Coordinate basis

In mathematics and mathematical physics, a coordinate basis or holonomic basis for a differentiable manifold $M$ is a set of basis vector fields ${e α}$ defined at every point $P$ of a region of the manifold as

where $s$ is the infinitesimal 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 $u = u α e α$ , 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

Since we have that $u = u α e α$ , the identification is often made between a coordinate basis vector $e α$ and the partial derivative operator $\partial / \partial x α$ , under the interpretation of all vector relations as equalities between operators acting on scalar quantities.

...
Wikipedia