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# Coordinate-free

A coordinate-free, or component-free, treatment of a scientific theory or mathematical topic develops its concepts on any form of manifold without reference to any particular coordinate system.

Benefits

Coordinate-free treatments generally allow for simpler systems of equations and inherently constrain certain types of inconsistency, allowing greater mathematical elegance at the cost of some abstraction from the detailed formulae needed to evaluate these equations within a particular system of coordinates.

History

Coordinate-free treatments were the only available approach to geometry (and are now known as synthetic geometry) before the development of analytic geometry by Descartes. After several centuries of generally coordinate-based exposition, the modern tendency is generally to introduce students to coordinate-free treatments early on, and then to derive the coordinate-based treatments from the coordinate-free treatment, rather than vice versa.

Applications

Fields that are now often introduced with coordinate-free treatments include vector calculus, tensors, and differential geometry.

In physics, the existence of coordinate-free treatments of physical theories is a corollary of the principle of general covariance.

See also

Axiomatic geometry

Change of basis

Coordinate conditions

Component-free treatment of tensors

Background independence

Undergraduate Texts in Mathematics

Graduate Studies in Mathematics

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