Holonomic basis


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
where is the infinitesimal displacement vector between the point and a nearby point
whose coordinate separation from is along the coordinate curve .
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
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 scalar quantities.
A local condition for a basis to be holonomic is that all mutual Lie derivatives vanish:
A basis that is not holonomic is called a 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.