From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: HELP: computing GRADIENT of f:(R x T2) --> R
Date: 29 Jan 1996 16:28:22 GMT
Keywords: gradient, manifolds

In article <4e33gf$et5@decaxp.harvard.edu>,
Zorro  <berriz@husc.harvard.edu> wrote:
>I'm working with a differentiable mapping f from the set R x T2 (the
>Cartesian product of the real line with the 2-torus) to R.  I need the
>gradient of f, but I don't know how to compute it.  

I suppose I would say that the _derivative_ is a certain linear map, which
makes it conceptually clear but computationally murky; the _gradient_ is
a matrix representation of the derivative, making it computationally
straightforward but conceptually awkward. If you really want the
gradient, you need to choose local coordinates for your manifolds; then grad f
is just the ordinary gradient. But of course you could choose different
coordinate systems and change the result of your computation.

One coordinate system for  M=R x T2  is the universal covering  p: R^3 -> M;
if you think of the torus as the product of two unit circles each in its
own copy of  R^2, then  M is the subset of  R^5  which is the image of
	p(t, u, v)=(t, cos(u), sin(u), cos(v), sin(v) ).
So now if you have a function  f: M -> R, it is perhaps a function of 5
variables which is defined at least on this subset. (Or perhaps it is
defined in some other manner; it doesn't really matter. What does matter is:)
You can then take the composite of  f  with  p  to get a function
g = f o p  defined on  (a region of)  R^3  and with codomain = R. Then
the gradient of  f  is the ordinary gradient of  g.

Of course this assumes you're willing to identify the tangent space of  M
at a point with the image of the tangent space at the corresponding point
in  R^3. As I said, the computations are straightforward, but if you want
to assign some meaning to the computations you're perhaps better off just
thinking about the derivative, a linear map between tangent spaces.

dave

==============================================================================

From: Nils Goesche <goesche@sfb288.math.tu-berlin.de>
Newsgroups: sci.math
Subject: Re: HELP: computing GRADIENT of f:(R x T2) --> R
Date: Tue, 30 Jan 1996 11:46:12 +0100

Dave Rusin wrote:
> 
> In article <4e33gf$et5@decaxp.harvard.edu>,
> Zorro  <berriz@husc.harvard.edu> wrote:
> >I'm working with a differentiable mapping f from the set R x T2 (the
> >Cartesian product of the real line with the 2-torus) to R.  I need the
> >gradient of f, but I don't know how to compute it.
> 
If you really want to define a general notion of gradient on a manifold
you have to choose a semi-Riemannian metric tensor g_ij.
Then you may define grad f := g^ij * df/du^j * d/du^i , which is a
vector uniquely determined by the equation grad f . X = X(f), where
X is any tangentvector, regarded as a derivation in the space of germs
of functions.
-- 
Nils Goesche