Editing Curvilinear coordinates (section)

===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>
}}