# What is the difference between the following definitions of Vector Functions and Parametric Curves?

The definitions seem exactly the same to me. We have a(t) to be a vector function and x(t) to be a parametric curve where the tip of the position vector traces out a curve in an $$n$$-dimensional space as the input varies throughout it's domain values of t.

Vector Functions:

Vector functions are given by a: I $$\rightarrow\mathbb{R}^n$$ with the domain I, an interval of $$\mathbb{R}$$, and the range being a subset of two or three dimensional space. For example, with $$n = 3$$, with respect to the ﬁxed orthonormal basis $${\bf{e}}_1$$, $${\bf{e}}_2$$ and $${\bf{e}}_3$$. The vector $${\bf{a}}(t)$$ can be written as $${\bf{a}}(t)=a_1(t){\bf{e}}_1 + a_2(t){\bf{e}}_2 + a_3(t){\bf{e}}_3.$$

Parametric Curves:

A parametric curve is a function x : I$$\mathbb{R}^n$$, x : t $$\mapsto$$ x(t) where I is an interval. We will consider $$n = 2$$ (two-dimesional parametric curves, or plane curves) and $$n = 3$$ (three-dimensional parametric curves, space curves). Some deﬁnitions further require the function x to be diﬀerentiable. As the parameter t varies in the interval I, the point with position vector x(t) traces out a curve.

• What's "an interval of $\mathbb R^n$"? – Calvin Khor Jan 12 at 14:09
• That was an error. I've corrected it to $\mathbb{R}$. Thanks. – user503154 Jan 12 at 14:17
• In which case it looks the same to me, except that curves usually have some amount of smoothness to them. Maps $\mathbb R^n \to \mathbb R^m$ (which is what I previously thought you called vector functions) are of course not the same – Calvin Khor Jan 12 at 14:26