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.
- http://en.wikipedia.org/wiki/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() 3
-