1. Curvilinear Co-ordinates

Lets start with some notes on the curvilinear system.
I am assuming that we are all familiar with the ordinary cartesian coordinate system and the respective axioms of the Euclids on which the cartesian sytem is based. Cartesian coordinates are the foundation of analytic geometry and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more. Thus we cannot avoid but have good knowledge on the basics.
Suppose the rectangular co-ordinates $\left ( x,y,z \right )$ of any point in space are expressed as functions of  $\left ( u_{1},u_{2},u_{3}\right )$.

Say, $x=x\left ( u_{1},u_{2},u_{3}\right)$,  $y=y\left ( u_{1},u_{2},u_{3}\right)$,  $z=z\left ( u_{1},u_{2},u_{3}\right)$ ... $\left (1\right)$

Suppose that above can be solved for $u_{1},u_{2},u_{3}$ in terms of $ x,y,z$ that is,

$u_{1}=u_{1}\left (x,y,z\right)$, $u_{2}=u_{2}\left (x,y,z\right)$, $u_{3}=u_{3}\left (x,y,z\right)$ ... $\left (2\right)$

The above function transformation relation is an important one. The function $\left (1\right)$ & $\left (2\right)$ are assumed to be single valued and to have continuous derivatives so that the correnspondence between $\left ( x,y,z \right )$ and  $\left ( u_{1},u_{2},u_{3}\right )$ is unique. In practice, this assumption may not apply at certain points and special consideration is required.

Given a point $P$ with rectangular co-ordinates $\left ( x,y,z \right )$ we can from $\left (2\right)$, associate a unique set of co-ordinates $\left ( u_{1},u_{2},u_{3}\right )$ called the curvilinear co-ordinates of $P$. The sets of equation $\left (1\right)$ & $\left (2\right)$ define a transformation of co-ordinates.

A sample curvilinear in two dimensional space.
 (source Wikipedia)

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