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D.15.2.44 derivationContraction
Procedure from library difform.lib (see difform_lib).
- Usage:
- derivationContraction(phi,df); phi derivation, df difform
- Return:
- the image of the contraction map i_phi applied to df
- Remarks:
- Since the contraction map is linear, it is only applied to the generators:
So the image of df under i_phi is a sum, where the coefficients are multiplied
by the image of the generators.
- Note:
- over the basering, the contraction map is the 0-map
Example:
| LIB "difform.lib";
ring R = 0,(x,y,z),lp;
diffAlgebra();
==> // The differential algebra Omega_R was constructed and the differential \
forms dDx, dDy, dDz, dx, dy, dz are available.
/////////////////////////////////
// Construction of derivations //
/////////////////////////////////
list L_1; L_1[1] = list(dx,dy,dz); L_1[2] = list(x,y,z);
derivation phi_1 = L_1; phi_1;
==> Omega_R^1 --> R
==> dx |--> x
==> dy |--> y
==> dz |--> z
==>
==>
list L_2; L_2[1] = list(dx,dy,dz); L_2[2] = list(y-x,z-y,x-z);
derivation phi_2 = L_2; phi_2;
==> Omega_R^1 --> R
==> dx |--> -x+y
==> dy |--> -y+z
==> dz |--> x-z
==>
==>
/////////////////////////////////
// Contractions of derivations //
/////////////////////////////////
derivationContraction(phi_1,dx+dy+dz);
==> x+y+z
==>
derivationContraction(phi_1,x2*y4-z);
==> 0
==>
derivationContraction(phi_1,x2*dx*dy + dx*dy*dz);
==> (-x2y)*dx+(-x3)*dy+z*dx*dy+y*dx*dz+x*dy*dz
==>
derivationContraction(phi_2,dx+dy+dz);
==> 0
==>
derivationContraction(phi_2,dx*dy*dz - dx*dy + dx*dz);
==> (-x-y+2z)*dx+(-x+y)*dy+(x-y)*dz+(x-z)*dx*dy+(-y+z)*dx*dz+(-x+y)*dy*dz
==>
kill Omega_R,dx,dy,dz,L_1,L_2,phi_1,phi_2;
| See also:
derivationContraction;
derivationLie.
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