Coordinate-free
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A coordinate-free, or component-free, treatment of a
Benefits
Coordinate-free treatments generally allow for simpler systems of equations and inherently constrain certain types of inconsistency, allowing greater
In addition to elegance, coordinate-free treatments are crucial in certain applications for proving that a given definition is well formulated. For example, for a vector space with basis , it may be tempting to construct the dual space as the formal span of the symbols with bracket , but it is not immediately clear that this construction is independent of the initial coordinate system chosen. Instead, it is best to construct as the space of
Nonetheless it may sometimes be too complicated to proceed from a coordinate-free treatment, or a coordinate-free treatment may guarantee uniqueness but not existence of the described object, or a coordinate-free treatment may simply not exist. As an example of the last situation, the mapping indicates a general isomorphism between a finite-dimensional vector space and its dual, but this isomorphism is not attested to by any coordinate-free definition. As an example of the second situation, a common way of constructing the fiber product of schemes involves gluing along affine patches.[1] To alleviate the inelegance of this construction, the fiber product is then characterized by a convenient universal property, and proven to be independent of the initial affine patches chosen.
History
Coordinate-free treatments were the only available approach to
Applications
Fields that are now often introduced with coordinate-free treatments include vector calculus, tensors, differential geometry, and computer graphics.[2]
In physics, the existence of coordinate-free treatments of physical theories is a corollary of the principle of general covariance.
See also
- General covariance
- Foundations of geometry
- Change of basis
- Coordinate conditions
- Component-free treatment of tensors
- Background independence
- Pointless topology
References
- ISBN 978-0387902449.
- ^ DeRose, Tony D. Three-Dimensional Computer Graphics: A Coordinate-Free Approach. Retrieved 25 September 2017.