14
Vector Functions
14.1
Vector Functions and Space Curves
Definition 14.1.1
Let
r
:
R
→
R
3
(or
R
2
) be given by
r
(
t
) =
< f
(
t
)
, g
(
t
)
, h
(
t
)
>
=
f
(
t
)
i
+
g
(
t
)
j
+
h
(
t
)
k
,
where
f
,
g
, and
h
are realvalued functions.
r
(
t
)
is called a vectorvalued function.
Definition 14.1.2
Let
r
(
t
)
be given by
r
(
t
) =
< f
(
t
)
, g
(
t
)
, h
(
t
)
>
, where
f
,
g
and
h
are realvalued func
tions of
t
. The domain of
r
is the largest set of real numbers
t
so that
r
(
t
)
is defined.
Definition 14.1.3
Let
r
(
t
)
be given by
r
(
t
) =
< f
(
t
)
, g
(
t
)
, h
(
t
)
>
, where
f
,
g
and
h
are realvalued func
tions of
t
. We say that the limit of
r
as
t
approaches
a
is the vector
v
if for any
>
0
, there is a number
δ >
0
so that whenever
0
<

t

a

< δ
then

r
t

v

<
.
Theorem:
Let
r
(
t
) =
< f
(
t
)
, g
(
t
)
, h
(
t
)
>
be a vectorvalued function with realvalued component functions
f
,
g
, and
h
. If the component functions all have limits as
t
approaches
a
, i.e.,
lim
t
→
a
f
(
t
) =
L
1
,
lim
t
→
a
g
(
t
) =
L
2
,
lim
t
→
a
h
(
t
) =
L
3
,
then lim
t
→
a
r
(
t
) exists and
lim
t
→
a
r
(
t
) = lim
t
→
a
< f
(
t
)
, g
(
t
)
, h
(
t
)
>
=
D
lim
t
→
a
f
(
t
)
,
lim
t
→
a
g
(
t
)
,
lim
t
→
a
h
(
t
)
E
=
< L
1
, L
2
, L
3
> .
Definition 14.1.4
We say that the vectorvalued function
r
(
t
)
is continuous at
a
if (1)
r
(
a
)
is defined,
i.e.,
a
is in the domain of
r
, (2)
lim
t
→
a
r
(
t
)
exists, and (3)
lim
t
→
a
r
(
t
) =
r
(
a
)
. More simply put,
r
(
t
)
is continuous
at
a
if
lim
t
→
a
r
(
t
) =
r
(
a
)
. (Why?)
Example 14.1.1
Let
r
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 Spring '07
 Chung
 Derivative, Multivariable Calculus, Graph of a function, Vectorvalued function

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