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3.1 - Vector-Valued Functions and Space Curves - Coggle Diagram
3.1 - Vector-Valued Functions and Space Curves
Definition of a Vector-Valued Function
A
vector valued function
is a function in the form of:
r
(
t
) = f(
t
)
i
+ g(
t
)
j
r
(
t
) = f(
t
)
i
+ g(
t
)
j
+ h(
t
)
k
Where the
component functions
f
,
g
, and
h
, are real-valued functions of the parameter
t
. They are also written in this form:
r
(
t
) = < f(
t
)
i
, g(
t
)
j
>
r
(
t
) = < f(
t
)
i
, g(
t
)
j
, h(
t
)
k
>
The parameter
t
can lie:
Between real numbers:
a
</
t
</
b
ε
t
might take on all
real numbers
Domain restrictions that enforce restrictions on
t
Graphing Vector-Valued Functions
The graph of a vector-valued function of the form
r
(
t
) = f(
t
)
i
+ g(
t
)
j
consists of the set of all (
t
,
r
(
t
) ), and the path it traces is called a
plane curve
The graph of a vector-valued function of the form
r
(
t
) = f(
t
)
i
+ g(
t
)
j
+ h(
t
)
k
consists of the set of all (
t
,
r
(
t
) ), and the path it traces is called a
space curve
Any curve has an infinite number of reparametrizations
r
(
t
) = f(
t
)
i
+ g(
t
)
j
can be defined as x =
f
(
t
) and y =
f
(
t
)
Limits and Continuity of a Vector-Valued Function
A vector-valued function
r
approaches the limit
L
as
t
approaches
a
.
lim
r
(
t
) =
L
t --> a
Provided:ε0
lim ||
r
(
t
) -
L
|| = 0
t --> a
Limit of a Vector-Valued Function
Let
f
,
g
, and
h
be functions of
t
. Then the limit of the vector valued function
r
(
t
) = f(
t
)
i
+ g(
t
)
j
as
t
approaches
a
is given by
lim
r
(
t
) = [ lim
f
(
t
) ]
i
+ [ lim
g
(
t
) ]
j
t --> a
lim
r
(
t
) = [ lim
f
(
t
) ]
i
+ [ lim
g
(
t
) ]
j
+ [ lim
h
(
t
) ]
k
t --> a
Continuity
Let
f
,
g
, and
h
be functions of
t
. The the vectors
r
(
t
) = f(
t
)
i
+ g(
t
)
j
and
r
(
t
) = f(
t
)
i
+ g(
t
)
j
+ h(
t
)
k
are continuous at point
t = a
if the following three conditions hold
lim
r
(
t
) exists
t --> a
r
(
a
) exists
lim
r
(
t
) =
r
(
a
)
t --> a