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{{for|other topics related to '''Einstein'''|Einstein (disambiguation)}} | |||
In [[mathematics]], especially in applications of [[linear algebra]] to [[physics]], the '''Einstein notation''' or '''Einstein summation convention''' is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving notational brevity. As part of mathematics it is a notational subset of [[Ricci calculus]]; however, it is often used in applications in physics that do not distinguish between [[Tangent space|tangent]] and [[cotangent space|cotangent]] spaces. It was introduced to physics by [[Albert Einstein]] in 1916.<ref name = Ein1916> {{cite journal|last = Einstein |first = Albert |authorlink = Albert Einstein |coauthors = |title = The Foundation of the General Theory of Relativity |journal = Annalen der Physik |volume = |issue = |pages = |year = 1916 |publisher = |url = http://www.alberteinstein.info/gallery/gtext3.html | doi = 10.1002/andp.19163540702 |format = [[PDF]] |id = |accessdate = 2006-09-03 |archiveurl = http://web.archive.org/web/20060829045130/http://www.alberteinstein.info/gallery/gtext3.html <!--Added by H3llBot--> |archivedate = 2006-08-29 |bibcode = 1916AnP...354..769E }} </ref> | |||
== Introduction == | |||
===Statement of convention=== | |||
According to this convention, when an index variable appears twice in a single term it implies summation of that term over all the values of the index. So where the indices can range over the [[Set (mathematics)|set]] {{math|{1, 2, 3}|}}, | |||
:<math> y = \sum_{i=1}^3 c_i x^i = c_1 x^1 + c_2 x^2 + c_3 x^3 </math> | |||
is reduced by the convention to: | |||
:<math> y = c_i x^i \,.</math> | |||
The upper indices are not [[Exponentiation|exponents]] but are indices of coordinates, [[coefficient]]s or [[basis vector]]s. For example, {{math|''x''<sup>2</sup>}} should be read as "x-two", not "x squared", and typically {{math|(''x''<sup>1</sup>, ''x''<sup>2</sup>, ''x''<sup>3</sup>)}} would be equivalent to the traditional {{math|(''x'', ''y'', ''z'')}}. | |||
In [[general relativity]], a common convention is that | |||
*the [[Greek alphabet]] is used for space and time components, where indices take values 0,1,2,3 (frequently used letters are {{math|''μ'', ''ν'', ...}}), | |||
*the [[Latin alphabet]] is used for spatial components only, where indices take values 1,2,3 (frequently used letters are {{math|''i'', ''j'', ...}}), | |||
In general, indices can range over any [[Indexed family|indexing set]], including an [[infinite set]]. This should not be confused with a typographically similar convention used to distinguish between [[tensor index notation]] and the closely related but distinct basis-independent [[abstract index notation]]. | |||
An index that is summed over is a ''summation index'', in this case ''i''. It is also called a ''dummy index'' since any symbol can replace ''i'' without changing the meaning of the expression, provided that it does not collide with index symbols in the same term. | |||
An index that is not summed over is a ''free index'' and should be found in each term of the equation or formula if it appears in any term. Compare dummy indices and free indices with [[free variables and bound variables]]. | |||
===Application=== | |||
Einstein notation can be applied in slightly different ways. Typically, each index occurs once in an upper (superscript) and once in a lower (subscript) position in a term; however, the convention can be applied more generally to any repeated indices within a term.<ref name="wolfram">{{cite web |url=http://mathworld.wolfram.com/EinsteinSummation.html |title=Einstein Summation|accessdate=13 April 2011 |last= |first= |year= |work= |publisher=Wolfram Mathworld }}</ref> When dealing with [[Covariance and contravariance of vectors|covariant and contravariant]] vectors, where the position of an index also indicates the type of vector, the first case usually applies; a covariant vector can only be contracted with a contravariant vector, corresponding to summation of the products of coefficients. On the other hand, when there is a fixed coordinate basis (or when not considering coordinate vectors), one may choose to use only subscripts; [[#Superscripts and subscripts vs. only subscripts|see below]]. | |||
