Coordinate System

VECTOR REPRESENTATION

3 PRIMARY COORDINATE SYSTEMS:

• RECTANGULAR

• CYLINDRICAL

• SPHERICAL

Choice is based on symmetry of problem

Examples:

Sheets - RECTANGULAR

Wires/Cables - CYLINDRICAL

Spheres - SPHERICAL

Orthogonal Coordinate Systems: (coordinates mutually perpendicular)

Spherical Coordinates

Cylindrical Coordinates

Cartesian Coordinates

P (x,y,z)

P (r, Θ, Φ)

P (r, Θ, z)

x

y

zP(x,y,z)

θ

z

rx y

z

P(r, θ, z)

θ

Φ

r

z

yx

P(r, θ, Φ)

Page 108

Rectangular Coordinates

Cartesian CoordinatesP(x,y,z)

Spherical CoordinatesP(r, θ, Φ)

Cylindrical CoordinatesP(r, θ, z)

x

y

zP(x,y,z)

θ

z

rx y

z

P(r, θ, z)

θ

Φ

r

z

yx

P(r, θ, Φ)

Coordinate Transformation

• Cartesian to Cylindrical(x, y, z) to (r,θ,Φ)

(r,θ,Φ) to (x, y, z)

• Cartesian to CylindricalVectoral Transformation

Coordinate Transformation

Coordinate Transformation

• Cartesian to Spherical(x, y, z) to (r,θ,Φ)

(r,θ,Φ) to (x, y, z)

• Cartesian to Spherical Vectoral Transformation

Coordinate Transformation

Page 109

x

y

z

Z plane

y planex plane

xyz

x1

y1

z1

Ax

Ay

Unit vector properties

0ˆˆˆˆˆˆ

1ˆˆˆˆˆˆ

xzzyyx

zzyyxx

yxz

xzy

zyx

ˆˆˆ

ˆˆˆ

ˆˆˆ

Vector Representation

Unit (Base) vectors

A unit vector aA along A is a vector whose magnitude is unity

A

Aa

zyx AzAyAxA ˆˆˆ

Page 109

x

y

z

Z plane

y planex plane

222zyx AAAAAA

xyz

x1

y1

z1

Ax

Ay

Az

Vector representation

Magnitude of A

Position vector A

),,( 111 zyxA

111 ˆˆˆ zzyyxx

Vector Representation

x

y

z

Ax

Ay

AzA

B

Dot product:

zzyyxx BABABABA

Cross product:

zyx

zyx

BBB

AAA

zyx

BA

ˆˆˆ

Back

Cartesian Coordinates

Page 108

x

z

y

VECTOR REPRESENTATION: UNIT VECTORS

yaxa

zaUnit Vector

Representation for Rectangular

Coordinate System

xaThe Unit Vectors imply :

ya

za

Points in the direction of increasing x

Points in the direction of increasing y

Points in the direction of increasing z

Rectangular Coordinate System

r

f

z

P

x

z

y

VECTOR REPRESENTATION: UNIT VECTORS

Cylindrical Coordinate System

za

a

ra

The Unit Vectors imply :

za

Points in the direction of increasing r

Points in the direction of increasing j

Points in the direction of increasing z

ra

a

BaseVectors

A1

ρ radial distance in x-y plane

Φ azimuth angle measured from the positive x-axis

Z

r0

20

z

Cylindrical Coordinates

ˆˆˆ

,ˆˆˆ

,ˆˆˆ

z

z

z

zAzAAAaA ˆˆˆˆ

Pages 109-112Back

( ρ, Φ, z)

Vector representation

222zAAAAAA

Magnitude of A

Position vector A

Base vector properties

11 ˆˆ zz

Dot product:

zzrr BABABABA

Cross product:

zr

zr

BBB

AAA

zr

BA

ˆˆˆ

B A

Back

Cylindrical Coordinates

Pages 109-111

VECTOR REPRESENTATION: UNIT VECTORS

Spherical Coordinate System

r

f

P

x

z

y

q

a

a

ra

The Unit Vectors imply :

Points in the direction of increasing r

Points in the direction of increasing j

Points in the direction of increasing q

ra

aa

ˆˆˆ,ˆˆˆ,ˆˆˆ RRR

Spherical Coordinates

Pages 113-115Back

(R, θ, Φ)

AAARA Rˆˆˆ

Vector representation

222 AAAAAA R

Magnitude of A

Position vector A

1ˆRR

Base vector properties

Dot product:

BABABABA RR

Cross product:

BBB

AAA

R

BA

R

R

ˆˆˆ

Back

B A

Spherical Coordinates

Pages 113-114

zr aaa ˆˆˆ aaar ˆˆˆ zyx aaa ˆˆˆ

RECTANGULAR Coordinate Systems

CYLINDRICAL Coordinate Systems

SPHERICAL Coordinate Systems

NOTE THE ORDER!

r,f, z r, q ,f

Note: We do not emphasize transformations between coordinate systems

VECTOR REPRESENTATION: UNIT VECTORS

Summary

METRIC COEFFICIENTS

1. Rectangular Coordinates:

When you move a small amount in x-direction, the distance is dx

In a similar fashion, you generate dy and dz

Unit is in “meters”

zz

yx

yxr

ˆˆ

cosˆsinˆˆ

sinˆcosˆˆ

zz

yx

yxr

AA

AAA

AAA

cossin

sincos

Back

Cartesian to Cylindrical Transformation

zz

xy

yxr

)/(tan 1

22

Page 115