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{{main|Curvilinear coordinates}}
 
'''[[Curvilinear coordinates]]''' can be formulated in [[tensor calculus]], with important applications in [[physics]] and [[engineering]], particularly for describing transportation of physical quatities and deformation of matter in [[fluid mechanics]] and [[continuum mechanics]].
 
==Vector and tensor algebra in three-dimensional curvilinear coordinates==
{{Einstein_summation_convention}}
Elementary vector and tensor algebra in curvilinear coordinates is used in some of the older scientific literature in [[mechanics]] and [[physics]] and can be indispensable to understanding work from the early and mid 1900s, for example the text by Green and Zerna.<ref name=Green>{{cite book | last1=Green | first1=A. E. | last2=Zerna | first2=W. | year=1968 | title=Theoretical Elasticity | publisher=Oxford University Press | isbn=0-19-853486-8 }}</ref>  Some useful relations in the algebra of vectors and second-order tensors in curvilinear coordinates are given in this section.  The notation and contents are primarily from Ogden,<ref name=Ogden00/> Naghdi,<ref name=Naghdi>{{cite book | first1=P. M. | last1=Naghdi | year=1972 | contribution=Theory of shells and plates | editor=S. Flügge | title=Handbook of Physics | volume=VIa/2 | pages=425–640}}</ref> Simmonds,<ref name=Simmonds>{{cite book | last=Simmonds | first=J. G. | year=1994 | title=A brief on tensor analysis | publisher=Springer | isbn=0-387-90639-8}}</ref> Green and Zerna,<ref name=Green/>  Basar and Weichert,<ref name=Basar>{{cite book | last1=Basar | first1=Y. | last2=Weichert | first2=D. | year=2000 | title=Numerical continuum mechanics of solids: fundamental concepts and perspectives | publisher=Springer}}</ref> and Ciarlet.<ref name=Ciarlet>{{cite book | last=Ciarlet | first=P. G. | year=2000 | title=Theory of Shells | volume=1 | publisher=Elsevier Science }}</ref>
 
===Vectors in curvilinear coordinates===
Let ('''b'''<sub>1</sub>, '''b'''<sub>2</sub>, '''b'''<sub>3</sub>) be an arbitrary basis for three-dimensional Euclidean space.  In general, the basis vectors are '''neither unit vectors nor mutually orthogonal'''.  However, they are required to be linearly independent.  Then a vector '''v''' can be expressed as<ref name=Simmonds/>{{rp|page=27}}
:<math>
    \mathbf{v} = v^k~\mathbf{b}_k
</math>
The components ''v<sup>k</sup>'' are the '''contravariant''' components of the vector '''v'''.
 
The '''reciprocal basis''' ('''b'''<sup>1</sup>, '''b'''<sup>2</sup>, '''b'''<sup>3</sup>) is defined by the relation <ref name=Simmonds/>{{rp|pages=28–29}}
:<math>
  \mathbf{b}^i\cdot\mathbf{b}_j = \delta^i_j
</math>
where ''δ<sup>i</sup> <sub>j</sub>'' is the [[Kronecker delta]].
 
The vector '''v''' can also be expressed in terms of the reciprocal basis:
:<math>
    \mathbf{v} = v_k~\mathbf{b}^k
</math>
The components ''v<sub>k</sub>'' are the '''covariant''' components of the vector <math>\mathbf{v}</math>.
 
===Second-order tensors in curvilinear coordinates===
A second-order tensor can be expressed as
:<math>
  \boldsymbol{S} = S^{ij}~\mathbf{b}_i\otimes\mathbf{b}_j = S^{i}_{~j}~\mathbf{b}_i\otimes\mathbf{b}^j = S_{i}^{~j}~\mathbf{b}^i\otimes\mathbf{b}_j = S_{ij}~\mathbf{b}^i\otimes\mathbf{b}^j
</math>
The components ''S<sup>ij</sup>'' are called the '''contravariant''' components, ''S<sup>i</sup> <sub>j</sub>'' the '''mixed right-covariant''' components, ''S<sub>i</sub> <sup>j</sup>'' the '''mixed left-covariant''' components, and ''S<sub>ij</sub>'' the '''covariant''' components of the second-order tensor.
 
====Metric tensor and relations between components====
The quantities ''g<sub>ij</sub>'', ''g<sup>ij</sup>'' are defined as<ref name=Simmonds/>{{rp|page=39}}
 
:<math>
    g_{ij} = \mathbf{b}_i \cdot \mathbf{b}_j = g_{ji} ~;~~ g^{ij} = \mathbf{b}^i \cdot \mathbf{b}^j = g^{ji}
</math>
From the above equations we have
:<math>
  v^i = g^{ik}~v_k ~;~~ v_i = g_{ik}~v^k ~;~~ \mathbf{b}^i = g^{ij}~\mathbf{b}_j ~;~~ \mathbf{b}_i = g_{ij}~\mathbf{b}^j
</math>
 
The components of a vector are related by<ref name=Simmonds/>{{rp|pages=30–32}}
:<math>
  \mathbf{v}\cdot\mathbf{b}^i = v^k~\mathbf{b}_k\cdot\mathbf{b}^i = v^k~\delta^i_k = v^i </math>
:<math>
  \mathbf{v}\cdot\mathbf{b}_i = v_k~\mathbf{b}^k\cdot\mathbf{b}_i = v_k~\delta_i^k = v_i </math>
Also,
:<math>
  \mathbf{v}\cdot\mathbf{b}_i = v^k~\mathbf{b}_k\cdot\mathbf{b}_i = g_{ki}~v^k </math>
:<math>
  \mathbf{v}\cdot\mathbf{b}^i = v_k~\mathbf{b}^k\cdot\mathbf{b}^i = g^{ki}~v_k
</math>
 
The components of the second-order tensor are related by
:<math>
  S^{ij} = g^{ik}~S_k^{~j} = g^{jk}~S^i_{~k} = g^{ik}~g^{jl}~S_{kl}
</math>
 
===The alternating tensor===
In an orthonormal right-handed basis, the third-order [[Levi-Civita symbol|alternating tensor]] is defined as
:<math>
  \boldsymbol{\mathcal{E}} = \varepsilon_{ijk}~\mathbf{e}^i\otimes\mathbf{e}^j\otimes\mathbf{e}^k
</math>
In a general curvilinear basis the same tensor may be expressed as
:<math>
  \boldsymbol{\mathcal{E}} = \mathcal{E}_{ijk}~\mathbf{b}^i\otimes\mathbf{b}^j\otimes\mathbf{b}^k
  = \mathcal{E}^{ijk}~\mathbf{b}_i\otimes\mathbf{b}_j\otimes\mathbf{b}_k
</math>
It can be shown that
:<math>
  \mathcal{E}_{ijk} = \left[\mathbf{b}_i,\mathbf{b}_j,\mathbf{b}_k\right] =(\mathbf{b}_i\times\mathbf{b}_j)\cdot\mathbf{b}_k ~;~~
  \mathcal{E}^{ijk} = \left[\mathbf{b}^i,\mathbf{b}^j,\mathbf{b}^k\right]
</math>
Now,
:<math>
  \mathbf{b}_i\times\mathbf{b}_j = J~\varepsilon_{ijp}~\mathbf{b}^p = \sqrt{g}~\varepsilon_{ijp}~\mathbf{b}^p
</math>
Hence,
:<math>
  \mathcal{E}_{ijk} = J~\varepsilon_{ijk} = \sqrt{g}~\varepsilon_{ijk}
</math>
Similarly, we can show that
:<math>
  \mathcal{E}^{ijk} = \cfrac{1}{J}~\varepsilon^{ijk} = \cfrac{1}{\sqrt{g}}~\varepsilon^{ijk}
</math>
 
===Vector operations===
 
<ol>
<li>'''Identity map'''
The identity map '''I''' defined by <math>\mathsf{I}\cdot\mathbf{v} = \mathbf{v}</math> can be shown to be<ref name=Simmonds/>{{rp|page=39}}
:<math>
  \mathsf{I} = g^{ij}~\mathbf{b}_i\otimes\mathbf{b}_j = g_{ij}~\mathbf{b}^i\otimes\mathbf{b}^j = \mathbf{b}_i\otimes\mathbf{b}^i = \mathbf{b}^i\otimes\mathbf{b}_i
</math></li>
 
<li>'''Scalar (dot) product'''
The scalar product of two vectors in curvilinear coordinates is<ref name=Simmonds/>{{rp|page=32}}
:<math>
  \mathbf{u}\cdot\mathbf{v} = u^i~v_i = u_i~v^i = g_{ij}~u^i~v^j = g^{ij}~u_i~v_j
</math></li>
 
<li>'''Vector (cross) product'''
The [[cross product]] of two vectors is given by<ref name=Simmonds/>{{rp|pages=32–34}}
:<math>
  \mathbf{u}\times\mathbf{v} = \varepsilon_{ijk}~{u}_j~{v}_k~\mathbf{e}_i
</math>
where ε<sub>''ijk''</sub> is the [[permutation symbol]] and '''e'''<sub>''i''</sub> is a Cartesian basis vector.  In curvilinear coordinates, the equivalent expression is
:<math>
  \mathbf{u}\times\mathbf{v} = [(\mathbf{b}_m\times\mathbf{b}_n)\cdot\mathbf{b}_s]~u^m~v^n~\mathbf{b}^s
    = \mathcal{E}_{smn}~u^m~v^n~\mathbf{b}^s
</math>
where <math>\mathcal{E}_{ijk}</math> is the [[Curvilinear_coordinates#The_alternating_tensor|third-order alternating tensor]].
 
