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 {e_1, ..., e_n} defined at every point P of a region of the manifold as 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 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 {e_1, ..., e_n} to be holonomic is that all mutual Lie derivatives vanish: 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 R^n considered as a manifold with g being the Euclidean metric δ_ij e^i ⊗ e^j at every point.