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Chapter 1.Vector analysis

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Mathematics -Glossary• Scalar : a quantity defined by one number (eg. Temperature, mass, density, voltage, ... )

• Vector : a quantity defined by a set of numbers. It can be represented by a magnitude and a direction. (velocity, acceleration, …)

• Field : a scalar or vector as a function of a position in the space. (scalar field, vector field, …)

Scalar field Vector field

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Scalar field Vector field

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Vector algebra

BABA

BABABAs

CBACBA

ABBA

ssrr

srr

)()())((

)()(

• Unit vector : ‘^’ means the unit vector.

A

AAAAA

ˆ,,ˆA A

•Components of a vector

zyx

Azyx

ˆA,ˆA,ˆA

AAA,ˆAˆAˆA

zyx

2z

2y

2xzyx

AAA

A

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5Products of vectors

• scalar product : the result of a scalar product is a scalar.

ABBA

BABA

ABcos

A

B

• vector product : the result of a vector product is a vector which is normal to the original vectors.

ABBA

BAnBA

ABsinˆ

x

yz

The magnitude of C is the area of a parallelogram made by two vectors A and B.

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Example of a scalar product

gm

sinmg

hs

gF

sF

m

dWs

0

mgh

mgs

dsmgdmWss

sin

sin00

sg

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yz

x

ˆˆ

ˆ

)(ˆ)(ˆ)(ˆ

ˆˆˆ

ˆˆˆ

xyyxzxxzzyzy

yx

yx

zx

zx

zy

zy

zyx

zyx

BABABABABABA

BB

AA

BB

AA

BB

AA

BBB

AAA

zyx

zyx

zyx

BA

Calculation of vector products

yz

x

ˆˆ

ˆ (+)

(+)

(+)

(-)

(-)(-)

(+) (-) (+)

• If rotated in the direction of x,y,z, the sign of the result is (+). If rotated in the opposite direction, the sign is (-).

• The same rule applied to other coordinate systems with change of unit vectors.

ˆˆ

r

ˆˆ

ˆ

z

(+)

(+)

(+)

(-)

(-)(-)

ˆˆ

r

ˆˆ

ˆ

z

(+)

(+)

(+)

(-)

(-)(-)

yzxzxyzyx ˆˆˆ,ˆˆˆ,ˆˆˆ

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8Example of vector product

ro

F

Frτ

torque

FrT

F

Angular acceleration is proportional to the applied torque.

Torque is proportional to the product of radius and force.

If the sum of torques due to A and B has nonzero value, the seesaw is rotated.

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9Vector identity

CBACBA

)()(

zyx

zyx

zyx

CCC

BBB

AAA

A

B

C • The order of operators and can be inter-

changed.

• The scalar triple product is equal to the volume of the parallelogram defined by the three vectors.

• Scalar triple product

The result is the same despite the change of the order of • and x interchanged.

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10Quantities represented by vectors

• Position vector : a position can be represented by a vector whose origin is specified by O.

• Field vector : A vector that represents physical quantities at the position specified by a position vector.

Position vec-tor

Field vector

Origin

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O

Origin

Position vector

Example : Velocity

Velocity vector

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Coordinate system

• Position vector : A position can be specified with a vector whose origin is at a point O.

R

R

O

Observation position is writ-ten by R.

Source position is repre-sented by a primed vec-tor.

1. A coordinate system is only an appara-tus that describes positions and physi-cal phenomena.

2. Physical laws are independent of co-ordinate systems adopted by any ob-servers.

R

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We can choose a set of unit vectors which fits a specific problem.

Some orthogonal coordinate systems and their unit vectors

y

x

z

y

x

z

ρ

rθ

x

y

constant unit vectors

The directions of two unit vectors change with ob-servation positions.

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),,( zyxR

O

x

y

z),,( zR

O

x

y

z

ρ

zz ˆ

zz ˆˆ R

zzyyxx ˆˆˆ R

),,( rR

O

x

y

z

rR ˆr

Position vectors

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z

1ρ

1

z

2ρ 2

For cylindrical coordinate systems, unit vectors ρ and φ are functions of an observer’s position.

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r

θ

2r

2θ

2

For spherical coordinate systems, unit vectors r, θ, and φ are functions of a position.

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•Circular cylindrical coordinate

sin

cos

tan

ˆˆˆ

ˆˆ

1

22

y

x

x

y

yx

z

zzR

Cylindrical coordinate

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Example D1.5.