The [[cross product]] of two vectors is given by
:<math>
  \mathbf{u}\times\mathbf{v} = \varepsilon_{ijk}~\hat{u}_j~\hat{v}_k~\mathbf{e}_i
</math>
where ε<sub>''ijk''</sub> is the [[permutation symbol]] and <math>\mathbf{e}_i</math> is a Cartesian basis vector.  Therefore,
:<math>
  \mathbf{e}_p\times\mathbf{e}_q = \varepsilon_{ipq}~\mathbf{e}_i
</math>
and
:<math>
\begin{align}
  \mathbf{b}_m\times\mathbf{b}_n & = \frac{\partial \mathbf{x}}{\partial q^m}\times\frac{\partial \mathbf{x}}{\partial q^n}
    = \frac{\partial (x_p~\mathbf{e}_p)}{\partial q^m}\times\frac{\partial (x_q~\mathbf{e}_q)}{\partial q^n} \\[8pt]
& = \frac{\partial x_p}{\partial q^m}~\frac{\partial x_q}{\partial q^n}~\mathbf{e}_p\times\mathbf{e}_q
    = \varepsilon_{ipq}~\frac{\partial x_p}{\partial q^m}~\frac{\partial x_q}{\partial q^n}~\mathbf{e}_i
\end{align}
</math>
 
Hence,
 
:<math>
  (\mathbf{b}_m\times\mathbf{b}_n)\cdot\mathbf{b}_s =
    \varepsilon_{ipq}~\frac{\partial x_p}{\partial q^m}~\frac{\partial x_q}{\partial q^n}~\frac{\partial x_i}{\partial q^s}
</math>
Returning back to the vector product and using the relations
:<math>
  \hat{u}_j = \frac{\partial x_j}{\partial q^m}~u^m ~;~~
  \hat{v}_k = \frac{\partial x_k}{\partial q^n}~v^n ~;~~
  \mathbf{e}_i = \frac{\partial x_i}{\partial q^s}~\mathbf{b}^s
</math>
 
gives us
 
:<math>
\begin{align}
  \mathbf{u}\times\mathbf{v} & = \varepsilon_{ijk}~\hat{u}_j~\hat{v}_k~\mathbf{e}_i
    = \varepsilon_{ijk}~\frac{\partial x_j}{\partial q^m}~\frac{\partial x_k}{\partial q^n}~\frac{\partial x_i}{\partial q^s}~ u^m~v^n~\mathbf{b}^s \\[8pt]
  & = [(\mathbf{b}_m\times\mathbf{b}_n)\cdot\mathbf{b}_s]~u^m~v^n~\mathbf{b}^s
    = \mathcal{E}_{smn}~u^m~v^n~\mathbf{b}^s
\end{align}
</math></li>
</ol>
 
===Tensor operations===
<ol>
<li>'''[[Identity function|Identity map]]:'''
The identity map <math>\mathsf{I}</math> defined by <math>\mathsf{I}\cdot\mathbf{v} = \mathbf{v}</math> can be shown to be<ref name=Simmonds/>{{rp|page=39}}
:<math>
  \mathsf{I} = g^{ij}\mathbf{b}_i\otimes\mathbf{b}_j = g_{ij}\mathbf{b}^i\otimes\mathbf{b}^j = \mathbf{b}_i\otimes\mathbf{b}^i = \mathbf{b}^i\otimes\mathbf{b}_i
</math></li>
<li>'''Action of a second-order tensor on a vector:'''
The action <math>\mathbf{v} = \boldsymbol{S}\cdot\mathbf{u}</math> can be expressed in curvilinear coordinates as
:<math>
  v^i\mathbf{b}_i = S^{ij}u_j\mathbf{b}_i = S^i_{j}u^j\mathbf{b}_i ;\qquad v_i\mathbf{b}^i = S_{ij}u^i\mathbf{b}^i = S_{i}^{j}u_j\mathbf{b}^i
</math></li>
 
<li>'''[[Inner product]] of two second-order tensors:'''
The inner product of two second-order tensors <math>\boldsymbol{U} = \boldsymbol{S}\cdot\boldsymbol{T}</math> can be expressed in curvilinear coordinates as
:<math>
  U_{ij}\mathbf{b}^i\otimes\mathbf{b}^j = S_{ik}T^k_{.j} \mathbf{b}^i\otimes\mathbf{b}^j= S_i^{.k}T_{kj}\mathbf{b}^i\otimes\mathbf{b}^j
</math>
Alternatively,
:<math>
  \boldsymbol{U} = S^{ij}T^m_{.n}g_{jm}\mathbf{b}_i\otimes\mathbf{b}^n = S^i_{.m}T^m_{.n}\mathbf{b}_i\otimes\mathbf{b}^n
    = S^{ij}T_{jn}\mathbf{b}_i\otimes\mathbf{b}^n
</math></li>
 
<li>'''[[Determinant]] of a second-order tensor:'''
If <math>\boldsymbol{S}</math> is a second-order tensor, then the determinant is defined by the relation
:<math>
  \left[\boldsymbol{S}\cdot\mathbf{u}, \boldsymbol{S}\cdot\mathbf{v}, \boldsymbol{S}\cdot\mathbf{w}\right] = \det\boldsymbol{S}\left[\mathbf{u}, \mathbf{v}, \mathbf{w}\right]
</math>
where <math>\mathbf{u}, \mathbf{v}, \mathbf{w}</math> are arbitrary vectors and
:<math>
  \left[\mathbf{u},\mathbf{v},\mathbf{w}\right] := \mathbf{u}\cdot(\mathbf{v}\times\mathbf{w}).
</math></li>
</ol>
 
===Relations between curvilinear and Cartesian basis vectors===
Let ('''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub>) be the usual Cartesian basis vectors for the Euclidean space of interest and let
:<math>
  \mathbf{b}_i = \boldsymbol{F}\cdot\mathbf{e}_i
</math>
where '''''F'''<sub>i</sub>'' is a second-order transformation tensor that maps '''e'''<sub>''i''</sub> to '''b'''<sub>''i''</sub>.  Then,
:<math>
  \mathbf{b}_i\otimes\mathbf{e}_i = (\boldsymbol{F}\cdot\mathbf{e}_i)\otimes\mathbf{e}_i = \boldsymbol{F}\cdot(\mathbf{e}_i\otimes\mathbf{e}_i) = \boldsymbol{F}~.
</math>
From this relation we can show that
:<math>
  \mathbf{b}^i = \boldsymbol{F}^{-\rm{T}}\cdot\mathbf{e}^i ~;~~ g^{ij} = [\boldsymbol{F}^{-\rm{1}}\cdot\boldsymbol{F}^{-\rm{T}}]_{ij} ~;~~ g_{ij} = [g^{ij}]^{-1} = [\boldsymbol{F}^{\rm{T}}\cdot\boldsymbol{F}]_{ij}
</math>
Let <math>J := \det\boldsymbol{F}</math> be the Jacobian of the transformation.  Then, from the definition of the determinant,
:<math>
  \left[\mathbf{b}_1,\mathbf{b}_2,\mathbf{b}_3\right] = \det\boldsymbol{F}\left[\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3\right] ~.
</math>
Since
:<math>
  \left[\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3\right] = 1
</math>
we have
:<math>
  J = \det\boldsymbol{F} = \left[\mathbf{b}_1,\mathbf{b}_2,\mathbf{b}_3\right] = \mathbf{b}_1\cdot(\mathbf{b}_2\times\mathbf{b}_3)
</math>
A number of interesting results can be derived using the above relations.
 
First, consider
:<math>
  g := \det[g_{ij}]\,
</math>
Then
:<math>
  g = \det[\boldsymbol{F}^{\rm{T}}]\cdot\det[\boldsymbol{F}] = J\cdot J = J^2
</math>
Similarly, we can show that
:<math>
  \det[g^{ij}] = \cfrac{1}{J^2}
</math>
Therefore, using the fact that <math>[g^{ij}] = [g_{ij}]^{-1}</math>,
:<math>
  \cfrac{\partial g}{\partial g_{ij}} = 2~J~\cfrac{\partial J}{\partial g_{ij}} = g~g^{ij}
</math>
 
Another interesting relation is derived below.  Recall that
:<math>
  \mathbf{b}^i\cdot\mathbf{b}_j = \delta^i_j \quad \Rightarrow \quad \mathbf{b}^1\cdot\mathbf{b}_1 = 1,~\mathbf{b}^1\cdot\mathbf{b}_2=\mathbf{b}^1\cdot\mathbf{b}_3=0 \quad \Rightarrow \quad \mathbf{b}^1 = A~(\mathbf{b}_2\times\mathbf{b}_3)
</math>
where ''A'' is a, yet undetermined, constant. Then
:<math>
  \mathbf{b}^1\cdot\mathbf{b}_1 =  A~\mathbf{b}_1\cdot(\mathbf{b}_2\times\mathbf{b}_3) = AJ = 1 \quad \Rightarrow \quad A = \cfrac{1}{J}
</math>
This observation leads to the relations
:<math>
  \mathbf{b}^1 = \cfrac{1}{J}(\mathbf{b}_2\times\mathbf{b}_3) ~;~~
  \mathbf{b}^2 = \cfrac{1}{J}(\mathbf{b}_3\times\mathbf{b}_1) ~;~~
  \mathbf{b}^3 = \cfrac{1}{J}(\mathbf{b}_1\times\mathbf{b}_2)
</math>
In index notation,
:<math>
  \varepsilon_{ijk}~\mathbf{b}^k = \cfrac{1}{J}(\mathbf{b}_i\times\mathbf{b}_j) = \cfrac{1}{\sqrt{g}}(\mathbf{b}_i\times\mathbf{b}_j)
</math>
where <math>\varepsilon_{ijk}\,</math> is the usual [[permutation symbol]].
 
We have not identified an explicit expression for the transformation tensor '''''F''''' because an alternative form of the mapping between curvilinear and Cartesian bases is more useful.  Assuming a sufficient degree of smoothness in the mapping (and a bit of abuse of notation), we have
:<math>
  \mathbf{b}_i = \cfrac{\partial\mathbf{x}}{\partial q^i} =  \cfrac{\partial\mathbf{x}}{\partial x_j}~\cfrac{\partial x_j}{\partial q^i} = \mathbf{e}_j~\cfrac{\partial x_j}{\partial q^i}
</math>
Similarly,
:<math>
  \mathbf{e}_i = \mathbf{b}_j~\cfrac{\partial q^j}{\partial x_i}
</math>
From these results we have
:<math>
  \mathbf{e}^k\cdot\mathbf{b}_i = \frac{\partial x_k}{\partial q^i} \quad \Rightarrow \quad
  \frac{\partial x_k}{\partial q^i}~\mathbf{b}^i = \mathbf{e}^k\cdot(\mathbf{b}_i\otimes\mathbf{b}^i) = \mathbf{e}^k
</math>
and
:<math>
  \mathbf{b}^k = \frac{\partial q^k}{\partial x_i}~\mathbf{e}^i
</math>
 
==Vector and tensor calculus in three-dimensional curvilinear coordinates==
{{Einstein_summation_convention}}
Simmonds,<ref name=Simmonds/> in his book on [[tensor analysis]], quotes [[Albert Einstein]] saying<ref name=Lanczos>{{cite book | last=Einstein | first=A. | year=1915 | contribution=Contribution to the Theory of General Relativity | editor=Laczos, C. | title=The Einstein Decade | page=213 | isbn=0-521-38105-3 }}</ref>
<blockquote>
  The magic of this theory will hardly fail to impose itself on anybody who has truly understood it; it represents a genuine triumph of the method of absolute differential calculus, founded by Gauss, Riemann, Ricci, and Levi-Civita.
</blockquote>
Vector and tensor calculus in general curvilinear coordinates is used in tensor analysis on four-dimensional curvilinear [[manifold]]s in [[general relativity]],<ref name=Misner>{{cite book | last1=Misner | first1=C. W. | last2=Thorne | first2=K. S. | last3=Wheeler | first3=J. A. | year=1973 | title=Gravitation | publisher=W. H. Freeman and Co. | isbn=0-7167-0344-0}}</ref> in the [[solid mechanics|mechanics]] of curved [[Plate theory|shells]],<ref name=Ciarlet/> in examining the [[invariant (mathematics)|invariance]] properties of [[Maxwell's equations]] which has been of interest in [[metamaterials]]<ref name=Greenleaf>{{cite journal | doi=10.1088/0967-3334/24/2/353 | last1=Greenleaf | first1=A. | last2=Lassas | first2=M. | last3=Uhlmann | first3=G. | year=2003 | title=Anisotropic conductivities that cannot be detected by EIT | journal=Physiological measurement | volume=24 | issue=2 | pages=413–419 | pmid=12812426}}</ref><ref name=Leonhardt>{{cite journal | last1=Leonhardt | first1=U. | last2=Philbin | first2=T.G. | year=2006 | title=General relativity in electrical engineering | journal=New Journal of Physics | volume=8 | page=247 |arxiv = cond-mat/0607418 |bibcode = 2006NJPh....8..247L |doi = 10.1088/1367-2630/8/10/247 }}</ref> and in many other fields.
 
