A while ago I suddenly realised that although I could routinely perform changes of variables in simplifying a wide variety of problems, I had no real understanding of what I was doing, besides having some intuition about "changing my point of view", or "changing coordinate systems", to make things more tractable. Examples would be changing variables in evaluating multivariable integrals, changing to a different set of coordinates (picking a basis) in linear algebra, and making variable substitutions to show that an algebraic variety is equivalent to another one. For concreteness, I will use $\mathbf{R}^{n}$ as an illustrative example throughout.
To start things off, I am pretty comfortable with what "change of coordinates" means in linear algebra. It simply means moving from a basis $B = \{e_{1}, \dots, e_{n}\}$ of $\mathbf{R}^{n}$, to a new basis $B' = \{e'_{1},\dots,e'_{n}\}$. The "coordinates" of a vector $\mathbf{v} \in \mathbf{R}^{n}$ with respect to $B$ will transform accordingly in $B'$.
But I need not restrict myself to only linear, or affine change of variables; in multivariable analysis one writes things like $x = r \cos \theta, y = r \sin \theta$ all the time, which clearly isn't linear. This means that there needs to be a more advanced notion of what a coordinate change means. There's already a great answer here about what a "coordinate system" means; the machinery of smooth manifolds makes this very precise, and it doesn't seem too difficult to make sense of what $x = r \cos \theta, y = r \sin \theta$ means in terms of charts, atlases, parametrisations and all that. In this case I am also quite happy, because it's clear how to make sense of introducing coordinates on $\mathbf{R}^{n}$, and switching between different coordinate systems.
In elementary algebraic geometry (elementary study of plane curves), there exists a related notion of "change of variables", or "change of coordinates". For example, in a book I'm reading, there's an exercise which asks the reader to show that there's a map from the real algebraic variety $V(x^{2} -y)$ to $V(9v^{2} -4u)$ (these varieties are simply sets of points of $\mathbf{R}^{2}$ which make the polynomials vanish).
And here come the questions:
In multivariable analysis, one usually does not make any claims as to what coordinate system one has chosen for $\mathbf{R}^{n}$ in the context of specific problems (e.g. evaluating integrals). I find this confusing; because $\mathbf{R}^{n}$ by itself does not come with a coordinate system, we need to make a choice first (although the choice is always the trivial one), before one can talk meaningfully about "changing variables" by the substitution $x = r \cos \theta, y = r \sin \theta$, for example. Can one meaningfully talk about coordinate changes without the language of smooth manifolds?
In a similar vein, changing coordinates in algebraic geometry does not make a lot of sense unless one talks about smooth manifolds (or does it?). I suppose I could interpret a substitution like $x=u+1, y=v+1$ by first noting that $\mathbf{R}^{2}$ is a smooth manifold, and talk about change of coordinates in that context, but this seems overkill. But without that language, it's not clear to me if the variety $V(x^{2} -y)$ refers to the set of points $(x,y) \in \mathbf{R}^{2}$ satisfying $x^{2}-y =0$, or the set of coordinates $[x,y]$ of points of $\mathbf{R}^{2}$ that satisfy the polynomial equation after picking a suitable basis (or is there no difference) ? This confusion is further complicated by the fact that some authors talk about transformations $\mathbf{R}^{2} \to \mathbf{R}^{2}$ of the plane to itself in establishing equivalence of conics, and others simply of "coordinate change", without messing with the plane $\mathbf{R}^{2}$. They might be in some sense equivalent or dual to each other (though I can't make this precise at the moment), but to me they are certainly different, because we can talk about transformations of $\mathbf{R}^{2}$ to itself without introducing a coordinate system! In very old books I see concepts like "active transformations" and "passive transformations" which make this distinction, but I don't see this language used in any contemporary mathematical textbook.
Can one even make a sharp distinction between the maps $\mathbf{R}^{2} \to \mathbf{R}^{2}$, and changing from one coordinate system on $\mathbf{R}^{2}$ to another? Is it ever helpful to make this distinction?
Before the the modern concept of a smooth manifold was invented, how did mathematicians think about magic tricks like $x = r \cos \theta, y = r \sin \theta$?
