Can someone please confirm whether my intuitive notions behind what vector-valued functions and parameterisation is correct. Below are some questions.
Are vector-valued functions like functions of a single variable with domain of real values and range of vectors in an $n$-dimensional space? Does the vector-valued function involve a formula for a position vector which traces out a curve when the range of input values varies?
Do we use parameterisation's to describe curves which cannot be expressed as functions like $y=f(x)$ and $x=g(y)$? How are vector-valued functions and parameterisation's related?
I'm studying physics and these are some math preliminary topics for when I study motion of point Particles, dynamics and conservation laws.
Best Answer
Yes, but a vector-valued function can have any number of variables, e.g., ${\bf v}(r, s, t, w)$.
Yes, but again, there may be several variables involved. For a continuous vector-valued function the "tip of the vector" in general sweeps out a line, surface, volume, etc., depending upon the number of variables.
One can parameterize a vector-valued function as ${\bf v} = (v_1, v_2, \ldots, v_k) = (f_1(t), f_2(t), \ldots, f_k(t))$.