Dr. Hugh Blanton ENTC 3331 Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 2 Fields and Waves VECTORS and VECTOR CALCULUS.

Dr. Hugh Blanton

ENTC 3331

Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 2

Fields and Waves

VECTORS and VECTOR CALCULUS


VECTORS

Today’s Class will focus on:

• vectors - description in 3 coordinate systems

• vector operations - DOT & CROSS PRODUCT

• vector calculus - AREA and VOLUME INTEGRALS


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 CoordinatesP (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


-Parabolic Cylindrical Coordinates (u,v,z)-Paraboloidal Coordinates (u, v, Φ)-Elliptic Cylindrical Coordinates (u, v, z)-Prolate Spheroidal Coordinates (ξ, η, φ)-Oblate Spheroidal Coordinates (ξ, η, φ)-Bipolar Coordinates (u,v,z)-Toroidal Coordinates (u, v, Φ)-Conical Coordinates (λ, μ, ν)-Confocal Ellipsoidal Coordinate (λ, μ, ν)-Confocal Paraboloidal Coordinate (λ, μ, ν)


Parabolic Cylindrical CoordinatesParabolic Cylindrical Coordinates


Paraboloidal CoordinatesParaboloidal Coordinates


Elliptic Cylindrical CoordinatesElliptic Cylindrical Coordinates


Prolate Spheroidal CoordinatesProlate Spheroidal Coordinates


Oblate Spheroidal CoordinatesOblate Spheroidal Coordinates


Bipolar CoordinatesBipolar Coordinates


Toroidal CoordinatesToroidal Coordinates


Conical CoordinatesConical Coordinates


Confocal Ellipsoidal CoordinateConfocal Ellipsoidal Coordinate


Confocal Paraboloidal CoordinateConfocal Paraboloidal Coordinate


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)


VECTOR NOTATION

VECTOR NOTATION:

zzyyxx aAaAaAA ˆˆˆ Rectangular or

Cartesian Coordinate

System

x

z

y

zzyyxx BABABABA

Dot Product

zyx

zyx

zyx

BBB

AAA

aaa

BA

ˆˆˆ

Cross Product

2

1222zyx AAAA

Magnitude of vector

(SCALAR)

(VECTOR)


Cartesian Coordinates

zyx AzAyAxA ˆˆˆ

Page 109

x

y

z

Z plane

y planex plane

222zyx AAAAAA

xyz

x1

y1

z1

Ax

Ay

Az

( x, y, z)Vector representation

Magnitude of A

Position vector A

),,( 111 zyxA

111 ˆˆˆ zzyyxx

Base vector properties

0ˆˆˆˆˆˆ

1ˆˆˆˆˆˆ

xzzyyx

zzyyxx

yxz

xzy

zyx

ˆˆˆ

ˆˆˆ

ˆˆˆ


x

y

z

Ax

Ay

AzA

B

Dot product:

zzyyxx BABABABA

Cross product:

zyx

zyx

BBB

AAA

zyx

BA

ˆˆˆ

Back


Page 108


VECTOR REPRESENTATION: CYLINDRICAL COORDINATES

Cylindrical representation uses: r ,, z

zzrr aAaAaAA ˆˆˆ

zzrr BABABABA

UNIT VECTORS:

zr aaa ˆˆˆ

Dot Product(SCALAR)

r

z

P

x

z

y


VECTOR REPRESENTATION: SPHERICAL COORDINATES

r

P

x

z

y

Spherical representation uses: r ,, UNIT VECTORS:

aaar ˆˆˆ

aAaAaAA rr ˆˆˆ

BABABABA rr

Dot Product(SCALAR)


x

z

y

VECTOR REPRESENTATION: UNIT VECTORS

yaxa

za Unit 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

z

P

x

z

y


Cylindrical Coordinate System

za

a

ra

The Unit Vectors imply :

za

Points in the direction of increasing r

Points in the direction of increasing

Points in the direction of increasing z

ra

a



Spherical Coordinate System

r

P

x

z

y

a

a

ra

The Unit Vectors imply :

Points in the direction of increasing r



ra

aa


zr aaa ˆˆˆ aaar ˆˆˆ zyx aaa ˆˆˆ

RECTANGULAR Coordinate Systems

CYLINDRICAL Coordinate Systems

SPHERICAL Coordinate Systems

NOTE THE ORDER!

r,, z r,,

Note: We do not emphasize transformations between coordinate systems


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”



Differential quantities:

Length:

Area:

Volume:

dzzdyydxxld ˆˆˆ

dxdyzsd

dxdzysd

dydzxsd

z

y

x

ˆ

ˆ

ˆ

dxdydzdv

Page 109


METRIC COEFFICIENTS

2. Cylindrical Coordinates:

Distance = r d

x

y

d

r

Differential Distances:

( dr, rd, dz )


3. Spherical Coordinates:

Distance = r sin d

x

y

d

r sin

Differential Distances:

( dr, rd, r sind )

r

P

x

z

y

METRIC COEFFICIENTS


Representation of differential length dl in coordinate systems:

zyx adzadyadxld ˆˆˆ

zr adzadradrld ˆˆˆ

adrardadrld r ˆsinˆˆ

rectangular

cylindrical

spherical

METRIC COEFFICIENTS


AREA INTEGRALS

• integration over 2 “delta” distances

dx

dy

Example:

x

y

2

6

3 7

AREA = 7

3

6

2

dxdy = 16

Note that: z = constant

In this course, area & surface integrals will be on similar types of surfaces e.g. r =constant or = constant or = constant et c….


Representation of differential surface element:

zadydxsd ˆ

Vector is NORMAL to surface

SURFACE NORMAL


DIFFERENTIALS FOR INTEGRALS

Example of Line differentials

or or

Example of Surface differentials

zadydxsd ˆradzrdsd ˆ

or

Example of Volume differentials dzdydxdv

xadxld ˆ

radrld ˆ

ardld ˆ


BaseVectors

A1

r radial distance in x-y plane

Φ azimuth angle measured from the positive x-axis

Z

r0

20

z


ˆˆˆ

,ˆˆˆ

,ˆˆˆ

rz

rz

zr

zr AzAArAaA ˆˆˆˆ

Pages 109-112Back

( r, θ, z)

Vector representation

222zr AAAAAA

Magnitude of A

Position vector A


11 ˆˆ zzrr


Dot product:

zzrr BABABABA

Cross product:

zr

zr

BBB

AAA

zr

BA

ˆˆˆ

B A

Back


Pages 109-111




Length:

Area:

Volume:

dzzrddrrld ˆˆˆ

rdrdzsd

drdzsd

dzrdrsd

z

r

ˆ

ˆ

ˆ

dzrdrddv

Pages 109-112


ˆˆˆ,ˆˆˆ,ˆˆˆ RRR


Pages 113-115Back

(R, θ, Φ)

AAARA Rˆˆˆ

Vector representation

222 AAAAAA R

Magnitude of A

Position vector A

1ˆRR



Dot product:

BABABABA RR

Cross product:

BBB

AAA

R

BA

R

R

ˆˆˆ

Back

B A


Pages 113-114




Length:

Area:

Volume:

dRRddRR

dldldlRld R

sinˆˆˆ

ˆˆˆ

RdRddldlsd

dRdRdldlsd

ddRRdldlRsd

R

R

R

ˆˆ

sinˆˆ

sinˆˆ 2

ddRdRdv sin2

dRdl

Rddl

dRdlR

sin

Pages 113-115Back


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

Dr. Hugh Blanton ENTC 3331 Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 2 Fields and Waves VECTORS and VECTOR CALCULUS.

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