projectivize -- projectivize a differential form or vector field

Synopsis

• Usage:

  projectivize e

• Inputs:
  • e, an instance of the type DiffAlgElement, a form or a vector field
• Outputs:
  • an instance of the type DiffAlgElement, the projectivization of e with respect to a new variable. The resulting form or vector field descends to projective space.

Description

This returns the unique differential form that extends the given one from affine space to projective space.

i1 : w = newForm("2*x_0*dx_0+x_1^2*dx_1")

               2
o1 = 2x dx  + x dx
       0  0    1  1

o1 : DiffAlgForm

i2 : r = radial 2

o2 = x ax  + x ax  + x ax
      0  0    1  1    2  2

o2 : DiffAlgField

i3 : projectivize w

         2       2            3     2
o3 = 2x x dx  + x x dx  + (- x  - 2x x )dx
       0 2  0    1 2  1       1     0 2   2

o3 : DiffAlgForm

i4 : r_oo

o4 = 0

o4 : DiffAlgForm

i5 : projectivize newField ("ax_0+x_1*ax_2+a*ax_1")  

o5 = x ax  + a*x ax  + x ax
      3  0      3  1    1  2

o5 : DiffAlgField

Caveat

See also

• homogenize -- homogenize with respect to a variable
• isHomogeneous -- whether something is homogeneous (graded)