newField -- constructor of a vector field

Synopsis

• Usage:

  newField(n,d,varName)

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

Description

This function defines homogeneous vector fields with generic scalar coefficients. By default, the affine coordinates will be x_0,...,x_n and the partial derivatives are denoted as ax_0,...,ax_n, respectively.

In this example we define a homogeneous vector field 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 : X = newField(2,2,"a")

         2               2                          2           2           
o1 = (a x  + a x x  + a x  + a x x  + a  x x  + a  x )ax  + (a x  + a x x  +
       0 0    3 0 1    9 1    6 0 2    12 1 2    15 2   0     1 0    4 0 1  
     ------------------------------------------------------------------------
         2                          2           2                2           
     a  x  + a x x  + a  x x  + a  x )ax  + (a x  + a x x  + a  x  + a x x  +
      10 1    7 0 2    13 1 2    16 2   1     2 0    5 0 1    11 1    8 0 2  
     ------------------------------------------------------------------------
                   2
     a  x x  + a  x )ax
      14 1 2    17 2   2

o1 : DiffAlgField

i2 : ring X

      QQ[i]
o2 = ------[][a ..a  ][x ..x ][ax ..ax ]
      2        0   17   0   2    0    2
     i  + 1

o2 : PolynomialRing

Caveat

See also

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