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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?

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  • $\begingroup$ 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. $\endgroup$ Commented Dec 8, 2018 at 14:17
  • $\begingroup$ @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" $\endgroup$
    – Number4
    Commented Dec 8, 2018 at 14:27
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    $\begingroup$ 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. $\endgroup$ Commented Dec 8, 2018 at 14:40

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"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.

In linear algebra, a change of coordinates comes up in linear transformations and diagonalization. In many contexts, we change the coordinates to make calculations easier. In exchange, there's a little bit of work to change the coordinates, either by using the Jacobian or finding a change of basis matrix.

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  • $\begingroup$ So my example with the function is also called a change of coordinates? $\endgroup$
    – Number4
    Commented Dec 9, 2018 at 8:08
  • $\begingroup$ Not quite. A transformation in general changes the image to another image. A change of coordinates, in my mind, keeps the image the same, but the elements have different "names." $\endgroup$ Commented Dec 9, 2018 at 16:03

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