Holonomic basis

${\displaystyle {\mathbf{e}}_{\alpha }=\underset{\delta x^{\alpha }\to 0}{lim}\frac{\delta \mathbf{s}}{\delta x^{\alpha }},}$

where δs is the displacement vector between the point P and a nearby point Q whose coordinate separation from P is δx^α along the coordinate curve x^α (i.e. the curve on the manifold through P for which the local coordinate x^α 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 C on the manifold defined by x^α(λ) with the tangent vector u = u^αe_α, where u^α = ⁠dx^α/dλ⁠, and a function f(x^α) defined in a neighbourhood of C, the variation of f along C can be written as

${\displaystyle \frac{df}{d\lambda }=\frac{d{x}^{\alpha }}{d\lambda }\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}^{\alpha }}=u^{\alpha }\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{\alpha }}f.}$

Since we have that u = u^αe_α, the identification is often made between a coordinate basis vector e_α and the partial derivative operator ⁠∂/∂x^α⁠, under the interpretation of vectors as operators acting on functions.

A local condition for a basis {e₁, ..., e_n} to be holonomic is that all mutual Lie derivatives vanish:

${\displaystyle \left[{\mathbf{e}}_{\alpha },{\mathbf{e}}_{\beta }\right]={\mathcal{L}}_{{\mathbf{e}}_{\alpha }}{\mathbf{e}}_{\beta }=0.}$

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

Given a metric tensor g on a manifold M, it is in general not possible to find a coordinate basis that is orthonormal in any open region U of M. An obvious exception is when M is the real coordinate space Rⁿ considered as a manifold with g being the Euclidean metric δ_ij eⁱ ⊗ e^j at every point.