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atlas™ - modern differential geometry for Maple™
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atlas[`&$`] - generalized interior product operator

Calling Sequence:

     `&$`(A1, A2, ..., An, expr)

Parameters:

         expr - any expression.
        A1, A2, ..., An - vectors or 1-forms.

Description:

  • The `&$`  - procedure calculates the generalized interior product (see atlas[iota] ) of given expression and vector or 1-forms fields.
  • Let X be a vector and Tbe a tensor then under definition: `&$`(X,T) = iota[X](T)  
  • Let omegabe 1-form and Xbe a vector then under definition: `&$`(omega,X) = iota[X](omega)  
  • For any vectors or 1-forms X[1], X[2], `` .. ``, X[j]multiple `&$` operator defined as follows:   `&$`(X[1],X[2] .. X[j],T) = `&$`(X[j],`` .. ``,`&$`(X[2],`&$`(X[1],T)))
  • For any vector or 1-form Xand any tensors Tand Omegathe following rule takes place: `&$`(X,`&.`(T,Omega)) = `&.`(`&$`(X,T),Omega)

Examples:
restart:
with(atlas):

Declare constants:
Constants(alpha);

{Catalan, _Z, Pi, I, alpha, -I}

Declare p-forms:
Forms(e[i]=1,omega=2,omega[1]=p,omega[2]=q);

{omega, e[i], omega[1], omega[2]}

Declare vectors:
Vectors(X,Y,Z);

{X, Y, Z}

Using `&$` - procedure:

Just definition for "long" operator:
'`&$`(X,Y,Z,omega[1])'=`&$`(X,Y,Z,omega[1]);
'`&$`(e[i],e[j],e[k],`&.`(X,Y,Z,Y))'=`&$`(e[i],e[j],e[k],`&.`(X,Y,Z,Y));

`&$`(X,Y,Z,omega[1]) = iota[Z](iota[Y](iota[X](omega[1])))

`&$`(e[i],e[j],e[k],`&.`(X,Y,Z,Y)) = iota[X](e[i])*iota[Y](e[j])*iota[Z](e[k])*Y

Generalize interior product is linear with respect to any argument:
'`&$`(alpha*X+F*Y+x*Z,e[j])'=`&$`(alpha*X+F*Y+x*Z,e[j]);

`&$`(alpha*X+F*Y+x*Z,e[j]) = alpha*iota[X](e[j])+F*iota[Y](e[j])+x*iota[Z](e[j])

'`&$`(alpha*e[1]+F*e[2]+x*e[3],X)'=`&$`(alpha*e[1]+F*e[2]+x*e[3],X);

`&$`(alpha*e[1]+F*e[2]+x*e[3],X) = alpha*iota[X](e[1])+F*iota[X](e[2])+x*iota[X](e[3])

And
'`&$`(X,F*e[j]+alpha*e[k]+x*e[l])'=`&$`(X,F*e[j]+alpha*e[k]+x*e[l]);

`&$`(X,F*e[j]+alpha*e[k]+x*e[l]) = F*iota[X](e[j])+alpha*iota[X](e[k])+x*iota[X](e[l])

'`&$`(F*e[j]+alpha*e[k]+x*e[l],X)'=`&$`(F*e[j]+alpha*e[k]+x*e[l],X);

`&$`(F*e[j]+alpha*e[k]+x*e[l],X) = F*iota[X](e[j])+alpha*iota[X](e[k])+x*iota[X](e[l])

'`&$`(e[k],Y,`&.`(X,e[i],e[j]))'=`&$`(e[k],Y,`&.`(X,e[i],e[j]));

`&$`(e[k],Y,`&.`(X,e[i],e[j])) = iota[X](e[k])*iota[Y](e[i])*e[j]

Example 1
restart:
with(atlas):

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

{e[j], xi}

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

{X, Y, Z, E[j]}

Declare coframe:
Coframe(e[1]=x*d(x)+y*d(y),e[2]=x*d(y)-y*d(x));

[e[1] = x*d(x)+y*d(y), e[2] = x*d(y)-y*d(x)]

Declare frame:
Frame(E[i]);

[E[1] = 1/(y^2+x^2)*x*Diff(``,x)+1/(y^2+x^2)*y*Diff(``,y), E[2] = -1/(y^2+x^2)*y*Diff(``,x)+1/(y^2+x^2)*x*Diff(``,y)]

Connection definition:
omega[1,1]:=x*e[1];

omega[1,1] := x*e[1]

omega[2,2]:=y*e[2];

omega[2,2] := e[2]*y

omega[1,2]:=y*e[1];

omega[1,2] := y*e[1]

omega[2,1]:=-x*e[2];

omega[2,1] := -x*e[2]

Connection declaration:
Connection(omega);

omega[i,j]

Curvature calculation:
Curvature(Omega);

