Re: sufficient condition for smooth map to be rigid motion
- From: israel@xxxxxxxxxxx (Robert Israel)
- Date: 14 Mar 2006 09:09:58 GMT
In article <441605AC.8020306@xxxxxxxxxxxxxxxxx>,
Stephen Montgomery-Smith <stephen@xxxxxxxxxxxxxxxxx> wrote:
Dave Rusin wrote:
In article <wICQf.19297$oL.5314@attbi_s71> you write:
Suppose that I have a smooth map F on some open set in R^n, such that
the derivative DF(x) is always an orthogonal matrix. Can one deduce
that F is a rigid motion (translation combined with rotation or reflection)?
FWIW, Maple reports that this is true when n=2 :
pdsolve((diff(u(x,y),x))^2+(diff(u(x,y),y))^2 = 1, u(x,y));
(u(x, y) = _F1(x) + _F2(y)) &where
d /d \2 2
[{-- _F2(y) = _c[2], |-- _F1(x)| = -_c[2] + 1}]
dy \dx /
pdsolve does not claim to find the most general solution to a PDE.
It tries to find closed-form solutions (if there are any) of various
forms, e.g. separation of variables solutions of the form F1(x)*F2(y)
or F1(x) + F2(y).
Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.
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- Re: sufficient condition for smooth map to be rigid motion
- From: Stephen Montgomery-Smith
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