Problem 1.
In cylindrical coordinate system:
±
2
r
v
T
π
φ
=
K
where
2
T
π
ω
=
So
±
v
r
ω φ
=
K
Then transform it into spherical coordinate:
±
sin
v
R
ω
θφ
=
K
Problem 2.
In cylindrical coordinate system
±
2
v
r
ω
φ
=
K
by applying the curl in cylindrical coordinate using the formula on page 13 in lecture
note
=>
3
v
z
r
ω
∇×
=
K
±
Then convert P1, P2, P3 into cylindrical coordinate system
=>
,
1
(0,0,0)
P
=
2
(0,
/3,
/ 2)
o
P
R
π
=
,
3
(
3
/ 4,
/3,
/ 4)
o
o
P
R
R
π
=
in cylindrical coordinate (r,
φ
, z).
Then by noticing that the amplitude of the curl of this vector field is only proportional
to r, the rank should be
P3>P2=P1
Problem 3.
a.
at R=R
m
R
S
v
ds
•
∫
K
v
o
=
l
l
2
2
0
0
sin
(
sin
)
9
o
o
R
R
R
d d
π π
R
θ
θ θ φ
•
∫ ∫
=
2
3
2
0
0
sin
9
o
R
d d
π π
θ θ φ
∫ ∫
=
2
3
9
o
R
π

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