Algebra of differential forms

Algebra of differential forms defined on a CoordinatePatch (an open subset of Euclidian space, see CoordinatePatch for details).

AUTHORS:

  • Joris Vankerschaver (2010-05-26)

Todo

  • Allow for forms with values in a vector space
  • Incorporate Kahler differentials

REFERENCES:

  • R. Abraham, J. E. Marsden, and T. S. Ratiu: Manifolds, tensor analysis, and applications. Springer-Verlag 1988, texts in Applied Mathematical Sciences, volume 75, 2nd edition.
  • Wikipedia article Differential_form
class sage.tensor.differential_forms.DifferentialForms(coordinate_patch=None)

Bases: sage.rings.ring.Algebra

The algebra of all differential forms on an open subset of Euclidian space of arbitrary dimension.

EXAMPLES:

To define an algebra of differential forms, first create a coordinate patch:

sage: p, q = var('p, q')
sage: U = CoordinatePatch((p, q)); U
Open subset of R^2 with coordinates p, q
sage: F = DifferentialForms(U); F
Algebra of differential forms in the variables p, q

If no coordinate patch is supplied, a default one (using the variables x, y, z) will be used:

sage: F = DifferentialForms(); F
Algebra of differential forms in the variables x, y, z
Element

alias of DifferentialForm

base_space()

Return the coordinate patch on which this algebra is defined.

EXAMPLES:

sage: x, y, z = var('x, y, z')
sage: U = CoordinatePatch((x, y, z)); U
Open subset of R^3 with coordinates x, y, z
sage: F = DifferentialForms(U); F
Algebra of differential forms in the variables x, y, z
sage: F.base_space()
Open subset of R^3 with coordinates x, y, z
gen(i=0)

Return the \(i^{th}\) generator of self. This is a one-form, more precisely the exterior derivative of the i-th coordinate.

INPUT:

  • i - integer (optional, default 0)

EXAMPLES:

sage: x, y, z = var('x, y, z')
sage: U = CoordinatePatch((x, y, z)); U
Open subset of R^3 with coordinates x, y, z
sage: F = DifferentialForms(U); F
Algebra of differential forms in the variables x, y, z
sage: F.gen(0)
dx
sage: F.gen(1)
dy
sage: F.gen(2)
dz
gens()

Return a list of the generators of self.

EXAMPLES:

sage: x, y, z = var('x, y, z')
sage: U = CoordinatePatch((x, y, z)); U
Open subset of R^3 with coordinates x, y, z
sage: F = DifferentialForms(U); F
Algebra of differential forms in the variables x, y, z
sage: F.gens()
(dx, dy, dz)
ngens()

Return the number of generators of this algebra.

EXAMPLES:

sage: x, y, z = var('x, y, z')
sage: U = CoordinatePatch((x, y, z)); U
Open subset of R^3 with coordinates x, y, z
sage: F = DifferentialForms(U); F
Algebra of differential forms in the variables x, y, z
sage: F.ngens()
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