Holonomic basis explained

In mathematics and mathematical physics, a coordinate basis or holonomic basis for a differentiable manifold is a set of basis vector fields defined at every point of a region of the manifold as

e _{\alpha} = \lim _{\delta x ^{\alpha} \to 0} (\delta s)/(\delta x ^{\alpha}) , 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

(df)/(d λ) = (dx ^{\alpha})/(d λ) (\partial f)/(\partial x ^{\alpha}) = u ^{\alpha} (\partial)/(\partial x ^{\alpha}) 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:

\left[ e _{\alpha} , e _{\beta} \right] = { l{L}} _{e _{\alpha}} e _{\beta} = 0 .

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

Given a metric tensor 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.

See also

• Jet bundle
• Tetrad formalism
• Ricci calculus