Why are curves in the plane of the form R -> R^2?

[Simfish]

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...

 

[tiny-tim]

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...

Hi simfish!

Because a curve can be defined by one parameter - it's one-dimensional.

The parameter could be length, or angle, or anything convenient.

Usually, it's the length, s.

Then f(s) is the position (on a plane or in space) of the point whose distance along the curve is s.

So f maps the real numbers (R) into the plane or space.

You could use two parameters, but they wouldn't be independent.

Essentially, using (x,y) to define a curve in R2 would be using a function from s to (x,y) and then from (x,y) to R2!

A surface is two-dimensional, and needs two parameters. For example, points on a sphere are specified by latitude and longitude, so the "function for a sphere" in space would be a map from R2 to R3, specifying a point (x,y,z) for every point (theta,phi).