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Section 13.1: Vector fields (concluded)
Group Work: Sketching vector fields, 1 and 2 only (page 7
of Instructor’s Guide)
Section 13.2: Line integrals
Key ideas?
..?.
.
•
It’s a misnomer: the “lines” are actually paths
•
Line integral of a scalar field w.r.t. arclength
o
Definition as a limit of Riemann sums: formula 2
o
Geometric meaning: Figure 2
o
How to compute it: Example 2
•
Line integral of a scalar field w.r.t. coordinate
variables
o
∫
C
f
(
x
,
y
)
dx
o
∫
C
f
(
x
,
y
)
dy
o
∫
C
P dx
+
Q dy
•
Line integral of a vector field
o
Definitions as a limit of Riemann sums
o
Geometric/physical meaning
o
How to compute it
•
Vector fields and work:
W
=
∫
C
F
•
d
r
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f
is nonnegative, then
∫
C
f
(
x
,
y
)
ds
can be
interpreted as the area of a curved wall whose base follows
the curve
C
and whose height above the basepoint (
x
,
y
) is
f
(
x
,
y
).
How can we write the line integral of a scalar field (with
respect to arc length) in terms of an ordinary firstyear
calculus integral?
..?.
.
Formula (3):
∫
C
f
(
x
,
y
)
ds
=
∫
a
b
f
(
x
(
t
),
y
(
t
)) sqrt((
x
′
(
t
))
2
+(
y
′
(
t
))
2
)
dt
What are
a
and
b
?
..?.
.
They arise from a parametrization of the curve
C
as the set
of points (
x
(
t
),
y
(
t
)) as
t
goes from
a
to
b
.
Isn’t there more than one way to parametrize the curve
C
in
this way?
Which parametrization should one use?
..?.
.
Whichever parametrization is most convenient, because it
doesn’t matter which one you use
; formula (3) will give the
same answer no matter what parametrization one uses, as
long as…
..?.
.
every point along
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 Fall '11
 Staff
 Integrals, Scalar

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