Gauss-Bonnet formula - ορισμός. Τι είναι το Gauss-Bonnet formula
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Τι (ποιος) είναι Gauss-Bonnet formula - ορισμός

Chern-Gauss-Bonnet theorem; Generalized Gauss-Bonnet theorem; Gauss–Bonnet term; Gauss-Bonnet term; Chern–Gauss–Bonnet formula; Chern-Gauss-Bonnet formula; Chern theorem; Generalized Gauss–Bonnet theorem; Chern-Gauss–Bonnet theorem; Chern formula; Chen-Gauss-Bonnet theorem; Chern-Gauss-Bonnet Theorem; Gauss-Bonnet equation

Chern–GaussBonnet theorem         
In mathematics, the Chern theorem (or the Chern–GaussBonnet theorem after Shiing-Shen Chern, Carl Friedrich Gauss, and Pierre Ossian Bonnet) states that the Euler-Poincaré characteristic (a topological invariant defined as the alternating sum of the Betti numbers of a topological space) of a closed even-dimensional Riemannian manifold is equal to the integral of a certain polynomial (the Euler class) of its curvature form (an analytical invariant).
GaussBonnet gravity         
In general relativity, GaussBonnet gravity, also referred to as Einstein–GaussBonnet gravity, is a modification of the Einstein–Hilbert action to include the GaussBonnet term (named after Carl Friedrich Gauss and Pierre Ossian Bonnet)
Shoelace formula         
  • Shoelace scheme for determining the area of a polygon with point coordinates <math>(x_1,y_1),...,(x_n,y_n)</math>
  • Manipulations of a polygon
  • Example
  • Basic idea: Any polygon edge determines the ''signed'' area of a trapezoid. All these areas sum up to the polygon area.
  • Deriving the trapezoid formula
  • Horizontal shoelace form for the example.
  • Triangle form: The color of the edges indicate, which triangle area  is positive (green) and negative (red) respectively.
MATHEMATICAL ALGORITHM TO DETERMINE THE AREA OF A SIMPLE POLYGON
Surveyor's formula; Shoelace algorithm; Shoelace Method; Gauss' area formula; Gauss's area formula; Gauss area formula
The shoelace formula, shoelace algorithm, or shoelace method (also known as Gauss's area formula and the surveyor's formula) is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. It is called the shoelace formula because of the constant cross-multiplying for the coordinates making up the polygon, like threading shoelaces.

Βικιπαίδεια

Chern–Gauss–Bonnet theorem

In mathematics, the Chern theorem (or the Chern–Gauss–Bonnet theorem after Shiing-Shen Chern, Carl Friedrich Gauss, and Pierre Ossian Bonnet) states that the Euler–Poincaré characteristic (a topological invariant defined as the alternating sum of the Betti numbers of a topological space) of a closed even-dimensional Riemannian manifold is equal to the integral of a certain polynomial (the Euler class) of its curvature form (an analytical invariant).

It is a highly non-trivial generalization of the classic Gauss–Bonnet theorem (for 2-dimensional manifolds / surfaces) to higher even-dimensional Riemannian manifolds. In 1943, Carl B. Allendoerfer and André Weil proved a special case for extrinsic manifolds. In a classic paper published in 1944, Shiing-Shen Chern proved the theorem in full generality connecting global topology with local geometry.

Riemann–Roch and Atiyah–Singer are other generalizations of the Gauss–Bonnet theorem.