Represent the position D(x= -3.1, y= 2.6, z= -3) using circular cylindrical coordinate system.

z

z

x

y

yx

ˆ3ˆ05.4

3

140tan

05.4

1

22

R

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Example

)3,2,1(P

zyxr ˆ3ˆ2ˆ P

zyxE ˆ1ˆ2ˆ3)( P

zzE)EρρEE ˆ)ˆ(ˆˆ(ˆ)ˆ()( P

5

ˆ2ˆˆˆˆ,

5

ˆ2ˆˆ

xyρzφ

yxρ

ρzzzrr

zzrrρ ˆˆˆ,

ˆ)ˆ(

ˆ)ˆ(ˆ

zρzρr ˆˆˆˆ 22 zyxzP

zρE ˆˆ5

4ˆ

5

7)( P

For an observation point P(1,2,3) in rectangular coordinate, express the posi-tion vector and E-field vector using unit vectors of cylindrical coordinate.

(1) Position vector

(2) Field vector

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•Spherical coordinate

cos

sinsin

cossin

tan

cos

ˆˆˆ

ˆ

1

222

1

222

rz

ry

rx

x

y

zyx

z

zyxr

r

rr

R

Spherical coordinate

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rθrz

rz

r

rr ˆˆˆ,

ˆˆ

ˆˆˆ,ˆ

Example

)3,2,1(P

zyxr ˆ3ˆ2ˆ P

zyxE ˆ1ˆ2ˆ3)( P

ˆˆ(ˆ)ˆ(ˆ)ˆ()( )EθθErrEE P

5

ˆ5ˆ6ˆ3ˆˆˆ

5

ˆˆ2

145

14

ˆ2ˆˆˆ

ˆˆˆ,

14

ˆ3ˆ2ˆˆ

zyxrφθ

yxxy

rz

rzφ

zyxr

rrrr ˆ14ˆ941ˆ rP

For an observation point P(1,2,3) in rectangular coordinate, express the posi-tion vector and E-field vector using unit vectors of spherical coordinate.

(1) Position vector

(2) Field vector

14

4ˆ14

16ˆ

14

10)( θrE P

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O

Origin

Position vector changed

Displacement vector

Displacement vector

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24Rectangular coordinate

y

x

)(tr

zyxr ˆˆˆ zyx

zyxr ˆ)(ˆ)(ˆ)()( tztytxt

zyxr ˆˆˆ dzdydxd

If the position can be described by line segments, rectangular coordinate systems are convenient.

Unit vectors are parallel to x, y, z axes. Their directions are fixed. (The unit vectors are constant ones.)

0 xdx

Position vector changed

Displacement vector

O

Origin

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Cylindrical coordinate

zρr ˆˆ z

zρρr ˆˆˆ dzddd

•The direction of ρ is defined as that points away from z-axis.

•Φ-direction is orthogonal to ρ.

y

x

ρr

y

x

z

ρ

φ

rzz

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y

x

ρr

φ

zρr ˆˆ zy

x

)(ˆ ρ

φ

d

)(ˆ dρ

φρρρ ˆ)(ˆ)(ˆˆ ddd

?ˆ ρd

cosˆsinˆˆ

ˆ

sinˆcosˆˆ

yxρ

φ

yxρ

zφρr ˆˆˆ dzddd

Unit circle : the magnitude of unit vectors are unity.

Unit circle : the magnitude of unit vectors are unity.

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27Spherical coordinate

rθ

x

y

R : points away from the origin.

rr ˆr

drdrdr

ddrdr

drdrrdd

sinˆˆˆ

ˆˆˆ

ˆˆ)ˆ(

φθr

rrr

rrrr

zθ : longitudinal direction

Φ: latitude di-rection

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z

x

d

),(ˆ r

θrrr ˆ),(ˆ),(ˆˆ dd

?ˆ

dd r

),(ˆ dr

y

)(ˆ r

φ

d

)(ˆ dr

φrrr ˆsin),(ˆ),(ˆˆ dd

x

sin :radius

drdrdrd sinˆˆˆ φθrr

θ

Unit circle : the magnitude of unit vectors are unity.

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29Line integrals

y

x

)(tr

s

dW0

rF

F

Line integral

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y

x

)(tr

)(rB

y

x

S danrB ˆ)(

Normal vector to the surface

n

: a unit vector normal to the inte-gration surface

Surface integral

z

dxx

dyy

dxdydydxda zyxn ˆˆˆˆ

daBda nrB ˆ)(cos

n)(rB

n

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dz

dy

dx

The coordinate of the center of the cube is (x,y,z)

y

z

x

n

Volume integral

differential volume in rectangular co-ordinate

VVtotal dxdydzfdfm )()( rr

dxdydzd

Density

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zyx

zzyyxx

ˆˆˆ

ˆˆˆ r •To represent arbitrary physical vector quantities, the number of unit vectors in a rectangular coordinate is three.

