Holonomic basis

(Redirected from Coordinate basis)

In mathematics and mathematical physics, a holonomic basis or coordinate basis for a differentiable manifold is a set of basis vector fields {ek} such that some coordinate system {xk} exists for which

${\displaystyle e_{k}={\partial \over \partial x^{k}}}$

A local condition for a basis {ek} to be holonomic is that all mutual Lie derivatives vanish:[1]

${\displaystyle [e_{i},e_{j}]=0}$

A basis that is not holonomic is called a non-holonomic or non-coordinate basis.

References

1. ^ Roger Penrose; Wolfgang Rindler, Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields, Cambridge University Press, pp. 197–199