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Maxima 5.9.3 http://maxima.sourceforge.net
Using Lisp GNU Common Lisp (GCL) GCL 2.6.7 (aka GCL)
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
This is a development version of Maxima. The function bug_report()
provides bug reporting information.
(%i1) batch("new_cartan_test4.bat");
batching #p/home/furuya/sagyo/new_cartan_test4.bat
(%i2) load(cartan_new.lisp)
(%i3) infix(@)
(%i4) infix(&)
(%i5) infix(|)
(%i6) coords : read(Input new coordinate)
Input new coordinate
[a,b];
(%i7) dim : extdim : length(coords)
(%i8) basis : VR : extsub : extsubb : []
1 1
(%i9) for i thru dim do basis : endcons(concat(D, coords ), basis)
i
(%i10) for i thru dim do (extsub : cons(basis = - basis , extsub ),
1 + i i i i
extsubb : cons(basis = 0, extsub ), ci : concat(v, i), VR : endcons(ci, VR))
i i i
(%i11) basis
(%o11) [Da, Db]
(%i12) cliffordtype : read(please input metric type,for example [1,1,1],if E3)
please input metric type,for example [1,1,1],if E3
[1,1];
(%o12) [1, 1]
(%i13) translist : read(represent the standard coordinates with new one)
represent the standard coordinates with new one
[(2+cos(a))*cos(b),(2+cos(a))*sin(b),sin(a)];
(%o13) [(cos(a) + 2) cos(b), (cos(a) + 2) sin(b), sin(a)]
(%i14) norm_table : []
(%i15) scale_factor : []
(%i16) _l : []
(%i17) for i thru dim do (_l : map(lambda([x], diff(x, coords )), translist),
i
2
_l : map(lambda([x], x ), _l), _p : ratsimp(trigsimp(apply(+, _l))),
cliffordtype
i
norm_table : endcons(-------------, norm_table))
_p
(%i18) norm_table
1
(%o18) [1, ----------------------]
2
cos (a) + 4 cos(a) + 4
norm_table
i
(%i19) for i thru dim do extsubb2 : cons(basis = -----------, extsub )
i i basis i
i
2 1
(%i20) for i thru dim do (a_ : solve(x_ - -----------, [x_]),
norm_table
i
scale_factor : cons(rhs(a_ ), scale_factor))
2
(%i21) scale_factor : reverse(scale_factor)
(%i22) scale_factor
(%o22) [1, cos(a) + 2]
(%i23) nest2(_f, _x) := block([_a : [_x], i],
if listp(_f) then (_f : reverse(_f), for i thru length(_f)
do _a : map(_f , _a)) else _a : map(_f, _a), _a )
i 1
(%i24) nest3(_f, _x, _n) := block([_a, i], _a : [_x],
for i thru _n do _a : map(_f, _a), _a)
2
(%i25) aa_ : solve(x_ - apply(*, norm_table), [x_])
(%i26) volume : rhs(aa_ )
2
1
(%i27) volume : ------
volume
(%i28) matrix_element_mult : lambda([x, y], x @ y)
(%i29) load(hodge_test3.mac)
(%i30) load(f_star_test4.mac)
(%i31) load(helpfunc.mac)
(%i32) load(coeflist.lisp)
(%i33) load(format.lisp)
(%i34) load(diag)
(%i35) load(poisson.mac)
(%i36) load(frobenius.mac)
(%i37) load(curvture2.mac)
(%o38) new_cartan_test4.bat
/*[x,y,z]=translist,this case parametrized torous by [a,b] */
(%i38) d(translist);
(%o38) [- cos(a) sin(b) Db - 2 sin(b) Db - sin(a) cos(b) Da,
cos(a) cos(b) Db + 2 cos(b) Db - sin(a) sin(b) Da, cos(a) Da]
(%i39) m:coefmatrix(%,basis);
[ - sin(a) cos(b) (- cos(a) - 2) sin(b) ]
[ ]
(%o39) [ - sin(a) sin(b) (cos(a) + 2) cos(b) ]
[ ]
[ cos(a) 0 ]
(%i40) trigsimp(m.diag([1,1/(cos(a)+2)]));
[ - sin(a) cos(b) - sin(b) ]
[ ]
(%o40) [ - sin(a) sin(b) cos(b) ]
[ ]
[ cos(a) 0 ]
(%i41) trigsimp(add_tan(%));
[ - sin(a) cos(b) - sin(b) - cos(a) cos(b) ]
[ ]
(%o41) [ - sin(a) sin(b) cos(b) - cos(a) sin(b) ]
[ ]
[ cos(a) 0 - sin(a) ]
(%i42) transpose(%).d(%)$
(%i43) trigsimp(%);
/*this is in Flanders P41 but transposed */
[ 0 sin(a) Db - Da ]
[ ]
(%o43) [ - sin(a) Db 0 - cos(a) Db ]
[ ]
[ Da cos(a) Db 0 ]
(%i44) map("*",scale_factor,basis);
/*this is [sigma1,sigma2] in Flanders P41 but transposed */
(%o44) [Da, (cos(a) + 2) Db]
(%i45) basis;
(%o45) [Da, Db]
(%i46) scale_factor;
(%o46) [1, cos(a) + 2]
/*Gaussian curvature K is K*sigma1@aigma2=w1@w2
now w1=-Da,w2=-cos(a)*Db,and %o44 so K=cos(a)/(2+cos(a))
0<=a<2%pi,0<=b<2%pi,integrate K on this torous,
cos(a)/(2+cos(a)) *(2+cos(a))*Da@Db is equal cos(a)Da@Db
on 0<=a<2%pi,0<=b<2%pi ,apparently this is 0.
Theory says integrate K on total surface is 2*%pi*
is euler character,torous's character is 0 */
/* d()=-. ,this style is Darling,Flanderse style is transpose
transpose(d(omega))=d(transpose(omega))=-(transpose(omega).transpose(omega))
obviously AL:transpose(omega)=-omega,d(AL)=AL.AL */
(%i47) d(%o43)+%o43.%o43;
[ 0 0 0 ]
[ ]
(%o47) [ 0 0 0 ]
[ ]
[ 0 0 0 ]
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