Holonomic basis

 ( Differential Geometry ) 

In mathematics and mathematical physics, a coordinate basis or holonomic basis for a differentiable manifold is a set of basis } defined at every point of a region of the manifold as

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

where is the displacement vector between the point and a nearby point

whose coordinate separation from  is  along the coordinate curve  (i.e. the curve on the manifold through  for which the local coordinate  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 on the manifold defined by with the tangent vector , where , and a function defined in a neighbourhood of , the variation of along can be written as

$\backslash frac\{df\}\{d\backslash lambda\}\; =\; \backslash frac\{dx^\{\backslash alpha\}\}\{d\backslash lambda\}\backslash frac\{\backslash partial\; f\}\{\backslash partial\; x^\{\backslash alpha\}\}\; =\; u^\{\backslash alpha\}\; \backslash frac\{\backslash partial\; \}\{\backslash partial\; x^\{\backslash alpha\}\}\; f\; .$

Since we have that , the identification is often made between a coordinate basis vector and the partial derivative operator , under the interpretation of vectors as operators acting on functions.

A local condition for a basis } to be holonomic is that all mutual vanish:

$\backslash left\; =\; \{\backslash mathcal\{L\}\}\_\{\backslash mathbf\{e\}\_\{\backslash alpha\}\}\; \backslash mathbf\{e\}\_\{\backslash beta\}\; =\; 0\; .$

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

Given a metric tensor on a manifold , it is in general not possible to find a coordinate basis that is orthonormal in any open region of . An obvious exception is when is the Real number coordinate space considered as a manifold with being the Euclidean metric at every point.

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See also

• Jet bundle
• Tetrad formalism
• Ricci calculus

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