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:
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
alias of DifferentialForm
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
Return the generator of self. This is a one-form,
more precisely the exterior derivative of the i-th coordinate.
INPUT:
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
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)
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()
3