* Gauss-Bonnet theorem for an elementary application of curvature

The difference between initial and final orientations is, in which case the Gauss-Bonnet theorem applies.

An example of complex region where Gauss-Bonnet theorem can apply.

# Gauss – Bonnet theorem The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to 2πχ ( M ) where χ ( M ) denotes the Euler characteristic of M. This theorem has a generalization to any compact even-dimensional Riemannian manifold, see generalized Gauss-Bonnet theorem.

The characteristic class approach to curvature invariants was a particular reason to make a theory, to prove a general Gauss-Bonnet theorem.

The Gauss-Bonnet theorem links total curvature of a surface to its topological properties.

It includes areas currently fashionable ( the Chern-Simons theory arising from a 1974 paper written jointly with Jim Simons ), perennial ( the Chern-Weil theory linking curvature invariants to characteristic classes from 1944, after the Allendoerfer-Weil paper of 1943 on the Gauss-Bonnet theorem ), the foundational ( Chern classes ), and some areas such as projective differential geometry and webs that have a lower profile.

Witten's formula might be understood as an infinite-dimensional analogue of the Gauss-Bonnet theorem.

The exponential of its vacuum expectation value determines the coupling constant g, as for compact worldsheets by the Gauss-Bonnet theorem and the Euler characteristic, where g is the genus that counts the number of handles and thus the number of loops or string interactions described by a specific worldsheet.

The analogous result holds for hyperbolic triangles, with " excess " replaced by " defect "; these are both special cases of the Gauss-Bonnet theorem.

In the special case of a sphere of radius 1, the area simply equals the excess angle: A = E. One can also use Girard's formula to obtain the discrete Gauss-Bonnet theorem.

He made some important contributions to the differential geometry of surfaces, including the Gauss-Bonnet theorem.

A formula for the line integral of the geodesic curvature along a closed curve is known as the Gauss-Bonnet theorem.

By the Gauss-Bonnet theorem, the area of the surface is

The angle of rotation in both cases is determined by the area integral of curvature in agreement with the Gauss-Bonnet theorem.

Because the proof uses the Gauss-Bonnet theorem, it does not generalize to higher dimensions.

* The integral of the Gaussian curvature K of a 2-dimensional Riemannian manifold ( M, g ) is invariant under changes of the Riemannian metric g. This is the Gauss-Bonnet Theorem.

See Gauss-Bonnet gravity for more details.

Bayes ' theorem then links the degree of belief in a proposition before and after accounting for evidence.

Bayes ' theorem links these probabilities, which are in general different.

The one shape that is perfectly convex and symmetrical is the circle, although this, in itself, does not represent a rigorous proof of the isoperimetric theorem ( see external links ).

The case i = n has also been noted already, and is an easy consequence of the Hurewicz theorem: this theorem links homotopy groups with homology groups, which are generally easier to calculate ; in particular, it shows that for a simply-connected space X, the first nonzero homotopy group π < sub > k </ sub >( X ), with k > 0, is isomorphic to the first nonzero homology group H < sub > k </ sub >( X ).

Thurston's theorem on hyperbolic Dehn surgery states that, provided a finite collection of filling slopes are avoided, the remaining Dehn fillings on hyperbolic links are hyperbolic 3-manifolds.

Many of these appear to be fairly uniform, so by the virial theorem, the total kinetic energy should be half the total gravitational binding energy of the galaxies.

Experimentally, however, the total kinetic energy is found to be much greater: in particular, assuming the gravitational mass is due to only the visible matter of the galaxy, stars far from the center of galaxies have much higher velocities than predicted by the virial theorem.

In mechanics, the virial theorem provides a general equation relating the average over time of the total kinetic energy,, of a stable system consisting of N particles, bound by potential forces, with that of the total potential energy,, where angle brackets represent the average over time of the enclosed quantity.

The significance of the virial theorem is that it allows the average total kinetic energy to be calculated even for very complicated systems that defy an exact solution, such as those considered in statistical mechanics ; this average total kinetic energy is related to the temperature of the system by the equipartition theorem.

Although the virial theorem depends on averaging the total kinetic and potential energies, the presentation here postpones the averaging to the last step.

This probability approaches 1 as the total string approaches infinity, and thus the original theorem is correct.

The discrete analog of curvature, corresponding to curvature being concentrated at a point and particularly useful for polyhedra, is the ( angular ) defect ; the analog for the Gauss – Bonnet theorem is Descartes ' theorem on total angular defect.

By taking the square root of the sum of the squares of the components the total magnetic field strength ( also called total magnetic intensity, TMI ) can be calculated by Pythagoras's theorem.

While Hilbert's tenth problem is not a formal mathematical statement as such the nearly universal acceptance of the ( philosophical ) identification of a decision algorithm with a total computable predicate allows us to use the MRDP theorem to conclude the tenth problem is unsolvable.

The geodesic curvature of geodesics being zero, and the Euler characteristic of T being 1, the theorem then states that the sum of the turning angles of the geodesic triangle is equal to 2π minus the total curvature within the triangle.

Descartes ' theorem on total angular defect of a polyhedron is the polyhedral analog:

The divergence theorem is thus a conservation law which states that the volume total of all sinks and sources, the volume integral of the divergence, is equal to the net flow across the volume's boundary.

The second recursion theorem can be applied to any total recursive function.

An old result in this direction is the Fary – Milnor theorem states that if the total curvature of a knot K in satisfies

* One of the results of Fourier analysis is Parseval's theorem which states that the area under the energy spectral density curve is equal to the area under the square of the magnitude of the signal, the total energy:

In physics, Green's theorem is mostly used to solve two-dimensional flow integrals, stating that the sum of fluid outflows at any point inside a volume is equal to the total outflow summed about an enclosing area.

For functions of more than one variable, the theorem states that if the total derivative of a continuously differentiable function F defined from an open set U of R < sup > n </ sup > into R < sup > n </ sup > is invertible at a point p ( i. e., the Jacobian determinant of F at p is non-zero ), then F is an invertible function near p. That is, an inverse function to F exists in some neighborhood of F ( p ).

This led in 1828 to an important theorem, the Theorema Egregium ( remarkable theorem ), establishing an important property of the notion of curvature.

Informally, the theorem says that the curvature of a surface can be determined entirely by measuring angles and distances on the surface.

On curved surfaces, the formula for C ( r ) will be different, and the Gaussian curvature K at the point P can be computed by the Bertrand – Diquet – Puiseux theorem as

The integral of the Gaussian curvature over the whole surface is closely related to the surface's Euler characteristic ; see the Gauss – Bonnet theorem.

The curvature of space is a mathematical description of whether or not the Pythagorean theorem is valid for spatial coordinates.

* If the curvature is zero, then Ω = 1, and the Pythagorean theorem is correct ;

* Theorema Egregium − The " remarkable theorem " discovered by Gauss which showed there is an intrinsic notion of curvature for surfaces.

The Gauss – Bonnet theorem or Gauss – Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry ( in the sense of curvature ) to their topology ( in the sense of the Euler characteristic ).

Thinking of curvature as a measure, rather than as a function, Descartes ' theorem is Gauss – Bonnet where the curvature is a discrete measure, and Gauss – Bonnet for measures generalizes both Gauss – Bonnet for smooth manifolds and Descartes ' theorem.

Informally, the theorem says that the curvature of a surface can be determined entirely by measuring distances along paths on the surface.

The theorem has also been generalized to triangles on other surfaces of constant curvature ( Masal ' tsev 1994 ).