==Vector representations== | |||
=== Superscripts and subscripts vs. only subscripts === | |||
In terms of [[covariance and contravariance of vectors]], | |||
*upper indices represent components of [[Covariance and contravariance of vectors|contravariant vectors]] ([[coordinate vector|vector]]s), | |||
*lower indices represent components of [[covariant vector|covariant]] vectors ([[covector]]s): they transform covariantly (resp., contravariantly) with respect to change of basis. | |||
In recognition of this fact, the following notation uses the same symbol both for a (co)vector and its ''components'', as in: | |||
:<math>\, v = v^i e_i = \begin{bmatrix}e_1&e_2&\cdots&e_n\end{bmatrix}\begin{bmatrix}v^1\\v^2\\\vdots\\v^n\end{bmatrix} </math> | |||
:<math>\, w = w_i e^i = \begin{bmatrix}w_1 & w_2 & \cdots & w_n\end{bmatrix}\begin{bmatrix}e^1\\e^2\\\vdots\\e^n\end{bmatrix} </math> | |||
where ''v'' is the vector and ''v<sup>i</sup>'' are its components (not the ''i''th covector ''v'') ''w'' is the covector and ''w''<sub>''i''</sub> are its components. | |||
In the presence of a non-degenerate form (an isomorphism <math>V \to V^*</math>, for instance a [[Riemannian metric]] or [[Minkowski metric]]), one can [[raising and lowering indices|raise and lower indices]]. | |||
A basis gives such a form (via the [[dual basis]]), hence when working on '''R'''<sup>n</sup> with a Euclidian metric and a fixed orthonormal basis, one can work with only subscripts. | |||
However, if one changes coordinates, the way that coefficients change depends on the variance of the object, and one cannot ignore the distinction; see [[covariance and contravariance of vectors]]. | |||
===Mnemonics=== | |||
In the above example, vectors are represented as ''n''×1 matrices (column vectors), while covectors are represented as 1×''n'' matrices (row covectors). | |||
When using the column vector convention | |||
* "'''Up'''per indices go '''up''' to down; '''l'''ower indices go '''l'''eft to right" | |||
* Vectors can be stacked (column matrices) side-by-side: | |||
:<math>\begin{bmatrix}v_1 & \cdots & v_k\end{bmatrix}.</math> | |||
:Hence the lower index indicates which ''column'' you are in. | |||
* You can stack covectors (row matrices) top-to-bottom: | |||
:<math>\begin{bmatrix}w^1 \\ \vdots \\ w^k\end{bmatrix}</math> | |||
:Hence the upper index indicates which ''row'' you are in. | |||
=== Abstract description === | |||
The virtue of Einstein notation is that it represents the invariant quantities with a simple notation. | |||
Any scalar is invariant under transformations of [[basis (mathematics)|basis]]; the individual terms in the sum are not. When the basis is changed, the ''components'' of a vector change by a linear transformation described by a matrix. This led Einstein to propose the convention that repeated indices imply the summation is to be done. | |||
As for covectors, they change by the inverse matrix. This is designed to guarantee that the linear function associated with the covector, the sum above, is the same no matter what the basis is. | |||
The value of the Einstein convention is that it applies to other vector spaces built from ''V'' using the [[tensor product]] and [[dual space|duality]]. For example, <math>V\otimes V</math>, the tensor product of ''V'' with itself, has a basis consisting of tensors of the form <math>\mathbf{e}_{ij} = \mathbf{e}_i \otimes \mathbf{e}_j</math>. Any tensor <math>\mathbf{T}</math> in <math>V\otimes V</math> can be written as: | |||
:<math>\mathbf{T} = T^{ij}\mathbf{e}_{ij}</math>. | |||
<math>V^*</math>, the dual of <math>V</math>, has a basis '''e'''<sup>1</sup>, '''e'''<sup>2</sup>, ..., '''e'''<sup>''n''</sup> which obeys the rule | |||
:<math>\mathbf{e}^i (\mathbf{e}_j) = \delta^i_j.</math> | |||
where δ is the [[Kronecker delta]]. As | |||
:<math>\mathrm{Hom}(V,W) = V^* \otimes W</math> | |||
the row-column coordinates on a matrix correspond to the upper-lower indices on the tensor product. | |||
== Common operations in this notation == | |||
In Einstein notation, the usual element reference <math>A_{mn}</math> for the ''m''th row and ''n''th column of matrix '''A''' becomes <math>A^n{}_m</math>. We can then write the following operations in Einstein notation as follows. | |||
