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Section 13.2: Line integrals (concluded)
In analyzing
∫
C
F
•
d
r
=
∫
C
(
F
•
T
)
ds
, the sign of
F
•
T
can
be determined visually by looking at the angle
θ
between
F
and
T
.
Since
r
′
(
t
) and
T
(
t
) are parallel and point in the
same direction, the angle
is is also the angle between
F
and
r
′
(
t
).
Text question (page 936)
Section 13.3: The fundamental theorem for line integrals
Key ideas?
..?.
.
•
The fundamental theorem:
∫
C
∇
f
•
d
r
=
f
(
r
(
b
)) –
f
(
r
(
a
))
•
The pathindependence of
∫
C
∇
f
•
d
r
(when
C
is
smooth,
f
is differentiable, and
∇
f
is continuous)
•
Path independence
⇔
the condition that
∫
C
F
•
d
r
= 0
for every closed curve
C
in the domain of
F
•
The equivalence of the following three conditions on a
simplyconnected domain:
o
Path independence
o
F
=
P
i
+
Q
j
being conservative (i.e.,
F
=
∇
f
)
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∂
P
/
∂
y
=
∂
Q
/
∂
x
•
Conservation of energy
The analogy beween the Fundamental Theorem of the
Calculus and the fundamental theorem of line integrals:
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This note was uploaded on 02/13/2012 for the course MATH 241 taught by Professor Staff during the Fall '11 term at UMass Lowell.
 Fall '11
 Staff
 Integrals

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