Holonomic basis

In mathematics and mathematical physics, a coordinate basis or holonomic basis for a differentiable manifold $M$ is a set of basis vector fields ${e1, ..., en}$ defined at every point $P$ of a region of the manifold as
 * $$\mathbf{e}_{\alpha} = \lim_{\delta x^{\alpha} \to 0} \frac{\delta \mathbf{s}}{\delta x^{\alpha}} ,$$

where $δs$ is the displacement vector between the point $P$ and a nearby point $Q$ whose coordinate separation from $P$ is $δxα$ along the coordinate curve $xα$ (i.e. the curve on the manifold through $P$ for which the local coordinate $xα$ 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 $xα(λ)$ with the tangent vector $u = uαeα$, where $uα = dxα⁄dλ$, and a function $f(xα)$ defined in a neighbourhood of $C$, the variation of $f$ along $C$ can be written as
 * $$\frac{df}{d\lambda} = \frac{dx^{\alpha}}{d\lambda}\frac{\partial f}{\partial x^{\alpha}} = u^{\alpha} \frac{\partial }{\partial x^{\alpha}} f .$$

Since we have that $u = uαeα$, the identification is often made between a coordinate basis vector $eα$ and the partial derivative operator $∂⁄∂xα$, under the interpretation of vectors as operators acting on functions.

A local condition for a basis ${e1, ..., en}$ to be holonomic is that all mutual Lie derivatives vanish:
 * $$ \left[ \mathbf{e}_{\alpha}, \mathbf{e}_{\beta} \right] = {\mathcal{L}}_{\mathbf{e}_{\alpha}} \mathbf{e}_{\beta} = 0 .$$

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 $δij&thinsp;ei ⊗ e$ at every point.