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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.mac");
batching #p/home/furuya/sagyo/new_cartan_test4.mac
(%i2) load(cartan_new.lisp)
(%i3) infix(@)
(%i4) infix(&)
(%i5) infix(|)
(%i6) coords : read(Input new coordinate)
Input new coordinate
[x,y,z,t];
(%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) [Dx, Dy, Dz, Dt]
(%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
[1,1,1,-1];
(%o12) [1, 1, 1, - 1]
(%i13) translist : read(represent the standard coordinates with new one)
represent the standard coordinates with new one
[x,y,z,%i*c*t];
(%o13) [x, y, z, %i c t]
(%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, 1, 1, --]
2
c
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, 1, 1, c]
(%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
(%i38) h_st(Dx@c*Dt);
(%o38) Dy Dz
(%i39) h_st(Dx@Dy);
(%o39) c Dt Dz
(%i40) h_st(Dz@c*Dt);
(%o40) Dx Dy
(%i41) h_st(Dy@c*Dt);
(%o41) - Dx Dz
(%i42) h_st(Dz@Dx);
(%o42) c Dt Dy
(%i43) h_st(Dy@Dz);
(%o43) c Dt Dx
/* [e1,e2,e3] is electric field,[b1,b2,b3]is magnetic field
in free space Maxwell equation d(F)=0 and d(h_st(F))=0 */
(%i44) depends([e1,e2,e3],[x,y,z,t]);
(%o44) [e1(x, y, z, t), e2(x, y, z, t), e3(x, y, z, t)]
(%i45) depends([b1,b2,b3],[x,y,z,t]);
(%o45) [b1(x, y, z, t), b2(x, y, z, t), b3(x, y, z, t)]
(%i46) F:(e1*Dx+e2*Dy+e3*Dz)@(c*Dt)+(b1*Dy@Dz+b2*Dz@Dx+b3*Dx@Dy);
(%o46) c Dt Dz e3 + c Dt Dy e2 + c Dt Dx e1 + b1 Dy Dz - b2 Dx Dz + b3 Dx Dy
(%i47) d(F)$
(%i48) format(%o47,%poly(Dx,Dy,Dz),factor);
de3 de2 db1 de3 de1 db2
(%o48) Dt Dy Dz (c --- - c --- + ---) + Dt Dx Dz (c --- - c --- - ---)
dy dz dt dx dz dt
de2 de1 db3 db3 db2 db1
+ Dt Dx Dy (c --- - c --- + ---) + (--- + --- + ---) Dx Dy Dz
dx dy dt dz dy dx
/*to translate this relation to vector ,we introduce d_2 operator which is exterior
derivative with space variables only. it is easy to define this.
or like as %i152,can use local d operator,maybe this is more confortable */
(%i49) d_2(_pform):=sum(basis[i]@(diff(_pform,coords[i])),i,1,dim-1);
(%o49) d_2(_pform) := sum(basis @ diff(_pform, coords ), i, 1, dim - 1)
i i
(%i50) d_2(e1*Dx+e2*Dy+e3*Dz);
de3 de3 de2 de2 de1 de1
(%o50) Dy Dz --- + Dx Dz --- - Dy Dz --- + Dx Dy --- - Dx Dz --- - Dx Dy ---
dy dx dz dx dz dy
(%i51) format(%,%poly(Dx,Dy,Dz),factor);
de3 de2 de3 de1 de2 de1
(%o51) Dy Dz (--- - ---) + Dx Dz (--- - ---) + Dx Dy (--- - ---)
dy dz dx dz dx dy
/* rot(E) <---->h_st(d(E)),div(B) <----> d(h_st(B))
and X=0 <--->h_st(X)=0 */
(%i52) fstar_with_clf([x,y,z],[x,y,z],d(e1*Dx+e2*Dy+e3*Dz));
de3 de3 de2 de2 de1 de1
(%o52) Dy Dz --- + Dx Dz --- - Dy Dz --- + Dx Dy --- - Dx Dz --- - Dx Dy ---
dy dx dz dx dz dy
(%i53) fstar_with_clf([x,y,z],[x,y,z],h_st(d(e1*Dx+e2*Dy+e3*Dz)))$
(%i54) format(%,%poly(Dx,Dy,Dz),factor);
de3 de2 de3 de1 de2 de1
(%o54) Dx (--- - ---) - Dy (--- - ---) + Dz (--- - ---)
dy dz dx dz dx dy
(%i55) h_st(%);
de3 de3 de2 de2
(%o55) c Dt Dy Dz --- + c Dt Dx Dz --- - c Dt Dy Dz --- + c Dt Dx Dy ---
dy dx dz dx
de1 de1
- c Dt Dx Dz --- - c Dt Dx Dy ---
dz dy
(%i56) format(%,%poly(Dx,Dy,Dz),factor);
de3 de2 de3 de1 de2 de1
(%o56) c Dt Dy Dz (--- - ---) + c Dt Dx Dz (--- - ---) + c Dt Dx Dy (--- - ---)
dy dz dx dz dx dy
(%i57) diff(b1*Dx+b2*Dy+b3*Dz,t);
db3 db2 db1
(%o57) --- Dz + --- Dy + --- Dx
dt dt dt
(%i58) h_st(%);
db1 db2 db3
(%o58) --- c Dt Dy Dz - --- c Dt Dx Dz + --- c Dt Dx Dy
dt dt dt
/* from d(F)=0,See %o48,and we find two part
1/c*%o58+%o56 ,div(B).each of them is zero. 1/c*%o58+%o56 is equal
rot(E)+1/c*d(B)/dt (Faraday's law),%o60 is equal div(B) (=0,nonexistence magnetism) */
(%i59) %o48 -(1/c*%o58+%o56)$
(%i60) format(%,%poly(Dx,Dy,Dz),factor);
db3 db2 db1
(%o60) (--- + --- + ---) Dx Dy Dz
dz dy dx
(%i61) fstar_with_clf([x,y,z],[x,y,z],d(h_st(b1*Dx+b2*Dy+b3*Dz)));
db3 db2 db1
(%o61) --- Dx Dy Dz + --- Dx Dy Dz + --- Dx Dy Dz
dz dy dx
/*you may continue d(h_st(F)),see Flanders "Differential Forms with Applications
to the Physical Sciences"4.6 Maxwell's Field Equations P45~P47.
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