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16.360 Lecture 14
• Today
Orthogonal coordinate systems
1. The Cartesian (rectangular) coordinate system2. The cylindrical coordinate system3. The spherical coordinate system
16.360 Lecture 14
• Cartesian coordinate system
x
y
z
dl
xl
yl
zl
^^^^^^
zdzydyxdxzdlydlxdlld zyx
dydzxdldlxsd zyx
^^
dxdzydldlysd zxy
^^
dxdyzdldlzsd yxz
^^
dxdydzdv
16.360 Lecture 14
• Cartesian coordinate system
x
y
z
xs
s
directions of area
16.360 Lecture 14
• Cylindrical coordinate system
z
x
y
),,( 111 zrP
^
r
^
^
z
^^^
zr ^^^
rz ^^^
rz
0^^
rr 1^^
rr
zr AzAArA^^^
222zr AAAA
1R
1
^
1
^
1 zzrrR
16.360 Lecture 14
• the differential areas and volume
^
drdzsd
^
rdzrdsd r
^
zdrrdsd z
z
x
y
dr rd
drdzrddv
16.360 Lecture 14
Example: cylindrical area
z
x
y
5
3
30
60
3/
6/
3
0
60
30
35
ddzrdS
^
rdzrdsd r
16.360 Lecture 14
• Spherical coordinate system
^^^
r^^^
R^^^
R
0^^
rr 1^^
rr
AAARA R
^^^
222 AAAA R
16.360 Lecture 14
• differential volume in Spherical coordinate system
dRRddRR
dldldlRld R
sin^^^
^^^
ddRRdldlRsd R sin2^^
dRdRdldlsd R sin^^
dRdRdldlsd R
^^
ddRdRdldRdldv sin2
16.360 Lecture 14
• Examples
(1) Find the area of the strip
60
30
360
0
2 sin ddRSdS R
(2) A sphere of radius 2cm contains a volume charge density 2cos4v
Find the total charge contained in the sphere
100/2
0
260
0
2180
0
2
100/2
0
2260
0
180
0
2
100/2
0
2260
0
180
0
))(sincos4(
))(sincos4(
)sin(
dddRR
dddRR
RdddRdvQ vv
