Theorema egregium

The Theorema egregium is a set of differential geometry, a branch of mathematics. He was found by Carl Friedrich Gauss, and in a concise formulation it is:

The Gaussian curvature is one of the main curvature quantities in classical differential geometry.

History

While Gauss in the years the Kingdom of Hanover was measured from 1821 to 1825, he suspected that the curvature of the earth's surface can be determined solely by the length and angle measurement. In fact, Gauss took some time to prove this statement. His evidence was anything but straightforward and easy. For this reason, he described the sentence as egregium Theorema, " outstandingly important theorem ".

Classification in modern differential geometry

The differential geometry has undergone substantial impetus by Gauss. This meant that the later considered by Gauss curvature was also called the Gaussian curvature. In addition, it can be considered that the length and angle measurement is induced on a surface by the coefficients of the first fundamental form precisely this. In the language of differential geometry, the statement of the Theorema is egregium:

In this sense, the Gaussian curvature of the internal geometry of a size, that is the geometry that is induced only by the first fundamental form. Other sizes of inner geometry to measure the length of a curve of the surface, the surface area and the geodesic curvature of a curve.

Derivation

Gauss himself has this set, as already mentioned, can determine only after a lengthy statement. Later, you could simplify these calculations. For example, apply the formula of Brioschi:

The Theorema egregium it follows obviously as corollary.