Editing Curvilinear coordinates (section)

==Transformation of coordinates==

From a more general and abstract perspective, a curvilinear coordinate system is simply a [[Atlas (topology)|coordinate patch]] on the [[differentiable manifold]] '''E'''<sup>n</sup> (n-dimensional [[Euclidean space]]) that is [[Diffeomorphism|diffeomorphic]] to the [[Cartesian coordinate system|Cartesian]] coordinate patch on the manifold.<ref>{{cite book | last=Boothby | first=W. M. | year=2002 | title=An Introduction to Differential Manifolds and Riemannian Geometry | edition=revised | publisher=Academic Press | location=New York, NY }}</ref> Two diffeomorphic coordinate patches on a differential manifold need not overlap differentiably. With this simple definition of a curvilinear coordinate system, all the results that follow below are simply applications of standard theorems in [[differential topology]].

The transformation functions are such that there's a one-to-one relationship between points in the "old" and "new" coordinates, that is, those functions are [[bijection]]s, and fulfil the following requirements within their [[domain of a function|domain]]s:
{{ordered list
|1= They are [[smooth function]]s: q<sup>''i''</sup> = q<sup>''i''</sup>('''x''')

|2= The inverse [[Jacobian matrix and determinant|Jacobian]] determinant
:<math> J^{-1}=\begin{vmatrix}
\dfrac{\partial q^1}{\partial x_1} & \dfrac{\partial q^1}{\partial x_2} & \cdots & \dfrac{\partial q^1}{\partial x_n} \\
\dfrac{\partial q^2}{\partial x_1} & \dfrac{\partial q^2}{\partial x_2} & \cdots & \dfrac{\partial q^2}{\partial x_n} \\
\vdots & \vdots & \ddots & \vdots \\
\dfrac{\partial q^n}{\partial x_1} & \dfrac{\partial q^n}{\partial x_2} & \cdots & \dfrac{\partial q^n}{\partial x_n}
\end{vmatrix} \neq 0 </math>

is not zero; meaning the transformation is [[invertible]]: ''x<sub>i</sub>''('''q''') according to the [[inverse function theorem]]. The condition that the Jacobian determinant is not zero reflects the fact that three surfaces from different families intersect in one and only one point and thus determine the position of this point in a unique way.<ref>{{cite book | last=McConnell | first=A. J. | year=1957 | publisher=Dover Publications, Inc. | location=New York, NY | at=Ch. 9, sec. 1 | title=Application of Tensor Analysis | url=https://archive.org/details/applicationoften0000mcco | url-access=registration | isbn=0-486-60373-3 }}</ref>
}}