WORKSHEET XVIII
Fundamental Theorem of Line Integrals
1. Let F(x, y) = (2xy + cos(2y)) i + (x2 – 2x sin(2y)) j. Is F
conservative? For each curve C below, compute
F
C
(a) C consists of the line segment from (0, 2) to (7, 11)
followed by the line segment from (7, 11) to (3, 0).
(b) C is the curve parameterized by:
(t) = (t2 – t, t (2 + cos t)(1 + sin t)), 0 ≤ t ≤ 
(c) C consists of one circuit of the elliptical path 9x2 + 16y2 =
144 in the counter-clockwise direction.
2.
Let G(x, y) = 2xy i + (x2 + z2) j + 2yz k. Is G conservative?
For each curve C below, compute
G
C
(a) C is the path parameterized by r(t) = (t2, t3, t5), 0 ≤ t ≤ 
(b) C is the straight line path from (0, 0, 0) to (1, 0, 0) followed
by the straight-line path from (1, 0, 0) to (1, 1, 1).
3. For each vector field below, find a potential function or show
that no potential function exists.
(a) F(x, y) = (e-y – y sin(xy)) i + (x e-y + xsin(xy) j
(b) F(x, y, z) = (2xz – y2 + yz exyz) i – (2xy + xz exyz) j + (x2
+ xy exyz) k
(c) F(x, y, z) = 3x2 i + (z2/y) j + (2z ln y) k
(d) F(x, y, z) = 6z i + y2 j + 12x k,
r(t) = (sin t) i + (cos t) j + t) k, 0  t  2.
4. (Colley) Of the two vector fields
F(x, y, z) = xy2z3 i + 2x2y j + 3x2y2z2 k
and G(x, y, z) = 2xy i + (x2 + 2yz) j + y2 k,
one is conservative and one is not. Determine which is which,
and, for the conservative field, find a scalar potential function.
5.
Discuss the relationship among the concepts: pathindependence, gradient vector field, conservative vector field,
potential function.
6. What is a simply-connected region? In a simply connected
region, what is the relationship between conservative vector
field and the curl of the vector field?
7. (Colley) Find all functions N(x, y) such that the vector field
F(x, y) = (ye2x + 3x2ey) i + N(x, y) j is conservative.
8. (Colley) For which values of the constants a and b will the
vector field
F(x, y, z) = (3x2 + 3y2 z sin(xz) i + (ay cos(xz) + bz) j +
(3xy2 sin(xz) + 5y) k
be conservative?
9.
Evaluate

C
xdx  ydy
x2  y2
where C is the semicircular arc of x2 + y2 = 4 from (2, 0) to (-2, 0).
10. Consider the vector field
x  xy2
x2  1
F ( x, y) 
i 3 j
y2
y
(a) On which regions is the field conservative?
(b) Find a potential function for F or show that none exists.
(c) Find the work done by F in moving a particle along the
parabolic curve y = 1 + x – x2 from (0, 1) to (1, 1).
This web of time – the strands of which approach one another,
bifurcate, intersect or ignore each other through the centuries –
embrace every possibility.
– Borges, The Garden of Forking Paths
Course Home Page
Department Home Page
Loyola Home Page