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Section2.6CylindricalandSpherical

Coordinates

A)ReviewonthePolarCoordinates

µ<2¼:

r

P(x,y)

O X Y

PolarCoord.toCartesianCoord.:

x=rcosµ y=rsinµ

CartesianCoord.toPolarCoord.:

r= p x 2 +y 2 tanµ= y x tangentfunctionis 2 Sotheangleshouldbedeterminedby 8 arctan y x ;ifx>0 arctan y x +¼;ifx<0 2 ;ifx=0;y>0 2 ;ifx=0;y<0 2; 3

Sol.(a)

x=2cos 3 =1; y=2sin 3 p 3: (b) r= p 1+1= p 2 tanµ= ¡1 ¡1 =1=)µ= 4 orµ= 4 5¼ 4 5¼ 4 so p 2; 5¼ 4 isPolarCoord. curvesthatformPolargrid: r=constantcirclecenteredatOwithradiusr

µ=constantraywithangleµ:

2

B)CylindricalCoordinateSystem

ontoxy¡plane: r

P(x,y,z)

O Y Z X

Q(x,y,0)

z 3

Thus,wereadilyhavetheconversionformula:

x=rcosµ y=rsinµ z=z: nates: r 2 =x 2 +y 2

µ=arctan

y x orarctan y x xy¡plane:r=2;µ=2¼=3=120 o :Thenweraiseitupvertically1unit. x=rcosµ=2cos 2¼ 3 =2 1 2 =¡1 y=rsinµ=2sin 2¼ 3 =2 p 3 2 p 3

P(-1, 3 ,1)

Q(-1, 3 ,0)

= 2/3 O Y Z X 4 (b) r= p x 2 +y 2 p 9+9= p 18 tanµ= y x ¡3 3 =¡1;arctan(¡1)=¡ 4 coord(3;¡3;¡7)is p

18;¡¼=4;¡7

asfollows: itssymmetricaxis. r=1 withtheconstantangletozx¡planey=x 5

µ=0;µ=

4 therectangularsystem.

CylindricalGrids

6 thecylindricalcoordinatesystem. z=r: 2 +4y 2 +z 2 =1. 2 +4y 2 +z 2 =1:

Solution:(a)

z=r=)z 2 =r 2 z 2 =x 2 +y 2

Thisaconewithitsaxisonz¡axis:

(b) 4x 2 +4y 2 +z 2 =1=) 4r 2 +z 2 =1 get x 2 +4y 2 +z 2 =1 r 2 cos 2

µ+4r

2 sin 2

µ+z

2 =1 or r 2 +3r 2 sin 2

µ+z

2 =1: i.e., z=rcos' x=rsin' y=y; theequationbetween x 2 +4y 2 +z 2 =1=) 4y 2 +r 2 =1:

C)SphericalCoordinateSystem.

7 aredenedasfollows.

²½=dist(P;O)

z¡axisandcontainingP OP: r

P(x,y,z)

O Y Z X

Q(x,y,0)

z

NotethatwhenPisonz¡axis;Á=0;and

Áincreasesfrom0to

2 asPmovesclosertoxy¡plane; 8 x=rcosµ=½sinÁcosµ y=rsinµ=½sinÁsinµ z=rcotÁ=½cosÁ p x 2 +y 2 +z 2 tanµ= y x cosÁ= z p x 2 +y 2 +z 2 pointisin.Moreprecisely, 8 arctan y x ;ifx>0 arctan y x +¼;ifx<0 2 ;ifx=0;y>0 2 ;ifx=0;y<0

However,Ácanbedetermineduniquelyas

Á=arccos

z p x 2 +y 2 +z 2 thepointwithrectangularcoordinates 0;2 p

3;¡2

(2;¼=4):Wethenrotate reaches¼=3: 9 r = 2 = 2 = /3 Q = /4

P(x,y,z)

O Y Z X x=½sinÁcosµ=2sin 3 cos 4 p 6 2 y=½sinÁsinµ=2sin 3 sin 4 p 6 2 z=½cosÁ=2cos 3 =1: (b) p x 2 +y 2 +z 2 r 2 p 3 2 +4= p 16=4 2 ;since 0;2 p

3;¡2

isonpositivey¡axis cosÁ= z ¡2 4 1 2 2¼ 3

Ans:Sphericalcoord.=

4; 2 2¼ 3

Thecoordinatesurfacesare:

0 0 0 0 10 0 axisandtheopeningangleÁ 0

Infact,ifweconvertÁ=Á

0 intotherectangularcoordinatesystem,we have x=½sinÁ 0 cosµ y=½sinÁ 0 sinµ z=½cosÁ 0

Wecaneliminateµby

x 2 +y 2 2 sin 2 0 z 2 2 cos 2 0 x 2 +y 2 z 2 =tan 2 0 or x 2 tan 2 0 y 2 tan 2 0 z 2 1 whichisacone.

SphericalGrids

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