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