mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

and

mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...

, a coordinate basis or holonomic basis for a

differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

is a set of basis vector fields defined at every point of a

region In geography, regions, otherwise referred to as zones, lands or territories, are areas that are broadly divided by physical characteristics ( physical geography), human impact characteristics ( human geography), and the interaction of humanity an ...

of the manifold as :

\mathbf_ = \lim_ \frac ,

where is the displacement vector between the point and a nearby point whose coordinate separation from is along the coordinate curve (i.e. the curve on the manifold through for which the local coordinate varies and all other coordinates are constant). It is possible to make an association between such a basis and directional derivative operators. Given a parameterized curve on the manifold defined by with the tangent vector , where , and a function defined in a neighbourhood of , the variation of along can be written as :

\frac = \frac\frac = u^ \frac f .

Since we have that , the identification is often made between a coordinate basis vector and the partial derivative operator , under the interpretation of vectors as operators acting on functions. A local condition for a basis to be holonomic is that all mutual Lie derivatives vanish: :

= _ \mathbf_ = 0 .

A basis that is not holonomic is called an anholonomic, non-holonomic or non-coordinate basis. Given a

metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...

on a manifold , it is in general not possible to find a coordinate basis that is orthonormal in any open region of . An obvious exception is when is the real coordinate space considered as a manifold with being the Euclidean metric at every point.

References

See also