CHAPTER 14 Vectors in three space

Team 6:Bhanu Kuncharam

Tony Rocha-ValadezWei Lu

The position vector R from the origin of Cartesian coordinate system to the point (x(t), y(t), z(t)) is given by the expression

k)t(zj)t(yi)t(x)t(R

k)t(zj)t(yi)t(x)t(R)t(v

i

j

A Cartesian coordinate system (by MIT OCW)

 

14.6 Non-Cartesian Coordinates

k)t(zj)t(yi)t(x)t(R)t(a

The vector expression for velocity is given by

The vector expression for acceleration is given by

http://www.wepapers.com/Papers/4521/1_Newton's_Laws,_Cartesian_and_Polar_Coordinates,_Dynamics_of_a_Single_Particle

14.6.1 Plane polar coordinate

To define the Polar Coordinates of a plane we need first to fix a point which will be called the Pole (or the origin) and a half-line starting from the pole. This half-line is called the Polar Axis.

Polar Angles: The Polar Angle θ of a point P, P ≠ pole, is the angle between the Polar Axis and the line connecting the point P to the pole. Positive values of the angle indicate angles measured in the counterclockwise direction from the Polar Axis.

The Polar Coordinates (r,θ) of the point P, P ≠ pole, consist of the distance r of the point P from the Pole and of the Polar Angle θ of the point P. Every (0, θ) represents the pole.

θ

Polar Axis

rP(r, θ)

Definitions:

http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node5.html

Plane polar coordinate

More than one coordinate pair can refer to the same point.

150o

30o210o

2

2,30o

2,210o

2, 150o

All of the polar coordinates of this point are:

2,30 360

2, 150 360 0, 1, 2 ...

o o

o o

n

n n

http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node5.html

Plane polar coordinate

Difference quotient method to get rr e

d

edande

d

edˆ

ˆˆ

ˆ

))((ˆ)( tetrR r

rr ererRtv ˆˆ)(

d

ed

dt

d

d

edte

dt

de rr

rr

ˆˆ))((ˆˆ

)(ˆ)(ˆlim

ˆ0

rrr ee

d

ed

e

e

d

ed r ˆˆ)1(

limˆ

0

erertv r ˆˆ)( ererererer)t(v)t(a rr

What is ? d

ed rˆ

Greenberg, M. D. (1998). Advanced Engineering Mathematics (2nd ed.): Prentice Hall.

Plane polar coordinate

Difference quotient method to get rr e

d

edande

d

edˆ

ˆˆ

ˆ

What is ?

d

ed ˆ

d

ed

dt

d

d

edte

dt

de

ˆˆ)(ˆˆ

)(ˆ)(ˆlim

ˆ0

ee

d

ed

rr ee

d

edˆ

)ˆ)(1(lim

ˆ0

ree ˆˆ

errerrta r ˆ)2(ˆ)()( 2 Greenberg, M. D. (1998). Advanced Engineering Mathematics (2nd ed.): Prentice Hall.

Transform method to get rr e

d

edande

d

edˆ

ˆˆ

ˆ

Plane polar coordinate

jid

ed

jid

ed

jie

jie

r

r

ˆsinˆcosˆ

ˆcosˆsinˆ

ˆcosˆsinˆ

ˆsinˆcosˆ

eej

eei

r

r

ˆcosˆsin

ˆsinˆcos

r22

rr

22rr

r

e)sin(cos)e cose (sinsin)e sine (coscosd

ed

ee)cos(sin)e cose (sincos)e sine (cossind

ed

e)r2r(e)rr()t(a

ererR)t(v

))t((e)t(rR

r2

r

r

x

y

yxr

ry

rx

1

22

tan

sin

cos

ktzjtyitxtRtvta

ktzjtyitxtRtv

ktzjtyitxtR

ˆ)(ˆ)(ˆ)()()()(

ˆ)(ˆ)(ˆ)()()(

ˆ)(ˆ)(ˆ)()(

e re

A polar coordinate system (by MIT OCW)

 

rr

rr

r

erererererdt

dva

ererdt

edre

dt

dr

dt

dRv

ˆˆˆˆˆ

ˆˆˆ

ˆ

2

The expressions of R, v, a in polar coordinates

http://www.wepapers.com/Papers/4521/1_Newton's_Laws,_Cartesian_and_Polar_Coordinates,_Dynamics_of_a_Single_Particle

r

r

(r,,z)

Cylindrical coordinates are a generalization of two-dimensional polar coordinates to three dimensions by superposing a height z axis.

14.6.2 Cylindrical coordinates

A cylindrical coordinate system

 

)2(

http://mathworld.wolfram.com/CylindricalCoordinates.html

Cylindrical coordinates

x

y

yxr

ry

rx

1tan

22

sin

cos

The relations between cylindrical coordinates and Cartesian coordinates.

2 2 2

tan( )

r x y

y

xz z

Definitions:

zr

zr

zr

ezerrerrtRta

ezerertRtv

ezerR

ˆˆ)2(ˆ)()()(

ˆˆˆ)()(

ˆˆ

2

The expressions of position R, velocity v, and acceleration a in Cylindrical coordinates are

given by

Greenberg, M. D. (1998). Advanced Engineering Mathematics (2nd ed.): Prentice Hall.

Cylindrical coordinates

Example1:

Find the cylindrical coordinates of the point whose Cartesian coordinates are (1, 2, 3)

Answer:

3

1071.1

5

z

r

Example2:

Find the Cartesian coordinates of the point whose cylindrical coordinates are (2, Pi/4, 3)

Answer:

3

22

22

z

y

x

http://mathworld.wolfram.com/CylindricalCoordinates.html

14.6.3 Spherical coordinates

(x,y,z) r

z

Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear coordinates that are natural for describing positions on a sphere or spheroid. Define to be the azimuthal angle in the -plane from the x-axis with (denoted when referred to as the longitude), to be the polar angle (also known as the zenith angle and colatitude, with where is the latitude) from the positive z-axis with , and to be distance (radius) from a point to the origin.