Some useful relations in the calculus of vectors and second-order tensors in curvilinear coordinates are given in this section.  The notation and contents are primarily from Ogden,<ref name=Ogden00/> Simmonds,<ref name=Simmonds /> Green and Zerna,<ref name=Green/>  Basar and Weichert,<ref name=Basar/> and Ciarlet.<ref name=Ciarlet/>
 
===Basic definitions===
Let the position of a point in space be characterized by three coordinate variables <math>(q^1, q^2, q^3)</math>.
 
The coordinate curve ''q''<sup>1</sup> represents a curve on which ''q''<sup>2</sup>, ''q''<sup>3</sup> are constant.  Let '''x''' be the [[position vector]] of the point relative to some origin.  Then, assuming that such a mapping and its inverse exist and are continuous, we can write <ref name=Ogden00>{{cite book | last=Ogden | first=R. W. | year=2000 | title=Nonlinear elastic deformations | publisher=Dover}}</ref>{{rp|page=55}}
:<math>
  \mathbf{x} = \boldsymbol{\varphi}(q^1, q^2, q^3) ~;~~ q^i = \psi^i(\mathbf{x}) = [\boldsymbol{\varphi}^{-1}(\mathbf{x})]^i
</math>
The fields ψ<sup>''i''</sup>('''x''') are called the '''curvilinear coordinate functions''' of the '''curvilinear coordinate system''' '''ψ'''('''x''') = '''ψ'''<sup>−1</sup>('''x''').
 
The ''q<sup>i</sup>'' '''coordinate curves''' are defined by the one-parameter family of functions given by
:<math>
  \mathbf{x}_i(\alpha) = \boldsymbol{\varphi}(\alpha, q^j, q^k) ~,~~ i\ne j \ne k
</math>
with ''q<sup>j</sup>, q<sup>k</sup>'' fixed.
 
===Tangent vector to coordinate curves===
The '''tangent vector''' to the curve '''x'''<sub>''i''</sub> at the point '''x'''<sub>''i''</sub>(α) (or to the coordinate curve ''q<sub>i</sub>'' at the point '''x''') is
:<math>
  \cfrac{\rm{d}\mathbf{x}_i}{\rm{d}\alpha} \equiv \cfrac{\partial\mathbf{x}}{\partial q^i}
</math>
 
===Gradient===
 
====Scalar field====
Let ''f''('''x''') be a scalar field in space. Then
:<math>
  f(\mathbf{x}) = f[\boldsymbol{\varphi}(q^1,q^2,q^3)] = f_\varphi(q^1,q^2,q^3)
</math>
The gradient of the field ''f'' is defined by
:<math>
  [\boldsymbol{\nabla}f(\mathbf{x})]\cdot\mathbf{c} = \cfrac{\rm{d}}{\rm{d}\alpha} f(\mathbf{x}+\alpha\mathbf{c})\biggr|_{\alpha=0}
</math>
where '''c''' is an arbitrary constant vector.  If we define the components ''c<sup>i</sup>'' of '''c''' are such that
:<math>
  q^i + \alpha~c^i = \psi^i(\mathbf{x} + \alpha~\mathbf{c})
</math>
then
:<math>
  [\boldsymbol{\nabla}f(\mathbf{x})]\cdot\mathbf{c} = \cfrac{\rm{d}}{\rm{d}\alpha} f_\varphi(q^1 + \alpha~c^1, q^2 + \alpha~c^2, q^3 + \alpha~c^3)\biggr|_{\alpha=0} = \cfrac{\partial f_\varphi}{\partial q^i}~c^i = \cfrac{\partial f}{\partial q^i}~c^i
</math>
 
If we set <math>f(\mathbf{x}) = \psi^i(\mathbf{x})</math>, then since <math>q^i = \psi^i(\mathbf{x})</math>, we have
:<math>
  [\boldsymbol{\nabla}\psi^i(\mathbf{x})]\cdot\mathbf{c} = \cfrac{\partial \psi^i}{\partial q^j}~c^j = c^i
</math>
which provides a means of extracting the contravariant component of a vector '''c'''.
 
If '''b'''<sub>''i''</sub> is the covariant (or natural) basis at a point, and if '''b'''<sup>''i''</sup> is the contravariant (or reciprocal) basis at that point, then
:<math>
  [\boldsymbol{\nabla}f(\mathbf{x})]\cdot\mathbf{c} = \cfrac{\partial f}{\partial q^i}~c^i = \left(\cfrac{\partial f}{\partial q^i}~\mathbf{b}^i\right)
  \left(c^i~\mathbf{b}_i\right) \quad \Rightarrow \quad \boldsymbol{\nabla}f(\mathbf{x}) = \cfrac{\partial f}{\partial q^i}~\mathbf{b}^i
</math>
A brief rationale for this choice of basis is given in the next section.
 
====Vector field====
A similar process can be used to arrive at the gradient of a vector field '''f'''('''x''').  The gradient is given by
:<math>
  [\boldsymbol{\nabla}\mathbf{f}(\mathbf{x})]\cdot\mathbf{c} = \cfrac{\partial \mathbf{f}}{\partial q^i}~c^i
</math>
If we consider the gradient of the position vector field '''r'''('''x''') = '''x''', then we can show that
:<math>
  \mathbf{c} = \cfrac{\partial\mathbf{x}}{\partial q^i}~c^i = \mathbf{b}_i(\mathbf{x})~c^i ~;~~ \mathbf{b}_i(\mathbf{x}) := \cfrac{\partial\mathbf{x}}{\partial q^i}
</math>
The vector field '''b'''<sub>''i''</sub> is tangent to the ''q<sup>i</sup>'' coordinate curve and forms a '''natural basis''' at each point on the curve.  This basis, as discussed at the beginning of this article, is also called the '''covariant''' curvilinear basis.  We can also define a '''reciprocal basis''', or '''contravariant''' curvilinear basis, '''b'''<sup>''i''</sup>.  All the algebraic relations between the basis vectors, as discussed in the section on tensor algebra, apply for the natural basis and its reciprocal at each point '''x'''.
 
Since '''c''' is arbitrary, we can write
:<math>
  \boldsymbol{\nabla}\mathbf{f}(\mathbf{x}) = \cfrac{\partial \mathbf{f}}{\partial q^i}\otimes\mathbf{b}^i
</math>
 
Note that the contravariant basis vector '''b'''<sup>''i''</sup> is perpendicular to the surface of constant ψ<sup>''i''</sup> and is given by
:<math>
  \mathbf{b}^i = \boldsymbol{\nabla}\psi^i
</math>
 
====Christoffel symbols of the first kind====
The [[Christoffel symbols]] of the first kind are defined as
:<math>
  \mathbf{b}_{i,j} = \frac{\partial \mathbf{b}_i}{\partial q^j} := \Gamma_{ijk}~\mathbf{b}^k \quad \Rightarrow \quad
  \mathbf{b}_{i,j} \cdot \mathbf{b}_l = \Gamma_{ijl}
</math>
To express Γ<sub>''ijk''</sub> in terms of ''g<sub>ij</sub>'' we note that
:<math>
  \begin{align}
    g_{ij,k} & = (\mathbf{b}_i\cdot\mathbf{b}_j)_{,k} = \mathbf{b}_{i,k}\cdot\mathbf{b}_j + \mathbf{b}_i\cdot\mathbf{b}_{j,k}
        = \Gamma_{ikj} + \Gamma_{jki}\\
    g_{ik,j} & = (\mathbf{b}_i\cdot\mathbf{b}_k)_{,j} = \mathbf{b}_{i,j}\cdot\mathbf{b}_k + \mathbf{b}_i\cdot\mathbf{b}_{k,j}
        = \Gamma_{ijk} + \Gamma_{kji}\\
    g_{jk,i} & = (\mathbf{b}_j\cdot\mathbf{b}_k)_{,i} = \mathbf{b}_{j,i}\cdot\mathbf{b}_k + \mathbf{b}_j\cdot\mathbf{b}_{k,i}
        = \Gamma_{jik} + \Gamma_{kij}
  \end{align}
</math>
Since '''b'''<sub>''i,j''</sub> = '''b'''<sub>''j,i''</sub> we have Γ<sub>''ijk''</sub> = Γ<sub>''jik''</sub>.  Using these to rearrange the above relations gives
:<math>
  \Gamma_{ijk} = \frac{1}{2}(g_{ik,j} + g_{jk,i} - g_{ij,k})
      = \frac{1}{2}[(\mathbf{b}_i\cdot\mathbf{b}_k)_{,j} + (\mathbf{b}_j\cdot\mathbf{b}_k)_{,i} - (\mathbf{b}_i\cdot\mathbf{b}_j)_{,k}]
</math>
 
====Christoffel symbols of the second kind====
The [[Christoffel symbol]]s of the second kind are defined as
:<math> \Gamma_{ij}^k = \Gamma_{ji}^k </math>
in which
 
:<math>\cfrac{\partial \mathbf{b}_i}{\partial q^j} = \Gamma_{ij}^k~\mathbf{b}_k
</math>
 
This implies that
:<math>
  \Gamma_{ij}^k = \cfrac{\partial \mathbf{b}_i}{\partial q^j}\cdot\mathbf{b}^k = -\mathbf{b}_i\cdot\cfrac{\partial \mathbf{b}^k}{\partial q^j}
</math>
Other relations that follow are
:<math>
    \cfrac{\partial \mathbf{b}^i}{\partial q^j} = -\Gamma^i_{jk}~\mathbf{b}^k ~;~~
    \boldsymbol{\nabla}\mathbf{b}_i = \Gamma_{ij}^k~\mathbf{b}_k\otimes\mathbf{b}^j ~;~~
    \boldsymbol{\nabla}\mathbf{b}^i = -\Gamma_{jk}^i~\mathbf{b}^k\otimes\mathbf{b}^j
</math>
 
Another particularly useful relation, which shows that the Christoffel symbol depends only on the metric tensor and its derivatives, is
:<math>
  \Gamma^k_{ij} = \frac{g^{km}}{2}\left(\frac{\partial g_{mi}}{\partial q^j} + \frac{\partial g_{mj}}{\partial q^i} - \frac{\partial g_{ij}}{\partial q^m} \right)
</math>
 
====Explicit expression for the gradient of a vector field====
The following expressions for the gradient of a vector field in curvilinear coordinates are quite useful.
:<math>
  \begin{align}
  \boldsymbol{\nabla}\mathbf{v} & = \left[\cfrac{\partial v^i}{\partial q^k} + \Gamma^i_{lk}~v^l\right]~\mathbf{b}_i\otimes\mathbf{b}^k \\[8pt]
    & = \left[\cfrac{\partial v_i}{\partial q^k} - \Gamma^l_{ki}~v_l\right]~\mathbf{b}^i\otimes\mathbf{b}^k
  \end{align}
</math>
 
====Representing a physical vector field====
The vector field '''v''' can be represented as
:<math>
  \mathbf{v} = v_i~\mathbf{b}^i = \hat{v}_i~\hat{\mathbf{b}}^i
</math>
where <math>v_i\,</math> are the covariant components of the field, <math>\hat{v}_i</math> are the physical components, and (no [[Einstein notation|summation]])
:<math>
  \hat{\mathbf{b}}^i = \cfrac{\mathbf{b}^i}{\sqrt{g^{ii}}}
</math>
is the normalized contravariant basis vector.
 
