Simfish

Gold Member

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## Main Question or Discussion Point

So here are some functions of the following types...

f: R -> R^2 (curves in the plane)

f: R -> R^3 (curves in space)

f: R^2 -> R (functions f(x,y) of 2 vars)

f: R^3 -> R: (functions f(x,y,z) of 3 vars)

f: R^2 -> R^2 (vector fields v(x,y) in the plane)

The question is - why are curves in the plane of the form R -> R^2? My intuition tells me R^2 -> R^2 (since after all, curves in the plane are based on x and y coordinates...). And R^2 is a cartesian product of two sets. For any curve, I'd expect x AND y input values...

f: R -> R^2 (curves in the plane)

f: R -> R^3 (curves in space)

f: R^2 -> R (functions f(x,y) of 2 vars)

f: R^3 -> R: (functions f(x,y,z) of 3 vars)

f: R^2 -> R^2 (vector fields v(x,y) in the plane)

The question is - why are curves in the plane of the form R -> R^2? My intuition tells me R^2 -> R^2 (since after all, curves in the plane are based on x and y coordinates...). And R^2 is a cartesian product of two sets. For any curve, I'd expect x AND y input values...