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 allows one to declare coframe 1-forms. If the third type of calling sequence is used, then the local coordinates, coframe 1-forms and dimension of the 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.

 

Warning! Only indexed variables can be used as a coframe 1-forms. 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).  

Examples: 

Example 1 

>	restart: with(atlas):

 

Declare forms: 

>	Forms(e[j]=1,phi[i]=1);

 

(2.1.1)

 

Declare vectors: 

>	Vectors(X,Y,Z,E[j]);

 

(2.1.2)

 

Declare functions: 

>	Functions(f=f(x,y,z),F=F(z[i]));

 

(2.1.3)

 

Declare another coframe with 1-forms: (dim = n): 

>	Coframe(e[i],i=1..n);

 

(2.1.4)

 

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

(2.1.5)

 

For function f we have:
'd(f)'=d(f); 

(2.1.6)

 

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

(2.1.7)

 

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

(2.1.8)

 

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

(2.1.9)

 

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

(2.1.10)

 

Example 2 

>	restart: with(atlas):

 

Declare forms: 

>	Forms(e[j]=1,phi[i]=1);

 

(2.2.1)

 

Declare vectors: 

>	Vectors(X,Y,Z,E[j]);

 

(2.2.2)

 

Declare functions: 

>	Functions(F=F(z[i]),z[1]=z[1](x,y));

 

(2.2.3)

 

Declare coframe with 1-forms: (dim = 3): 

>	Coframe(e[j],j=1..3);

 

(2.2.4)

 

Functions F, are declared as follows and ; now : 

>	'd(F)'=d(F);

 

(2.2.5)

 

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

(2.2.6)

 

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

(2.2.7)

 

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

(2.2.8)

 

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

(2.2.9)

 

Example 3 

>	restart: with(atlas):

 

Declare forms: 

>	Forms(e[j]=1);

 

(2.3.1)

 

Declare vectors: 

>	Vectors(X,Y,Z,E[j],Phi[i]);

 

(2.3.2)

 

Declare functions: 

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

 

(2.3.3)

 

Declare constant :
Constants(lambda); 

(2.3.4)

 

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

(2.3.5)

 

Declare frame
Frame(E[i]); 

(2.3.6)

 

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); 

(2.3.7)

 

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

(2.3.8)

 

>

 

Example 4 

>	restart: with(atlas):

 

Declare functions: 

>	Functions(f=f(r,theta,phi));

 

(2.4.1)

 

Declare constant
Constants(r[g]); 

(2.4.2)

 

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

(2.4.3)

 

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

(2.4.4)

 

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]); 

(2.4.5)

 

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))); 

(2.4.6)

 

Declare frame:
Frame(E[i]); 

(2.4.7)

 

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

(2.4.8)

 

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

(2.4.9)

 

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

 

(2.4.10)

 

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

 

(2.4.11)

 

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

 

(2.4.12)

 

Example 5 

>	restart: with(atlas):

 

Declare forms: 

>	Forms(e[j]=1);

 

(2.5.1)

 

Declare vectors: 

>	Vectors(E[j]);

 

(2.5.2)

 

Declare functions: 

>	Functions(f=f(x,y[k],z));

 

(2.5.3)

 

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

(2.5.4)

 

Declare frame:
Frame(E[i]); 

(2.5.5)

 

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

(2.5.6)

 

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

 

(2.5.7)

 

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

 

(2.5.8)

 

Example 6 

>	restart: with(atlas):

 

Declare constant :
Constants(lambda); 

(2.6.1)

 

Declare forms: 

>	Forms(e[j]=1);

 

(2.6.2)

 

Declare vectors: 

>	Vectors(E[j]);

 

(2.6.3)

 

Declare functions: 

>	Functions(h=h(x[k]));

 

(2.6.4)

 

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

(2.6.5)

 

Declare frame:
Frame(E[i]); 

(2.6.6)

 

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

(2.6.7)

 

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

 

(2.6.8)

 

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

 

(2.6.9)

 

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

 

(2.6.10)

 

>

 

See Also:  

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