The Gauss-Bonnet Theorem
Reproduced here is part of what is written in section 74 of the book 'Tensor Analysis' by I. S. Sokolnikoff (Second Edition, 1964).
'The description of surfaces with the aid of differential equations has local character, since relations among the derivatives describe properties of surfaces only in the neighborhood of a point. To obtain results valid for the entire surface one must perform integrations. Because of the complex structure of differential equations of the theory of surfaces, relatively few global results have been obtained, and the available results in global geometry are largely concerned with a special class of convex surfaces. There is one important classical result, however, that relates the integral of the Gaussian curvature evaluated over the area of an arbitrary smooth surface to the line integral of the geodesic curvature computed over the curve that bounds the area. Gauss viewed this result as the most elegant theorem of geometry of surfaces in the large.'