From: foltinek@math.utexas.edu (Kevin Foltinek)
Subject: Re: Help needed with Lie Derivatives.
Date: 04 Feb 2000 12:46:08 -0600
Newsgroups: sci.math
Summary: [missing]

In article <389869FC.6895BFFC@dartmouth.edu> "Ashok. R"
<navigator@dartmouth.edu> writes:

> Can someone explain to me what a Lie derivative in really simple
> terms? Say I have a function that is f(t) = sin(10t), how do I take
> the Lie Derivative of it?

The Lie derivative (it's pronounced "LEE") involves differentiation
with respect to a vector field.  Without going into details, and not
stating all the technical assumptions:
Any vector field X gives rise to a flow F_t.  This flow is, for any t,
a diffeomorphism; and F_0 is the identity mapping.  The fact that it's
a diffeomorphism means that you can use it to compare something at two
different points.
The idea is the following.  Suppose you want to compute the Lie
derivative of some object (function, tensor) G, at the point p.  Then
you compute q=F_t(p), and look at the value G(q).  Because F_t is a
diffeomorphism, you can "pull back" G(q) to p, and compare it with
G(p).  Then take the difference and divide by t, let t->0, and this is
the Lie derivative of G with respect to X at p.  It's usually denoted
L_X G(p).

There are nice formulas for actually computing these things.  One of
them is that, for any real-valued function f, L_X f = df(X).  But
don't be misled: in general, the Lie derivative depends on the
derivative of X as well as of what you are differentiating.  For an
exterior form G, for example, L_X G = d(X_|G) + X_|dG (where "_|" is
the interior product, or contraction).

Kevin.