===Second-order tensor field===
The gradient of a second order tensor field can similarly be expressed as
:<math>
  \boldsymbol{\nabla}\boldsymbol{S} = \cfrac{\partial \boldsymbol{S}}{\partial q^i}\otimes\mathbf{b}^i
</math>
 
====Explicit expressions for the gradient====
If we consider the expression for the tensor in terms of a contravariant basis, then
:<math>
  \boldsymbol{\nabla}\boldsymbol{S} = \cfrac{\partial}{\partial q^k}[S_{ij}~\mathbf{b}^i\otimes\mathbf{b}^j]\otimes\mathbf{b}^k
  = \left[\cfrac{\partial S_{ij}}{\partial q^k} - \Gamma^l_{ki}~S_{lj} - \Gamma^l_{kj}~S_{il}\right]~\mathbf{b}^i\otimes\mathbf{b}^j\otimes\mathbf{b}^k
</math>
We may also write
:<math>
  \begin{align}
  \boldsymbol{\nabla}\boldsymbol{S} & = \left[\cfrac{\partial S^{ij}}{\partial q^k} + \Gamma^i_{kl}~S^{lj} + \Gamma^j_{kl}~S^{il}\right]~\mathbf{b}_i\otimes\mathbf{b}_j\otimes\mathbf{b}^k \\[8pt]
  & = \left[\cfrac{\partial S^i_{~j}}{\partial q^k} + \Gamma^i_{kl}~S^l_{~j} - \Gamma^l_{kj}~S^i_{~l}\right]~\mathbf{b}_i\otimes\mathbf{b}^j\otimes\mathbf{b}^k \\[8pt]
  & = \left[\cfrac{\partial S_i^{~j}}{\partial q^k} - \Gamma^l_{ik}~S_l^{~j} + \Gamma^j_{kl}~S_i^{~l}\right]~\mathbf{b}^i\otimes\mathbf{b}_j\otimes\mathbf{b}^k
  \end{align}
</math>
 
====Representing a physical second-order tensor field====
The physical components of a second-order tensor field can be obtained by using a normalized contravariant basis, i.e.,
:<math>
  \boldsymbol{S} = S_{ij}~\mathbf{b}^i\otimes\mathbf{b}^j = \hat{S}_{ij}~\hat{\mathbf{b}}^i\otimes\hat{\mathbf{b}}^j
</math>
where the hatted basis vectors have been normalized.  This implies that (again no summation)
 
:<math>
  \hat{S}_{ij} = S_{ij}~\sqrt{g^{ii}~g^{jj}}
</math>
 
===Divergence===
 
====Vector field====
The [[divergence]] of a vector field (<math>\mathbf{v}</math>)is defined as
:<math>
  \operatorname{div}~\mathbf{v} = \boldsymbol{\nabla}\cdot\mathbf{v} = \text{tr}(\boldsymbol{\nabla}\mathbf{v})
</math>
In terms of components with respect to a curvilinear basis
:<math>
  \boldsymbol{\nabla}\cdot\mathbf{v} = \cfrac{\partial v^i}{\partial q^i} + \Gamma^i_{\ell i}~v^\ell
    = \left[\cfrac{\partial v_i}{\partial q^j} - \Gamma^\ell_{ji}~v_\ell\right]~g^{ij}
</math>
 
An alternative equation for the divergence of a vector field is frequently used. To derive this relation recall that
:<math>
  \boldsymbol{\nabla} \cdot \mathbf{v} = \frac{\partial v^i}{\partial q^i} + \Gamma_{\ell i}^i~v^\ell
</math>
Now,
:<math>
  \Gamma_{\ell i}^i = \Gamma_{i\ell}^i = \cfrac{g^{mi}}{2}\left[\frac{\partial g_{im}}{\partial q^\ell} +
    \frac{\partial g_{\ell m}}{\partial q^i} - \frac{\partial g_{il}}{\partial q^m}\right]
</math>
Noting that, due to the symmetry of <math>\boldsymbol{g}</math>,
:<math>
  g^{mi}~\frac{\partial g_{\ell m}}{\partial q^i}  = g^{mi}~ \frac{\partial g_{i\ell}}{\partial q^m}
</math>
we have
:<math>
  \boldsymbol{\nabla} \cdot \mathbf{v} = \frac{\partial v^i}{\partial q^i} +  \cfrac{g^{mi}}{2}~\frac{\partial g_{im}}{\partial q^\ell}~v^\ell
</math>
Recall that if [''g<sub>ij</sub>''] is the matrix whose components are ''g<sub>ij</sub>'', then the inverse of the matrix is <math>[g_{ij}]^{-1} = [g^{ij}]</math>. The inverse of the matrix is given by
:<math>
  [g^{ij}] = [g_{ij}]^{-1} = \cfrac{A^{ij}}{g} ~;~~ g := \det([g_{ij}]) = \det\boldsymbol{g}
</math>
where ''A<sup>ij</sup>'' are the [[Cofactor matrix]] of the components ''g<sub>ij</sub>''. From matrix algebra we have
:<math>
  g = \det([g_{ij}]) = \sum_i g_{ij}~A^{ij} \quad \Rightarrow \quad
  \frac{\partial g}{\partial g_{ij}} = A^{ij}
</math>
Hence,
:<math>
  [g^{ij}] = \cfrac{1}{g}~\frac{\partial g}{\partial g_{ij}}
</math>
Plugging this relation into the expression for the divergence gives
:<math>
  \boldsymbol{\nabla} \cdot \mathbf{v} = \frac{\partial v^i}{\partial q^i} +  \cfrac{1}{2g}~\frac{\partial g}{\partial g_{mi}}~\frac{\partial g_{im}}{\partial q^\ell}~v^\ell = \frac{\partial v^i}{\partial q^i} +  \cfrac{1}{2g}~\frac{\partial g}{\partial q^\ell}~v^\ell
</math>
A little manipulation leads to the more compact form
:<math>
  \boldsymbol{\nabla} \cdot \mathbf{v} = \cfrac{1}{\sqrt{g}}~\frac{\partial }{\partial q^i}(v^i~\sqrt{g})
</math>
 
====Second-order tensor field====
 
The [[divergence]] of a second-order tensor field is defined using
:<math>
  (\boldsymbol{\nabla}\cdot\boldsymbol{S})\cdot\mathbf{a} = \boldsymbol{\nabla}\cdot(\boldsymbol{S}\cdot\mathbf{a})
</math>
where '''a''' is an arbitrary constant vector.
<ref>{{cite web | url=http://en.wikiversity.org/wiki/Introduction_to_Elasticity/Tensors#The_divergence_of_a_tensor_field | publisher=[[Wikiversity]] | work=Introduction to Elasticity/Tensors | title=The divergence of a tensor field | accessdate=2010-11-26 }}</ref>
In curvilinear coordinates,
:<math>
  \begin{align}
  \boldsymbol{\nabla}\cdot\boldsymbol{S} & = \left[\cfrac{\partial S_{ij}}{\partial q^k} - \Gamma^l_{ki}~S_{lj} - \Gamma^l_{kj}~S_{il}\right]~g^{ik}~\mathbf{b}^j \\[8pt]
  & = \left[\cfrac{\partial S^{ij}}{\partial q^i} + \Gamma^i_{il}~S^{lj} + \Gamma^j_{il}~S^{il}\right]~\mathbf{b}_j \\[8pt]
  & = \left[\cfrac{\partial S^i_{~j}}{\partial q^i} + \Gamma^i_{il}~S^l_{~j} - \Gamma^l_{ij}~S^i_{~l}\right]~\mathbf{b}^j \\[8pt]
  & = \left[\cfrac{\partial S_i^{~j}}{\partial q^k} - \Gamma^l_{ik}~S_l^{~j} + \Gamma^j_{kl}~S_i^{~l}\right]~g^{ik}~\mathbf{b}_j
  \end{align}
</math>
 
===Laplacian ===
 
====Scalar field====
The Laplacian of a scalar field φ('''x''') is defined as
:<math>
  \nabla^2 \varphi := \boldsymbol{\nabla} \cdot (\boldsymbol{\nabla} \varphi)
</math>
Using the alternative expression for the divergence of a vector field gives us
:<math>
  \nabla^2 \varphi =  \cfrac{1}{\sqrt{g}}~\frac{\partial }{\partial q^i}([\boldsymbol{\nabla} \varphi]^i~\sqrt{g})
</math>
Now
:<math>
  \boldsymbol{\nabla} \varphi = \frac{\partial \varphi}{\partial q^l}~\mathbf{b}^l = g^{li}~\frac{\partial \varphi}{\partial q^l}~\mathbf{b}_i
  \quad \Rightarrow \quad
  [\boldsymbol{\nabla} \varphi]^i = g^{li}~\frac{\partial \varphi}{\partial q^l}
</math>
Therefore,
:<math>
  \nabla^2 \varphi =  \cfrac{1}{\sqrt{g}}~\frac{\partial }{\partial q^i}\left(g^{li}~\frac{\partial \varphi}{\partial q^l}
~\sqrt{g}\right)
</math>
 
===Curl of a vector field===
The curl of a vector field '''v''' in covariant curvilinear coordinates can be written as
:<math>
  \boldsymbol{\nabla}\times\mathbf{v} = \mathcal{E}^{rst} v_{s|r}~ \mathbf{b}_t
</math>
where
:<math>
  v_{s|r} = v_{s,r} - \Gamma^i_{sr}~v_i
</math>
 
==Orthogonal curvilinear coordinates==
Assume, for the purposes of this section, that the curvilinear coordinate system is [[orthogonal]], i.e.,
: <math> \mathbf{b}_i\cdot\mathbf{b}_j =
  \begin{cases} g_{ii} & \text{if } i = j \\
                0 & \text{if } i \ne j,
  \end{cases}
</math>
or equivalently,
: <math> \mathbf{b}^i\cdot\mathbf{b}^j =
  \begin{cases} g^{ii} & \text{if } i = j \\
                0 & \text{if } i \ne j,
  \end{cases}
</math>
where <math>g^{ii} = g_{ii}^{-1}</math>. As before, <math>\mathbf{b}_i, \mathbf{b}_j</math> are covariant basis vectors and '''b'''<sup>''i''</sup>, '''b'''<sup>''j''</sup> are contravariant basis vectors.  Also, let ('''e'''<sup>1</sup>, '''e'''<sup>2</sup>, '''e'''<sup>3</sup>) be a background, fixed, [[Cartesian coordinate system|Cartesian]] basis.  A list of orthogonal curvilinear coordinates is given below.
 
