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[MUG] coordinate-free or expression simplification
| [MUG] coordinate-free or expression simplification |
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Author: Charles James Leonardo Quarra Cappiello
Posted: Fri, 08 Nov 2002 14:31:21 -0400
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>> From: "Charles James Leonardo Quarra Cappiello" "charsquarra"
Hi,
Im total newbie to Maple. I'm trying to do a vector calculation that keeps
results expressed in terms of geometric invariants like a.b or ||v||. This
seems to be tricky since Maple define vectors as < , ..> objects, which are
inherently coordinate based. I've tried to do this in the following two
approachs:
1 ) Define an alternate dot product law:
> define ( Multip(x,y) , multilinear, orderless );
now, i want to define it homogeneous in the coefficient field, ie:
> definemore ( Multip(x,y) , Multip( (t::real)*x,y ) = t*Multip( x,y);
which looks good, but i soon (not too soon) realized that i wasnt defining
in any special way x and y, but im not sure how to define a vector space
(where im defining Multip as a function from ( MyVectorSpace X MyVectorSpace
) -> CoefficientField)
Q#1: How i define a vector space or any other entities objects in Maple?
The second way im trying to do this is:
2 ) put to good use the simplify functionality after the coordinate-based
calculation has been done:
In this method i first do all my calculations on an defined but otherway
arbitrary basis with the < Ax, Ay , Az> formalism
Then i try to use simplify adding equations that can be used by the simplify
routine to substitute Ax*Bx + Ay*By + Az*Bz with the variable dotAB, but i
must be doing something naive here, because i get this error:
> simplify ( result , dotAB = Ax*Bx + Ay*By + Az*Bz );
Error, (in simplify/do) invalid simplification command
What im doing wrong?
Greetings,
Charles Quarra
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| [MUG] Re: coordinate-free or expression simplification |
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Author: Bertfried Fauser
Posted: Thu, 14 Nov 2002 10:08:43 +0100
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>> From: Bertfried Fauser "fauser"
Dear Charles!
>> From: "Charles James Leonardo Quarra Cappiello" "charsquarra"
> define ( Multip(x,y) , multilinear, orderless );
Note: The define facility of maple miscomputes at least in mapleV, maple6
and 7 (we are still testing maple 8). Hence it is dangerouse to use this
device. But you may look at he maple8 group at yahoogroups.com where you
may find a patched define (which was however tested mainly for the
associative,multilinear case only)
> now, i want to define it homogeneous in the coefficient field, ie:
> > definemore ( Multip(x,y) , Multip( (t::real)*x,y ) = t*Multip( x,y);
The patched define allows you also to specify a domain (ring) you want to
calculate over. Hence you have not to fiddle arround with additional
things like making your function linear in the first argument as you did
above.
> (where im defining Multip as a function from ( MyVectorSpace X MyVectorSpace
> ) -> CoefficientField)
> Q#1: How i define a vector space or any other entities objects in Maple?
As far as I know, there is no method in maple to compute with abstract
algebra elements not tied to a basis. We are on the way to develop a
package in that direction, but still there are mathematical and software
problems.
If you seek for a well documented and well tested package which is able to
handle computations as you try to do, and you don't mind to formulate them
in a Clifford language, hence vectors are represented by algebra elements,
you may have a look at the url:
http://math.tntech.edu/rafal/
and look for the package for your maple version. Untill now there are
packages for mapleV,6,7 maple 8 will follow soon. However, this package
internally uses still bases :(
The package comes with a function clisolve, which is able to do
what you desire, you can specify algebraic equations in arbitrary (but in
a basis expanded) element and solve, the equations may be even non-linear,
but then solving is of course not guaranteed.
best
BF.
% Bertfried Fauser Fachbereich Physik Fach M 678
% Universit"at Konstanz 78457 Konstanz Germany
% Phone : +49 7531 883786 FAX : +49 7531 88-4864 or 4266
% E-mail: "Bertfried.Fauser"
% Web : http://clifford.physik.uni-konstanz.de/~fauser
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