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LECTURE 1 slide2

Coordinates and Position Vector in RCS

example

1 2 3P x y zOP = + +R a a aJJJG

2 2 1Q x y z OQ = +R a a aJJJG

x

y

z

OPJJJG

Pz

PxP

y

( , , )P P PP x y z

0

POP RJJJG

position coordinates: ( , , )P P Px y zposition vector:

P P x P y P zx y z= + +R a a a

P x P

P y PP z P

xyz

= = =

R aR aR a

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LECTURE 1 slide3

Principal Planes and Principal Lines in RCS

principal planes satisfy:...

x consty constz const

===

principal lines are intersections

of principal planes:., . ( varies)., . ( varies)., . ( varies)

x const y const zy const z const xz const x const y

= == == =

lineli

line 0,

0,ne 0,

0

00

y z

x

y

z

x=

=

==

=

=Where are these lines?

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LECTURE 1 slide4

Principal Lines and Base Unit Vectors in RCS

base unit vectors always right-hand triplets

x y z

y z x

z x y

=

=

=

a a a

a a a

a a a

cyclic substitutions apply x y z x

base unit vectors are along principal lines pointing in the directionof coordinate increase

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LECTURE 1 slide5

Coordinates and Position Vector in CCS

0P P P zz = + +R aa a

x

z( , , )

P P PP z

PP

Pz

position coordinates: ( , , )P P Pz

position vector:1

1

0P

P

P z

= = =

R aR aR a

1 10P zz= + +aR a a

PR

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LECTURE 1 slide7

Coordinates and Position Vector in SCS

P

Pr

P

( , , )P P PP r

x

y

z

Pr

P

P

0 0P P rr = + +aR aa

position coordinates: ( , , )P P Pr

position vector:00

P r P

P

P

r

= = =

R aR aR a

ra

a

a

PR

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LECTURE 1 slide8

(half-plane)

Principal Surfaces and Principal Lines in SCS

Principal Surfaces:

...

r const constconst

==

=

Principal Lines:., . ( varies)., . ( varies)., . ( varies)

r const const const const r const r const

= == == =

line

line

line 1, 9

0

line

, any

145 , 1

c

80 ,

onst.

any cons .0

t

r

r

r

=

= =

==

===

D

D

D

Where are these lines?1

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LECTURE 1 slide9

Principal Lines and Base Unit Vectors in SCS

r

r

r

=

=

=

a a a

a a a

a a a

base unit vectors:

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LECTURE 1 slide11

Q1: In cylindrical CS, the surface = const. is a half-plane.

Q2: In spherical CS, the surface = const. is a sphere.

TRUE OR FALSE?

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LECTURE 1 slide12

cos

sin

x

yz z

=

==

Rectangular Cylindrical

2 2

arctan

x yy

x

z z

= +

=

=

Transformation of Coordinates

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LECTURE 1 slide13

2 2 2

2 2 2arccos

arctan

r x y z

z

x y zy

x

= + +

=

+ +

=

Rectangular Spherical

sin cos

sin sincos

x r

y rz r

=

==

Transformation of Coordinates 2

sin

r

sin sinr

x

y

z

cos

r

sin

cos

r

rP

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LECTURE 1 slide14

Cylindrical Spherical

sin

cos

r

z r

=

==

2 2

2 2arccos

r z

z

z

= +=

= +

Transformation of Coordinates 3

sin

r

cos

r

z

r

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LECTURE 1 slide15

Q1: The point (0,0,0) in a rectangular CS when transformed

into spherical coordinates result in a coordinate r= 1.

Q2: The point (r,0,0), r 0, in spherical coordinates lies on thex axis.

TRUE OR FALSE?

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LECTURE 1 slide16

unit vectors of the rectangular coordinate system same atany point of space

More on Unit Vectors 1

x y0

zayaxa

za

yaxaza

ya

xa

1P

2P

3P

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LECTURE 1 slide17

unit vectors a and a of the CCS NOT the same at differentpoints of space: they depend on


x

y

z

za

1P

2P

a

aza

aa

explain

sin

1

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LECTURE 1 slide18

unit vectors of the SCS NOT the same at different points ofspace: they depend on and


x

y

z

ra

ra

a

aa

a1P

2P

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LECTURE 1 slide19

Vectors in RCS

vector components are projections onto the unit vectors of therespective CS at the given position

vector components in general depend on position

( , , ) ( , , ) ( , , )x x y y z zx y z A x y z A x y z= + +A a a a

Aza

y yA a

0x

x

yy

AA+

a

a

xa

ya z za

x xA a

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LECTURE 1 slide20

Vectors in CCS

a

x

zA

za

a

z

( , , ) ( , , ) ( , , )z zz A z A z = + +A a a a

do not forget that the orientation ofa and a depends on the

position as well

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LECTURE 1 slide21

Vectors in SCS

( , , ) ( , , ) ( , , )r rr A r A r = + +A a a a

do not forget that the orientation ofar, a, and a depends on the

position as well

x

y

z

A

ra

aa

r

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LECTURE 1 slide22

cos sin

sin cosx

y

z z

A A A

A A A

A A

= = +

=cos sin

sin cosx y

x y

z z

A A A

A A A

A A

= += +=

vector components depend on the angular position

RCS CCS

Vector Transformations 1

sin

1

/ , ,x x y y z z zA A = + + A a a a a a a

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LECTURE 1 slide23

RCS SCS

sin cos sin sin cos

cos cos cos sin sin

sin cos

r x y z

x y z

x y

A A A A

A A A A

A A A

= + += +

= +

sin cos cos cos sin

sin sin cos sin cos

cos sin

x r

y r

z r

A A A A

A A A A

A A A

= + = + +

=

vector components depend on the angular coordinates


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LECTURE 1 slide24

CCS SCS


Let us derivesin cos cos cos sin cos sinx r = + = a a a a a a

sin cosr = +a a a

(same result if the ay equation is used)

cos sinz r z = =a a a a

sin cos sin cos cos sincos sin

r r z

z

z r

= + = += = = =

a a a a a aa a a a aa a a a a

sin 0 coscos 0 sin

0 1 0

z

r

a a aaaa

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LECTURE 1 slide25

CCS SCS


sin 0 coscos 0 sin

0 1 0

z

r

a a aaa

a

sin cos

cos sin

r

z r

A A A

A AA A A

= +

=

=

sin cos

cos sin

r z

z

A A A

A A AA A

= +

= =

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LECTURE 1 slide26

A vectorA = 5a exists at the point P( = 1, = 90,

z= 0).

(a) Give the coordinates ofPin RCS.

x = y = z=

(b) What are the components ofA in RCS?

Ax

= Ay

= Az

=

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LECTURE 1 slide27

Summary

we will be using 3 orthogonal coordinate systems: RCS, CCS, SCS

your can easily transform position coordinates and vector

components formula sheet will be provided in exams papers

the rules of vector multiplication are the same in all orthogonal CSs

the choice of the CS depends on the symmetry of the problem

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