===Metric tensor in orthogonal curvilinear coordinates===
{{Main|Metric tensor}}
 
Let '''r'''('''x''') be the [[position vector]] of the point '''x''' with respect to the origin of the coordinate system.  The notation can be simplified by noting that '''x''' = '''r'''('''x''').  At each point we can construct a small line element d'''x'''.  The square of the length of the line element is the scalar product d'''x''' • d'''x''' and is called the [[Metric (mathematics)|metric]] of the [[space]]. Recall that the space of interest is assumed to be [[Euclidean space|Euclidean]] when we talk of curvilinear coordinates. Let us express the position vector in terms of the background, fixed, Cartesian basis, i.e.,
:<math>
  \mathbf{x} = \sum_{i=1}^3 x_i~\mathbf{e}_i
</math>
 
Using the [[chain rule]], we can then express d'''x''' in terms of three-dimensional orthogonal curvilinear coordinates (''q''<sup>1</sup>, ''q''<sup>2</sup>, ''q''<sup>3</sup>) as
:<math>
  \mathrm{d}\mathbf{x} = \sum_{i=1}^3 \sum_{j=1}^3 \left(\cfrac{\partial x_i}{\partial q^j}~\mathbf{e}_i\right)\mathrm{d}q^j
</math>
Therefore the metric is given by
:<math>
  \mathrm{d}\mathbf{x}\cdot\mathrm{d}\mathbf{x} = \sum_{i=1}^3 \sum_{j=1}^3 \sum_{k=1}^3 \cfrac{\partial x_i}{\partial q^j}~\cfrac{\partial x_i}{\partial q^k}~\mathrm{d}q^j~\mathrm{d}q^k
  </math>
 
The symmetric quantity
:<math>
  g_{ij}(q^i,q^j) = \sum_{k=1}^3 \cfrac{\partial x_k}{\partial q^i}~\cfrac{\partial x_k}{\partial q^j} = \mathbf{b}_i\cdot\mathbf{b}_j
</math>
is called the [[metric tensor|fundamental (or metric) tensor]] of the [[Euclidean space]] in curvilinear coordinates.
 
Note also that
:<math>
  g_{ij} = \cfrac{\partial\mathbf{x}}{\partial q^i}\cdot\cfrac{\partial\mathbf{x}}{\partial q^j}
          = \left(\sum_{k} h_{ki}~\mathbf{e}_k\right)\cdot\left(\sum_{m} h_{mj}~\mathbf{e}_m\right)
          = \sum_{k} h_{ki}~h_{kj}
</math>
where ''h<sub>ij</sub>'' are the Lamé coefficients.
 
If we define the scale factors, ''h<sub>i</sub>'',  using
:<math>
  \mathbf{b}_i\cdot\mathbf{b}_i = g_{ii} = \sum_{k} h_{ki}^2 =: h_i^2
  \quad \Rightarrow \quad \left|\cfrac{\partial\mathbf{x}}{\partial q^i}\right| = \left|\mathbf{b}_i\right| = \sqrt{g_{ii}} = h_i
</math>
we get a relation between the fundamental tensor and the Lamé coefficients.
 
====Example: Polar coordinates====
If we consider polar coordinates for '''R'''<sup>2</sup>, note that
: <math> (x, y)=(r \cos \theta, r \sin \theta) \,\!</math>
(r, θ) are the curvilinear coordinates, and the Jacobian determinant of the transformation (''r'',θ) → (''r'' cos θ, ''r'' sin θ) is ''r''.
 
The [[orthogonal]] basis vectors are '''''b'''''<sub>''r''</sub> = (cos θ, sin θ), '''''b'''''<sub>θ</sub> = (−''r'' sin θ, ''r'' cos θ).  The normalized basis vectors are '''''e'''''<sub>''r''</sub> = (cos θ, sin θ), '''''e'''''<sub>θ</sub> = (−sin θ, cos θ) and the scale factors are ''h''<sub>''r''</sub> = 1 and ''h''<sub>θ</sub>= ''r''. The fundamental tensor is ''g''<sub>11</sub> =1, ''g''<sub>22</sub> =''r''<sup>2</sup>, ''g''<sub>12</sub> = ''g''<sub>21</sub> =0.
 
===Line and surface integrals===
If we wish to use curvilinear coordinates for [[vector calculus]] calculations, adjustments need to be made in the calculation of line, surface and volume integrals.  For simplicity, we again restrict the discussion to three dimensions and orthogonal curvilinear coordinates.  However, the same arguments apply for <math>n</math>-dimensional problems though there are some additional terms in the expressions when the coordinate system is not orthogonal.
 
====Line integrals====
Normally in the calculation of [[line integral]]s we are interested in calculating
: <math> \int_C f \,ds = \int_a^b f(\mathbf{x}(t))\left|{\partial \mathbf{x} \over \partial t}\right|\; dt</math>
where '''''x'''''(''t'') parametrizes C in Cartesian coordinates.
In curvilinear coordinates, the term
: <math> \left|{\partial \mathbf{x} \over \partial t}\right| = \left| \sum_{i=1}^3 {\partial \mathbf{x} \over \partial q^i}{\partial q^i \over \partial t}\right|</math>
by the [[chain rule]].  And from the definition of the Lamé coefficients,
: <math> {\partial \mathbf{x} \over \partial q^i} = \sum_{k} h_{ki}~ \mathbf{e}_{k} </math>
and thus
: <math>
\begin{align}
\left|{\partial \mathbf{x} \over \partial t}\right| & = \left| \sum_k\left(\sum_i h_{ki}~\cfrac{\partial q^i}{\partial t}\right)\mathbf{e}_k\right| \\[8pt]
& = \sqrt{\sum_i\sum_j\sum_k h_{ki}~h_{kj}\cfrac{\partial q^i}{\partial t}\cfrac{\partial q^j}{\partial t}} = \sqrt{\sum_i\sum_j g_{ij}~\cfrac{\partial q^i}{\partial t}\cfrac{\partial q^j}{\partial t}}
\end{align}
</math>
Now, since <math>g_{ij} = 0\,</math> when <math> i \ne j </math>, we have
: <math>
\left|{\partial \mathbf{x} \over \partial t}\right| =  \sqrt{\sum_i g_{ii}~\left(\cfrac{\partial q^i}{\partial t}\right)^2} = \sqrt{\sum_i h_{i}^2~\left(\cfrac{\partial q^i}{\partial t}\right)^2}
</math>
and we can proceed normally.
 
====Surface integrals====
Likewise, if we are interested in a [[surface integral]], the relevant calculation, with the parameterization of the surface in Cartesian coordinates is:
: <math>\int_S f \,dS = \iint_T f(\mathbf{x}(s, t)) \left|{\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}\right| \, ds \, dt</math>
Again, in curvilinear coordinates, we have
: <math>
\left|{\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}\right| = \left|\left(\sum_i {\partial \mathbf{x} \over \partial q^i}{\partial q^i \over \partial s}\right) \times \left(\sum_j {\partial \mathbf{x} \over \partial q^j}{\partial q^j \over \partial t}\right)\right|
</math>
and we make use of the definition of curvilinear coordinates again to yield
: <math>
{\partial \mathbf{x} \over \partial q^i}{\partial q^i \over \partial s} = \sum_k \left(\sum_{i=1}^3 h_{ki}~{\partial q^i \over \partial s}\right) \mathbf{e}_k ~;~~
{\partial \mathbf{x} \over \partial q^j}{\partial q^j \over \partial t} = \sum_m \left(\sum_{j=1}^3 h_{mj}~{\partial q^j \over \partial t}\right) \mathbf{e}_{m}
</math>
 
Therefore,
: <math>
  \begin{align}
\left|{\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}\right|
  & = \left|
    \sum_k \sum_m \left(\sum_{i=1}^3 h_{ki}~{\partial q^i \over \partial s}\right)\left(\sum_{j=1}^3 h_{mj}~{\partial q^j \over \partial t}\right) \mathbf{e}_k\times\mathbf{e}_m
    \right| \\[8pt]
  & = \left|\sum_p \sum_k \sum_m \mathcal{E}_{kmp}\left(\sum_{i=1}^3 h_{ki}~{\partial q^i \over \partial s}\right)\left(\sum_{j=1}^3 h_{mj}~{\partial q^j \over \partial t}\right) \mathbf{e}_p \right|
  \end{align}
  </math>
where <math>\mathcal{E}</math> is the [[permutation symbol]].
 
