4.
Line Integrals in the Plane
4A.
Plane Vector Fields
4A1
Describe geometrically how the vector fields determined by each of the following
vector functions looks. Tell for each what the largest region in which
F is continuously
differentiable is.
a) a i
+
b
j
,
a, b constants
b) xi

y
j
4A2
Write down the gradient field Vw for each of the following:
a)w=ax+by
b)w=lnr
c)w=f(r)
4A3
Write down an explicit expression for each of the following fields:
a) Each vector has the same direction and magnitude
as
i
+
2
j
.
b) The vector at (x, y) is directed radially in towards the origin, with magnitude
r2.
c) The vector at (x, y) is tangent to the circle through (x, y) with center at the origin,
clockwise direction, magnitude l/r2.
d) Each vector is parallel to i
+
j
,
but the magnitude varies.
4A4
The electromagnetic force field of a long straight wire along the zaxis, carrying a
uniform current, is a twodimensional field, tangent to horizontal circles centered along the
zaxis, in the direction given by the righthand rule (thumb pointed in positive zdirection),
and with magnitude k/r. Write an expression for this field.
4B. Line Integrals in the Plane
4B1
For each of the fields
F and corresponding curve C or curves Ci, evaluate
L
F
.
dr.
Use any convenient parametrization of C, unless one is specified.
.
Begin by writing the
r
integral in the differential form
M dx
+
N dy.
b
a)
F
=
(x2

y) i
+
2x
j
;
C1 and C2 both run from
(
1,O) to (1,O)
:
C1
:
the xaxis
C2: the parabola y
=
1

x2
b)
F
=
xy i

x2
j
;
C: the quarter of the unit circle running from (0,l) to (1,O).
C)
F
=
y i

x
j
;
C: the triangle with vertices at (0, O), (0, I), (1, 0), oriented clockwise.
d)
F
=
y i
;
C is the ellipse x
=
2 cost, y
=
sin t, oriented counterclockwise.
e)
F
=
6y i
+
x
j
;
C is the curve x
=
t2, y
=
t3, running from (1,l) to
(4,8).
f)
F
=
(x
+
y) i
+
xy
j
;
C is the broken line running from (0,O) to (0,2) to (1,2).
4B2
For the following fields
F and curves C, evaluate
Jc
F

dr without any formal
calculation, appealing instead to the geometry of
F and C.
F
=
xi
+
y
j
;
C is the counterclockwise circle, center at (0, 0), radius a.
b)
F
=
y i

x
j
;
C is the counterclockwise circle, center at (O,O), radius a.
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E. 18.02 EXERCISES
4B3
Let
F
=
i
+
j
.
How would you place a directed line segment
C of length one so
that the value of
Jc
F
.
dr would be
a) a maximum;
b) a minimum;
c) zero;
d) what would the maximum and minimum values of the integral be?
4C.
Gradient Fields and Exact Differentials
4C1
Let f (x, y)
=
x3
+
y3, and
C be
y2
=
x, between (1, 1) and (1, I), directed upwards.
a) Calculate
F
=
V
f.
b) Calculate the integral
F
.
dr three different ways:
(i) directly;
(ii) by using pathindependence to replace
C by a simpler path
(iii) by using the Fundamental Theorem for line integrals.
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 Spring '08
 Auroux
 Integrals, Vector Calculus, 2W, 2J, 1 4G, 5 4G

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