# Changing variables or coordinates?

While posing another question I got stuck on the distinction of the following two concepts;

The components of a vector is usually referred to as it's coordinates, while trying to understand what "change of variables" means when it comes to systems of ODE's, I saw several sourse talked about change of variables as "a change of coordiinates".

Sure a function has a graph $$(x,f(x))$$

where if we do some kind of transformation we get a new graph. But this should be the same kind of coordinates as in a linear space.

Is this the thing that one refers to when one says "change of coordinates" as in change to polar coordiantes?

• If you edit the question to provide direct quotes from the "several sources" we may be able to help. As posed now the too vague to answer. Commented Dec 8, 2018 at 14:17
• @Ethan Bolker so there is no consensus how to use these words? It is just a system of ODEs that should undergo change to polar "coordiates" Commented Dec 8, 2018 at 14:27
• Both are general phrases whose precise meaning depends on context. If you have an ODE vocabulary question just ask it directly. That's all the help I can give you. Commented Dec 8, 2018 at 14:40

"Change of coordinates" comes up in many contexts in mathematics. Even in calculus, there are multiple contexts. For example, the graph $$x^2+y^2=k^2$$ is not a function in x. However, when we change from rectangular to polar coordinates, we get a function r = k. Then it's easier to compute tangent lines and area. In integral calculus, u-substitution is a "change of coordinates" to make integration easier.