What does holonomic basis mean?
Definitions for holonomic basis
holo·nom·ic ba·sis
This dictionary definitions page includes all the possible meanings, example usage and translations of the word holonomic basis.
Wikipedia
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 e α = lim δ x α → 0 δ s δ x α , {\displaystyle \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 d f d λ = d x α d λ ∂ f ∂ x α = u α ∂ ∂ x α f . {\displaystyle {\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.
Wikidata
Holonomic basis
In mathematics and mathematical physics, a holonomic basis or coordinate basis for a differentiable manifold is a set of basis vector fields {ek} such that some coordinate system {x} exists for which A local condition for a basis {ek} to be holonomic is that all mutual Lie derivatives vanish: A basis that is not holonomic is called a non-holonomic or non-coordinate basis.
Numerology
Chaldean Numerology
The numerical value of holonomic basis in Chaldean Numerology is: 7
Pythagorean Numerology
The numerical value of holonomic basis in Pythagorean Numerology is: 1
Translations for holonomic basis
From our Multilingual Translation Dictionary
- أساس holonomicArabic
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"holonomic basis." Definitions.net. STANDS4 LLC, 2024. Web. 26 Apr. 2024. <https://www.definitions.net/definition/holonomic+basis>.
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