Transcript Slide 1
Chapter 16 – Vector Calculus
16.3 The Fundamental Theorem for Line Integrals Objectives: Understand The Fundamental Theorem for line integrals Determine conservative vector fields 16.3 The Fundamental Theorem for Line Integrals 1
FTC – Part 2
Recall from Section 5.3 that Part 2 of the Fundamental Theorem of Calculus (FTC2) can be written as:
a b
where
F′
is continuous on [
a
,
b
].
16.3 The Fundamental Theorem for Line Integrals 2
Fundamental Theorem for Line Integrals
Let
C
be a smooth curve given by the vector function
r
(
t
),
a
≤
t
≤
b
.
Let
f
be a differentiable function of two or three variables whose gradient vector is continuous on
C
. Then,
C f d
r
f
r
f
r
16.3 The Fundamental Theorem for Line Integrals 3
Note
Theorem 2 says that we can evaluate the line integral of a conservative vector field (the gradient vector field of the potential function
f
) simply by knowing the value of
f
at the endpoints of
C
.
◦ In fact, it says that the line integral of net change in
f
.
f is the 16.3 The Fundamental Theorem for Line Integrals 4
Note:
If f is a function of two variables and
C
is a plane curve with initial point
A
(
x
1 ,
y
1 ) and terminal point
B
(
x
2 ,
y
2 ), Theorem 2 becomes:
C f d
r
2 , 2 1 , 1 16.3 The Fundamental Theorem for Line Integrals 5
Note:
If f is a function of three variables and
C
is a space curve joining the point
A
(
x
1 ,
y
1 ,
z
1 ) to the point
B
(
x
2 ,
y
2 ,
z
2 ), we have:
C f d
r
2 , 2 , 2 1 , 1 , 1 16.3 The Fundamental Theorem for Line Integrals 6
Paths
Suppose
C
1 and
C
same initial point 2 are two piecewise-smooth curves (which are called paths) that have the
A
and terminal point
B
.
We know from Example 4 in Section 16.2 that, in general,
C
1
F
d
r
C
2
F
d
r
16.3 The Fundamental Theorem for Line Integrals 7
Conservative Vector Field
However, one implication of Theorem 2 is that
C
1
r
C
2
f d
r
whenever f is continuous.
◦ That is, the line integral of a conservative vector field depends only on the initial point and terminal point of a curve.
16.3 The Fundamental Theorem for Line Integrals 8
Independence of a Path
In general, if domain
D
F
is a continuous vector field with independent of path if
C
F
d
r
is
C
1
F
d
r
C
2
F
d
r
for any two paths
C
1 and
C
2 in
D
same initial and terminal points.
that have the ◦ This means that line integrals of conservative vector fields are independent of path.
16.3 The Fundamental Theorem for Line Integrals 9
Closed Curve
A curve is called closed if its terminal point coincides with its initial point, that is,
r
(
b
) =
r
(
a
) 16.3 The Fundamental Theorem for Line Integrals 10
Theorem 3
F
d
r
is independent of path in
D C
if and only if:
C
F
d
r
0 for every closed path
C
in
D
.
16.3 The Fundamental Theorem for Line Integrals 11
Physical Interpretation
The physical interpretation is that: ◦ The work done by a conservative force field (such as the gravitational or electric field in Section 16.1) as it moves an object around a closed path is 0.
16.3 The Fundamental Theorem for Line Integrals 12
Theorem 4
Suppose
F
is a vector field that is continuous on an open, connected region
D
.
F
d
r
conservative vector field on
D .
D
, then
F
is a ◦ That is, there exists a function
f
such that f=
F
.
16.3 The Fundamental Theorem for Line Integrals 13
Determining Conservative Vector Fields
The question remains: ◦ How is it possible to determine whether or not a vector field is conservative?
16.3 The Fundamental Theorem for Line Integrals 14
Theorem 5
If
F
(
x
,
y
) =
P
(
x
,
y
)
i
+
Q
(
x
,
y
)
j
throughout
D
, we have: is a conservative vector field, where
P
and
Q
have continuous first order partial derivatives on a domain
D
, then,
P
y
Q
x
16.3 The Fundamental Theorem for Line Integrals 15
Conservative Vector Fields
The converse of Theorem 5 is true only for a special type of region, specifically simply-connected region.
16.3 The Fundamental Theorem for Line Integrals 16
Simply Connected Region
A simply-connected region in the plane is a connected region
D
such that every simple closed curve in
D
encloses only points in
D
.
Intuitively, it contains no hole and can’t consist of two separate pieces.
16.3 The Fundamental Theorem for Line Integrals 17
Theorem 6
Let
F
= P i + Q j be a vector field on an open simply-connected region
D
. Suppose that
P
and
Q
have continuous first-order derivatives and throughout
D .
◦ Then,
F
is conservative.
16.3 The Fundamental Theorem for Line Integrals 18
Example 1 – pg. 1106
3.
Determine whether or not
F
is a conservative vector field. If it is, find a function
f
such that
F
= f .
F
2
x
3
y x
4
y
8
j
4.
F
( , )
e x
cos
y
i
e x
sin
y
j F
xy
cos
xy
sin
xy
i
x
2 cos
xy
j
16.3 The Fundamental Theorem for Line Integrals 19
Example 2
a) Find a function
f
such that
F
=
f
.
b)
F
given curve
C
.
C
F
d
r
2
xz
y
2
i
2
xy
j
x
2 3
z
2
k
,
t
2 ,
y
1,
z
1, 0 1 16.3 The Fundamental Theorem for Line Integrals 20
Example 3 – pg. 1107 # 24
Find the work done by the force field F in moving an object from P to Q.
F
e
y
i
xe
y
j
;
P
(0,1),
Q
(2, 0) 16.3 The Fundamental Theorem for Line Integrals 21
Example 4
Determine whether or not the given set is a) b) c) open, connected, and simply-connected.
x
0
16.3 The Fundamental Theorem for Line Integrals 22
Example 5 – pg. 1106 # 12
a) Find a function
f
such that
F
=
f
.
b) given curve
C
.
C
F
d
r F
x
2
i
y
2
j
,
C
is the arc of the parabola
y
from
2
x
2 16.3 The Fundamental Theorem for Line Integrals 23
Example 6 – pg. 1107 # 16
a) Find a function
f
such that
F
=
f
.
F
b) given curve
C
.
C
F
d
r
2
y z
2
xz
2
i
2
xyz
j
xy
2 2 2
x z
k
,
t
,
y
1,
z
t
2 , 0 1 16.3 The Fundamental Theorem for Line Integrals 24
Example 7 – pg. 1107 # 17
a) Find a function
f
such that
F
=
f
.
F
C
:
r
b) given curve
C
.
C
F
d
r
t
2
yze
xz
i
1
t
2
e xz
j
xye xz
k
, 1
t
2 2
t
k
, 0 2 16.3 The Fundamental Theorem for Line Integrals 25
Example 8 – pg. 1107 # 18
a) Find a function
f
such that
F
=
f
.
F
b)
C
:
r
given curve
C
.
C
F
d
r
sin
i
x
cos
y
cos
z
j
y
i
t
j
k
0 2
k
16.3 The Fundamental Theorem for Line Integrals 26