Calculus of vector valued functions

In the previous lectures we had been dealing with functions from a subset of R to R. In this
lecture we will deal with the functions whose domain is a subset of R and whose range is in R3 (or
Rn). Such functions are called vector valued functions of a real variable.
If the values of a function F are in R3, then each F(t) has 3 components, for example F(t) =
((f1(t); f2(t); f3(t)). Therefore, each vector valued function F is associated with 3 real valued
functions f1; f2 and f3 and in this case we write F = (f1; f2; f3).
Let us see some examples of vector valued functions.
Examples: 1. Let X0; P 2 R3 and P 6= 0. Consider the vector valued function F(t) = X0 + tP .
It is clear that the range of the vector valued function is the line though the point X0 parallel to
the vector P.
2. Consider the vector valued functions F1(t) = (cost; sint); 0 · t · 2¼ and F2(t) = (cost; sint; t),
1 < t < 1. We can geometrically visualize the ranges of F1 and F2 as t varies. In fact F1(t) varies
on a circle and F2(t) varies on a helix. Both these curves are particular cases of parametric curves.
Parametric curves: Let I be an interval and F : I ! R3. The set of points fF(t) : t 2 Ig is
called the graph of the function F. If F is continuous (for the de¯nition see below) then such a
graph is called a curve or parametric curve with the parameter t.