newForm(String) -- constructor of a differential form

Function: newForm
Usage:

newForm(expression)
Inputs:
- expression, a string, the expression to be evaluated
Outputs:
- an instance of the type DiffAlgForm, the differential form written in expression

Description

This function defines the particular differential form written in the given expression as elements of type DiffAlgForm. Notice that the exterior product must be written as the ordinary product of variables *. If any parameters are founded in the given expression, they are automatically included in the ring of scalar coefficients.

In the following example we define two particular differential forms, w and z, and compute the exterior product w^z, see DiffAlgForm ^ DiffAlgForm. In the definition of w we are introducing a scalar parameter named a. Notice that the variable x_2 is missing from the ring of w. But when computing w^z, the rings of both vector fields are automatically merged.

i1 : w = newForm("a * x_1 * dx_0 * dx_1")

o1 = a*x dx dx
        1  0  1

o1 : DiffAlgForm

i2 : ring w

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

o2 : PolynomialRing, 2 skew commutative variable(s)

i3 : z = newForm("x_0^2 * dx_2 - x_2^2 * dx_0")

        2       2
o3 = - x dx  + x dx
        2  0    0  2

o3 : DiffAlgForm

i4 : ring z

      QQ[i]
o4 = ------[][x ..x ][dx ..dx ]
      2        0   2    0    2
     i  + 1

o4 : PolynomialRing, 3 skew commutative variable(s)

i5 : w ^ z

        2
o5 = a*x x dx dx dx
        0 1  0  1  2

o5 : DiffAlgForm

i6 : ring (w+z)

      QQ[i]
o6 = ------[][a][x ..x ][dx ..dx ]
      2           0   2    0    2
     i  + 1

o6 : PolynomialRing, 3 skew commutative variable(s)

In this example we show that the variables will always start from the index 0 and go up to the highest index encountered in the expression defining the differential form.

i7 : v = newForm("dx_5")

o7 = dx
       5

o7 : DiffAlgForm

i8 : ring v

      QQ[i]
o8 = ------[][x ..x ][dx ..dx ]
      2        0   5    0    5
     i  + 1

o8 : PolynomialRing, 6 skew commutative variable(s)

Caveat

By default, the affine coordinates will be x_0,...,x_n and the differentials are denoted as dx_0,...,dx_n, respectively. The coefficient i is the imaginary unit.

Ways to use this method:

newForm(String) -- constructor of a differential form

The source of this document is in DiffAlg.m2:622:0.

newForm(String) -- constructor of a differential form

Description

Caveat

See also

Ways to use this method: