Equation of state (cosmology)
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Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among them being symplectic geometry. The theorem is named after Jean Gaston Darboux[1] who established it as the solution of the Pfaff problem.[2]
One of the many consequences of the theorem is that any two symplectic manifolds of the same dimension are locally symplectomorphic to one another. That is, every 2n-dimensional symplectic manifold can be made to look locally like the linear symplectic space Cn with its canonical symplectic form. There is also an analogous consequence of the theorem as applied to contact geometry.
Statement and first consequences
The precise statement is as follows.[3] Suppose that θ is a differential 1-form on an n dimensional manifold, such that dθ has constant rank p. If
- θ ∧ (dθ)p = 0 everywhere,
then there is a local system of coordinates x1,...,xn-p, y1, ..., yp in which
- θ = x1 dy1 + ... + xp dyp.
If, on the other hand,
- θ ∧ (dθ)p ≠ 0 everywhere,
then there is a local system of coordinates x1,...,xn-p, y1, ..., yp in which
- θ = x1 dy1 + ... + xp dyp + dxp+1.
In particular, suppose that ω is a symplectic 2-form on an n=2m dimensional manifold M. In a neighborhood of each point p of M, by the Poincaré lemma, there is a 1-form θ with dθ=ω. Moreover, θ satisfies the first set of hypotheses in Darboux's theorem, and so locally there is a coordinate chart U near p in which
- θ = x1 dy1 + ... + xm dym.
Taking an exterior derivative now shows
- ω = dθ = dx1 ∧ dy1 + ... + dxm ∧ dym.
The chart U is said to be a Darboux chart around p.[4] The manifold M can be covered by such charts.
To state this differently, identify R2m with Cm by letting zj = xj + i yj. If φ : U → Cn is a Darboux chart, then ω is the pullback of the standard symplectic form ω0 on Cn:
Comparison with Riemannian geometry
This result implies that there are no local invariants in symplectic geometry: a Darboux basis can always be taken, valid near any given point. This is in marked contrast to the situation in Riemannian geometry where the curvature is a local invariant, an obstruction to the metric being locally a sum of squares of coordinate differentials.
The difference is that Darboux's theorem states that ω can be made to take the standard form in an entire neighborhood around p. In Riemannian geometry, the metric can always be made to take the standard form at any given point, but not always in a neighborhood around that point.
See also
- Carathéodory-Jacobi-Lie theorem, a generalization of this theorem.
Notes
References
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A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
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