Vector Identities, Redux and more questions

hi all

We can vector analysis with differential form.
but maxima cartan package is rather insufficient.
It can't do induced mapping nor hodge star operator,so on.
I have rewrited and add these features since three years.
ya,I almost completed it,so at next month i want to add contrib
dir ,newcartan or diff_form.
Neiles vector equations can be treated with exterior derivative.

1.example
(Pontrjaguin index)
(%i74)
f_star([phi,theta],(pu1:sin(phi)*cos(theta),pu2:sin(phi)*sin(theta),pu3:cos(phi),
    pu1*d(pu2)@d(pu3)+pu2*d(pu3)@d(pu1)+pu3*d(pu1)@d(pu2)))$
(%i76) trigsimp(%o74);
(%o76)                       Dphi Dtheta sin(phi)
So this map induce (x,y)-->(pu1,pu2,pu3)  IxI--->S2,boundary 1X1 -->one
point.
%o76 is volume form  in S2,if we can pull back IxI,we integrate this and say

this integral value is n*4pai. n is a integer.
(%i79)
f_star([x,y],(depends([p1,p2,p3],[x,y]),p1*d(p2)@d(p3)+p2*d(p3)@d(p1)+p3*d(p1)@d(p2)));

                dp2 dp3         dp2 dp3              dp1 dp3         dp1 dp3
(%o79) p1 (Dx Dy --- --- - Dx Dy --- ---) + p2 (Dx Dy --- --- - Dx Dy ---
---)
                 dx  dy          dy  dx               dy  dx          dx  dy
                                                    dp1 dp2         dp1 dp2
                                           + (Dx Dy --- --- - Dx Dy --- ---)
p3
                                                    dx  dy          dx  dy

%o79 is written by using vector,(p1,p2,p3).{(d/dx(p1,p2,p3)Xd/dy(p1,p2,p3))*
Dx at Dy}
integrate with IxI,this is homotopy invariant.
generaly speaking,in exchaing coordinates, vector equations are not always
invariant but differntial forms are invariant. so we can calculate forms
with
some suitable or fit coordinate,then by pulling back it  with general ones.

2.differntial geometry with moving frame
we can use matrix with
 matrix_element_mult:lambda([x,y],x at y)$
and exterior derivative d(some matrix). we can do with
Harley Flanders's Differntial Forms with Applications  to the Physical
Sciences
IV,VI,VIII . we can induce Stucture equatons and Integrability conditions
as this book and can solve problems (exercise) with MAXIMA.
thank Barton's pdiff package.

3
 changing integral variable more than two variables,such like
(%i170)
f_star([u,v],(s:u/(v+1),t:u*v/(v+1),exp(-(s+t))*s^(-z)*t^(z-1)*d(s)@d(t)))$
this is from ?(z)*?(1-z)
(%i171) load("coeflist.lisp")$
(%i172) load("format.lisp")$
(%i173) format(%o170,%poly(u,v),factor);
                                          u v      u
                                       - ----- - -----
                               z - 1     v + 1   v + 1
                        Du Dv v      %e
(%o175)                 ------------------------------
                                    v + 1

thank format package.
4.others we can calc integrating factor (but locally)
more physical aprecations will  be made by  college.


conclusion:
|(a,b,c)|^2    <---> (a*Dx+b*Dy+c*Dz)&(a*Dx+b*Dy+c*Dz) (& is clifford
product)
    (a,b,c).(f,g,h)<--->
1/2((a*Dx+b*Dy+c*Dz)&(f*Dx+g*Dy+h*Dz)+(f*Dx+g*Dy+h*Dz)&(a*Dx+b*Dy+c*Dz))
    (a,b,c)X(f,g,h)<--->h_st((a*Dx+b*Dy+c*Dz)@(f*Dx+g*Dy+h*Dz))
                       or -J((a*Dx+b*Dy+c*Dz)@(f*Dx+g*Dy+h*Dz))
    grad(a)        <---> d(a)
    rot(V)         <---> h_st(d(U)) or nest2([h_st,d],U)
    div(V)         <---> h_st(d(h_st(U))) or nest2([h_st,D,h_st],U)
    laplacian(a)   <---> d(h_st(d(a))) or nest2([D,h_st,D],a)
    many equations <---> d(d(...))=0 ,poincare's lennma
    almost every thing <--- differntial form calculation with hodge star
                            operator (or pseudo scalar J in clifford
algebra)

Gosei Furuya



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