;[[Inner product]] (hence also [[Vector dot product]]) | |||
Using an [[orthogonal basis]], the inner product is the sum of corresponding components multiplied together: | |||
:<math> \mathbf{u} \cdot \mathbf{v} = u_j v^j </math> | |||
This can also be calculated by multiplying the covector on the vector. | |||
;[[Vector cross product]] | |||
Again using an orthogonal basis (in 3d) the cross product intrisically involves summations over permutations of components: | |||
:<math> \mathbf{u} \times \mathbf{v}= u^j v^k\epsilon^i{}_{jk} \mathbf{e}_i </math> | |||
where | |||
:<math>\epsilon^i{}_{jk}=\delta^{il}\epsilon_{ljk}</math> | |||
and <math> \epsilon_{ijk}</math> is the [[Levi-Civita symbol]]. | |||
;[[Matrix multiplication]] | |||
The [[Matrix multiplication#matrix product|matrix product]] of two matrices <math>A_{ij}</math> and <math>B_{jk}</math> is: | |||
:<math> \mathbf{C}_{ik} = (\mathbf{A} \, \mathbf{B})_{ik} =\sum_{j=1}^N A_{ij} B_{jk}</math> | |||
equivalent to | |||
:<math>C^i{}_k = A^i{}_j \, B^j{}_k </math> | |||
;[[Trace (linear algebra)|Trace]] | |||
For a square matrix <math>A^i{}_j</math>, the trace is the sum of the diagonal elements, hence the sum over a common index <math>A^i{}_i</math>. | |||
;[[Outer product]] | |||
The outer product of the column vector <math>u^i</math> by the row vector <math>v_j</math> yields an ''m''×''n'' matrix '''A''': | |||
:<math>A^i{}_j = u^i \, v_j = (uv)^i{}_j </math> | |||
Since ''i'' and ''j'' represent two ''different'' indices, there is no summation and the indices are not eliminated by the multiplication. | |||
;[[Tensor contraction]] | |||
The contraction of a tensor with the metric tensor, <math>g_{\mu \nu}</math>, is a common operation used to raise or lower indices (typically for convenience) on a tensor. For example, the index in superscript of <math>\Lambda^\sigma{}_{\nu}</math>, can be lowered like so: | |||
:<math>\Lambda_{\mu \nu} = g_{\mu \sigma} \Lambda^\sigma{}_{\nu}.</math> | |||
In this case, only summation over the "<math>\sigma</math>" index is implied. | |||
== See also == | |||
{{wikibooks|General relativity|Einstein Summation Notation}} | |||
* [[Ricci calculus]] | |||
* [[Tensor]] | |||
* [[Abstract index notation]] | |||
* [[Bra–ket notation]] | |||
* [[Penrose graphical notation]] | |||
* [[Kronecker delta]] | |||
* [[Levi-Civita symbol]] | |||
==Notes== | |||
#{{note label|a}}This applies only for numerical indices. The situation is the opposite for [[abstract indices]]. Then, vectors themselves carry upper abstract indices and covectors carry lower abstract indices, as per the example in the [[#Introduction|introduction]] of this article. Elements of a basis of vectors may carry a lower ''numerical'' index and an upper ''abstract'' index. | |||
==References== | |||
{{reflist}} | |||
==Bibliography== | |||
* {{springer|id=E/e035220|first=L.P.|last=Kuptsov|title=Einstein rule}}. | |||
==External links== | |||
*{{cite news | |||
|last=Rawlings | |||
|first=Steve | |||
|url=http://www-astro.physics.ox.ac.uk/~sr/lectures/vectors/lecture10final.pdfc | |||
|title=Lecture 10 - Einstein Summation Convention and Vector Identities | |||
|publisher=Oxford University | |||
|date=2007-02-01 }} | |||
{{tensors}} | |||
[[Category:Mathematical notation]] | |||
[[Category:Multilinear algebra]] | |||
[[Category:Tensors]] | |||
[[Category:Riemannian geometry]] | |||
[[Category:Mathematical physics]] | |||
[[Category:Albert Einstein]] | |||
Latest revision as of 21:27, 12 March 2013
28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving notational brevity. As part of mathematics it is a notational subset of Ricci calculus; however, it is often used in applications in physics that do not distinguish between tangent and cotangent spaces. It was introduced to physics by Albert Einstein in 1916.[1]
Introduction
Statement of convention
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is reduced by the convention to:
The upper indices are not exponents but are indices of coordinates, coefficients or basis vectors. For example, Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. should be read as "x-two", not "x squared", and typically Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. would be equivalent to the traditional Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer..
