Editing Curvilinear coordinates (section)

==Vector and tensor calculus in three-dimensional curvilinear coordinates==
{{Einstein_summation_convention}}

Adjustments need to be made in the calculation of [[line integral|line]], [[surface integral|surface]] and [[volume integral|volume]] [[integration (mathematics)|integrals]].  For simplicity, the following restricts to three dimensions and orthogonal curvilinear coordinates.  However, the same arguments apply for ''n''-dimensional spaces. When the coordinate system is not orthogonal, there are some additional terms in the expressions.

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 | doi=10.1088/1367-2630/8/10/247 | issue=10 | arxiv=cond-mat/0607418 | bibcode=2006NJPh....8..247L }}</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>Ogden</ref> Simmonds,<ref name=Simmonds /> Green and Zerna,<ref name=Green/>  Basar and Weichert,<ref name=Basar/> and Ciarlet.<ref name=Ciarlet/>

Let φ = φ('''x''') be a well defined scalar field and '''v''' = '''v'''('''x''') a well-defined vector field, and ''λ''<sub>1</sub>, ''λ''<sub>2</sub>... be parameters of the coordinates

===Geometric elements===

{{ordered list
|1= '''[[Tangent vector]]:''' If '''x'''(''λ'') parametrizes a curve ''C'' in Cartesian coordinates, then

:<math> {\partial \mathbf{x} \over \partial \lambda} = {\partial \mathbf{x} \over \partial q^i}{\partial q^i \over \partial \lambda} = \left( h_{ki}\cfrac{\partial q^i}{\partial \lambda}\right)\mathbf{b}_k </math>

is a tangent vector to ''C'' in curvilinear coordinates (using the [[chain rule]]). Using the definition of the Lamé coefficients, and that for the metric ''g<sub>ij</sub>'' = 0 when ''i'' ≠ ''j'', the magnitude is:

:<math> \left|{\partial \mathbf{x} \over \partial \lambda} \right| = \sqrt{h_{ki}h_{kj}\cfrac{\partial q^i}{\partial \lambda}\cfrac{\partial q^j}{\partial \lambda}} = \sqrt{ g_{ij}\cfrac{\partial q^i}{\partial \lambda}\cfrac{\partial q^j}{\partial \lambda}} = \sqrt{h_{i}^2\left(\cfrac{\partial q^i}{\partial \lambda}\right)^2} </math>
|2= '''[[Tangent plane]] element:''' If '''x'''(''λ''<sub>1</sub>, ''λ''<sub>2</sub>) parametrizes a surface ''S'' in Cartesian coordinates, then the following cross product of tangent vectors is a normal vector to ''S'' with the magnitude of infinitesimal plane element, in curvilinear coordinates. Using the above result,

:<math> {\partial \mathbf{x} \over \partial \lambda_1}\times {\partial \mathbf{x} \over \partial \lambda_2} =\left({\partial \mathbf{x} \over \partial q^i}{\partial q^i \over \partial \lambda_1}\right) \times \left({\partial \mathbf{x} \over \partial q^j}{\partial q^j \over \partial \lambda_2}\right) = \mathcal{E}_{kmp}\left( h_{ki}{\partial q^i \over \partial \lambda_1}\right)\left(h_{mj}{\partial q^j \over \partial \lambda_2}\right) \mathbf{b}_p </math>

where <math>\mathcal{E}</math> is the [[permutation symbol]]. In determinant form:

:<math>{\partial \mathbf{x} \over \partial \lambda_1}\times {\partial \mathbf{x} \over \partial \lambda_2}
=\begin{vmatrix}
\mathbf{e}_1 & \mathbf{e}_2 & \mathbf{e}_3 \\
h_{1i} \dfrac{\partial q^i}{\partial \lambda_1} & h_{2i} \dfrac{\partial q^i}{\partial \lambda_1} & h_{3i} \dfrac{\partial q^i }{\partial \lambda_1} \\
h_{1j} \dfrac{\partial q^j}{\partial \lambda_2} & h_{2j} \dfrac{\partial q^j}{\partial \lambda_2} & h_{3j} \dfrac{\partial q^j }{\partial \lambda_2}
\end{vmatrix}</math>
}}