Omega[i,j]

Riemann calculation:
Riemann(R);

R = -1/2*y*(-1+x*y^2+x^3)/(y^2+x^2)*`&.`(E[1],e[1],`&^`(e[1],e[2]))+1/2*x*(-3+x*y^2+x^3)/(y^2+x^2)*`&.`(E[2],e[1],`&^`(e[1],e[2]))+1/2*(-x+y^4+y^2*x^2)/(y^2+x^2)*`&.`(E[1],e[2],`&^`(e[1],e[2]))+1/2*y*(...
R = -1/2*y*(-1+x*y^2+x^3)/(y^2+x^2)*`&.`(E[1],e[1],`&^`(e[1],e[2]))+1/2*x*(-3+x*y^2+x^3)/(y^2+x^2)*`&.`(E[2],e[1],`&^`(e[1],e[2]))+1/2*(-x+y^4+y^2*x^2)/(y^2+x^2)*`&.`(E[1],e[2],`&^`(e[1],e[2]))+1/2*y*(...
 

`&$`(e[i],R);

-1/2*y*(-1+x*y^2+x^3)/(y^2+x^2)*delta[1,i]*`&.`(e[1],`&^`(e[1],e[2]))+1/2*x*(-3+x*y^2+x^3)/(y^2+x^2)*delta[2,i]*`&.`(e[1],`&^`(e[1],e[2]))+1/2*(-x+y^4+y^2*x^2)/(y^2+x^2)*delta[1,i]*`&.`(e[2],`&^`(e[1],...
-1/2*y*(-1+x*y^2+x^3)/(y^2+x^2)*delta[1,i]*`&.`(e[1],`&^`(e[1],e[2]))+1/2*x*(-3+x*y^2+x^3)/(y^2+x^2)*delta[2,i]*`&.`(e[1],`&^`(e[1],e[2]))+1/2*(-x+y^4+y^2*x^2)/(y^2+x^2)*delta[1,i]*`&.`(e[2],`&^`(e[1],...

`&$`(e[i],E[j],R);

-1/2*y*(-1+x*y^2+x^3)/(y^2+x^2)*delta[1,i]*delta[1,j]*`&^`(e[1],e[2])+1/2*x*(-3+x*y^2+x^3)/(y^2+x^2)*delta[2,i]*delta[1,j]*`&^`(e[1],e[2])+1/2*(-x+y^4+y^2*x^2)/(y^2+x^2)*delta[1,i]*delta[2,j]*`&^`(e[1]...
-1/2*y*(-1+x*y^2+x^3)/(y^2+x^2)*delta[1,i]*delta[1,j]*`&^`(e[1],e[2])+1/2*x*(-3+x*y^2+x^3)/(y^2+x^2)*delta[2,i]*delta[1,j]*`&^`(e[1],e[2])+1/2*(-x+y^4+y^2*x^2)/(y^2+x^2)*delta[1,i]*delta[2,j]*`&^`(e[1]...

 `&$`(e[i],E[j],E[k],E[l],R);

-1/2*y*(-1+x*y^2+x^3)/(y^2+x^2)*delta[1,i]*delta[1,j]*(delta[1,k]*delta[2,l]-delta[2,k]*delta[1,l])+1/2*x*(-3+x*y^2+x^3)/(y^2+x^2)*delta[2,i]*delta[1,j]*(delta[1,k]*delta[2,l]-delta[2,k]*delta[1,l])+1/...
-1/2*y*(-1+x*y^2+x^3)/(y^2+x^2)*delta[1,i]*delta[1,j]*(delta[1,k]*delta[2,l]-delta[2,k]*delta[1,l])+1/2*x*(-3+x*y^2+x^3)/(y^2+x^2)*delta[2,i]*delta[1,j]*(delta[1,k]*delta[2,l]-delta[2,k]*delta[1,l])+1/...
-1/2*y*(-1+x*y^2+x^3)/(y^2+x^2)*delta[1,i]*delta[1,j]*(delta[1,k]*delta[2,l]-delta[2,k]*delta[1,l])+1/2*x*(-3+x*y^2+x^3)/(y^2+x^2)*delta[2,i]*delta[1,j]*(delta[1,k]*delta[2,l]-delta[2,k]*delta[1,l])+1/...
-1/2*y*(-1+x*y^2+x^3)/(y^2+x^2)*delta[1,i]*delta[1,j]*(delta[1,k]*delta[2,l]-delta[2,k]*delta[1,l])+1/2*x*(-3+x*y^2+x^3)/(y^2+x^2)*delta[2,i]*delta[1,j]*(delta[1,k]*delta[2,l]-delta[2,k]*delta[1,l])+1/...

See Also:

atlas , atlas[Constants] , atlas[Functions] , atlas[Forms] , atlas[iota] , atlas[`&^`] .
 
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