In general relativity, a common convention is that
- the Greek alphabet is used for space and time components, where indices take values 0,1,2,3 (frequently used letters are Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.), - the Latin alphabet is used for spatial components only, where indices take values 1,2,3 (frequently used letters are Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.),
In general, indices can range over any indexing set, including an infinite set. This should not be confused with a typographically similar convention used to distinguish between tensor index notation and the closely related but distinct basis-independent abstract index notation.
An index that is summed over is a summation index, in this case i. It is also called a dummy index since any symbol can replace i without changing the meaning of the expression, provided that it does not collide with index symbols in the same term.
An index that is not summed over is a free index and should be found in each term of the equation or formula if it appears in any term. Compare dummy indices and free indices with free variables and bound variables.
Application
Einstein notation can be applied in slightly different ways. Typically, each index occurs once in an upper (superscript) and once in a lower (subscript) position in a term; however, the convention can be applied more generally to any repeated indices within a term.[2] When dealing with covariant and contravariant vectors, where the position of an index also indicates the type of vector, the first case usually applies; a covariant vector can only be contracted with a contravariant vector, corresponding to summation of the products of coefficients. On the other hand, when there is a fixed coordinate basis (or when not considering coordinate vectors), one may choose to use only subscripts; see below.
Vector representations
Superscripts and subscripts vs. only subscripts
In terms of covariance and contravariance of vectors,
- upper indices represent components of contravariant vectors (vectors),
- lower indices represent components of covariant vectors (covectors): they transform covariantly (resp., contravariantly) with respect to change of basis.
In recognition of this fact, the following notation uses the same symbol both for a (co)vector and its components, as in:
where v is the vector and vi are its components (not the ith covector v) w is the covector and wi are its components.
In the presence of a non-degenerate form (an isomorphism , for instance a Riemannian metric or Minkowski metric), one can raise and lower indices.
A basis gives such a form (via the dual basis), hence when working on Rn with a Euclidian metric and a fixed orthonormal basis, one can work with only subscripts.
However, if one changes coordinates, the way that coefficients change depends on the variance of the object, and one cannot ignore the distinction; see covariance and contravariance of vectors.
Mnemonics
In the above example, vectors are represented as n×1 matrices (column vectors), while covectors are represented as 1×n matrices (row covectors).
When using the column vector convention
- "Upper indices go up to down; lower indices go left to right"
- Vectors can be stacked (column matrices) side-by-side:
- You can stack covectors (row matrices) top-to-bottom:
Abstract description
The virtue of Einstein notation is that it represents the invariant quantities with a simple notation.
Any scalar is invariant under transformations of basis; the individual terms in the sum are not. When the basis is changed, the components of a vector change by a linear transformation described by a matrix. This led Einstein to propose the convention that repeated indices imply the summation is to be done.
As for covectors, they change by the inverse matrix. This is designed to guarantee that the linear function associated with the covector, the sum above, is the same no matter what the basis is.
