# [Maxima] differential forms

Gosei Furuya go.maxima
Wed Nov 1 01:52:56 CST 2006

```hi all
maybe I can't post new topic this mailing -list?
what has happend.

PREFACE

This is a differential form package slightly singular.
This package is intended to provide a working konwledge
through calulating differntial forms for students finished advanced calc.
In short,not to gather formulars, but to calc and induce them from
calculation.
In this field we need various other knowledge about math and physics.
I hope this package is useful for maxima people.
In detail you shoud read nice books such that....

Differential Forms with Applications to the Physical Sciences
written by Harley Flanders ISBN 0-486-66169-5 Dover (reprint)

Differential Forms and Connections
written by R.W.R.Darling ISBN 0-521-46800-0 Cambridge university Press

The Geometry of Physics An Introduction
written by Theodore Frankel ISBN 0-521-53927-7 Cambridge university Press

Differential Forms
written by Henri Cartan ISBN 0-486-45010-4 Dover (reprint)

Geometrical methods of mathematical physics
written by Bernand F.Schutz  Cambridge university Press

I think this package is intermediate between vect package and tensor package

in maxima.  So in this time,I did not implement bundle spaces, though
especially
need S1-bundle. I don't like too formalism such as other CAS's diff-form
package.
Because nonusability of these package prevents from using bundle spaces,as
bundle is a natural tool for peoples using it for thinking.
My todo-list is to implement bundles with natural usability.
I welcome All coments, advices,bug reports.to maxima mailing list
or me.

A little new thing I implemented is to use clifford operator for seeking
integral factor.
If exsist,(f^-1 is integral factor),d(w)=f^(-1)df @ w. then @--->&(clifford)
d(w)/w is possible,becase we calculate w & w -->A (some numbers),in clifford
algebra
1/w=w/A. d(w)/w---> d(w)&w/A. when  &-->@,use some quantization,@ is
independent from metric.
when calculating with &,we introduce metric parameta automatically
for example u1,u2,u3,such that dx&dx=u1,...then u1-->0,u2-->0,u3=\=0,&-->@.

This calculation is illegal only within  differential forms,
but regal clifford-grassman algebra. Usually in such a case we use an Ideal
on differntial forms.
But I think it is essential that limit of metric breakes clifford
structure,but cannot affect
grassman structure,so after this limit, we can change & to @ .
See example.txt.

hodge star operator is written with clifford algebra in definition,but user
need not to
recognize that. hodge star is named h_st().
poisson braket with symplectic form is much useful,if you may explicitly
give hamilton
operator.
To multiple unit pseudo scalar is almost same hodge star.(at most,differnt
sign + or -)
this J() is definded and used to define antidifferential operater antid().

INTRODUCTION

There is two way for starting.
One way is batch("cartan_init.bat"). This mean to use no grobal coords,basis
others.
As it were clean start. All work may be done in f_star() or
fstar_with_clf().
you  can change coords freely.
Other way is batch("new_cartan_test4.bat").This mean to use global
coords,basis.
But to change coords locally is allways possible in f_star() or
fstar_with_clf().
A example of this grobal way is seen in lorentz_example.txt

It is important for using this package well that we awake to distinction
between
global variables and local variables. for example
(%i21) fstar_with_clf([x,y,z],[x,y,z],(r:x^2+y^2+z^2,d((x*Dy at Dz+y*Dz at Dx+z*
Dx at Dy)/(sqrt(r)))));

for usage
f_star(coordinates,calc area)
fstar_with_clf(new coordinates,representing standered coordinate with new
one,calc area)

now basis is[Dx,Dy,Dz],norm_table,scale_factor ,all this local. but
r:x^2+y^2+z^2 is not
local,only r is global. (%i22) format(%,%poly(Dx,Dy,Dz),factor);
after this,if foget that r was global,
(%i23)
fstar_with_clf([r,phi,th],[r*sin(th)*cos(phi),r*sin(th)*sin(phi),r*cos(th)],
d(x)@d(y)@d(z)));
error #0: f_star(newcoords=[z^2+y^2+x^2,phi,th],.......
so (%i24) kill(r); then (%i23) is OK.
nest2([d,h_st,d],f*d(x)) is equal d(h_st(d(f*d(x))))
d(x) is equal Dx,so Dx at Dy is d(x)@d(y),but internaly in d(x)@d(y) exterior
derivative are done.

ALL files
cartan_init.bat one initial file ,(only local coordinate environment)

new_cartan_test4.bat another initial file,(global coordinate and local
coords environment)

cartan_new.lisp
derived from share/calculus/cartan.lisp add & operator and others (for
future use)

f_star_test4.mac
main functions f_star(),fstar_with_clf(),inner(),Lie(),these functions run
under any dimension

hodge_test3.mac
hodge star operator h_st() under any dimension

helpfunc.mac
utilities or help function vtof1(),vtof2(),J(),antid(),others

poisson.mac
poisson bracket on simplectic manifold (dim 2*n) p_braket()

frobenius.mac
calculate integral factor.trans_toexact(),see example.txt

curvture2.mac
add_tan(),cross(),make_tan(),only dim 2,or 3,this is experimental,see
surface_example.txt

lorentz_example.txt
surface_example.txt
example.txt

thanks
Gosei Furuya
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