newForm -- constructor of a differential form

Synopsis

• Usage:

  newForm(n,r,d,varName)

• Inputs:
  • n, an integer, number of variables minus one
  • r, an integer, degree of the differential form
  • d, an integer, degree of the polynomial coefficients of the differential form
  • varName, a string, name of the generic scalar coefficients of the differential form
• Outputs:
  • an instance of the type DiffAlgForm, a homogeneous differential r-form in (n+1)-dimensional affine space with polynomial coefficients of degree d

Description

This function defines homogeneous differential forms with generic scalar coefficients. By default, the affine coordinates will be x_0,...,x_n and their exterior derivatives are denoted as dx_0,...,dx_n, respectively.

In this example we define a homogeneous differential 1-form with linear polynomial coefficients in 3 variables. The scalar coefficients are chosen to be defined with the variable a. The index of the scalar coefficients will always start with 0.

i1 : w = newForm(2,1,1,"a")

o1 = (a x  + a x  + a x )dx  + (a x  + a x  + a x )dx  + (a x  + a x  +
       0 0    3 1    6 2   0     1 0    4 1    7 2   1     2 0    5 1  
     ------------------------------------------------------------------------
     a x )dx
      8 2   2

o1 : DiffAlgForm

i2 : ring w

      QQ[i]
o2 = ------[][a ..a ][x ..x ][dx ..dx ]
      2        0   8   0   2    0    2
     i  + 1

o2 : PolynomialRing, 3 skew commutative variables

Caveat

See also

• newForm(String) -- constructor of a differential form
• newField -- constructor of a vector field