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EEE 340 Lecture 04 1
2-4.3 Spherical Coordinates
EEE 340 Lecture 04 2
• A vector in spherical coordinates
• The local base vectors from a right –handed system
AaAaAaA RR
)652(
aaa
aaa
aaa
R
R
R
,
,
)642( c
)642( b
)642( a
EEE 340 Lecture 04 3
The differential length
The differential areas are
The differential volume
dRaRdadRald R sin
)662(
RdRdds
dRdRds
ddRdsR
sin
sin2
)672(
ddRdRdv sin2 )682(
EEE 340 Lecture 04 4
addRRds
addRRds
addRds RR
;sin
;sin2
On many occasions the differential
areas are vectors
EEE 340 Lecture 04 5
Table 2-1 Basic Orthogonal Coordinates
Cartesian Cylindrical Spherical
)682(
sin
)542()462(
)672()532()452(
)652(sinˆ
ˆˆ
)522(ˆ
ˆˆ
)442(ˆ
ˆˆ
)ˆ(
)ˆ(
)ˆ(
)ˆ(
)ˆ(
)ˆ,ˆ(
)ˆ(
)ˆ(
)ˆ()(
2
,
ddRdRdvdzrdrddvdxdydzdv
volume
aldifferenti
areas
aldifferenti
dR
RddRRld
dzz
rddrrld
dzz
dyydxxld
length
aldifferenti
ora
ora
Rora
zora
ora
raora
zora
yora
xora
vectors
baseunit R
z
r
z
y
x
EEE 340 Lecture 04 6
Cartesian coordinates
and are vectors.
is a scalar.
Differential displacement
Differential normal area
Differential volume
zyx adzadyadxd
z
y
x
adydx
adzdx
adzdydS
dzdydxdv
d dS
dv
EEE 340 Lecture 04 7
• The differential surface element may be defined as
• we need to remember only !
dS
d
nadSdS
zz adSadydxdS
EEE 340 Lecture 04 8
Cylindrical coordinates
dzrdrddzdddv Differential volume
zr adrdradzdradzdrdS ;;
Differential normal area
Differential displacement
zr adzadradrd
EEE 340 Lecture 04 9
Coordinate transforms
Example 2-11. Convert a vector in spherical coordinates (SPC)
into the Cartesian coordinates (CRT).
Solution. The general form of a vector in the CRT is
We need
In fact
AaAaAaA RR
zzyyxx AaAaAaA
xxxRRx
xx
aaAaaAaaAA
aAA
EEE 340 Lecture 04 10
The other eight dot-products can be worked out.
A faster and better way to represent the transformation is based on the del operator.
222
22222
22
cossin
zyx
x
yx
x
zyx
yx
aa xR
)722(
EEE 340 Lecture 04 11
Example 2-12
Sphare chell
ra=2 cm
rb=5 cm
The charge density
Find the total charge Q
2 cos103 24
8
mC
v R
EEE 340 Lecture 04 12
• Solution:
dvQ ρ θρθφππ
sinrdrdd2
0 0
r
r
2b
a
302103 8 π
C108.1 6 π
b
a
r
r
drr
ddQ2
0
2
0
28 1sincos103
EEE 340 Lecture 04 13
2-5 Integrals Containing Vector Functions
.
Scalar
Vector
integral Volume
integral Surface
integral Line
)812(
)802(
)792(
)782(
sdA
ldF
ldV
dvF
S
C
C
V
EEE 340 Lecture 04 14
The line integral around a close path C is denoted as
In the Cartesian coordinates (CRT)
Cz
C Cyx
C zyx
dzzyxVa
dyzyxVadxzyxVa
dzadyadxazyxVlVd
),,(
),,(),,(
),,(
C lVd
)822(
)832(
EEE 340 Lecture 04 15
• Example 2-13
• a) along the straight line OP, where P(1,1,0)
rdrP
O
2
P(1,1,0)y
x0
P1
P2
EEE 340 Lecture 04 16
b). Along path OP1P
Solution. Using (2-52) of cylindrical
a).
)(3
2
)45sin45cos(3
22
3
23
2 22
yx
yx
r
Or
P
O
aa
aa
a
drrardr
