the end-points can be considered as the integral of the function over the (zero-dimensional) boundary of the interval. The divergence theorem expresses the integral of a derivative of a vector field as a one-dimensional line ...

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