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I am currently reading about the topic alluded to in the topic of this thread. In Taylor's Classical Mechanics, the author appears to be making a requirement about any arbitrary coordinate system you employ in solving some particular problem. He says,

"Instead of the Cartesian coordinates r = (x,y,z), suppose that we wish to use some other coordinates. These could be spherical polar coordinates [itex](r, \theta, \phi)[/itex], r cylindrical coordinates [itex]\rho , \phi , z)[/itex], or any set of "generalized coordinates" q1, q2, q3, i the property that each position specifies a unique value of (q1, q2, q3); that is,

[itex]q_i = q_i(\vec{r})[/itex] for [itex]i = 1,2~and~3[/itex],

[itex]\vec{r}=\vec{r}(q_1,q_2,q_3)[/itex]

I understand that [itex]\vec{r}[/itex] is our position vector function, and that requiring coordinates (q1, q2, 3) in our arbitrary coordinate system to be able to be represented by a position vector makes sense,

**although I wouldn't mind someone telling me the exact reason we require this.**Again, it also makes sense that we would require (q1, q2, q3) and the position vector function representing that point to specify one unique point--if this weren't true, then a particle could occupy to positions at the same time,

**although I wouldn't mind someone telling me the exact reason we require this.**But I don't understand the requirement [itex]q_i = q_i(\vec{r})[/itex], what does this mathematical statement mean?