Coordinate-free treatment
Encyclopedia
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, tensor
s, and differential geometry.
In physics, the existence of coordinate-free treatments of physical theories is a corollary of the principle of general covariance
.
Scientific theory
A scientific theory comprises a collection of concepts, including abstractions of observable phenomena expressed as quantifiable properties, together with rules that express relationships between observations of such concepts...
or mathematical topic develops its ideas without reference to any particular coordinate system
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...
.
Coordinate-free treatments generally allow for simpler systems of equations, allowing greater mathematical elegance at the cost of some abstraction
Abstraction
Abstraction is a process by which higher concepts are derived from the usage and classification of literal concepts, first principles, or other methods....
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
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
before the development of analytic geometry
Analytic geometry
Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
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, tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
s, and differential geometry.
In physics, the existence of coordinate-free treatments of physical theories is a corollary of the principle of general covariance
General covariance
In theoretical physics, general covariance is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations...
.
See also
- Change of basisChange of basisIn linear algebra, change of basis refers to the conversion of vectors and linear transformations between matrix representations which have different bases.-Expression of a basis:...
- Differential geometry
- Coordinate conditionsCoordinate conditionsIn general relativity, the laws of physics can be expressed in a generally covariant form. In other words, the real world does not care about our coordinate systems. However, it is often useful to fix upon a particular coordinate system, in order to solve actual problems or make actual predictions...
- Component-free treatment of tensors