In determinant form,  the cross product in terms of curvilinear coordinates will be:
: <math>\begin{vmatrix}
\mathbf{e}_1                    & \mathbf{e}_2                    & \mathbf{e}_3 \\
&& \\
\sum_i h_{1i} {\partial q^i \over \partial s} & \sum_i h_{2i} {\partial q^i \over \partial s} & \sum_i h_{3i} {\partial q^i \over \partial s} \\
&& \\
\sum_j h_{1j} {\partial q^j \over \partial t} & \sum_j h_{2j} {\partial q^j \over \partial t} & \sum_j h_{3j} {\partial q^j \over \partial t} \end{vmatrix}</math>
 
===Grad, curl, div, Laplacian===
 
In [[orthogonality|orthogonal]] curvilinear coordinates of 3 dimensions, where
: <math>
        \mathbf{b}^i = \sum_k g^{ik}~\mathbf{b}_k ~;~~ g^{ii} = \cfrac{1}{g_{ii}} = \cfrac{1}{h_i^2}
</math>
one can express the [[gradient]] of a [[scalar (mathematics)|scalar]] or [[vector field]] as
:<math>
  \nabla\varphi = \sum_{i} {\partial\varphi \over \partial q^i}~ \mathbf{b}^i = \sum_{i} \sum_j {\partial\varphi \over \partial q^i}~ g^{ij}~\mathbf{b}_j = \sum_i \cfrac{1}{h_i^2}~{\partial f \over \partial q^i}~\mathbf{b}_i ~;~~
  \nabla\mathbf{v} = \sum_i \cfrac{1}{h_i^2}~{\partial \mathbf{v} \over \partial q^i}\otimes\mathbf{b}_i
  </math>
For an orthogonal basis
:<math>
  g = g_{11}~g_{22}~g_{33} = h_1^2~h_2^2~h_3^2 \quad \Rightarrow \quad \sqrt{g} = h_1 h_2 h_3
</math>
The [[divergence]] of a vector field can then be written as
:<math>
  \boldsymbol{\nabla} \cdot \mathbf{v} = \cfrac{1}{h_1 h_2 h_3}~\frac{\partial }{\partial q^i}(h_1 h_2 h_3~v^i)
</math>
Also,
:<math>
  v^i = g^{ik}~v_k \quad \Rightarrow v^1 = g^{11}~v_1 = \cfrac{v_1}{h_1^2} ~;~~ v^2 = g^{22}~v_2  = \cfrac{v_2}{h_2^2}~;~~ v^3 = g^{33}~v_3 = \cfrac{v_3}{h_3^2}
</math>
Therefore,
:<math>
  \boldsymbol{\nabla} \cdot \mathbf{v} = \cfrac{1}{h_1 h_2 h_3}~\sum_i \frac{\partial }{\partial q^i}\left(\cfrac{h_1 h_2 h_3}{h_i^2}~v_i\right)
</math>
We can get an expression for the [[Laplacian]] in a similar manner by noting that
:<math>
  g^{li}~\frac{\partial \varphi}{\partial q^l} =
  \left\{ g^{11}~\frac{\partial \varphi}{\partial q^1}, g^{22}~\frac{\partial \varphi}{\partial q^2},
    g^{33}~\frac{\partial \varphi}{\partial q^3} \right\} =
  \left\{ \cfrac{1}{h_1^2}~\frac{\partial \varphi}{\partial q^1}, \cfrac{1}{h_2^2}~\frac{\partial \varphi}{\partial q^2},
    \cfrac{1}{h_3^2}~\frac{\partial \varphi}{\partial q^3} \right\}
</math>
Then we have
:<math>
  \nabla^2 \varphi =  \cfrac{1}{h_1 h_2 h_3}~\sum_i\frac{\partial }{\partial q^i}\left(\cfrac{h_1 h_2 h_3}{h_i^2}~\frac{\partial \varphi}{\partial q^i}\right)
</math>
The expressions for the gradient, divergence, and Laplacian can be directly extended to ''n''-dimensions.
 
The [[Curl (mathematics)|curl]] of a [[vector field]] is given by
: <math>
    \nabla\times\mathbf{v} = \frac{1}{h_1h_2h_3} \sum_{i=1}^n  \mathbf{e}_i
\sum_{jk} \varepsilon_{ijk} h_i \frac{\partial (h_k v_k)}{\partial q^j}
</math>
where ε<sub>''ijk''</sub> is the [[Levi-Civita symbol]].
 
==Example: Cylindrical polar coordinates==
For [[cylindrical coordinate]]s we have
:<math>
  (x_1, x_2, x_3) = \mathbf{x} = \boldsymbol{\varphi}(q^1, q^2, q^3) = \boldsymbol{\varphi}(r, \theta, z)
    = \{r\cos\theta, r\sin\theta, z\}
</math>
and
:<math>
  \{\psi^1(\mathbf{x}), \psi^2(\mathbf{x}), \psi^3(\mathbf{x})\} = (q^1, q^2, q^3) \equiv (r, \theta, z)
  = \{ \sqrt{x_1^2+x_2^2}, \tan^{-1}(x_2/x_1), x_3\}
</math>
where
:<math>
0 < r < \infty ~, ~~ 0 < \theta < 2\pi ~,~~ -\infty < z < \infty
</math>
 
Then the covariant and contravariant basis vectors are
:<math>
  \begin{align}
  \mathbf{b}_1 & = \mathbf{e}_r = \mathbf{b}^1 \\
  \mathbf{b}_2 & = r~\mathbf{e}_\theta = r^2~\mathbf{b}^2 \\
  \mathbf{b}_3 & = \mathbf{e}_z = \mathbf{b}^3
  \end{align}
</math>
where <math>\mathbf{e}_r, \mathbf{e}_\theta, \mathbf{e}_z</math> are the unit vectors in the <math>r, \theta, z</math> directions.
 
Note that the components of the metric tensor are such that
:<math>
    g^{ij} = g_{ij} = 0 (i \ne j) ~;~~ \sqrt{g^{11}} = 1,~\sqrt{g^{22}} = \cfrac{1}{r},~\sqrt{g^{33}}=1
</math>
which shows that the basis is orthogonal.
 
The non-zero components of the Christoffel symbol of the second kind are
:<math>
  \Gamma_{12}^2 = \Gamma_{21}^2 = \cfrac{1}{r} ~;~~ \Gamma_{22}^1 = -r
</math>
 
===Representing a physical vector field===
The normalized contravariant basis vectors in cylindrical polar coordinates are
:<math>
  \hat{\mathbf{b}}^1 = \mathbf{e}_r ~;~~\hat{\mathbf{b}}^2 = \mathbf{e}_\theta ~;~~\hat{\mathbf{b}}^3 = \mathbf{e}_z
</math>
and the physical components of a vector '''v''' are
:<math>
  (\hat{v}_1, \hat{v}_2, \hat{v}_3) = (v_1, v_2/r, v_3) =: (v_r, v_\theta, v_z)
</math>
 
===Gradient of a scalar field===
The gradient of a scalar field, ''f''('''x'''), in cylindrical coordinates can now be computed from the general expression in curvilinear coordinates and has the form
:<math>
  \boldsymbol{\nabla}f = \cfrac{\partial f}{\partial r}~\mathbf{e}_r + \cfrac{1}{r}~\cfrac{\partial f}{\partial \theta}~\mathbf{e}_\theta +  \cfrac{\partial f}{\partial z}~\mathbf{e}_z
</math>
 
===Gradient of a vector field===
Similarly, the gradient of a vector field, '''v'''('''x'''), in cylindrical coordinates can be shown to be
:<math>
  \begin{align}
  \boldsymbol{\nabla}\mathbf{v} & = \cfrac{\partial v_r}{\partial r}~\mathbf{e}_r\otimes\mathbf{e}_r +
    \cfrac{1}{r}\left(\cfrac{\partial v_r}{\partial \theta} - v_\theta\right)~\mathbf{e}_r\otimes\mathbf{e}_\theta + \cfrac{\partial v_r}{\partial z}~\mathbf{e}_r\otimes\mathbf{e}_z \\[8pt]
& + \cfrac{\partial v_\theta}{\partial r}~\mathbf{e}_\theta\otimes\mathbf{e}_r +
    \cfrac{1}{r}\left(\cfrac{\partial v_\theta}{\partial \theta} + v_r \right)~\mathbf{e}_\theta\otimes\mathbf{e}_\theta + \cfrac{\partial v_\theta}{\partial z}~\mathbf{e}_\theta\otimes\mathbf{e}_z \\[8pt]
& + \cfrac{\partial v_z}{\partial r}~\mathbf{e}_z\otimes\mathbf{e}_r +
    \cfrac{1}{r}\cfrac{\partial v_z}{\partial \theta}~\mathbf{e}_z\otimes\mathbf{e}_\theta + \cfrac{\partial v_z}{\partial z}~\mathbf{e}_z\otimes\mathbf{e}_z
  \end{align}
</math>
 
===Divergence of a vector field===
Using the equation for the divergence of a vector field in curvilinear coordinates, the divergence in cylindrical coordinates can be shown to be
:<math>
  \begin{align}
  \boldsymbol{\nabla}\cdot\mathbf{v} & = \cfrac{\partial v_r}{\partial r} +
    \cfrac{1}{r}\left(\cfrac{\partial v_\theta}{\partial \theta} + v_r \right)
+ \cfrac{\partial v_z}{\partial z}
  \end{align}
</math>
 
===Laplacian of a scalar field===
The Laplacian is more easily computed by noting that <math>\boldsymbol{\nabla}^2 f = \boldsymbol{\nabla}\cdot\boldsymbol{\nabla}f</math>. In cylindrical polar coordinates
:<math>
  \mathbf{v} = \boldsymbol{\nabla}f = \left[v_r~~ v_\theta~~ v_z\right] = \left[\cfrac{\partial f}{\partial r}~~  \cfrac{1}{r}\cfrac{\partial f}{\partial \theta}~~ \cfrac{\partial f}{\partial z} \right]
</math>
Hence,
:<math>
  \boldsymbol{\nabla}\cdot\mathbf{v} = \boldsymbol{\nabla}^2 f = \cfrac{\partial^2 f}{\partial r^2} +
    \cfrac{1}{r}\left(\cfrac{1}{r}\cfrac{\partial^2f}{\partial \theta^2} + \cfrac{\partial f}{\partial r} \right)
+ \cfrac{\partial^2 f}{\partial z^2}
  = \cfrac{1}{r}\left[\cfrac{\partial}{\partial r}\left(r\cfrac{\partial f}{\partial r}\right)\right] + \cfrac{1}{r^2}\cfrac{\partial^2f}{\partial \theta^2} + \cfrac{\partial^2 f}{\partial z^2}
</math>
 
===Representing a physical second-order tensor field===
The physical components of a second-order tensor field are those obtained when the tensor is expressed in terms of a normalized contravariant basis. In cylindrical polar coordinates these components are
:<math>
  \begin{align}
  \hat{S}_{11} & = S_{11} =: S_{rr} ~;~~\hat{S}_{12} = \cfrac{S_{12}}{r} =: S_{r\theta} ~;~~ \hat{S}_{13} & = S_{13} =: S_{rz} \\
    \hat{S}_{21} & = \cfrac{S_{11}}{r} =: S_{\theta r} ~;~~\hat{S}_{22} = \cfrac{S_{22}}{r^2} =: S_{\theta\theta} ~;~~ \hat{S}_{23} & = \cfrac{S_{23}}{r} =: S_{\theta z} \\
    \hat{S}_{31} & = S_{31} =: S_{zr} ~;~~\hat{S}_{32} = \cfrac{S_{32}}{r} =: S_{z\theta} ~;~~ \hat{S}_{33} & = S_{33} =: S_{zz}
  \end{align}
</math>
 
