Coordinateindependent

Coordinate independent, also called coordinate-free, refers to descriptions, constructions, and laws that do not rely on a specific choice of coordinate system or chart. An object is coordinate independent if its definition and properties hold true after any smooth change of coordinates, so the essential content is invariant under coordinate transformations.

In differential geometry, most fundamental notions are formulated in a coordinate-free way. Manifolds provide a setting where local charts exist but the theory does not depend on a particular chart. Vector fields can be defined as derivations on the algebra of smooth functions, tensor fields as multilinear maps on tangent spaces, and differential forms as skew-symmetric covariant tensors. Operations such as the exterior derivative, Lie derivative, and covariant derivative are defined without reference to coordinates. A Riemannian metric defines an inner product on tangent spaces in a way that does not require coordinates, leading to geometric objects like lengths, angles, and curvature that are coordinate independent.

In physics, coordinate independence is closely related to general covariance: the laws of physics are expressed in a form valid in all coordinate systems. General relativity is a prime example, where spacetime geometry and physical laws are described without privileging any particular coordinate chart.

A common companion idea is the use of coordinates to perform calculations, while the underlying constructs remain coordinate free. This approach emphasizes intrinsic properties of geometric or physical structures. While it offers conceptual clarity and generality, it can be more abstract and may require translating results into coordinate expressions for explicit computations.