Spherical coordinates

The expressions of Spherical coordinates for velocity and acceleration

eeeee

ee

eee

ee

eee

ˆcosˆsinˆ

,0ˆ

,0ˆ

ˆcosˆ

,ˆˆ

,0ˆ

ˆsinˆ

,ˆˆ

,0ˆ

eR ˆ ))(),((ˆˆ ttee

)ˆsinˆ(ˆ

)ˆˆ

(ˆ

eee

dt

de

dt

deeRv

e

eeta

ˆ)cos2sin2sin(

ˆ)cossin2(ˆ)sin()( 2222

The expressions of R, v, a in Spherical coordinates

ktzjtyitxtRtvta

ktzjtyitxtRtv

ktzjtyitxtR

ˆ)(ˆ)(ˆ)()()()(

ˆ)(ˆ)(ˆ)()()(

ˆ)(ˆ)(ˆ)()(

e

eeta

eeetv

eR

ˆ)cos2sin2sin(

ˆ)cossin2(ˆ)sin()(

ˆsinˆˆ)(

ˆ

2222

cos

sinsin

cossin

z

y

x

Figure taken from reference: http://mathworld.wolfram.com/SphericalCoordinates.html

Example 3 Calculate the three components of the position, velocity and

acceleration vectors at t=3. The position of the point R is given by R=(t, exp(t), 3t ). Do this for the in Cartesian coordinates,

Cylindrical coordinates, and Spherical coordinates

Examples: The expressions of R, v, a in Non-Cartesian coordinates

Solution:In Cartesian Coordinates:

ktzjtyitxtR ˆ)(ˆ)(ˆ)()(

ktzjtyitxtRtv ˆ)(ˆ)(ˆ)()()(

ktzjtyitxtRtvta ˆ)(ˆ)(ˆ)()()()( jeta

kjeitv

ktjeittR

t

t

t

ˆ)(

ˆ3ˆˆ)(

ˆ3ˆˆ)(

0,08.20,0

3,08.20,1

9,08.20,3

,ˆ)(,ˆ3ˆˆ)(,ˆ9ˆˆ3)( 333

zyx

zyx

zyx

aaa

vvv

RRR

orjetakjeitvkjeitR

Solution:In Cylindrical Coordinates:

eej

eei

r

r

ˆcosˆsin

ˆsinˆcos

put into

jeta

kjeitv

ktittR

t

t

ˆ)(

ˆ3ˆˆ)(

ˆ3ˆ)(

1468.0sin1cos,989.03

sin 2

62

3

22

e

e

yx

y

0

ˆˆ

R

ezerR zr

)ˆcosˆ(sin)(

ˆ3)ˆcosˆ(sin)ˆsinˆ(cos)(

ˆ3ˆcos)(

eeeta

eeeeeetv

etettR

rt

zrt

r

zr

get

0,948.2,86.19

3,959.1,00.20

9,0,4404.0

zr

zr

zr

aaa

vvv

RRR

The expressions of R, v, a in Non-Cartesian coordinates

In Spherical Coordinates:

eek

eeej

eeei

ˆsinˆcosˆ

ˆcosˆsincosˆsinsinˆ

ˆsinˆcoscosˆcossinˆ

put

jeta

kjeitv

ittR

t

t

ˆ)(

ˆ3ˆˆ)(

ˆ)(

into

Solution: 0.0

ˆ

zRR

eR

get

)ˆcosˆsincosˆsin(sin

)ˆsinˆ(cos

)ˆcosˆsincosˆsin(sin

)ˆsinˆcoscosˆcos(sin

ˆcossin

eeeea

eet

eeee

eeev

etR

t

t

9142.0cos1sin,4051.093

9cos

1468.0sin1cos,989.03

sin

2

262222

2

62

3

22

ezyx

z

e

e

yx

y

045.8,948.2,15.18

361.5,936.3,49.19

0,0,4026.0

aaa

vvv

RRR

The expressions of R, v, a in Non-Cartesian coordinates

Using the omega method derive the space derivatives of base vectors

0

02

tan2

AA

AAAAAA

tconsAAA

Consider a rigid body B undergoing an arbitrary motion through 3-space. And let A be any fixed vector with B, that is, A is a vector from one material point in B to another so is constant with time, because b is rigid. Thus A=A(t)Fixed vector in B

 

There exists a vector such that

AA 1

1

There exists a vector such that 2 BB 1

Greenberg, M. D. (1998). Advanced Engineering Mathematics (2nd ed.): Prentice Hall.

A

14.6.4 Omega Method

Omega method

Since B is arbitrary:

0)( 21 ASince A is arbitrary:

021

0)(

0

0)()(

0

cos

21

21

21

BASo

BABA

BABA

BABA

BABA

So we get AA

Omega Method

Omega method

In cylindrical coordinates: ze

Let A be :re eeeedt

edrzr

r ˆˆˆˆˆ

Using chain differentiation to write:

z

ez

e

r

er

dt

dz

z

e

dt

de

dt

dr

r

etzttre

dt

d

rrr

rrrr

ˆˆˆ

ˆˆˆ))(),(),((ˆ

z

ez

e

r

erzer rrr

ˆˆˆ

0ˆ0

0ˆ

,ˆˆ

,0ˆ

z

ee

e

r

e rrr

Similarly, let A be :e 0ˆ

,ˆˆ

,0ˆ

z

ee

e

r

er

Let A be :ze 0ˆ

,0ˆ

,0ˆ

z

ee

r

e zzz

Omega Method

End of Chapter 14