A coordinatefree, or componentfree, treatment of a scientific theory or mathematical topic develops its ideas without reference to any particular coordinate system.
Coordinatefree treatments generally allow for simpler systems of equations, 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.
Coordinatefree treatments were the only possible approach to geometry before the development of analytic geometry by Descartes. After several centuries of generally coordinatebased exposition, the "modern" tendency is now generally to introduce students to coordinatefree treatments early on, and then to derive the coordinatebased treatments from the coordinatefree treatment, rather than viceversa.
Fields which are now often introduced with coordinatefree treatments include vector calculus, tensors, and differential geometry.
In physics, the existence of coordinatefree treatments of physical theories is a corollary of the principle of general covariance.
