Michael Corral: Vector Calculus

1.8 Vector-Valued Functions 51
1.8 Vector-Valued Functions
Now that we are familiar with vectors and their operations, we can begin discussing
functions whose values are vectors.
Definition 1.10. A vector-valued function of a real variable is a rule that associates
a vector f(t) with a real number t, where t is in some subset D of 1 (called the
domain of f). We write f : D→ 3 to denote that f is a mapping of D into 3 .
For example, f(t)=ti+t 2 j+t 3 k is a vector-valued function in 3 , defined for all real
numbers t. We would write f :→ 3 . At t=1the value of the function is the vector
i+j+k, which in Cartesian coordinates has the terminal point (1,1,1).
A vector-valued function of a real variable can be written in component form as
or in the form
f(t)= f 1 (t)i+ f 2 (t)j+ f 3 (t)k
f(t)=(f 1 (t), f 2 (t), f 3 (t))
for some real-valued functions f 1 (t), f 2 (t), f 3 (t), called the component functions of f. The
first form is often used when emphasizing that f(t) is a vector, and the second form is
usefulwhenconsideringjusttheterminalpointsofthevectors. Byidentifyingvectors
withtheirterminalpoints,acurveinspacecanbewrittenasavector-valuedfunction.
Example 1.35. Define f :→ 3 by f(t)=(cost,sint,t).
This is the equation of a helix (see Figure 1.8.1). As the value
of t increases, the terminal points of f(t) trace out a curve spiraling
upward. For each t, the x- and y-coordinates of f(t) are
x=cost and y=sint, so
x 2 +y 2 = cos 2 t+sin 2 t=1.
Thus, the curve lies on the surface of the right circular cylinder
x 2 +y 2 = 1.
f(2π)
f(0)
x
0
z
Figure 1.8.1
It may help to think of vector-valued functions of a real variable in 3 as a generalization
of the parametric functions in 2 which you learned about in single-variable
calculus. Much of the theory of real-valued functions of a single real variable can be
applied to vector-valued functions of a real variable. Since each of the three component
functions are real-valued, it will sometimes be the case that results from singlevariable
calculus can simply be applied to each of the component functions to yield
a similar result for the vector-valued function. However, there are times when such
generalizations do not hold (see Exercise 13). The concept of a limit, though, can be
extended naturally to vector-valued functions, as in the following definition.
y