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4.
Line Integrals in the Plane
4A.
Plane Vector Fields
4A1
a) All vectors in the field are identical; continuously differentiable everywhere.
b) The vector at
P has its tail at P and head at the origin; field is cont. diff. everywhere.
c) All vectors have unit length and point radially outwards; cont. diff. except at (0,O).
d) Vector at
P has unit length, and the clockwise direction perpendicular to OP.
4B.
Line Integrals in the Plane
a) On Cl: y
=
0, dy
=
0; therefore
(x2

y) dx
+
2% dy
=
x2 dx
=
l
=

2
3
1 3'
b) C: use the parametrization x
=
cos t, y
=
sin t; then dx
=

sin t dt, dy
=
cost dt
c) C=Cl+C2+C3;
C1:x=dx=O; C2: y=1x; C3:y=dy=0
1
L
ydxxdy
=
L1o+1
LO
(1x)~xx(dx)+
=
Jdldx
=
1.
d) C:x=2cost, y=sint;
dx=2sintdt
Lydx
=
JdZT2~i~~tdt
=
2~
e) C: x=t2, y=t3; dx=2tdt, dy=3t2dt
2
2
2
L
6y dx
+
x dy
=
1
6t3(2t dt)
+
t2(3t2 dt)
=
1
(15t4) dt
=
3t5]
=
3
.
31.
1
4B2
a) The field
F points radially outward, the unit tangent t to the circle is always
perpendicular to the radius; therefore
F
.
t
=
0 and
Jc
F
.
dr
=
F
.
t ds
=
0
b)
The field
F is always tangent to the circle of radius a, in the clockwise direction, and
of magnitude a. Therefore
F
=
at, so that
F
.
=
F
.
t ds
=

ads
=
2aa2.
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E. 18.02 EXERCISES
i+j
4B3 a) maximum if
C
is in the direction of the field:
C
=

JZ
b) minimum if
C
is in the opposite direction to the field:
C
=

.
.
4
ij
c) zero if
C
is perpendicular to the field:
C
=
f

JZ
d) max
=
4,
min
=
4
by (a) and (b), fir the max or min
F and
C
have
respectively the same or opposite constant direction, so
Jc
F
.
dr
=
f
IF1
. ICI
=
fa.
4C.
Gradient Fields and Exact Differentials
4C1 a)
F
=
Vf
=
3x2y
i
+
(x3
+
3y2)
j
b) (i) Using y as parameter,
C1
is: x
=
Y2,
y
=
y; thus dx
=
2y dy, and
b) (ii) Using y as parameter,
C2
is: x
=
1, y
=
y; thus dx
=
0, and
b) (iii) By the Fundamental Theorem of Calculus for line integrals,
Vf
.
dl
=
f
(B)

f
(A).
P
4C2 a)
F
=
Vf
=
(xyexY
+
exY)
i
+
(x2exy)j
.
b) (i) Using x as parameter,
C
is: x
=
x, y
=
l/x, so dy
=
dx/x2, and so
0
lo
F
.
=
(e
+
e) dx
+
(x2e)
(dx/x2)
=
(2ex

ex)]
=
e.
b) (ii) Using the F.T.C. for line integrals,
F
.
=
f
(1,l)

f
(0,
co)
=
0

e
=
e.
Ic
4C3 a)
F
=
=
(cosxcosy)i

(sinxsiny)j.
b) Since
/
F
.
dr is pathindependent, for any
C
connecting A
:
(xo, yo) to B
:
(xi, yi),
J
c
we have by the F.T.C. for line integrals,
b
F
.
=
sin
XI
cos yl

sin xo cos yo
This difference on the righthand side is maximized if sinxl cos yl is maximized, and
sin xo cos yo is minimized. Since
I
sin x cos yl
=
I
sin
I
5
1,the difference on the right
hand side has a maximum of 2, attained when sinxl cos yl
=
1and sinxo cos yo
=
1.
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 Spring '08
 Auroux
 Integrals, Vectors

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