James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

848 Chapter 13 Vector Functions
In general, a function is a rule that assigns to each element in the domain an element in
the range. A vector-valued function, or vector function, is simply a function whose
domain is a set of real numbers and whose range is a set of vectors. We are most interested
in vector functions r whose values are three-dimensional vectors. This means that
for every number t in the domain of r there is a unique vector in V 3 denoted by rstd. If
f std, tstd, and hstd are the components of the vector rstd, then f , t, and h are real-valued
functions called the component functions of r and we can write
rstd − k f std, tstd, hstdl − f std i 1 tstd j 1 hstd k
We use the letter t to denote the independent variable because it represents time in most
applications of vector functions.
Example 1 If
rstd − kt 3 , lns3 2 td, st l
then the component functions are
f std − t 3 tstd − lns3 2 td hstd − st
By our usual convention, the domain of r consists of all values of t for which the expression
for rstd is defined. The expressions t 3 , lns3 2 td, and st are all defined when
3 2 t . 0 and t > 0. Therefore the domain of r is the interval f0, 3d. ■
Limits and Continuity
The limit of a vector function r is defined by taking the limits of its component functions
as follows.
If lim t la rstd − L, this definition is
equivalent to saying that the length and
direction of the vector rstd approach the
length and direction of the vector L.
1 If rstd − k f std, tstd, hstdl, then
lim rstd −
t l a k lim f std, lim tstd, lim hstd
t l a t l a t l a
l
provided the limits of the component functions exist.
Equivalently, we could have used an «- definition (see Exercise 54). Limits of vector
functions obey the same rules as limits of real-valued functions (see Exercise 53).
Example 2 Find lim rstd, where rstd − s1 1 t 3 d i 1 te 2t j 1 sin t k.
t l 0 t
SOLUTION According to Definition 1, the limit of r is the vector whose components are
the limits of the component functions of r:
lim rstd −
t l 0 f lim s1 1 t 3 d
t l 0 g i 1 f lim
t l 0 g te2t j 1 sin t
Flim
t l 0
G k
t
− i 1 k (by Equation 3.3.2) ■
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