Part 1: 3 Dimensional Vector-Valued Functions
Most of the concepts for oriented curves in 2 dimensions carries over to
higher dimensions. Indeed, in R3, vector-valued functions are of
the form
r( t) =
á f( t) ,g( t),h( t)
ñ , t in [ a,b] |
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Moreover, the set of position vectors r( t) =
á f( t) ,g( t) ,h( t)
ñ , t in [ a,b] , forms an oriented curve and r( t) is called a parametrization of C.
EXAMPLE 1 Sketch the curve parametrized by r(t) =
á cos( t) ,sin( t),t
ñ for t in [ 0,4p] .
Solution: The computer algebra system Maple yields the
following curve
However, it does little good to eliminate t when working with
curves in R3. Instead, we must develop the calculus of
vector-valued functions in R3 as a means of studying these
functions directly.
To begin with, we define the limit of a vector-valued function r( t) =
á f( t) ,g( t) ,h(t)
ñ to be
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lim
t® p
|
r( t) = |
|
lim
t® p
|
f( t) , |
lim
t® p
|
g( t), |
lim
t® p
|
h( t) | ![](rangle.gif) |
|
| (1) |
when each of these limits exist. As a result, we define the derivative of r( t) =
á f( t) ,g( t),h( t)
ñ to be
which via the definition of the limit (1) yields
|
|
= | ![](langle.gif) |
|
lim
Dt® 0
|
|
|
|
, |
lim
Dt® 0
|
|
|
, |
lim
Dt® 0
|
|
|
| ![](rangle.gif) |
|
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Moreover, we often denote dr/dt by v and call it the
velocity of r( t) . Likewise, the derivative of
v( t) is often called the acceleration of r( t) . The computation above shows us that we have the
following:
Theorem 6.1: The velocity v( t) of
the function r( t) =
á f( t),g( t) ,h( t)
ñ is
v( t) = |
|
|
=
á f ¢( t) ,g¢( t) ,h¢( t)
ñ |
|
Likewise, the acceleration of r( t) is given
by
a( t) = |
|
|
=
á f ¢¢( t) ,g¢¢( t) ,h¢¢( t)
ñ |
|
In addition, we define antiderivatives of r(t) =
á f( t) ,g( t) ,h(t)
ñ to be
| ![](int.gif) |
r( t) dt = |
|
| ![](int.gif) |
f( t) dt, | ![](int.gif) |
g( t) dt, | ![](int.gif) |
h( t) dt |
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+
áC1,C2,C3
ñ |
|
where C1,C2,C3 are arbitrary constants.
EXAMPLE 2 Compute the velocity and acceleration of
Solution: To begin with, the velocity of r(t) is
v( t) = | ![](langle.gif) |
|
t2, |
|
e2t, |
|
t3 |
![](rangle.gif) |
|
=
á 2t,2e2t,3t2
ñ |
|
As a result, the acceleration of r(t) is
a( t) = | ![](langle.gif) |
|
2t, |
|
2e2t, |
|
3t2 |
![](rangle.gif) |
|
=
á2,4e2t,6t
ñ |
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Check your Reading: What is v( t)
if r( t) =
á t2,t2,t2
ñ ?