Can a Function in Polar Coordinates Fail in Cartesian Coordinates?

[Undoubtedly0]

Hi all. What does it mean that a function in polar coordinates may not be a function in Cartesian coordinates?

For example, $r(\theta) = 1 + \sin\theta$ is a function because each $\theta$ corresponds to a single value of $r$. However, in Cartesian coordinates, the graph of this function most clearly fails the vertical line test.

Therefore, functionality depends on the coordinate system?

 

[mfb]

It depends on the type of function you want.

• If you want a function y(x), you cannot get the same graph as your equation gives.
• You can find a function R->R² with (x,y)(t), however, which gives one point (x,y) for each value of t, and has the same graph as your function.
• You can find a function R²->R, f(x,y) where your graph corresponds to all points where f(x,y)=0.
• ...

 

[Stephen Tashi]

Therefore, functionality depends on the coordinate system?

I wouldn't call the property of being a function "functionality", but, yes, a set of ordered pairs representing certain information may be a function and when the same information is represented in a different way as a different set of ordered pairs, that other set of ordered pairs may fail to be a function.