MATH 2720
Winter 2012
Tutorial of 4 April 2012
RR
1. Evaluate S (∇ × F) • dS where F = (x2 − y − 4)i + 3xyj + (2xz + z 2 )k and S is the surface
x2 + y 2 + z 2 = 16, z ≥ 0, both directly and using Stokes’ theorem. (Let n be the upward pointing
unit normal.) [First part was Exercise 5 last week.]
2. Verify Stokes’ theorem for the upper hemisphere z =
field F(x, y, z) = xi + yj + zk.
p
1 − x2 − y 2 , z ≥ 0 and the radial vector
3. Let S be a surface and let F be perpendicular to the tangent to the boundary of S. Show that
ZZ
(∇ × F) • dS = 0.
S
4. For a surface S, a fixed vector v and r(x, y, z) = (x, y, z), prove that
Z
ZZ
(v × r) • ds.
v • n dS =
2
S
∂S
5. If f (x) is a smooth function of one variable, must F(x, y) = f (x)i + f (y)j be a gradient?
6. Let F = F1 i + F2 j + F3 k and suppose each Fk satisfies the homogeneity condition
Fk (tx, ty, tz) = tFk (x, y, z),
for k = 1, 2, 3 and all t ∈ R. (We say such a function is homogeneous of degree 1.) Suppose that
∇ × F = 0. Prove that F = ∇f , where
2f (x, y, z) = xF1 (x, y, z) + yF2 (x, y, z) + zF3 (x, y, z).
[Hint: We say that g defined on D is homogeneous of degree p if g(λx) = λp g(x) for all λ ∈ R
and all x ∈ D such that λx ∈ D. If such a function is differentiable at x, then x • ∇g(x) = pg(x).
You can prove this by defining h(λ) = g(λx) for a fixed x and computing h0 (1).]
7. Let F(x, y, z) = xyi + yj + zk. Can there exist a function f such that ∇f = F?
R
8. Show that C (x dy − y dx)/(x2 + y 2 ) = 2π, where C is the unit circle and conclude that the
associated vector field −y/(x2 + y 2 )i + x/(x2 + y 2 )j is not conservative. Does this contradict the
follwing result: “If F is a C 1 vector field on R2 of the form P i + Qj that satisfies ∂P/∂y = ∂Q/∂x,
then F = ∇f for some f on R2 ”? If not, why not?
9. Let F = (x − y)i + (y − z)j + (z − x)k. Evaluate
RR
∂W
F • dS for each of the following regions:
1. x2 + y 2 ≤ z ≤ 1
2. x2 + y 2 ≤ z ≤ 1 and x ≥ 0
3. x2 + y 2 ≤ z ≤ 1 and x ≤ 0
10. Evaluate
RR
∂S
F • dS, where F = 3xy 2 i + 3x2 yj + z 3 k and S is the surface of the unit sphere.