Why are notions of curvilinear coordinates in 'a course in differential geometry and topology' backward? I'm just starting to learn riemannian geometry and was using the book 'a course in differential geometry and topology' by Mischenko and Formenko as my primary text. 
In chapter I of the book, it briefly describes the concept of curvilinear coordinates. But when the author tries to show that the polar coordinate is curvilinear on a certain domain, he wrotes:

Consider an Euclidean Plane $\textbf{R}^2\left( r,\phi \right)$, where
  $y^1=r, y^2=\phi$, and take an infinite strip defined by $0<\phi<2\pi,
  0<r<+\infty$ as a domain $C$, and the domain A in the plane
  $\textbf{R}^1\left( x^1,x^2 \right)$ should be chosen as the entire
  plane except the ray $x^1\geq 0, x^2=0$. The mapping $f:C\to A$ is
  given by the relation $x^1=r \cos \phi, x^2=r\sin\phi $

Then he presented the readers with a diagram of an euclidean plane with the conventional cartesian coordinates but horizontal axis labelled by $r$, vertical axis labelled by $\phi$, then another plane with the polar coordinate, but with $x^1$ and $x^2$ as labels. This is very confusing to me as I don't really get what's the meaning of it. Is the author just interchanging the notations or is there anything deeper?
 A: It seems like you're asking what is the motivation for this example, but it's a little hard to tell, because the motivation would come from its context, which you haven't included in the quote.
He's giving an example of how a plane or parts of a plane can be described using more than one coordinate system. He's not discussing any notion of measurement (distance or angles), so these mappings are pretty much arbitrary except that he's making them up such that they're one-to-one and onto (bijective). If he hadn't cut out the r=0 point, for example, then the mapping from y's to x's would have been many-to-one. These mappings are also homeomorphisms, meaning that besides being bijections, they're also continuous and have continuous inverses. The general idea is that when you change coordinates, the transformation should have these properties.
People tend to have an intuitive idea that Cartesian coordinates are always the most normal or standard ones. Part of what you learn in a good differential geometry course is to get rid of this prejudice.
