Editing Curvilinear coordinates (section)

===Constructing a covariant basis in three dimensions===

Doing the same for the coordinates in the other 2 dimensions, '''b'''<sub>1</sub> can be expressed as:
:<math>
\mathbf{b}_1 = p^1\mathbf{e}_1 + p^2\mathbf{e}_2 + p^3\mathbf{e}_3 = \cfrac{\partial x_1}{\partial q^1} \mathbf{e}_1 + \cfrac{\partial x_2}{\partial q^1} \mathbf{e}_2 + \cfrac{\partial x_3}{\partial q^1} \mathbf{e}_3
</math>
Similar equations hold for '''b'''<sub>2</sub> and '''b'''<sub>3</sub> so that the standard basis {'''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub>} is transformed to a local (ordered and '''''normalised''''') basis {'''b'''<sub>1</sub>, '''b'''<sub>2</sub>, '''b'''<sub>3</sub>} by the following system of equations:

:<math>\begin{align}
   \mathbf{b}_1 & = \cfrac{\partial x_1}{\partial q^1} \mathbf{e}_1 + \cfrac{\partial x_2}{\partial q^1} \mathbf{e}_2 + \cfrac{\partial x_3}{\partial q^1} \mathbf{e}_3 \\
   \mathbf{b}_2 & = \cfrac{\partial x_1}{\partial q^2} \mathbf{e}_1 + \cfrac{\partial x_2}{\partial q^2} \mathbf{e}_2 + \cfrac{\partial x_3}{\partial q^2} \mathbf{e}_3 \\
   \mathbf{b}_3 & = \cfrac{\partial x_1}{\partial q^3} \mathbf{e}_1 + \cfrac{\partial x_2}{\partial q^3} \mathbf{e}_2 + \cfrac{\partial x_3}{\partial q^3} \mathbf{e}_3
\end{align}</math>

By analogous reasoning, one can obtain the inverse transformation from local basis to standard basis:
:<math>\begin{align}
   \mathbf{e}_1 & = \cfrac{\partial q^1}{\partial x_1} \mathbf{b}_1 + \cfrac{\partial q^2}{\partial x_1} \mathbf{b}_2 + \cfrac{\partial q^3}{\partial x_1} \mathbf{b}_3 \\
   \mathbf{e}_2 & = \cfrac{\partial q^1}{\partial x_2} \mathbf{b}_1 + \cfrac{\partial q^2}{\partial x_2} \mathbf{b}_2 + \cfrac{\partial q^3}{\partial x_2} \mathbf{b}_3 \\
   \mathbf{e}_3 & = \cfrac{\partial q^1}{\partial x_3} \mathbf{b}_1 + \cfrac{\partial q^2}{\partial x_3} \mathbf{b}_2 + \cfrac{\partial q^3}{\partial x_3} \mathbf{b}_3
\end{align}</math>