===Integration===

:{| class="wikitable"
|-
!scope=col width="10px"| Operator
!scope=col width="200px"| Scalar field
!scope=col width="200px"| Vector field
|-
|[[Line integral]]
||<math> \int_C \varphi(\mathbf{x}) ds = \int_a^b \varphi(\mathbf{x}(\lambda))\left|{\partial \mathbf{x} \over \partial \lambda}\right| d\lambda</math>
||<math> \int_C \mathbf{v}(\mathbf{x}) \cdot d\mathbf{s} = \int_a^b \mathbf{v}(\mathbf{x}(\lambda))\cdot\left({\partial \mathbf{x} \over \partial \lambda}\right) d\lambda</math>
|-
| [[Surface integral]]
|| <math>\int_S \varphi(\mathbf{x}) dS = \iint_T \varphi(\mathbf{x}(\lambda_1, \lambda_2)) \left|{\partial \mathbf{x} \over \partial \lambda_1}\times {\partial \mathbf{x} \over \partial \lambda_2}\right| d\lambda_1 d\lambda_2</math>
||<math>\int_S \mathbf{v}(\mathbf{x}) \cdot dS = \iint_T \mathbf{v}(\mathbf{x}(\lambda_1, \lambda_2)) \cdot\left({\partial \mathbf{x} \over \partial \lambda_1}\times {\partial \mathbf{x} \over \partial \lambda_2}\right) d\lambda_1 d\lambda_2</math>
|-
| [[Volume integral]]
|| <math>\iiint_V \varphi(x,y,z) dV = \iiint_V \chi(q_1,q_2,q_3) Jdq_1dq_2dq_3 </math>
|| <math>\iiint_V \mathbf{u}(x,y,z) dV = \iiint_V \mathbf{v}(q_1,q_2,q_3) Jdq_1dq_2dq_3 </math>
|-
|}

===Differentiation===

The expressions for the gradient, divergence, and Laplacian can be directly extended to ''n''-dimensions, however the curl is only defined in 3D.

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'''.

:{| class="wikitable"
|-
!scope=col width="10px"| Operator
!scope=col width="200px"| Scalar field
!scope=col width="200px"| Vector field
!scope=col width="200px"| 2nd order tensor field
|-
| [[Gradient]]
|| <math> \nabla\varphi = \cfrac{1}{h_i}{\partial\varphi \over \partial q^i} \mathbf{b}^i </math>
|| <math>\nabla\mathbf{v} = \cfrac{1}{h_i^2}{\partial \mathbf{v} \over \partial q^i}\otimes\mathbf{b}_i </math>
|| <math>\boldsymbol{\nabla}\boldsymbol{S} = \cfrac{\partial \boldsymbol{S}}{\partial q^i}\otimes\mathbf{b}^i</math>
|-
| [[Divergence]]
|| N/A
|| <math> \nabla \cdot \mathbf{v} = \cfrac{1}{\prod_j h_j} \frac{\partial }{\partial q^i}(v^i\prod_{j\ne i} h_j) </math>
|| <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.
In curvilinear coordinates,

<math>\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 </math>
|-
| [[Laplacian]]
||<math>
\nabla^2 \varphi =  \cfrac{1}{\prod _j h_j}\frac{\partial }{\partial q^i}\left(\cfrac{\prod _j h_j}{h_i^2}\frac{\partial \varphi}{\partial q^i}\right)
</math>
||
<math>
\nabla^2 \mathbf{v} \equiv \nabla \nabla\cdot \mathbf{v} - \nabla \times \nabla \times \mathbf{v} </math> 
<math>~~~ = \hat{\mathbf{x}}\nabla^2 v_x +  \hat{\mathbf{y}}\nabla^2 v_y +  \hat{\mathbf{z}}\nabla^2 v_z</math> (First equality in 3D only; second equality in Cartesian components only)
||
|-
| [[Curl (mathematics)|Curl]]
|| N/A
|| For vector fields in 3D only,
<math> \nabla\times\mathbf{v} = \frac{1}{h_1h_2h_3} \mathbf{e}_i \epsilon_{ijk} h_i \frac{\partial (h_k v_k)}{\partial q^j} </math>

where <math>\epsilon_{ijk}</math> is the [[Levi-Civita symbol]].
|| See [[Tensor derivative (continuum mechanics)#Curl of a tensor field|''Curl of a tensor field'']]
|}