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Math 224 – Calculus III
1
13.1 Vector Functions and Space Curves
Recommended Homework: #127, 35, 37, 41
A vector valued function
()
t
r
or
rt
has the form (in
3
):
( )
( ), ( ), ( )
t
f t g t h t
r
or
( )
( )
( )
( )
t
f t
g t
h t
r
i
j
k
This is really just a generalization of the idea of parametric equations, where the component
function need not be linear
A vector valued function
t
r
is a function whose inputs (domain) are real numbers and whose
outputs (range) are vectors.
It is convenient to think of the variable
t
(the parameter) as being “time”.
We can think that the vector function describes the position of a particle as it traverse a curve
over time.
Example:
Consider the vector function:
2
( )
1
,
, ln(2
1)
t
t t
t
r
What is the domain of
t
r
?
What is the position at time
3
t
?
Note
: the functions
2
1
,
and
ln(2
x
t
y
t
z
t
are called the
component functions
of
t
r
, and they make up the parametric form for the curve (the “
space
” curve).
Picture:
x
y
z
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View Full Document Math 224 – Calculus III
2
Examples:
Sketch the following space curves.
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This note was uploaded on 10/31/2010 for the course MATH 224 taught by Professor Harris during the Fall '09 term at Illinois Central.
 Fall '09
 Harris
 Calculus

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