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D.15.2.29 difformDiff
Procedure from library difform.lib (see difform_lib).
- Usage:
- difformDiff(df); df difform
- Return:
- the image of df under the differential
- Remark:
- To compute the image, the universal derivation is applied to each coefficient and multiplied with the
corresponding generator
- Note:
- - the differential is a map Omega_R^(p) -> Omega_R^(p+1) and this procedure applies
the differential to all homogeneous parts of df
- this procedure can also be applied to polynomials - in this case it is just the universal derivation
Example:
| | LIB "difform.lib";
ring R = 0,(x,y,z,a,b,c),lp;
diffAlgebra();
==> // The differential algebra Omega_R was constructed and the differential \
forms dDx, dDy, dDz, dDa, dDb, dDc, dx, dy, dz, da, db, dc are available.
////////////////////////////////////////
// Construction of differential forms //
////////////////////////////////////////
difform df_1 = y*dx + z*dy + x*dz + a*db + b*dc + c*da;
difform df_2 = -5*c4*dc*dz*dy + 3*dx*dz - 13*a4*da*db + 12*a4*da*db + x8*dx*dy + 12 + dy*da + dz*dx - (y4-y5)*x12*dx*db*dz - dx - dy + db + x2*db*dy;
poly f = 3x2y2 - z3*c;
///////////////////////////////////////
// Differential applied to the forms //
///////////////////////////////////////
difformDiff(df_1);
==> (-1)*dx*dy+dx*dz+(-1)*dy*dz+da*db+(-1)*da*dc+db*dc
==>
difformDiff(df_2);
==> (-2x)*dx*dy*db+(5x12y4-4x12y3)*dx*dy*dz*db
==>
difformDiff(f);
==> 6xy2*dx+6x2y*dy+(-3z2c)*dz+(-z3)*dc
==>
// The composition of differentials is the zero-map:
difformDiff(difformDiff(df_1));
==> 0
==>
kill Omega_R,df_1,df_2,f,dx,dy,dz,da,db,dc;
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See also:
difformUnivDer.
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