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EE1259: Electromagnetic assignment
1
Problem 5
Problem 6
EE1259 Assignment 3
(Due on Thursday, September 20, 2007)
1.
Let
R
r
be the distance vector from a fixed point (x’, y’, z’) to the point (x, y, z), let
R
to be its length. Show that
a)
()
R
R
r
2
2
=
∇
b)
R
R
a
R
a
R
R
R
R
R
/
/
/
1
2
3
r
r
r
r
=
−
=
−
=
∇
c) What is the general formula for
( )
n
R
∇
2.
Compute
r
r
a
a
r
r
∇
⋅
, where
r
a
r
is the unit vector in spherical coordinate, or you can write
(
)
2
2
2
z
y
x
a
z
a
y
a
x
a
z
y
x
r
+
+
+
+
=
r
r
r
r
3.
For a scalar function
f
and a vector function
G
, prove:
( ) ( ) ( )
G
f
G
f
G
f
r
r
r
×
∇
+
×
∇
=
×
∇
4.
A vector field
ρ
a
D
r
r
3
=
exists in the region between two concentric cylindrical surfaces defined by
=1
and
=2
,
with both cylinders extending between
z=0
, and
z=5
. Verify the divergence theorem by evaluating the following.
(a).
∫
⋅
S
S
d
D
r
r
(b).
( )
∫∫∫
⋅
∇
V
dv
D
r
5. Check the divergence theorem for the function
φ
θ
a
r
a
r
a
r
v
r
r
r
r
r
tan
cos
4
sin
2
2
2
+
+
=
using the volume of
the “icecream cone” shown below (the top surface is spherical with radius R and centered at the origin).
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 Fall '07
 Chen
 Electromagnet

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