Perhaps it's due to the fact that I'm currently tackling introductory material on parametric equations, but it seems to me at the moment that there is no real difference between a set of parametric equations and a vector function.
I'll give the standard example:
Let's say I have a parametric curve defined by the parametric equations $x=\cos(t)$ and $y=\sin(t)$. What stops me from defining $r(t) = \left\langle x, \ y \right\rangle = \left\langle \cos(t), \ \sin(t) \right\rangle$, which describes the exact same parametric curve (the unit circle) as the parametric equations for $x$ and $y$?
Now I know assume that there has to be some difference between parametric equations and vector functions, but with the material I'm currently working with I can't seem to find a counter-example, or cases where they differ.
I also realize that the concept of parameterization is critical to fields like Differential Geometry (based on what I've read so far in do Carmo's book), and proofs of the big Integral Theorems (generalized Stokes' Theorem etc) rely on it, and this concept is something I want to understand rock solid.
Can anyone give an example, as to why a set of parametric equations are different from vector functions of the form $f : \mathbb{R} \to \mathbb{R^m}$, and furthermore as to why they are so important to theorems in higher-dimensions?
Best Answer
$F(x,y)=\langle x,y \rangle$ is a vector function $F:\Bbb R^2\to \Bbb R^2$. It describes a vector field. It is not a parameterized curve.