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
As epsilon goes to zero
Well, about time to conclude my two weeks break and perhaps the semester that just passed. I actually know not of what to write. I wonder if I have really put myself to the test or the test is flawed itself. Perhaps I will never know. It takes a lot of confidence to stay in my condition after so long, but perhaps I have too much of it. Like the irony of larger stars burns faster for example.
I do not know of what the next step will be, but there are always choices to be made. There are times where I have doubts of my choice of path thus far. But maybe I have moved to0 far, or maybe I just deep down wants to stay on course. Whatever the case, I offer no resolution.
It seems I have reached the point where this entry shall end meaninglessly. Well I hope I can add some meaning to it in the near future. For now, as long as I know epsilon is not zero, that's all it matters.
http://pennystockinvestment.blogspot.com
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