2
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.