What is the difference between vector-valued functions and parametric equations?

Solution 1:

The difference is that a parametrization has some extra properties. A vector valued function is a map $$f:U\subset\mathbb R^m\to V\subset\mathbb R^n$$

And parametric equations for a [portion of a] submanifold $M$ in Euclidean space (it's rare to parametrize things other than manifolds) is a map $$\varphi:U\subset\mathbb R^m\to M\subset\mathbb R^n$$ Where:

• $U$ is open
• $\varphi$ is a homeomorphism onto its image
• $\operatorname{rank}D\varphi = m$ everywhere

What we could say then, is that a parametrization is always in the form of a vector valued function, but conversely, we use vector valued functions with nice properties to parametrize varieties.

Solution 2:

They are basically the same (different ways of expressing the same idea).

In the equation below, the LHS is a vector-valued function, and the RHS is a parametric form of the same function.

$$\vec{v} = \pmatrix{\sin{(t)} \\ \cos{(t)} \\ t} \equiv \matrix{x(t) = \sin{(t)} \\ y(t) = \cos{(t)} \\ z(t) = t}$$