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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 +zquotesdbs_dbs20.pdfusesText_26