Next: Functions and Variables for atensor, Previous: atensor, Up: atensor [Contents][Index]
atensor is an algebraic tensor manipulation package. To use atensor,
type load("atensor"), followed by a call to the init_atensor
function.
The essence of atensor is a set of simplification rules for the
noncommutative (dot) product operator ("."). atensor recognizes
several algebra types; the corresponding simplification rules are put
into effect when the init_atensor function is called.
The capabilities of atensor can be demonstrated by defining the
algebra of quaternions as a Clifford-algebra Cl(0,2) with two basis
vectors. The three quaternionic imaginary units are then the two
basis vectors and their product, i.e.:
i = v j = v k = v . v
1 2 1 2
Although the atensor package has a built-in definition for the
quaternion algebra, it is not used in this example, in which we
endeavour to build the quaternion multiplication table as a matrix:
(%i1) load("atensor");
(%o1) /share/tensor/atensor.mac
(%i2) init_atensor(clifford,0,0,2);
(%o2) done
(%i3) atensimp(v[1].v[1]);
(%o3) - 1
(%i4) atensimp((v[1].v[2]).(v[1].v[2]));
(%o4) - 1
(%i5) q:zeromatrix(4,4);
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
(%o5) [ ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
(%i6) q[1,1]:1;
(%o6) 1
(%i7) for i thru adim do q[1,i+1]:q[i+1,1]:v[i];
(%o7) done
(%i8) q[1,4]:q[4,1]:v[1].v[2];
(%o8) v . v
1 2
(%i9) for i from 2 thru 4 do for j from 2 thru 4 do
q[i,j]:atensimp(q[i,1].q[1,j]);
(%o9) done
(%i10) q;
[ 1 v v v . v ]
[ 1 2 1 2 ]
[ ]
[ v - 1 v . v - v ]
[ 1 1 2 2 ]
(%o10) [ ]
[ v - v . v - 1 v ]
[ 2 1 2 1 ]
[ ]
[ v . v v - v - 1 ]
[ 1 2 2 1 ]
atensor recognizes as base vectors indexed symbols, where the symbol
is that stored in asymbol and the index runs between 1 and adim.
For indexed symbols, and indexed symbols only, the bilinear forms
sf, af, and av are evaluated. The evaluation
substitutes the value of aform[i,j] in place of fun(v[i],v[j])
where v represents the value of asymbol and fun is
either af or sf; or, it substitutes v[aform[i,j]]
in place of av(v[i],v[j]).
Needless to say, the functions sf, af and av
can be redefined.
When the atensor package is loaded, the following flags are set:
dotscrules:true; dotdistrib:true; dotexptsimp:false;
If you wish to experiment with a nonassociative algebra, you may also
consider setting dotassoc to false. In this case, however,
atensimp will not always be able to obtain the desired
simplifications.
Categories: Tensors ·Share packages ·Package atensor
Next: Functions and Variables for atensor, Previous: atensor, Up: atensor [Contents][Index]