mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

and

mathematical physics Mathematical physics is the development of mathematics, 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 de ...

, 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 ...

is a set of basis

vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...

s defined at every point of a

region In geography, regions, otherwise referred to as areas, zones, lands or territories, are portions of the Earth's surface that are broadly divided by physical characteristics (physical geography), human impact characteristics (human geography), and ...

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 allows ...

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 In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

considered as a manifold with being the Euclidean metric at every point.

References

See also