Coordinate system

See Cartesian coordinate system or Coordinates (mathematics) for a more elementary introduction to this topic.

Related Topics:
Cartesian coordinate system - Coordinates (mathematics)

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In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. "Numbers" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other field. If the space or manifold is curved, it may not be possible to provide one consistent coordinate system for the entire space. In this case, a set of coordinate systems, called charts, are cobbled together to form an atlas for the space.

Related Topics:
Mathematics - Geometry - Physics - Engineering - Tuple - Number - Point - Dimension - Real number - Complex number - Field - Manifold - Atlas

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When the space has some additional algebraic structure, then the co-ordinates will also transform under rings or groups; a particularly famous example in this case are the Lie groups.

Related Topics:
Algebraic structure - Rings - Group - Lie group

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Although any specific coordinate system is useful for numerical calculations in a given space, the space itself is considered to exist independently of any particular choice of coordinates. By convention the origin of the coordinate system in Cartesian coordinates is the point (0, 0, ..., 0), which may be assigned to any given point of Euclidean space.

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In physics, a scalar is a physical quantity which assumes a single value which is a "real" quantity independent of the coordinate system. In this sense coordinates are not scalars (although, of course, a scalar field can be defined which for one particular coordinate system corresponds to a particular coordinate).

Related Topics:
Physics - Scalar - Physical quantity

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In some coordinate systems some points are associated with multiple tuples of coordinates, e.g. the origin in polar coordinates: r = 0 but θ can be any angle.

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