===Gradient of a second-order tensor field===
Using the above definitions we can show that the gradient of a second-order tensor field in cylindrical polar coordinates can be expressed as
:<math>
\begin{align}
\boldsymbol{\nabla} \boldsymbol{S} & = \frac{\partial S_{rr}}{\partial r}~\mathbf{e}_r\otimes\mathbf{e}_r\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{rr}}{\partial \theta} - (S_{\theta r}+S_{r\theta})\right]~\mathbf{e}_r\otimes\mathbf{e}_r\otimes\mathbf{e}_\theta +
\frac{\partial S_{rr}}{\partial z}~\mathbf{e}_r\otimes\mathbf{e}_r\otimes\mathbf{e}_z \\[8pt]
& + \frac{\partial S_{r\theta}}{\partial r}~\mathbf{e}_r\otimes\mathbf{e}_\theta\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{r\theta}}{\partial \theta} + (S_{rr}-S_{\theta\theta})\right]~\mathbf{e}_r\otimes\mathbf{e}_\theta\otimes\mathbf{e}_\theta +
\frac{\partial S_{r\theta}}{\partial z}~\mathbf{e}_r\otimes\mathbf{e}_\theta\otimes\mathbf{e}_z \\[8pt]
& + \frac{\partial S_{rz}}{\partial r}~\mathbf{e}_r\otimes\mathbf{e}_z\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{rz}}{\partial \theta} -S_{\theta z}\right]~\mathbf{e}_r\otimes\mathbf{e}_z\otimes\mathbf{e}_\theta +
\frac{\partial S_{rz}}{\partial z}~\mathbf{e}_r\otimes\mathbf{e}_z\otimes\mathbf{e}_z \\[8pt]
& + \frac{\partial S_{\theta r}}{\partial r}~\mathbf{e}_\theta\otimes\mathbf{e}_r\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{\theta r}}{\partial \theta} + (S_{rr}-S_{\theta\theta})\right]~\mathbf{e}_\theta\otimes\mathbf{e}_r\otimes\mathbf{e}_\theta +
\frac{\partial S_{\theta r}}{\partial z}~\mathbf{e}_\theta\otimes\mathbf{e}_r\otimes\mathbf{e}_z \\[8pt]
& + \frac{\partial S_{\theta\theta}}{\partial r}~\mathbf{e}_\theta\otimes\mathbf{e}_\theta\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{\theta\theta}}{\partial \theta} + (S_{r\theta}+S_{\theta r})\right]~\mathbf{e}_\theta\otimes\mathbf{e}_\theta\otimes\mathbf{e}_\theta +
\frac{\partial S_{\theta\theta}}{\partial z}~\mathbf{e}_\theta\otimes\mathbf{e}_\theta\otimes\mathbf{e}_z \\[8pt]
& + \frac{\partial S_{\theta z}}{\partial r}~\mathbf{e}_\theta\otimes\mathbf{e}_z\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{\theta z}}{\partial \theta} + S_{rz}\right]~\mathbf{e}_\theta\otimes\mathbf{e}_z\otimes\mathbf{e}_\theta +
\frac{\partial S_{\theta z}}{\partial z}~\mathbf{e}_\theta\otimes\mathbf{e}_z\otimes\mathbf{e}_z \\[8pt]
& + \frac{\partial S_{zr}}{\partial r}~\mathbf{e}_z\otimes\mathbf{e}_r\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{zr}}{\partial \theta} - S_{z\theta}\right]~\mathbf{e}_z\otimes\mathbf{e}_r\otimes\mathbf{e}_\theta +
\frac{\partial S_{zr}}{\partial z}~\mathbf{e}_z\otimes\mathbf{e}_r\otimes\mathbf{e}_z \\[8pt]
& + \frac{\partial S_{z\theta}}{\partial r}~\mathbf{e}_z\otimes\mathbf{e}_\theta\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{z\theta}}{\partial \theta} + S_{zr}\right]~\mathbf{e}_z\otimes\mathbf{e}_\theta\otimes\mathbf{e}_\theta +
\frac{\partial S_{z\theta}}{\partial z}~\mathbf{e}_z\otimes\mathbf{e}_\theta\otimes\mathbf{e}_z \\[8pt]
& + \frac{\partial S_{zz}}{\partial r}~\mathbf{e}_z\otimes\mathbf{e}_z\otimes\mathbf{e}_r +
\cfrac{1}{r}~\frac{\partial S_{zz}}{\partial \theta}~\mathbf{e}_z\otimes\mathbf{e}_z\otimes\mathbf{e}_\theta +
\frac{\partial S_{zz}}{\partial z}~\mathbf{e}_z\otimes\mathbf{e}_z\otimes\mathbf{e}_z
\end{align}
</math>
 
===Divergence of a second-order tensor field===
The divergence of a second-order tensor field in cylindrical polar coordinates can be obtained from the expression for the gradient by collecting terms where the scalar product of the two outer vectors in the dyadic products is nonzero.  Therefore,
:<math>
\begin{align}
\boldsymbol{\nabla}\cdot \boldsymbol{S} & = \frac{\partial S_{rr}}{\partial r}~\mathbf{e}_r
  + \frac{\partial S_{r\theta}}{\partial r}~\mathbf{e}_\theta
  + \frac{\partial S_{rz}}{\partial r}~\mathbf{e}_z  \\[8pt]
&  + \cfrac{1}{r}\left[\frac{\partial S_{\theta r}}{\partial \theta} + (S_{rr}-S_{\theta\theta})\right]~\mathbf{e}_r  +
\cfrac{1}{r}\left[\frac{\partial S_{\theta\theta}}{\partial \theta} + (S_{r\theta}+S_{\theta r})\right]~\mathbf{e}_\theta  +\cfrac{1}{r}\left[\frac{\partial S_{\theta z}}{\partial \theta} + S_{rz}\right]~\mathbf{e}_z  \\[8pt]
&  +
\frac{\partial S_{zr}}{\partial z}~\mathbf{e}_r +
\frac{\partial S_{z\theta}}{\partial z}~\mathbf{e}_\theta +
\frac{\partial S_{zz}}{\partial z}~\mathbf{e}_z
\end{align}
</math>
 
==See also==
* [[Covariance and contravariance]]
* [[Basic introduction to the mathematics of curved spacetime]]
* [[Orthogonal coordinates]]
* [[Frenet–Serret formulas]]
* [[Covariant derivative]]
* [[Tensor derivative (continuum mechanics)]]
* [[Curvilinear perspective]]
* [[Del in cylindrical and spherical coordinates]]
 
==References==
;Notes
{{reflist|2}}
 
;Further reading
{{refbegin}}
*{{Cite book| first=M. R. | last=Spiegel | title=Vector Analysis | publisher=Schaum's Outline Series | location=New York | year=1959| isbn=0-07-084378-3 }}
*{{Cite book| last=Arfken | first=George | title=Mathematical Methods for Physicists | publisher=Academic Press | year=1995| isbn=0-12-059877-9}}
{{refend}}
 
==External links==
* [http://planetmath.org/?method=l2h&from=collab&id=83&op=getobj Derivation of Unit Vectors in Curvilinear Coordinates]
* [http://mathworld.wolfram.com/CurvilinearCoordinates.html MathWorld's page on Curvilinear Coordinates]
* [http://www.mech.utah.edu/~brannon/public/curvilinear.pdf Prof. R. Brannon's E-Book on Curvilinear Coordinates]
 
{{Orthogonal coordinate systems}}
 
{{tensors}}
 
{{DEFAULTSORT:Curvilinear Coordinates}}
[[Category:Coordinate systems]]
[[Category:Metric tensors|*3]]
 
[[de:Krummlinige Koordinaten]]
[[fr:Système de coordonnées curvilignes]]
[[it:Coordinate curvilinee]]
[[ru:Криволинейная система координат]]
[[sl:Krivočrtni koordinatni sistem]]

Revision as of 05:36, 3 November 2013

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.

Curvilinear coordinates can be formulated in tensor calculus, with important applications in physics and engineering, particularly for describing transportation of physical quatities and deformation of matter in fluid mechanics and continuum mechanics.

Vector and tensor algebra in three-dimensional curvilinear coordinates

Template:Einstein summation convention Elementary vector and tensor algebra in curvilinear coordinates is used in some of the older scientific literature in mechanics and physics and can be indispensable to understanding work from the early and mid 1900s, for example the text by Green and Zerna.[1] Some useful relations in the algebra of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden,[2] Naghdi,[3] Simmonds,[4] Green and Zerna,[1] Basar and Weichert,[5] and Ciarlet.[6]

Vectors in curvilinear coordinates

Let (b1, b2, b3) be an arbitrary basis for three-dimensional Euclidean space. In general, the basis vectors are neither unit vectors nor mutually orthogonal. However, they are required to be linearly independent. Then a vector v can be expressed as[4]Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.

To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for

One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier

The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved

First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen

The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01

Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger

The components vk are the contravariant components of the vector v.

The reciprocal basis (b1, b2, b3) is defined by the relation [4]Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.

To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for

One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier

The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved

First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen

The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01

Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger

where δi j is the Kronecker delta.

The vector v can also be expressed in terms of the reciprocal basis:

The components vk are the covariant components of the vector .

Second-order tensors in curvilinear coordinates

A second-order tensor can be expressed as

The components Sij are called the contravariant components, Si j the mixed right-covariant components, Si j the mixed left-covariant components, and Sij the covariant components of the second-order tensor.

Metric tensor and relations between components

The quantities gij, gij are defined as[4]Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.

To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for

One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier

The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved

First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen

The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01

Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger

From the above equations we have

The components of a vector are related by[4]Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.

To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for

One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier

The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved

First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen

The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01

Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger

Also,

The components of the second-order tensor are related by

The alternating tensor

In an orthonormal right-handed basis, the third-order alternating tensor is defined as

In a general curvilinear basis the same tensor may be expressed as

It can be shown that

Now,

Hence,

Similarly, we can show that

Vector operations

  1. Identity map The identity map I defined by can be shown to be[4]Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.

    To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for

    One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier

    The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved

    First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen

    The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01

    Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger
  2. Scalar (dot) product The scalar product of two vectors in curvilinear coordinates is[4]Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.

    To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for

    One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier

    The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved

    First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen

    The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01

    Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger
  3. Vector (cross) product The cross product of two vectors is given by[4]Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.

    To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for

    One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier

    The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved

    First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen

    The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01

    Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger
    where εijk is the permutation symbol and ei is a Cartesian basis vector. In curvilinear coordinates, the equivalent expression is
    where is the third-order alternating tensor. The cross product of two vectors is given by
    where εijk is the permutation symbol and is a Cartesian basis vector. Therefore,
    and
    Hence,
    Returning back to the vector product and using the relations
    gives us

Tensor operations

  1. Identity map: The identity map defined by can be shown to be[4]Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.

    To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for

    One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier

    The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved

    First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen

    The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01

    Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger
  2. Action of a second-order tensor on a vector: The action can be expressed in curvilinear coordinates as
  3. Inner product of two second-order tensors: The inner product of two second-order tensors can be expressed in curvilinear coordinates as
    Alternatively,
  4. Determinant of a second-order tensor: If is a second-order tensor, then the determinant is defined by the relation
    where are arbitrary vectors and

Relations between curvilinear and Cartesian basis vectors

Let (e1, e2, e3) be the usual Cartesian basis vectors for the Euclidean space of interest and let

where Fi is a second-order transformation tensor that maps ei to bi. Then,

From this relation we can show that

Let be the Jacobian of the transformation. Then, from the definition of the determinant,

Since

we have

A number of interesting results can be derived using the above relations.

