Curvilinear coordinates are coordinate systems in Euclidean space based on some transformation that converts the standard Cartesian coordinate system into a coordinate system with the same number of coordinates in which coordinate lines, i.e. the level curves of coordinates, are curved. The transformation must be locally invertible (a one-to-one map) at each point. This means that one can convert a point given in one coordinate system to its curvilinear coordinates and back.

A well-known example of a curvilinear system on the plane is polar coordinates:

x = r cos(φ)
y = r sin(φ)

Less well known are the parabolic and elliptic coordinates. Try to derive their formulas using the model.