Use the result in Exercise 34(b). Let F x , y = y x 2 + y 2 i − x x 2 + y 2 j . (a) Show that ∫ C 1 F ⋅ d r ≠ ∫ C 2 F ⋅ d r if C 1 and C 2 are the semicircular paths from 1 , 0 to − 1 , 0 given by C 1 : x = cos t , y = sin t 0 ≤ t ≤ π C 2 : x = cos t , y = − sin t 0 ≤ t ≤ π (b) Show that the components of F satisfy Formula (9). (c) Do the result in parts (a) and (b) contradict Theorem 15.3.3? Example.
Use the result in Exercise 34(b). Let F x , y = y x 2 + y 2 i − x x 2 + y 2 j . (a) Show that ∫ C 1 F ⋅ d r ≠ ∫ C 2 F ⋅ d r if C 1 and C 2 are the semicircular paths from 1 , 0 to − 1 , 0 given by C 1 : x = cos t , y = sin t 0 ≤ t ≤ π C 2 : x = cos t , y = − sin t 0 ≤ t ≤ π (b) Show that the components of F satisfy Formula (9). (c) Do the result in parts (a) and (b) contradict Theorem 15.3.3? Example.
(a) Show that
∫
C
1
F
⋅
d
r
≠
∫
C
2
F
⋅
d
r
if
C
1
and
C
2
are the semicircular paths from
1
,
0
to
−
1
,
0
given by
C
1
:
x
=
cos
t
,
y
=
sin
t
0
≤
t
≤
π
C
2
:
x
=
cos
t
,
y
=
−
sin
t
0
≤
t
≤
π
(b) Show that the components of F satisfy Formula (9).
(c) Do the result in parts (a) and (b) contradict Theorem 15.3.3? Example.
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