The value of the Einstein convention is that it applies to other vector spaces built from V using the tensor product and duality. For example, , the tensor product of V with itself, has a basis consisting of tensors of the form . Any tensor in can be written as:
, the dual of , has a basis e1, e2, ..., en which obeys the rule
where δ is the Kronecker delta. As
the row-column coordinates on a matrix correspond to the upper-lower indices on the tensor product.
Common operations in this notation
In Einstein notation, the usual element reference for the mth row and nth column of matrix A becomes . We can then write the following operations in Einstein notation as follows.
- Inner product (hence also Vector dot product)
Using an orthogonal basis, the inner product is the sum of corresponding components multiplied together:
This can also be calculated by multiplying the covector on the vector.
Again using an orthogonal basis (in 3d) the cross product intrisically involves summations over permutations of components:
where
and is the Levi-Civita symbol.
The matrix product of two matrices and is:
equivalent to
For a square matrix , the trace is the sum of the diagonal elements, hence the sum over a common index .
The outer product of the column vector by the row vector yields an m×n matrix A:
Since i and j represent two different indices, there is no summation and the indices are not eliminated by the multiplication.
The contraction of a tensor with the metric tensor, , is a common operation used to raise or lower indices (typically for convenience) on a tensor. For example, the index in superscript of , can be lowered like so:
In this case, only summation over the "" index is implied.
See also
DTZ's auction group in Singapore auctions all types of residential, workplace and retail properties, retailers, homes, accommodations, boarding houses, industrial buildings and development websites. Auctions are at the moment held as soon as a month.
Whitehaven @ Pasir Panjang – A boutique improvement nicely nestled peacefully in serene Pasir Panjang personal estate presenting a hundred and twenty rare freehold private apartments tastefully designed by the famend Ong & Ong Architect. Only a short drive away from Science Park and NUS Campus, Jade Residences, a recent Freehold condominium which offers high quality lifestyle with wonderful facilities and conveniences proper at its door steps. Its fashionable linear architectural fashion promotes peace and tranquility living nestled within the D19 personal housing enclave. Rising workplace sector leads real estate market efficiency, while prime retail and enterprise park segments moderate and residential sector continues in decline International Market Perspectives - 1st Quarter 2014
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In case you are eligible to purchase landed houses (open solely to Singapore residents) it is without doubt one of the best property investment choices. Landed housing varieties solely a small fraction of available residential property in Singapore, due to shortage of land right here. In the long term it should hold its worth and appreciate as the supply is small. In truth, landed housing costs have risen the most, having doubled within the last eight years or so. However he got here back the following day with two suitcases full of money. Typically we've got to clarify to such folks that there are rules and paperwork in Singapore and you can't just buy a home like that,' she said. For conveyancing matters there shall be a recommendedLondon Regulation agency familiar with Singapore London propertyinvestors to symbolize you
Sales transaction volumes have been expected to hit four,000 units for 2012, close to the mixed EC gross sales volume in 2010 and 2011, in accordance with Savills Singapore. Nevertheless the last quarter was weak. In Q4 2012, sales transactions were 22.8% down q-q to 7,931 units, in line with the URA. The quarterly sales discount was felt throughout the board. When the sale just starts, I am not in a hurry to buy. It's completely different from a private sale open for privileged clients for one day solely. Orchard / Holland (D09-10) House For Sale The Tembusu is a singular large freehold land outdoors the central area. Designed by multiple award-profitable architects Arc Studio Architecture + Urbanism, the event is targeted for launch in mid 2013. Post your Property Condos Close to MRT
- Ricci calculus
- Tensor
- Abstract index notation
- Bra–ket notation
- Penrose graphical notation
- Kronecker delta
- Levi-Civita symbol
Notes
- 24 years old Art Manager or Manager Harry Kamm from West Hill, really likes ceramics, property developers in Singapore house for rent and walking. Recently has traveled to .This applies only for numerical indices. The situation is the opposite for abstract indices. Then, vectors themselves carry upper abstract indices and covectors carry lower abstract indices, as per the example in the introduction of this article. Elements of a basis of vectors may carry a lower numerical index and an upper abstract index.
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
Bibliography
- Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.
my web-site http://himerka.com/.
External links
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Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
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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 - ↑ Template:Cite web