I first learned about coordinate systems by Gelfand and I knew that we basically have two axis x and y with origin O and some unit vectors $\hat i$ and $\hat j$ and if $\vec{OA}=x\hat i+y\hat j$ then the point $A$ lies in $(x,y)$ in that coordinate system.
And then I began doing some research on my own about when the x and y axis are nonlinear and when the unit vectors are not constant over some particular region of the coordinate system, what do I mean? If you draw a circle whose center is O from which arise two axis, the x axis and the y axis then I experimented how in the circle whose surface area is $1$ for example a unit vector $\hat i$ behaved and how it differed from another region see this picture to understand 
but I wasn't smart enough to find anything that could be interesting enough..
Then when going around those lecture notes: http://www.lecture-notes.co.uk/ I found that nonlinear coordinate systems do exist 
this motivated me to look on it more so I started working on coordinate systems where the x axis is dependent upon another coordinate system and the relation between the graph of a function in that first coordinate system and in the other coordinate system like here (where we get the x' axis by drawing the line y=0.5x in the x coordinate system)
but again I didn't find something very interesting
That's why I come here to ask about if there already exists a theory in maths about the relation of multiple coordinate systems, the graphs in those coordinate systems, non linear coordinate systems ...
Thank you very much
Gravitation by Misner, Wheeler and Thorne, is a thorough introduction to curvilinear coordinates and their application to general relativity. The math is gradually developed in a manner that makes it clear that "coordinate labels" are an arbitrary convention that may be coerced to make one's computations easier. This is an application of differential geometry, as identified by Sanath Devalapurkar in the comments to the posted question.