system
Given the following post, respond as if you're commenting on Reddit:
human
Curvilinear coordinates and analytic functions

I have a function F(x,y) expressed in cartesian coordinates, and I make a curvilinear coordinate transformation to express this function in a new coordinate basis (x', y'). Let's call the function in the new coordinates G(x',y'). If F(x,y) is an analytic function, is G(x',y') also an analytic function?
gpt
Yes, as long as the transformation (x,y)->(x',y') is analytic.

A function being real analytic on a real-analytic manifold is an intrinsic property of the function, ie it is coordinate independent. Changing coordinates on a manifold, using appropriate functions, changes nothing about the manifold or the functions on it, aside from how they look.