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# 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

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