A coordinate-free, or component-free, treatment of a scientific theory or mathematical topic develops its ideas without reference to any particular coordinate system.
Coordinate-free 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.
Coordinate-free treatments were the only possible approach to geometry before the development of analytic geometry by Descartes. After several centuries of generally coordinate-based exposition, the "modern" tendency is now 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.
Fields which 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.