MaplePrimes - answers and comments on Question, solving equations from equating linear combinations of wedge products

By solve,identity

Markiyan Hirnyk — Mon, 01 Oct 2012 15:10:03 Z

How about this?
> restart; with(difforms):
> defform(a = scalar, b = scalar);
> a*`&^`(e1, e2)+b*`&^`(e2, e3) = `&^`(e1, e2);

#Creation of the differential form.

               a e1 &^ e2 + b e2 &^ e3 = e1 &^ e2
> A := eval(a*`&^`(e1, e2)+b*`&^`(e2, e3) = `&^`(e1, e2), [`&^`(e1, e2) = x]);
#Because the solve,identity command works only with variables.

                      a x + b e2 &^ e3 = x
> solve(identity(A, x), [a, b]);

                        [[a = 1, b = 0]]

Generalization

Markiyan Hirnyk — Mon, 01 Oct 2012 22:47:54 Z

> restart; with(difforms): defform(a = scalar, b = scalar, c = scalar);
> A := 2*a*`&^`(e1, e2)+b*`&^`(e2, e3)+c*`&^`(e4, e3) = `&^`(e1, e2);

2 a e1 &^ e2 + b e2 &^ e3 + c e4 &^ e3 = e1 &^ e2

> indets(A);

{a, b, c, e1, e2, e3, e4, e1 &^ e2, e2 &^ e3, e4 &^ e3}

> B := eval(A, [`&^`(e1, e2) = x, `&^`(e2, e3) = y, `&^`(e4, e3) = z]);

#This step can be programmed too.

                     2 a x + b y + c z = x
> indets(B);

                       {a, b, c, x, y, z}
> C := selectremove(c ->is(c, scalar), indets(B))[2];

                           {x, y, z}
> E := {seq(coeff(lhs(B), C[j]) = coeff(rhs(B), C[j]), j = 1 .. nops(C))};

                    {b = 0, c = 0, 2 a = 1}
> solve(E, selectremove(c -> is(c, scalar) , indets(B))[1]);

PS. > simpform(`&^`(e1, e2) -`&^`(e2, e1));

Many thanks. Could one also do this for

fredbel6 — Tue, 02 Oct 2012 00:43:50 Z

Many thanks.

Could one also do this for more than one variable?

I tried solve(identity(A,{x,y}),[a,b]) but that didn't work.

Can you

Markiyan Hirnyk — Tue, 02 Oct 2012 02:26:48 Z

present the entire code? Have you tried the approach from my comment "Generalization"?

Waiting for your reply. Regard, Markiyan Hirnyk

@Markiyan Hirnyk Oh I am sorry, I missed

fredbel6 — Tue, 02 Oct 2012 17:06:44 Z

@Markiyan Hirnyk Oh I am sorry, I missed your last reply.

It works, thanks! However, when I switch e1 and e2 in A, something goes wrong:

> restart; with(difforms); defform(a = scalar, b = scalar, c = scalar);
> A := 2*a*`&^`(e1, e2)+b*`&^`(e2, e3)+c*`&^`(e4, e3) = `&^`(e2, e1);
       2 a e1 &^ e2 + b e2 &^ e3 + c e4 &^ e3 = e2 &^ e1
> B := eval(A, [`&^`(e1, e2) = x, `&^`(e2, e3) = y, `&^`(e4, e3) = z]);
                  2 a x + b y + c z = e2 &^ e1
> C := selectremove(proc (c) options operator, arrow; is(c, scalar) end proc, indets(B))[2];
                  {e1, e2, x, y, z, e2 &^ e1}
> E := {seq(coeff(lhs(B), C[j]) = coeff(rhs(B), C[j]), j = 1 .. nops(C))};

> solve(E, selectremove(proc (c) options operator, arrow; is(c, scalar) end proc, indets(B))[1]);

He doesn't seem to know that e2&^e1 is the opposite of e1 &^e2, and use of defform does not help.

I am planning to consider many equations between trivectors in 6-dimensional space, so inevitably I will need Maple to know this.

Is this possible?

This works

Markiyan Hirnyk — Tue, 02 Oct 2012 19:04:19 Z

@fredbel6 Up to my PS

> restart; with(difforms): defform(a = scalar, b = scalar, c = scalar); A := 2*a*`&^`(e1, e2)+b*`&^`(e2, e3)+c*`&^`(e4, e3) = `&^`(e2, e1);

       2 a e1 &^ e2 + b e2 &^ e3 + c e4 &^ e3 = e2 &^ e1
> A := lhs(A) = simpform(simpform(-`&^`(e1, e2)+`&^`(e2, e1))+`&^`(e1, e2));

           2 a e1 &^ e2 + b e2 &^ e3 + c e4 &^ e3 =

             (-1)^(wdegree(e2) wdegree(e1)) e1 &^ e2
> B := eval(A, [`&^`(e1, e2) = x, `&^`(e2, e3) = y, `&^`(e4, e3) = z]);


      2 a x + b y + c z = (-1)^(wdegree(e2) wdegree(e1))x
> E := {seq(coeff(lhs(B), j) = coeff(rhs(B), j), j = {x, y, z})};

     { b = 0, c = 0, 2 a = (-1)^(wdegree(e2) wdegree(e1))}

> solve(E, selectremove(proc (c) options operator, arrow; is(c, scalar) end proc, indets(B))[1]);


     { a = 1/2* (-1)^(wdegree(e2) wdegree(e1)), b = 0, c = 0 }

Try to order the terms of your exterior products (You don't tell us how you form these.)
in such a way

> simpform(simpform(-`&^`(`&^`(`&^`(e1, e2), e3), e6)+`&^`(`&^`(`&^`(e6, e2), e1), e3))+
`&^`(`&^`(`&^`(e1, e2), e3), e6));

(-1)^(wdegree(e6) wdegree(e2) + wdegree(e6) wdegree(e1)

   + wdegree(e6) wdegree(e3) + wdegree(e2) wdegree(e1)) &^(e1,

e2, e3, e6).
Maybe, you will need to write a procedure to this end. Good luck!