Editing Curvilinear coordinates (section)

===Constructing a covariant basis in one dimension===

[[File:Local basis transformation.svg|thumb|upright=1.35|Fig. 3 – Transformation of local covariant basis in the case of general curvilinear coordinates]]

Consider the one-dimensional curve shown in Fig. 3.  At point ''P'', taken as an [[Origin (mathematics)|origin]], ''x'' is one of the Cartesian coordinates, and ''q''<sup>1</sup> is one of the curvilinear coordinates. The local (non-unit) basis vector is '''b'''<sub>1</sub> (notated '''h'''<sub>1</sub> above, with '''b''' reserved for unit vectors) and it is built on the ''q''<sup>1</sup> axis which is a tangent to that coordinate line at the point ''P''. The axis ''q''<sup>1</sup> and thus the vector '''b'''<sub>1</sub> form an angle <math>\alpha</math> with the Cartesian ''x'' axis and the Cartesian basis vector '''e'''<sub>1</sub>.

It can be seen from triangle ''PAB'' that
:<math> \cos \alpha = \cfrac{|\mathbf{e}_1|}{|\mathbf{b}_1|} \quad \Rightarrow \quad |\mathbf{e}_1| = |\mathbf{b}_1|\cos \alpha</math>
where |'''e'''<sub>1</sub>|, |'''b'''<sub>1</sub>| are the magnitudes of the two basis vectors, i.e., the scalar intercepts ''PB'' and ''PA''.  ''PA'' is also the projection of '''b'''<sub>1</sub> on the ''x'' axis.

However, this method for basis vector transformations using ''directional cosines'' is inapplicable to curvilinear coordinates for the following reasons:
#By increasing the distance from ''P'', the angle between the curved line ''q''<sup>1</sup> and Cartesian axis ''x'' increasingly deviates from <math>\alpha</math>.
#At the distance ''PB'' the true angle is that which the tangent '''at point C''' forms with the ''x'' axis and the latter angle is clearly different from <math>\alpha</math>.

The angles that the ''q''<sup>1</sup> line and that axis form with the ''x'' axis become closer in value the closer one moves towards point ''P'' and become exactly equal at ''P''.

Let point ''E'' be located very close to ''P'', so close that the distance ''PE'' is infinitesimally small. Then ''PE'' measured on the ''q''<sup>1</sup> axis almost coincides with ''PE'' measured on the ''q''<sup>1</sup> line. At the same time, the ratio ''PD/PE'' (''PD'' being the projection of ''PE'' on the ''x'' axis) becomes almost exactly equal to <math>\cos\alpha</math>.

Let the infinitesimally small intercepts ''PD'' and ''PE'' be labelled, respectively, as ''dx'' and d''q''<sup>1</sup>. Then
:<math>\cos \alpha = \cfrac{dx}{dq^1} = \frac{|\mathbf{e}_1|}{|\mathbf{b}_1|}</math>.

Thus, the directional cosines can be substituted in transformations with the more exact ratios between infinitesimally small coordinate intercepts.  It follows that the component (projection) of '''b'''<sub>1</sub> on the ''x'' axis is

:<math>p^1 = \mathbf{b}_1\cdot\cfrac{\mathbf{e}_1}{|\mathbf{e}_1|} = |\mathbf{b}_1|\cfrac{|\mathbf{e}_1|}{|\mathbf{e}_1|}\cos\alpha = |\mathbf{b}_1|\cfrac{dx}{dq^1} \quad \Rightarrow \quad \cfrac{p^1}{|\mathbf{b}_1|} = \cfrac{dx}{dq^1}</math>.

If ''q<sup>i</sup>'' = ''q<sup>i</sup>''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>) and ''x<sub>i</sub>'' = ''x<sub>i</sub>''(''q''<sup>1</sup>, ''q''<sup>2</sup>, ''q''<sup>3</sup>) are [[Smooth function|smooth]] (continuously differentiable) functions the transformation ratios can be written as <math>\cfrac{\partial q^i}{\partial x_j}</math> and <math>\cfrac{\partial x_i}{\partial q^j}</math>. That is, those ratios are [[partial derivative]]s of coordinates belonging to one system with respect to coordinates belonging to the other system.