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EE 1  Homework #1 Solutions
(1)
R
π
0
R
2
π
0
ˆ
R dφdθ
=? (R,
θ
, and
φ
deﬁne the spherical coordinate system)
Physically, the above integral is summing the unit vector
ˆ
R
over the
entire sphere. If we inspect the ﬁgure below, one can see that for every
vector, there is exactly one other vector in the exact opposite direction with
the same magnitude (

ˆ
R

= 1). Therefore, every vector will have one other
that will cancel it out. The sum total would be equal to zero:
Z
π
0
Z
2
π
0
ˆ
R dφdθ
= 0
.
Mathematically, the same can be obtained by revealing the dependence
of
ˆ
R
on
θ
and
φ
.
ˆ
R
= ˆ
rsinθ
+ ˆ
zcosθ
= ˆ
xcosφsinθ
+ ˆ
ysinφsinθ
+ ˆ
zcosθ
The integral would then be:
Z
π
0
Z
2
π
0
ˆ
R dφdθ
= ˆ
x
Z
π
0
Z
2
π
0
cosφsinθ dφdθ
+ˆ
y
Z
π
0
Z
2
π
0
sinφsinθ dφdθ
+ˆ
z
Z
π
0
Z
2
π
0
cosθ dφdθ
=

ˆ
xsinφ

2
π
0
cosθ

π
0
+ ˆ
ycosφ

2
π
0
cosθ

π
0
+ ˆ
zsinθ

π
0
= ˆ
x
(0

0)(2) + ˆ
y
(1

1)(

2) + ˆ
z
(0

0) = 0
.
(2) Draw the vectors: 3
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This note was uploaded on 06/06/2009 for the course EE EE 1 taught by Professor Ozcan during the Spring '09 term at UCLA.
 Spring '09
 Ozcan

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