Editing Curvilinear coordinates (section)

===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'']]
|}