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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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