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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 *δ***s** 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).^{[1]}

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 ∂/∂*x*^{α}, under the interpretation of vectors as operators acting on functions.^{[2]}

A local condition for a basis {**e**_{1}, ..., **e**_{n}} to be holonomic is that all mutual Lie derivatives vanish:^{[3]}

A basis that is not holonomic is called an anholonomic,^{[4]} 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*.^{[5]} 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.

**^**M. P. Hobson; G. P. Efstathiou; A. N. Lasenby (2006),*General Relativity: An Introduction for Physicists*, Cambridge University Press, p. 57**^**T. Padmanabhan (2010),*Gravitation: Foundations and Frontiers*, Cambridge University Press, p. 25**^**Roger Penrose; Wolfgang Rindler,*Spinors and Space–Time: Volume 1, Two-Spinor Calculus and Relativistic Fields*, Cambridge University Press, pp. 197–199**^**Charles W. Misner; Kip S. Thorne; John Archibald Wheeler (1970),*Gravitation*, p. 210**^**Bernard F. Schutz (1980),*Geometrical Methods of Mathematical Physics*, Cambridge University Press, pp. 47–49, ISBN 978-0-521-29887-2