Pivot language

In mathematics, a volume form on a differentiable manifold is a nowhere-vanishing top-dimensionial form (i.e., a differential form of top degree). Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle Ωⁿ(M) = Λⁿ(T^∗M), that is nowhere equal to zero. A manifold has a volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a non-vanishing function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density.

A volume form provides a means to define the integral of a function on a differentiable manifold. In other words, a volume form gives rise to a measure with respect to which functions can be integrated by the appropriate Lebesgue integral. The absolute value of a volume form is a volume element, which is also known variously as a twisted volume form or pseudo-volume form. It also defines a measure, but exists on any differentiable manifold, orientable or not.

Kähler manifolds, being complex manifolds, are naturally oriented, and so possess a volume form. More generally, the n^th exterior power of the symplectic form on a symplectic manifold is a volume form. Many classes of manifolds have canonical volume forms: they have extra structure which allows the choice of a preferred volume form. Oriented Riemannian manifolds and pseudo-Riemannian manifolds have an associated canonical volume form.

Orientation

A manifold is orientable if it has a coordinate atlas all of whose transition functions have positive Jacobian determinants. A selection of a maximal such atlas is an orientation on M. A volume form ω on M gives rise to an orientation in a natural way as the atlas of coordinate charts on M that send ω to a positive multiple of the Euclidean volume form $d{x}^1\wedge \cdots \wedge d{x}^n$.

A volume form also allows for the specification of a preferred class of frames on M. Call a basis of tangent vectors (X₁,...,X_n) right-handed if

$\omega (X_1,X_2,\dots ,X_n)>0.$

The collection of all right-handed frames is acted upon by the group GL⁺(n) of general linear mappings in n dimensions with positive determinant. They form a principal GL⁺(n) sub-bundle of the linear frame bundle of M, and so the orientation associated to a volume form gives a canonical reduction of the frame bundle of M to a sub-bundle with structure group GL⁺(n). That is to say that a volume form gives rise to GL⁺(n)-structure on M. More reduction is clearly possible by considering frames that have

Thus a volume form gives rise to an SL(n)-structure as well. Conversely, given an SL(n)-structure, one can recover a volume form by imposing (Template:EquationNote) for the special linear frames and then solving for the required n-form ω by requiring homogeneity in its arguments.

A manifold is orientable if and only if it has a volume form. Indeed, SL(n) → GL⁺(n) is a deformation retract since GL⁺ = SL × R⁺, where the positive reals are embedded as scalar matrices. Thus every GL⁺(n)-structure is reducible to an SL(n)-structure, and GL⁺(n)-structures coincide with orientations on M. More concretely, triviality of the determinant bundle ${\mathrm{\Omega }}^n(M)$ is equivalent to orientability, and a line bundle is trivial if and only if it has a nowhere-vanishing section. Thus the existence of a volume form is equivalent to orientability.

Relation to measures

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In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. Given a volume form ω on an oriented manifold, the density |ω| is a volume pseudo-form on the nonoriented manifold obtained by forgetting the orientation. Densities may also be defined more generally on non-orientable manifolds.

Any volume pseudo-form ω (and therefore also any volume form) defines a measure on the Borel sets by

${\mu }_{\omega }(U)=\underset{U}{\int }\omega .$

The difference is that while a measure can be integrated over a (Borel) subset, a volume form can only be integrated over an oriented cell. In single variable calculus, writing $\underset{b}{\overset{a}{\int }}fdx=-\underset{a}{\overset{b}{\int }}fdx$ considers $dx$ as a volume form, not simply a measure, and $\underset{b}{\overset{a}{\int }}$ indicates "integrate over the cell $[a,b]$ with the opposite orientation, sometimes denoted $\overline{[a,b]}$".

Further, general measures need not be continuous or smooth: they need not be defined by a volume form, or more formally, their Radon–Nikodym derivative with respect to a given volume form need not be absolutely continuous.

Divergence

Given a volume form ω on M, one can define the divergence of a vector field X as the unique scalar-valued function, denoted by div X, satisfying

$(\mathrm{÷}X)\omega =L_X\omega =d(X⌟\omega )$

where L_X denotes the Lie derivative along X. If X is a compactly supported vector field and M is a manifold with boundary, then Stokes' theorem implies

$\underset{M}{\int }(\mathrm{÷}X)\omega =\underset{\partial M}{\int }X⌟\omega ,$

which is a generalization of the divergence theorem.

The solenoidal vector fields are those with div X = 0. It follows from the definition of the Lie derivative that the volume form is preserved under the flow of a solenoidal vector field. Thus solenoidal vector fields are precisely those that have volume-preserving flows. This fact is well-known, for instance, in fluid mechanics where the divergence of a velocity field measures the compressibility of a fluid, which in turn represents the extent to which volume is preserved along flows of the fluid.

