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30 year-old Entertainer or Range Artist Wesley from Drumheller, really loves vehicle, property developers properties for sale in singapore singapore and horse racing. Finds inspiration by traveling to Works of Antoni Gaudí. In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator which assigns, to any two vector fields X and Y on a smooth manifold M, a third vector field denoted [X, Y].

Conceptually, the Lie bracket [X,Y] is the derivative of Y in the `direction' generated by X. It is a special case of the Lie derivative which allows to form the derivative of any tensor field in the direction generated by X. Indeed, [X,Y] equals the Lie derivative ${\mathcal{ℒ}}_{X}Y$.

The Lie bracket is an R-bilinear operation and turns the set of all vector fields on the manifold M into an (infinite-dimensional) Lie algebra.

The Lie bracket plays an important role in differential geometry and differential topology, for instance in the Frobenius theorem, and is also fundamental in the geometric theory for nonlinear control systems (Template:Harvnb, nonholonomic systems; Template:Harvnb, feedback linearization).

Definitions

There are three conceptually different but equivalent approaches to defining the Lie bracket:

Vector fields as derivations

Each vector field X on a smooth manifold M may be regarded as a differential operator acting on smooth functions on M. Indeed, each smooth vector field X becomes a derivation on the smooth functions C^∞(M) when we define X(f) to be the element of C^∞(M) whose value at a point p is the directional derivative of f at p in the direction X(p). Furthermore, it is known that any derivation on C^∞(M) arises in this fashion from a uniquely determined smooth vector field X.

In general, the commutator ${\delta }_1\circ {\delta }_2-{\delta }_2\circ {\delta }_1$ of any two derivations ${\delta }_1$ and ${\delta }_2$ is again a derivation. This can be used to define the Lie bracket of vector fields as follows.

The Lie bracket, [X,Y], of two smooth vector fields X and Y is the smooth vector field [X,Y] such that

$[X,Y](f)=X(Y(f))-Y(X(f))\text{ for all }f\in C^{\mathrm{\infty }}(M).$

Flows and limits

Let ${\mathrm{\Phi }}_t^X$ be the flow associated with the vector field X, and let d denote the tangent map derivative operator. Then the Lie bracket of X and Y at the point x∈M can be defined as

$[X,Y{]}_x:=\underset{t\to 0}{\lim }\frac{(\mathrm{d}{\mathrm{\Phi }}_{-t}^X)Y_{{\mathrm{\Phi }}_t^X(x)}-Y_x}{t}={\frac{\mathrm{d}}{\mathrm{d}t}|}_{t=0}(\mathrm{d}{\mathrm{\Phi }}_{-t}^X)Y_{{\mathrm{\Phi }}_t^X(x)}$

or equivalently

$[X,Y{]}_x:={\frac{1}{2}\frac{{\mathrm{d}}^2}{{\mathrm{d}\mathrm{t}}^2}|}_{t=0}({\mathrm{\Phi }}_{-t}^Y\circ {\mathrm{\Phi }}_{-t}^X\circ {\mathrm{\Phi }}_t^Y\circ {\mathrm{\Phi }}_t^X)(x)={\frac{\mathrm{d}}{\mathrm{d}t}|}_{t=0}({\mathrm{\Phi }}_{-\sqrt{t}}^Y\circ {\mathrm{\Phi }}_{-\sqrt{t}}^X\circ {\mathrm{\Phi }}_{\sqrt{t}}^Y\circ {\mathrm{\Phi }}_{\sqrt{t}}^X)(x)$

In coordinates

Though neither definition of the Lie bracket depends on a choice of coordinates, in practice one often wants to compute the bracket with respect to a coordinate system.

If we have picked a coordinate chart on M with local coordinate functions $\{x^i\}$, and we write ${\partial }_i=\frac{\partial }{\partial x^i}$ for the associated local basis for the tangent bundle, then the vector fields can be written as

$X=\sum _{i=1}^n{X}^i{\partial }_i$

and

$Y=\sum _{i=1}^n{Y}^i{\partial }_i$

with smooth functions $X^i:M\to \mathbb{ℝ}$ and $Y^i:M\to \mathbb{ℝ}$. Then the Lie bracket is given by

$[X,Y]:=\sum _{i=1}^n(X(Y^i)-Y(X^i)){\partial }_i=\sum _{i=1}^n\sum _{j=1}^n(X^j{\partial }_j{Y}^i-Y^j{\partial }_j{X}^i){\partial }_i$

If M is (an open subset of) Rⁿ, then the vector fields X and Y can be written as smooth maps of the form $X:M\to {\mathbb{ℝ}}^n$ and $Y:M\to {\mathbb{ℝ}}^n$, and the Lie bracket $[X,Y]:M\to {\mathbb{ℝ}}^n$ is given by

$[X,Y]:=J_{Y}X-J_{X}Y$

where $J_Y$ and $J_X$ are the Jacobian matrices of $Y$ and $X$, respectively. These n-by-n matrices are multiplied by the n-vectors X and Y.

Properties

The Lie bracket of vector fields equips the real vector space $V=\mathrm{\Gamma }(TM)$ of all vector fields on M (i.e., smooth sections of the tangent bundle $TM$ of $M$) with the structure of a Lie algebra, i.e., [·,·] is a map from $V\times V$ to $V$ with the following properties

• $[\cdot ,\cdot ]$ is R-bilinear
• $[X,Y]=-[Y,X]$
• $[X,[Y,Z]]+[Z,[X,Y]]+[Y,[Z,X]]=0.$ This is the Jacobi identity.

An immediate consequence of the second property is that $[X,X]=0$ for any $X$.

Furthermore, there is a "product rule" for Lie brackets. Given a smooth real-valued function f defined on M and a vector field Y on M, we have a new vector field fY, defined by multiplying the vector Y_x with the number f(x), at each point x∈M. The Lie bracket of X and fY is then given by

• $[X,fY]=X(f)Y+f[X,Y]$

(where on the right-hand side we multiply the function X(f) with the vector field Y, and the function f with the vector field [X,Y]). This turns the vector fields with the Lie bracket into a Lie algebroid.

We also have the following fact:

Theorem:

$[X,Y]=0$ iff the flows of X and Y commute locally, i.e. iff for every x∈M and all sufficiently small real numbers s, t we have $({\mathrm{\Phi }}_t^Y{\mathrm{\Phi }}_s^X)(x)=({\mathrm{\Phi }}_s^X{\mathrm{\Phi }}_t^Y)(x)$.

Examples

For a Lie group G, the corresponding Lie algebra is the tangent space at the identity, which can be identified with the left invariant vector fields on G. The Lie bracket of the Lie algebra is then the Lie bracket of the left invariant vector fields, which is also left invariant.

For a matrix Lie group, smooth vector fields can be locally represented in the corresponding Lie algebra. Since the Lie algebra associated with a Lie group is isomorphic to the group's tangent space at the identity, elements of the Lie algebra of a matrix Lie group are also matrices. Hence the Jacobi–Lie bracket corresponds to the usual commutator for a matrix group:

$[X,Y]=XY-YX$

where juxtaposition indicates matrix multiplication.

Applications

The Jacobi–Lie bracket is essential to proving small-time local controllability (STLC) for driftless affine control systems.

Generalizations

As mentioned above, the Lie derivative can be seen as a generalization of the Lie bracket. Another generalization of the Lie bracket (to vector-valued differential forms) is the Frölicher–Nijenhuis bracket.

References

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  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.
  
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  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 For generalizations to infinite dimensions.
• 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

• 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