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# Open subset of Euclidian space with coordinates

An open subset of Euclidian space with a specific set of coordinates. This is the background on which differential forms can be defined.

AUTHORS:

- Joris Vankerschaver (2010-07-25)

EXAMPLES:

    sage: x, y, z = var('x, y, z')
    sage: S = CoordinatePatch((x, y, z)); S
    Open subset of R^3 with coordinates x, y, z

    sage: u, v = var('u, v')
    sage: S = CoordinatePatch((u, v)); S
    Open subset of R^2 with coordinates u, v

TODO:

- Add functionality for metric tensors

class `sage.tensor.coordinate_patch.``CoordinatePatch`(coordinates, metric=None)

  Bases: `sage.structure.parent.Parent`

  Construct a coordinate patch, i.e. an open subset of Euclidian space with a given set of coordinates.

  EXAMPLES:

      sage: x, y, z = var('x, y, z')
      sage: S = CoordinatePatch((x, y, z)); S
      Open subset of R^3 with coordinates x, y, z

      sage: u, v = var('u, v')
      sage: T = CoordinatePatch((u, v)); T
      Open subset of R^2 with coordinates u, v
      sage: loads(T.dumps()) == T
      True

  In a future release, it will be possible to specify a metric tensor on a coordinate patch. For now, providing any kind of metric raises an exception:

      sage: x, y, z = var('x, y, z')
      sage: m = matrix(SR, 3)
      sage: S = CoordinatePatch((x, y, z), metric=m)
      Traceback (most recent call last):
      ...
      NotImplementedError: Metric geometry not supported yet.

  `coordinate`(i=0)

    Return the \(i^{th}\) coordinate on `self`

    INPUT:

    - `i` - integer (optional, default 0)

    EXAMPLES:

        sage: x, y, z = var('x, y, z')
        sage: S = CoordinatePatch((x, y, z)); S
        Open subset of R^3 with coordinates x, y, z
        sage: S.coordinate(0)
        x
        sage: S.coordinate(1)
        y
        sage: S.coordinate(2)
        z

  `coordinates`()

    Return coordinates on this coordinate patch.

    OUTPUT:

    - list - a list of coordinates on this space.

    EXAMPLES:

        sage: x, y, z = var('x, y, z')
        sage: S = CoordinatePatch((x, y, z)); S
        Open subset of R^3 with coordinates x, y, z
        sage: S.coordinates()
        (x, y, z)

  `dim`()

    Return the dimension of this coordinate patch, i.e. the dimension of the Euclidian space of which this coordinate patch is an open subset.

    EXAMPLES:

        sage: a, b, c, d, e = var('a, b, c, d, e')
        sage: U = CoordinatePatch((a, b, c, d, e)); U
        Open subset of R^5 with coordinates a, b, c, d, e
        sage: U.dim()
        5

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