# projectivize -- projectivize a differential form or vector field

## Synopsis

• Usage:
projectivize e
• Inputs:
• 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

The projectivization process of a form increases the polynomial degree by one if the original element did not descend to projective space.