Editing Curvilinear coordinates (section)

==Integration==
{{Main|Covariant transformation}}

===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.

===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>

===Jacobian of the transformation===

The above [[systems of linear equations]] can be written in matrix form using the Einstein summation convention as
:<math>\cfrac{\partial x_i}{\partial q^k} \mathbf{e}_i = \mathbf{b}_k, \quad \cfrac{\partial q^i}{\partial x_k} \mathbf{b}_i = \mathbf{e}_k</math>.

This [[coefficient matrix]] of the linear system is the [[Jacobian matrix]] (and its inverse) of the transformation. These are the equations that can be used to transform a Cartesian basis into a curvilinear basis, and vice versa.

In three dimensions, the expanded forms of these matrices are
:<math>
\mathbf{J} = \begin{bmatrix}
       \cfrac{\partial x_1}{\partial q^1} & \cfrac{\partial x_1}{\partial q^2} & \cfrac{\partial x_1}{\partial q^3} \\
       \cfrac{\partial x_2}{\partial q^1} & \cfrac{\partial x_2}{\partial q^2} & \cfrac{\partial x_2}{\partial q^3} \\
       \cfrac{\partial x_3}{\partial q^1} & \cfrac{\partial x_3}{\partial q^2} & \cfrac{\partial x_3}{\partial q^3} \\
     \end{bmatrix},\quad
\mathbf{J}^{-1} = \begin{bmatrix}
       \cfrac{\partial q^1}{\partial x_1} & \cfrac{\partial q^1}{\partial x_2} & \cfrac{\partial q^1}{\partial x_3} \\
       \cfrac{\partial q^2}{\partial x_1} & \cfrac{\partial q^2}{\partial x_2} & \cfrac{\partial q^2}{\partial x_3} \\
       \cfrac{\partial q^3}{\partial x_1} & \cfrac{\partial q^3}{\partial x_2} & \cfrac{\partial q^3}{\partial x_3} \\
     \end{bmatrix}
 </math>

In the inverse transformation (second equation system), the unknowns are the curvilinear basis vectors. For any specific location there can only exist one and only one set of basis vectors (else the basis is not well defined at that point). This condition is satisfied if and only if the equation system has a single solution. In [[linear algebra]], a linear equation system has a single solution (non-trivial) only if the determinant of its system matrix is non-zero:
:<math> \det(\mathbf{J}^{-1})  \neq 0</math>
which shows the rationale behind the above requirement concerning the inverse Jacobian determinant.