Standard Problems 15
APPM 2350
Spring 2014
1. Let F be a differentiable vector field defined on a region containng a smooth closed
oriented surface S and its interior. Let n be the outward unit normal to the surface S.
Suppose that S is composed of the union of two surfaces S1 and S2 joined along the
smooth simple closed curve C. Can anything be said about
ZZ
∇ × F · n dσ?
(1)
S
Give reasons for your answer.
Solution:
By Stokes’ theorem, we know that
ZZ
I
∇ × F · n dσ =
F · dr
S1
and
ZZ
C
I
∇ × F · n dσ =
I
F · dr = −
−C
S2
F · dr.
C
Adding these together gives zero for (1).
2. Prove or disprove that if ∇ · F = 0 and ∇ × F = 0 then F = 0.
Solution:
Consider F = h1, 2, 3i. For this vector field, ∇ · F = 0 and ∇ × F = 0 but F 6= 0, so
the claim has been disproven.
3. If f (x, y, z) and g(x, y, z) are continuously differentiable scalar functions defined over
the oriented surface S with boundary curve C, prove that
ZZ
I
(∇f × ∇g) · n dσ =
f ∇g · T ds.
(2)
S
C
Give reasons for your answer.
Solution:
In problem 13.5.24, we proved the identity
∇ × (f G) = f (∇ × G) + (∇f ) × G.
Choosing G = ∇g, we have
∇ × (f ∇g) = f (∇ × (∇g)) + (∇f ) × (∇g) = ∇f × ∇g.
Stokes’ theorem can be written
I
ZZ
(∇ × F) · n dσ = F · T ds.
S
(2) is just a special case of (3), with F = f ∇g.
C
(3)