This preview shows pages 1–8. Sign up to view the full content.
If a curve is closed, we write the line integral as
l
l
l
d
⋅
∫
F
r
is a closed curve
l
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document l
l
d
⋅
∫
F
r
is a closed curve
l
Fundamental Theorem for Line Integrals
((
)
)
))
)
)
))
0
lC
f
d
fd
fb fa
fa fa
=∇
⋅ = ∇⋅
=
=
∫∫
F
F
rr
&
is
if
for some
(
is called a
function for ).
ff
f
FF
F
conservative
potential
is a closed curve,
so have the same
initial point and
terminal point.
l
( ): terminal point
b
r
( ): initial point
a
r
Implications of Conservative Field
Fundamental Theorem for Line Integrals
((
)
)
))
CC
f
d
fd
fb fa
=∇
⋅ = ∇⋅
=
∫∫
F
F
rr
is
independent of path
C
d
⋅
∫
F
r
l
d
l
path
closed
any
for
0
=
⋅
∫
r
F
ve
conservati
is
F
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document Example
circle.
unit
the
(ii)
];
,
0
[
,
sin
cos
)
(
by
given
(i)
is
where
curve
over the
integral
this
evaluate
and
path
of
t
independen
is
integral
line
that the
Show
.
)
2
(
)
3
(
)
,
(
Let
2
2
p
∈
+
=
⋅
+
+
=
∫
t
t
e
t
t
C
C
d
xy
x
y
y
x
t
C
j
i
r
r
F
j
i
F
23
By our earlier example,
where
(,
)
is the potential function of .
So is conservative.
Hence, the line integral
is independent of path.
CC
f
f x
y
x
yx
d
fd
∇
=
=+
⋅ = ∇⋅
∫∫
F
FF
F
rr
ve.
conservati
is
2
.
2
and
3
Here,
2
2
F
⇒
∂
∂
=
=
∂
∂
=
+
=
y
P
y
x
Q
xy
Q
x
y
P
22
(,)
(
3)
(
2
x
yy
x
xy
=
++
F
ij
To show that the line integral
is independent of path.
C
d
⋅
∫
Fr
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document (i)
(
)
co
s
si
n
,
0
t
t
te
tt
p
=
+
≤≤
r ij
( ( ))
( ( ))
( 1,0)
(1,0)
2
CC
d
fd
fb fa
ff
⋅ = ∇⋅
=

∫
∫
F
r
r
r
r
Fundamental Theorem for Line Integrals
((
)
)
))
f
d
=∇
∫∫
F
F
rr
22
(,)
(
3)
(
2
)
x
yy
x
xy
=
++
F
ij
23
where
f
f x
y
x
yx
∇
=
=+
F
()
(co
s
) (
si
n
)
0
(
1,0)
e
p
p
pp
=
+
+ →
r i
jij
0
(0
)
(cos0
sin0
)
e
=
+
=
+→
j
(ii)
Since the unit circle is a closed path and
is conservative, so we have
0
C
d
⋅=
∫
F
Fr
Fundamental Theorem for Line Integrals
((
)
)
))
)
)
))
0
lC
f
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.
This note was uploaded on 11/01/2011 for the course MATH 1505 taught by Professor Yap during the Spring '11 term at National University of Singapore.
 Spring '11
 yap
 Math

Click to edit the document details