Gauss's Theorema Egregium
Gauss's theorema egregium states that the
Gaussian curvature of a surface embedded in three-space may be understood
intrinsically to that surface. "Residents" of the surface may observe the
Gaussian curvature of the surface without ever venturing into full
three-dimensional space; they can observe the curvature of the surface they
live in without even knowing about the three-dimensional space in which they
are embedded.
In particular, Gaussian curvature can be measured by checking how closely
the arc lengths of circles of small radii correspond to what they should be
in Euclidean space, . If the arc length of circles tends to be smaller than
what is expected in Euclidean space, then the space is positively curved; if
larger, negatively; if the same, 0 Gaussian curvature.
Gauss (effectively) expressed the theorema egregium by saying that the
Gaussian curvature at a point is given by -R(v,w)v,w where R is the Riemann
tensor, and v and w are an orthonormal basis for the tangent space.
Excerpt from MathWorld
Theorem of the Day -- 29/09/2005
Fermat's 4n+1 Theorem
Fermat's 4n+1 theorem, sometimes called Fermat's two-square theorem or simply "Fermat's theorem," states that a prime number p can be represented in an essentially unique manner (up to the order of addends) in the form x^2 + x^2 for integer x and y iff p=1 (mod 4) or p = 2 (which is a degenerate case with x =1 y = 1 ). The theorem was stated by Fermat, but the first published proof was by Euler.
(Excerpt from Mathworld)
Comments:
Giving credit to Euler, it seems that he proved a lot of theorems regarding different categories of primes. I wonder if there is a chance that he would have proven the Riemann Hypothesis given it was formulated at his time.
Links
Friends
ARCHIVES
| Powered by TagBoard Message Board |