•The unit vectors are parallel to x, y, z axes and the directions are constant.

Rectangular coordinate

dzzdyydxx

zzyyxxddd

dC

ˆˆˆ

)ˆˆˆ()(

)(

rr

rrf

dxdyzdzdxydydzxd

dS

ˆˆˆ

)(

a

arf

dxdydzd

dV

)(rf

•Line integral

•Surface integral

•Volume integral

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•Line integral

C zyxC

dzAdyAdxAdsA

2

2

0,2 zxy

222 ˆˆˆ zzyyxx A

3

228

222

1 2

0

222

2

0

22

0

22

dxx

xdxx

dxxx

xdxx

dxdx

dyxdxxdyydxxdzAdyAdxA

C

zyx

Example

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Cylindrical coordinate1. The unit vectors for a cylindrical coordi-

nate are rho, phi, z.

2. The unit vector z is constant vector.

3. The directions of rho and phi are chang-ing with positions.

z

zz

ˆˆˆ

ˆˆ

r

dzzdd

dzzdd

zzddd

dC

ˆˆˆ

ˆˆˆ

)ˆˆ()(

)(

rr

rrf

ddzddzdzdd

dS

ˆˆˆ

)(

a

arf

dzddd

dV

)(Rf

•Line integral

•Surface integral

•Volume integral

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2cosˆ)4sin(3ˆ F

3

3 dzzddd ˆˆˆ r

4

3)2cos1(

2

3)cos3(

cos2/

0

2/

0

22/

0

2

ddd

dd

rF

rF

Example Evaluate a line integral which has an integrand of F∙dr.

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Spherical coordinate

ˆˆˆ

ˆ

r

rrr •Unit vectors for a spherical coordinate are r, theta, phi.

•The directions of them all change with positions.

drdrdrr

dd

rdd

d

rdrdrr

rdrdrrrrddd

dC

sinˆˆˆ

ˆˆˆ

ˆˆ)ˆ()(

)(

rr

rrf

ddrrddrrddrrd

dS

ˆsinˆsinˆ

)(

2

a

arf

dddrrd

dV

sin

)(

2

rf

•Line in-tegral

•Surface integral

•Volume integral

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22 sinˆrrF

3

sin3

0

2

0

22

adrrd

drrd

aa

rF

rF

Example

x

y

zθ : longitudi-nal direction

drdrdrrd sinˆˆˆ ra

dadrdr

ar

sincossin

cossin

d

a

rdrrd

sin

sinˆˆ r

cossin

0cossinsin

0)2/(

)2/()2/(

)2/()2/(

22

222

2222

222

ar

arr

yaaxx

ayaaxx

ayax

a

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Summary

•The unit vectors for a rectangular coordinate system are constant. For spherical or cylindrical coordinates, the direction of unit vectors change with position.

•Cylindrical coordinate •Spherical coordinate

zyx

zzyyxx

ˆˆˆ

ˆˆˆ r

sin

cos

tan

ˆˆˆ

ˆˆ

1

22

y

x

x

y

yx

z

zzr

cos

sinsin

cossin

tan

cos

ˆˆˆ

ˆ

1

222

1

222

rz

ry

rx

x

y

zyx

z

zyxr

r

rr

r

•Rectangular coordinate

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Coordinate transformation

cosˆsinˆˆˆ

sinˆcosˆˆ

yx

yx

z

y

x

z ˆ

ˆ

ˆ

100

0cossin

0sincos

ˆ

ˆˆ

A TA•Cylindrical coordinate

zz

y

x

ˆ

ˆˆ

100

0cossin

0sincos

ˆ

ˆ

ˆ

cosˆsinˆˆ

sin

1ˆ

sinˆsincosˆcoscosˆˆˆ

cosˆsinsinˆcossinˆˆ

yxr

zyxr

zyxr

z

y

xr

ˆ

ˆ

ˆ

0cossin

sinsincoscoscos

cossinsincossin

ˆ

ˆˆ

ˆ

ˆˆ

0sincos

cossincossinsin

sincoscoscossin

ˆ

ˆ

ˆ r

z

y

x

A TA

•Coordinate transformation ma-trix A is an orthogonal matrix.

•If a matrix is orthogonal, A-1 is equal to AT.

•Spherical coordinate

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40Integral of vector function

yxxyx ˆ2ˆ F

3

3

0

2

100

0cossin

0sincos

x

xy

F

F

F

z

dzzddd ˆˆˆ r

2/

0

222/

0 219)cos6cossin9(3

)cos2sin(ˆ)sin2cos(ˆ

dd

xxyxxy

rF

F