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# newForm -- constructor of a differential form

- 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 variable(s)

## Caveat

The coefficient i is the imaginary unit.

## See also

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

## Ways to use newForm:

- `newForm(ZZ,ZZ,ZZ,String)`
- newForm(String) -- constructor of a differential form

## For the programmer

The object newForm is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/DiffAlg.m2:593:0.