If I'm going to spend lots of time explaining a concept in math
and physics to someone (even myself!), I may as well make the
explanation public so that others can benefit as well. There
isn't a lot here yet, but I hope to keep adding to this section
over time.
I call these "tutorials", but the name doesn't always fit
perfectly. The tutorials are at very different levels: each
assumes that you have learned enough background to be "ready" for
the topic. (Thus, I don't attempt to explain what a gradient is
in the Lagrange multipliers tutorial, and I don't attempt to
explain basic differential geometry in the "GR with Torsion"
document.)
Tutorials
- Lagrange multipliers: a method
for finding extrema of functions of several variables when the
solution must satisfy a set of constraints, and for the
analogous problem in the calculus of variations (often used in
physics when studying Lagrangian mechanics).
- An Introduction to
String Theory: Not precisely a tutorial, this is a talk
that I gave in Feb. 2004 to the Chicago chapter of the MIT
alumni club. It's aimed at an audience who've had a year or
two of college physics, possibly a long time ago, and who
want to see at least a few of the equations of string theory
(presumably without being overwhelmed). I don't know if
that's possible, but I tried.
- General Relativity with
Torsion (200 KB PDF; 18 pages): Most applications of
differential geometry, including general relativity, assume that
the connection is "torsion free": that vectors do not rotate
during parallel transport. Because some extensions of GR (such
as string theory) do include torsion, it is useful to see how
torsion appears in standard geometrical definitions and formulas
in modern language. In this review article, I step through
chapter 3, "Curvature", of Robert Wald's textbook General
Relativity and show what changes when the torsion-free
condition is relaxed.
Up to my teaching page.
Up to my professional page.
My personal site is also available.
Any questions or comments? Write to me: jensens@alma.edu
Copyright © 2006 by
Steuard Jensen.