First, consider

Then

Similarly, we can show that

Therefore, using the fact that ,

Another interesting relation is derived below. Recall that

where A is a, yet undetermined, constant. Then

This observation leads to the relations

In index notation,

where is the usual permutation symbol.

We have not identified an explicit expression for the transformation tensor F because an alternative form of the mapping between curvilinear and Cartesian bases is more useful. Assuming a sufficient degree of smoothness in the mapping (and a bit of abuse of notation), we have

Similarly,

From these results we have

and

Vector and tensor calculus in three-dimensional curvilinear coordinates

Template:Einstein summation convention Simmonds,[4] in his book on tensor analysis, quotes Albert Einstein saying[7]

The magic of this theory will hardly fail to impose itself on anybody who has truly understood it; it represents a genuine triumph of the method of absolute differential calculus, founded by Gauss, Riemann, Ricci, and Levi-Civita.

Vector and tensor calculus in general curvilinear coordinates is used in tensor analysis on four-dimensional curvilinear manifolds in general relativity,[8] in the mechanics of curved shells,[6] in examining the invariance properties of Maxwell's equations which has been of interest in metamaterials[9][10] and in many other fields.

Some useful relations in the calculus of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden,[2] Simmonds,[4] Green and Zerna,[1] Basar and Weichert,[5] and Ciarlet.[6]

Basic definitions

Let the position of a point in space be characterized by three coordinate variables .

The coordinate curve q1 represents a curve on which q2, q3 are constant. Let x be the position vector of the point relative to some origin. Then, assuming that such a mapping and its inverse exist and are continuous, we can write [2]Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.

To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for

One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier

The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved

First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen

The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01

Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger

The fields ψi(x) are called the curvilinear coordinate functions of the curvilinear coordinate system ψ(x) = ψ−1(x).

The qi coordinate curves are defined by the one-parameter family of functions given by

with qj, qk fixed.

Tangent vector to coordinate curves

The tangent vector to the curve xi at the point xi(α) (or to the coordinate curve qi at the point x) is

Gradient

Scalar field

Let f(x) be a scalar field in space. Then

The gradient of the field f is defined by

where c is an arbitrary constant vector. If we define the components ci of c are such that

then

If we set , then since , we have

which provides a means of extracting the contravariant component of a vector c.

If bi is the covariant (or natural) basis at a point, and if bi is the contravariant (or reciprocal) basis at that point, then

A brief rationale for this choice of basis is given in the next section.

Vector field

A similar process can be used to arrive at the gradient of a vector field f(x). The gradient is given by

If we consider the gradient of the position vector field r(x) = x, then we can show that

The vector field bi is tangent to the qi coordinate curve and forms a natural basis at each point on the curve. This basis, as discussed at the beginning of this article, is also called the covariant curvilinear basis. We can also define a reciprocal basis, or contravariant curvilinear basis, bi. All the algebraic relations between the basis vectors, as discussed in the section on tensor algebra, apply for the natural basis and its reciprocal at each point x.

Since c is arbitrary, we can write

Note that the contravariant basis vector bi is perpendicular to the surface of constant ψi and is given by

Christoffel symbols of the first kind

The Christoffel symbols of the first kind are defined as

To express Γijk in terms of gij we note that

Since bi,j = bj,i we have Γijk = Γjik. Using these to rearrange the above relations gives

Christoffel symbols of the second kind

The Christoffel symbols of the second kind are defined as

in which

This implies that

Other relations that follow are

Another particularly useful relation, which shows that the Christoffel symbol depends only on the metric tensor and its derivatives, is

Explicit expression for the gradient of a vector field

The following expressions for the gradient of a vector field in curvilinear coordinates are quite useful.

Representing a physical vector field

The vector field v can be represented as

where are the covariant components of the field, are the physical components, and (no summation)

is the normalized contravariant basis vector.

Second-order tensor field

The gradient of a second order tensor field can similarly be expressed as

Explicit expressions for the gradient

If we consider the expression for the tensor in terms of a contravariant basis, then

We may also write

Representing a physical second-order tensor field

The physical components of a second-order tensor field can be obtained by using a normalized contravariant basis, i.e.,

where the hatted basis vectors have been normalized. This implies that (again no summation)

Divergence

Vector field

The divergence of a vector field ()is defined as

In terms of components with respect to a curvilinear basis

An alternative equation for the divergence of a vector field is frequently used. To derive this relation recall that

Now,

Noting that, due to the symmetry of ,

we have

Recall that if [gij] is the matrix whose components are gij, then the inverse of the matrix is . The inverse of the matrix is given by

where Aij are the Cofactor matrix of the components gij. From matrix algebra we have

Hence,

Plugging this relation into the expression for the divergence gives

A little manipulation leads to the more compact form

Second-order tensor field

The divergence of a second-order tensor field is defined using

where a is an arbitrary constant vector. [11] In curvilinear coordinates,

Laplacian

Scalar field

The Laplacian of a scalar field φ(x) is defined as

Using the alternative expression for the divergence of a vector field gives us

Now

Therefore,

Curl of a vector field

The curl of a vector field v in covariant curvilinear coordinates can be written as

where

Orthogonal curvilinear coordinates

Assume, for the purposes of this section, that the curvilinear coordinate system is orthogonal, i.e.,

or equivalently,

where . As before, are covariant basis vectors and bi, bj are contravariant basis vectors. Also, let (e1, e2, e3) be a background, fixed, Cartesian basis. A list of orthogonal curvilinear coordinates is given below.

Metric tensor in orthogonal curvilinear coordinates

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Let r(x) be the position vector of the point x with respect to the origin of the coordinate system. The notation can be simplified by noting that x = r(x). At each point we can construct a small line element dx. The square of the length of the line element is the scalar product dx • dx and is called the metric of the space. Recall that the space of interest is assumed to be Euclidean when we talk of curvilinear coordinates. Let us express the position vector in terms of the background, fixed, Cartesian basis, i.e.,

Using the chain rule, we can then express dx in terms of three-dimensional orthogonal curvilinear coordinates (q1, q2, q3) as

Therefore the metric is given by

The symmetric quantity

is called the fundamental (or metric) tensor of the Euclidean space in curvilinear coordinates.

Note also that

where hij are the Lamé coefficients.

If we define the scale factors, hi, using

we get a relation between the fundamental tensor and the Lamé coefficients.

Example: Polar coordinates

If we consider polar coordinates for R2, note that

(r, θ) are the curvilinear coordinates, and the Jacobian determinant of the transformation (r,θ) → (r cos θ, r sin θ) is r.

The orthogonal basis vectors are br = (cos θ, sin θ), bθ = (−r sin θ, r cos θ). The normalized basis vectors are er = (cos θ, sin θ), eθ = (−sin θ, cos θ) and the scale factors are hr = 1 and hθ= r. The fundamental tensor is g11 =1, g22 =r2, g12 = g21 =0.

Line and surface integrals

If we wish to use curvilinear coordinates for vector calculus calculations, adjustments need to be made in the calculation of line, surface and volume integrals. For simplicity, we again restrict the discussion to three dimensions and orthogonal curvilinear coordinates. However, the same arguments apply for -dimensional problems though there are some additional terms in the expressions when the coordinate system is not orthogonal.

Line integrals

Normally in the calculation of line integrals we are interested in calculating

where x(t) parametrizes C in Cartesian coordinates. In curvilinear coordinates, the term

by the chain rule. And from the definition of the Lamé coefficients,

and thus

Now, since when , we have

and we can proceed normally.

Surface integrals

Likewise, if we are interested in a surface integral, the relevant calculation, with the parameterization of the surface in Cartesian coordinates is:

Again, in curvilinear coordinates, we have

and we make use of the definition of curvilinear coordinates again to yield

Therefore,

where is the permutation symbol.

In determinant form, the cross product in terms of curvilinear coordinates will be:

Grad, curl, div, Laplacian

In orthogonal curvilinear coordinates of 3 dimensions, where

one can express the gradient of a scalar or vector field as

For an orthogonal basis

The divergence of a vector field can then be written as

Also,

Therefore,

We can get an expression for the Laplacian in a similar manner by noting that

Then we have

The expressions for the gradient, divergence, and Laplacian can be directly extended to n-dimensions.

The curl of a vector field is given by

where εijk is the Levi-Civita symbol.

Example: Cylindrical polar coordinates

For cylindrical coordinates we have

and

where

Then the covariant and contravariant basis vectors are

where are the unit vectors in the directions.

Note that the components of the metric tensor are such that

which shows that the basis is orthogonal.

The non-zero components of the Christoffel symbol of the second kind are

Representing a physical vector field

The normalized contravariant basis vectors in cylindrical polar coordinates are

and the physical components of a vector v are

Gradient of a scalar field

The gradient of a scalar field, f(x), in cylindrical coordinates can now be computed from the general expression in curvilinear coordinates and has the form

Gradient of a vector field

Similarly, the gradient of a vector field, v(x), in cylindrical coordinates can be shown to be

Divergence of a vector field

Using the equation for the divergence of a vector field in curvilinear coordinates, the divergence in cylindrical coordinates can be shown to be

Laplacian of a scalar field

The Laplacian is more easily computed by noting that . In cylindrical polar coordinates

Hence,

Representing a physical second-order tensor field

The physical components of a second-order tensor field are those obtained when the tensor is expressed in terms of a normalized contravariant basis. In cylindrical polar coordinates these components are

Gradient of a second-order tensor field

Using the above definitions we can show that the gradient of a second-order tensor field in cylindrical polar coordinates can be expressed as

Divergence of a second-order tensor field

The divergence of a second-order tensor field in cylindrical polar coordinates can be obtained from the expression for the gradient by collecting terms where the scalar product of the two outer vectors in the dyadic products is nonzero. Therefore,

See also

References

Notes

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Further reading

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  • 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.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 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.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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External links

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de:Krummlinige Koordinaten fr:Système de coordonnées curvilignes it:Coordinate curvilinee ru:Криволинейная система координат sl:Krivočrtni koordinatni sistem

  1. ↑ 1.0 1.1 1.2 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.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  2. ↑ 2.0 2.1 2.2 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.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  3. ↑ 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.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  4. ↑ 4.00 4.01 4.02 4.03 4.04 4.05 4.06 4.07 4.08 4.09 4.10 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.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  5. ↑ 5.0 5.1 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.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  6. ↑ 6.0 6.1 6.2 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.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  7. ↑ 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.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  8. ↑ 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.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  9. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

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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
  10. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    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

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    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
  11. ↑ Template:Cite web
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