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
5 Squared and a Dozen
Well, I did feel hesitated to post on this particular day. Not so much because of the lack of math content, but more so of what the day signifies. Anyway, it's probably not to think too much on thoughts that doesn't lead to logical resolutions. Haha
Merry Christmas =)
End of a Semester
With only Algebra II left, this semester has came to an end. I can probably give the conclusion right now--really an unpleasant struggle. Not with school, but really with myself. It's perhaps a reflection on my personality. Or perhaps something for me to think beyond what's just before me. But, no matter how I look at it, it's still a failure.
I will remember this entry for quite awhile. For now, I should open my algebra books. Because, even a failure deserves a consolatory ending, especially a mathematical one.
Ah well...
Well, i hate to admit that today's circuit theory exam was quite challenging, beyond the fact that i made so many stupid mistakes. Ah well, it's done and gone. Not going to think abt it anymore. Perhaps a lesson in trusting practice does make a difference. Really need this Christmas break to reassure certain things.
I always believed in myself; I still do, and always will.
Uniqueness of this entry
Haha this entry is not about math. but I guess I shall still record it somewhere. so here is the place. just spent half an hr going through in my mind abt my trip in Singapore. Amazingly, I could remember every single day of it. In order too. Wow. Just glanced through my diay for confirmation. Haha, it's such a special feeling. Got to seriously read through my diary about exams are over.
For now, high in spirit, I shall get down to work.
Putnam
Ah today was the Putnam contest, regrettably I didn't write it. Oh well, definitely going to spend the next summer on it and hopefully have a chance for something next year. It has been one of my key goals in university and it will continue to be.
Exams starting next week, got to really get into study mood.
=)
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