December 2008 Archives

Bill DeRouchey (@billder) was the opening presentation at CyborgCamp and was wonderfully entertaining and informative. Here are some quick notes I took:

  • How did the "Play/Pause" button become univerally understood?
  • Typewriters choose the winning "letter symbols"
  • Symbols evolve.
  • Are we losing some types/modes of literacy? Is that such a bad thing?
  • @ - commerce symbol "each at", then location, then identity
  • # - means a group on twitter "context, topic"
  • Mouse pointer ~1984
  • Future directions in the evolution of tech languages?
  • Splintered groups of tech languages
  • Emotional bandwidth
  • Gatekeepers of knowledge, screening out less knowledgeable people

A metric tensor g_{mn} is used to measure distances based on a given coordinate system. In terms of the Jacobian, the metric tensor can be found from g_{mn} = (J^T J)_{mn} where J^T is the transpose of the Jacobian. Since J^T J is a symmetric matrix for any matrix J , the metric tensor is always symmetric. (In fancy-pants math lingo this is called a symmetric bilinear form.) What is the real-life consequences of this? The distance from a to b is always the same as the distance from b to a , no matter what kind of crazy coordinate system you are living in!
If we want to calculate the length of a parameterized curve x^r = x^r(u) where u is a parameter with respect to some coordinate system, then we can write an infintesmal displacement element as dx^r = p^r(u) du . The length of this displacement is ds = \sqrt{g_{mn} p^m p^n} du  and the length of the curve from u=u_1 to u=u_2 is L =  \int_{u_1}^{u_2} ds = \int_{u_1}^{u_2} \sqrt{g_{mn} p^m p^n} du .
So we need the metric tensor to define distance along a curve when we are in non-cartesian coordinate systems, such as spherical or toroidal. From the metric tensor one can then start to study the "curvature" of a coordinate system. More soon!

About this Archive

This page is an archive of entries from December 2008 listed from newest to oldest.

November 2008 is the previous archive.

January 2009 is the next archive.

Find recent content on the main index or look in the archives to find all content.

Clicky Web Analytics Screw you, spammers! 42