atlas[Coframe] - declaration of coframe 1-forms

Calling Sequence:

     Coframe(Id[j], j=1..n)
     Coframe(Id[j]=expr[j], j=1..n)
     Coframe(Id[1]=expr[1], Id[2]=expr[2], ...Id[n]=expr[n])

Parameters:

      Id - identifier for indexed variable - the coframe 1-forms
      n - dimension of working manifold  (a variable or integer)
      Id[i] = expr[i] - equation where Id[i] is indexed variable - coframe 1-form and expr[i] is decomposition of the 1-form on exact 1-forms.  

Description:

The Coframe  procedure declares coframe 1-forms. If third type of calling sequence is used then local coordinates, coframe 1-forms and dimension of working manifold have been defined automatically.

The Coframe procedure can be used in tree ways:

• Coframe(Id[j], j=1..n) - sets Id[j] as a coframe 1-forms e.g. Coframe(e[j],j=1..n) - sets 1-forms  as a coframe, here n is dimension of working manifold i.e. dim=n  (see atlas[dim] ); n can be variable or integer.
• Coframe(Id[j]=f[j], j=1..n) - sets Id[j] as a coframe 1-forms so that Id[j]=f[j] where f[j] is some 1-form or liner expression on 1-forms (see examples below), here n is dimension of working manifold i.e. dim=n ; n can be variable or integer.
• Coframe(Id[1]=expr1, Id[2]=expr2, ...Id[n]=exprn) -sets Id[1], Id[2], ..Id[n] as coframe 1-forms e.g. Coframe(e[1]=d(x), e[2]=d(y)) sets 1-forms  as a coframe which is  where x, y are 0-forms.
• Only indexed variable can be used as a coframe 1-form. The name of the indexed variable must be one and the same for all coframe 1-forms e.g.   or  and so on. To declare coframes such as  use procedure alias  (see example 4 below).

Example 1
restart:
with(atlas):

Declare forms:
Forms(e[j]=1,phi[i]=1);

Declare vectors:
Vectors(X,Y,Z,E[j]);

Declare functions:
Functions(f=f(x,y,z),F=F(z[i]));

Declare another coframe with 1-forms:  (dim = n):
Coframe(e[i],i=1..n);

Function F is declared as follows ; now :
'd(F)'=d(F);

For function f we have:
'd(f)'=d(f);

Declare frame vectors (see atlas[Frame] ):
Frame(E[i]);

"To basis" decomposition:
Y=ToBasis(Y);

Interior product of vector Y and 1-form e[j]:
iota[Y](e[j])=iota[ToBasis(Y)](e[j]);

"To basis" decomposition:
phi[j]=ToBasis(phi[j]);

Example 2
restart:
with(atlas):

Declare forms:
Forms(e[j]=1,phi[i]=1);

Declare vectors:
Vectors(X,Y,Z,E[j]);

Declare functions:
Functions(F=F(z[i]),z[1]=z[1](x,y));

Declare coframe with 1-forms:  (dim = 3):
Coframe(e[j],j=1..3);

Functions F,  are declared as follows  and ; now :
'd(F)'=d(F);

Declare frame vectors (see atlas[Frame] ):
Frame(E[i]);

"To basis" decomposition:
Z=ToBasis(Z);

Interior product of vector Z and 1-form e[j]:
iota[Z](e[j])=iota[ToBasis(Z)](e[j]);

"To basis" decomposition:
phi[j]=ToBasis(phi[j]);

Example 3
restart:
with(atlas):

Declare forms:
Forms(e[j]=1);

Declare vectors:
Vectors(X,Y,Z,E[j],Phi[i]);

Declare functions:
Functions(f=f(x,y,z),F=F(z[i]),z[1]=z[1](x,y),h=h(x[k]));

Declare constant :
Constants(lambda);

Declare coframe:
Coframe(e[1]=d(x),e[2]=d(y),e[3]=d(z));

Declare frame
Frame(E[i]);

Declare metric (see atlas[Metric] ):
Metric(g=4*(d(x)&.d(x)+d(y)&.d(y)+d(z)&.d(z))/(1+lambda*(x^2+y^2+z^2))^2);

Simple calculation:
'iota[E[k]](g)'=iota[E[k]](g);

Example 4
restart:
with(atlas):

Declare functions:
Functions(f=f(r,theta,phi));

Declare constant
Constants(r[g]);

Declare vectors:
Vectors(L1,N,M,K,E[j]);

Declare forms:
Forms(u[j]=1,l=1,n=1,m=1,k=1,x=1,y=1,z=1);

Using alias for vectors and forms:
alias(L1=E[1],N=E[2],M=E[3],K=E[4],l=u[1],n=u[2],m=u[3],k=u[4]);

Declare coframe:
Coframe(l=1/2*((1-r[g]/r)*d(t)+d(r)),
        n=d(t)-1/(1-r[g]/r)*d(r),
        m=r/sqrt(2)*(d(theta)-I*sin(theta)*d(phi)),
        k=r/sqrt(2)*(d(theta)+I*sin(theta)*d(phi)));

Declare frame:
Frame(E[i]);

Declare metric:
Metric( g=u[1]&.u[2]+u[2]&.u[1]-u[3]&.u[4]-u[4]&.u[3] );

Simple calculations:
'd(phi)&^d(theta)'=d(phi)&^d(theta);

'd(r)'=d(r);

'd(f)'=d(f);

'iota[E[k]](g)'=iota[E[k]](g);

Example 5
restart:
with(atlas):

Declare forms:
Forms(e[j]=1);

Declare vectors:
Vectors(E[j]);

Declare functions:
Functions(f=f(x,y[k],z));

Declare coframe:
Coframe(e[k]=d(y[k]),k=1..n);

Declare frame:
Frame(E[i]);

Simple calculations:
'd(e[i])'=d(e[i]);

'd(y[i]*y[j])'=d(y[i]*y[j]);

'd(f)'=d(f);

Example 6
restart:
with(atlas):

Declare constant :
Constants(lambda);

Declare forms:
Forms(e[j]=1);

Declare vectors:
Vectors(E[j]);

Declare functions:
Functions(h=h(x[k]));

Declare coframe:
Coframe(e[j]=d(x[j])/(1+lambda*Sum(x[i]^2,i=1..n)),j=1..n);

Declare frame:
Frame(E[i]);

Simple calculations:
'd(e[j])'=normal(d(e[j]));

'd(x[k])'=d(x[k]);

'd(h)'=d(h);

'iota[E[j]](d(e[k]))'=normal(iota[E[j]](d(e[k])));

See Also:

atlas , atlas[Frame] , atlas[ToBasis] , atlas[Metric] .