Special cases

Lie groups

For any Lie group, a natural volume form may be defined by translation. That is, if ω_e is an element of ${\bigwedge }^n{T}_e^{*}G$, then a left-invariant form may be defined by ${\omega }_g=L_{g^{-1}}^{*}{\omega }_e$, where L_g is left-translation. As a corollary, every Lie group is orientable. This volume form is unique up to a scalar, and the corresponding measure is known as the Haar measure.

Symplectic manifolds

Any symplectic manifold (or indeed any almost symplectic manifold) has a natural volume form. If M is a 2n-dimensional manifold with symplectic form ω, then ωⁿ is nowhere zero as a consequence of the nondegeneracy of the symplectic form. As a corollary, any symplectic manifold is orientable (indeed, oriented). If the manifold is both symplectic and Riemannian, then the two volume forms agree if the manifold is Kähler.

Riemannian volume form

Any oriented Riemannian (or pseudo-Riemannian) manifold has a natural volume (or pseudo volume) form. In local coordinates, it can be expressed as

$\omega =\sqrt{\left|g\right|}d{x}^1\wedge \dots \wedge d{x}^n$

where the $d{x}^i$ are the 1-forms providing an oriented basis for the cotangent bundle of the n-dimensional manifold. Here, $\left|g\right|$ is the absolute value of the determinant of the matrix representation of the metric tensor on the manifold.

The volume form is denoted variously by

$\omega ={\mathrm{v}\mathrm{o}\mathrm{l}}_n=\epsilon =*(1).$

Here, the ∗ is the Hodge dual, thus the last form, ∗(1), emphasizes that the volume form is the Hodge dual of the constant map on the manifold.

Although the Greek letter ω is frequently used to denote the volume form, this notation is hardly universal; the symbol ω often carries many other meanings in differential geometry (such as a symplectic form); thus, the appearance of ω in a formula does not necessarily mean that it is the volume form.

Invariants of a volume form

Volume forms are not unique; they form a torsor over non-vanishing functions on the manifold, as follows. Given a non-vanishing function f on M, and a volume form $\omega$, $f\omega$ is a volume form on M. Conversely, given two volume forms $\omega ,{\omega }^{\prime }$, their ratio is a non-vanishing function (positive if they define the same orientation, negative if they define opposite orientations).

In coordinates, they are both simply a non-zero function times Lebesgue measure, and their ratio is the ratio of the functions, which is independent of choice of coordinates. Intrinsically, it is the Radon–Nikodym derivative of ${\omega }^{\prime }$ with respect to $\omega$. On an oriented manifold, the proportionality of any two volume forms can be thought of as a geometric form of the Radon–Nikodym theorem.

No local structure

A volume form on a manifold has no local structure in the sense that it is not possible on small open sets to distinguish between the given volume form and the volume form on Euclidean space Template:Harv. That is, for every point p in M, there is an open neighborhood U of p and a diffeomorphism φ of U onto an open set in Rⁿ such that the volume form on U is the pullback of $d{x}^1\wedge \cdots \wedge d{x}^n$ along φ.

As a corollary, if M and N are two manifolds, each with volume forms ${\omega }_M,{\omega }_N$, then for any points $m\in M,n\in N$, there are open neighborhoods U of m and V of n and a map $f:U\to V$ such that the volume form on N restricted to the neighborhood V pulls back to volume form on M restricted to the neighborhood U: $f^{*}{\omega }_N{|}_V={\omega }_M{|}_U$.

In one dimension, one can prove it thus: given a volume form $\omega$ on $\mathbf{R}$, define

$f(x):=\underset{0}{\overset{x}{\int }}\omega .$

Then the standard Lebesgue measure $dx$ pulls back to $\omega$ under f: $\omega =f^{*}dx$. Concretely, $\omega =fdx$. In higher dimensions, given any point $m\in M$, it has a neighborhood locally homeomorphic to $\mathbf{R}\times {\mathbf{R}}^{n-1}$, and one can apply the same procedure.

Global structure: volume

A volume form on a connected manifold M has a single global invariant, namely the (overall) volume (denoted $\mu (M)$), which is invariant under volume-form preserving maps; this may be infinite, such as for Lebesgue measure on ${\mathbf{R}}^n$. On a disconnected manifold, the volume of each connected component is the invariant.

In symbol, if $f:M\to N$ is a homeomorphism of manifolds that pulls back ${\omega }_N$ to ${\omega }_M$, then

$\mu (N)=\underset{N}{\int }{\omega }_N=\underset{f(M)}{\int }{\omega }_N=\underset{M}{\int }{f}^{*}{\omega }_N=\underset{M}{\int }{\omega }_M=\mu (M)$

and the manifolds have the same volume.

Volume forms can also be pulled back under covering maps, in which case they multiply volume by the cardinality of the fiber (formally, by integration along the fiber). In the case of an infinite sheeted cover (such as $\mathbf{R}\to S^1$), a volume form on a finite volume manifold pulls back to a volume form on an infinite volume manifold.

See also

• Cylindrical coordinate system#Line and volume elements
• Measure (mathematics)
• Poincaré metric provides a review of the volume form on the complex plane
• Spherical coordinate system#Integration and differentiation in spherical coordinates

References

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  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.