(1) In two dimensions, consider the vector function
(−y, x − 1)
(1 − y, x)
(−y, x + 1)
F~ = a
+b 2
+c
,
(x − 1)2 + y 2
x + (y − 1)2
(x + 1)2 + y 2
where a, b and c are nonzero constants. Compute
I
F~ · d~r ,
C
x2
y2
where C is the ellipse 4 + 9 = 1 traversed counterclockwise. Justify your answer by drawing the
appropriate path of integration.
(2) Find the area of the two dimensional region D bound by y = x3 , y = 8, and x = 0 by evaluating the
line integral
I
F~ · d~r,
C
where C = ∂D is the boundary of D traversed counterclockwise, and F~ = (P, Q) is a vector
function to be found.
(3) (a) Let S be the part of the plane y + z = 4 for which y > 0 and y < 4 − x2 . The density of S is
given by ρ(x, y, z) = z + x2 . Find the mass of S.
(b) For F~ = (0, x, −x), evaluate
I
I = F~ · d~r ,
C
where C = ∂S is traversed counterclockwise when viewed from above. Here, S is the same
surface as in part (a). The value of I represents the work done by F~ on a particle traversing C.
Using Stokes’ theorem, give another meaning for I.
(4) (a) Show that if F~ is a vector field with continuous partial derivatives defined on all of R3 , and S is
any closed surface in R3 with unit normal n̂,
ZZ
∇ × F~ · n̂ dS = 0 .
S
~ = (0, 0, v0 ) for some nonzero constant v0 . By direct computation,
(b) Consider the vector field G
~ across the hemispherical surface x2 + y 2 + z 2 = a2 , z > 0 with normal
compute the flux of G
oriented in the direction away from the origin.
~ across the unit disk x2 + y 2 ≤ a2 , z = 0, with normal oriented in the positive
(c) Find the flux of G
z direction.
~ Show that ∇ × (0, v0 x, 0) = G.
~ Use 4a to explain why the
(d) Compute the divergence of G.
answers in 4b and 4c are the same (this can also be thought of from a physical perspective).
(5) For F~ = (2, x, y 2 ), verify Stokes’ theorem
I
ZZ
ZZ
F~ · d~r =
∇ × F~ · n̂1 dS =
∇ × F~ · n̂2 dS ,
C
S1
S2
1
where C is the circle x2 + y 2 = 1 with z = 1 traversed counterclockwise when view from above, S1
is the disk x2 + y 2 ≤ 1 with z = 1, and S2 is the surface z = x2 + y 2 with 0 < z < 1. Here, n̂1 and
n̂2 are the unit normals to the surfaces S1 and S2 , respectively, with orientation consistent with the
direction of traversal of C.
2