A coordinate system assigns numerical labels to positions so that the labels determine the point in the region described. A Cartesian coordinate system uses components along chosen axes; polar coordinates use a radius and an angle. For example, has polar coordinates on the usual angular branch. On a manifold, a coordinate chart gives a homeomorphism from an open region to an open subset of Euclidean space. Coordinates describe points but do not alone specify a metric.
A component function of a local coordinate chart on a smooth manifold. Its differential belongs to the coordinate cotangent space basis.
An integral coordinate is a coordinate value belonging to the integers. A point has integral coordinates when all its components do, relative to a specified coordinate system. In standard Cartesian coordinates, these points form the integer lattice: qualifies, whereas does not. Integrality depends on the axes and units; a general rotation need not preserve it.
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A coordinate system is a mathematical framework used to define the position of points in a space. It allows for the representation of geometric objects and their relationships in a consistent way. Depending on the dimensionality of the space, different types of coordinate systems can be used.