(b) Evaluate
R
dS · (∇ × A) over a rectangle in the (x, y) plane bounded by the lines
x = 0, x = a, y = 0, y = b.
(c) Evaluate
H
A·dr around the boundary of the rectangle and verify Stokes’ theorem
for this case.
ANSWER:
∇ × A = (0, 0, 4y). For the surface integral dS =
ˆ
kdxdy, so answer is 2ab
2
.
For the line integral, split into 4 parts all with dz = 0: (0, 0) → (a, 0), with dy = 0 and
y = 0, giving a
3
/3; (a, 0) → (a, b), with dx = 0 and x = a, giving ab
2
; (a, b) → (0, b),
with dy = 0 and y = b, giving −a
3
/3 + ab
2
(remember integrating “backwards” in x);
(0, b) → (0, 0), with dx = 0 and x = 0, giving 0. Total for line integral is same as
surface integral, as per Stokes’ Theorem.
7. Using a vector field V = r, verify the divergence theorem for a cylinder of radius a,
centred on the z-axis and extending from z = 0 to z = h.
(a) Calculate the divergence and show the volume integral is 3πa
2
h (preferably with-
out integrating anything).
ANSWER:
∇ · V = 3, so volume integral is thrice the volume of cylinder.
(b) Show that ˆn =
ˆ
k on the top surface, and that the surface integral gives πa
2
h
(again, avoid doing integrals if you can).
ANSWER:
dS · V = zdA = hdA so surface integral is h times area of circular top.
(c) Show that the surface integral is zero on the other flat surface (where ˆn = −
ˆ
k).
ANSWER:
dS · V = −zdA = 0 as z = 0.
(d) On the curved surface, show the answer is a
R
dS = a.2πah, and hence verify the
divergence theorem.
ANSWER:
Normal to surface is ˆn = (x, y, 0)/
p
x
2
+ y
2
so ˆn · V =
p
x
2
+ y
2
= a. Surface
element is dS = adφdz, so surface integral is a.a.2π.h.
Add all surface integrals to get same result.
8. Calculate
R
dS · r over the whole surface of a cylinder bounded by x
2
+ y
2
= 1, z = 0
and z = 3. r = (x, y, z).
ANSWER:
Trick question: see above.
9. The electric field from a point charge of charge q centred at the origin is E =
qr/(4πε
0
r
3
). Find the electric flux across a spherical surface r = a.
ANSWER:
dS = ˆrdS and ˆr · r = r. On the surface of a sphere, dS = a
2
dΩ where
R
dΩ =
R
π
0
dθ
R
2π
0
dφ sin θ = 4π.
Flux is
R
E · dS =
q
4πε
0
R
dΩ =
q
ε
0
.
10. Evaluate the line integrals
(a)
R
C
(x, y, z) · dl