Next we introduce the notion of an oriented curve.
Defnition 95
Given a curve
#
with parametric representations
!
:
I
*
R
N
and
"
:
J
*
R
N
, we say that
!
and
"
have the
same orientation
if the para
meter change
h
:
I
*
J
is increasing and
opposite orientation
if the parameter
change
h
:
I
*
J
is decreasing. If
!
and
"
have the same orientation, we write
!
*
4
"
.
Exercise 96
Prove that
*
4
is an equivalence relation.
Defnition 97
An
oriented curve
#
is an equivalence class of parametric rep
resentations with the same orientation.
Note that any curve
#
gives rise to two oriented curves. Indeed, it is enough
to Fx a parametric representation
!
:
I
*
R
N
and considering the equivalence
class
#
+
of parametric representations with the same orientation of
!
and the
equivalence class
#
#
of parametric representations with the opposite orientation
of
!
.
Let
E
)
R
N
and let
g
:
E
*
R
N
be a continuous function. Given a piecewise
C
1
oriented curve
#
with parametric representation
!
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '14
 Topology, Trigraph, Continuous function, Equivalence relation, Topological space

Click to edit the document details