A coordinate-free, or component-free, treatment of a
scientific theory
A scientific theory is an explanation of an aspect of the natural world and universe that has been repeatedly tested and corroborated in accordance with the scientific method, using accepted protocols of observation, measurement, and evaluatio ...
or
mathematical topic develops its concepts on any form of
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
without reference to any particular
coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
.
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.
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
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
as the formal span of the symbols
with
bracket
A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
, 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
linear functionals
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
with bracket
, and then derive the coordinate-based formulae from this construction.
Nonetheless it is 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. 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
geometry (and are now known as
synthetic geometry) before the development of
analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineerin ...
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,
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, and
computer graphics.
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
{{Reflist
Coordinate systems