1

April 14 Triple product

6.3 Triple products

Triple scalar product:

Chapter 6 Vector Analysis

zyx

zyx

zyx

zyx

zyxzyx

CCC

BBB

AAA

CCC

BBBAAA

kji

CBA ),,()(

CBA

CBA

and ,by formed ipedparallelep theof Volume

cossin|)(|

ABC

)()()(

)()()(

ABCCABBCA

BACACBCBA

A

BC

+_

2

Triple vector product:

),0,0(

0

00

),,(A ),0,,( ),0,0,(

thatso system coordinate Choosing

.??)( figure, theFrom

yx

yx

x

zyxyxx

CB

CC

B

AAACCB

kji

CB

CB

CBCBA

B)C(AC)B(A

B)C(AC)B(A

kji

CBA

)0,,()0,0,)((

)0,,(

00

)(

yxxxxyyxx

yxxyxy

yx

zyx

CCBABCACA

CBACBA

CB

AAA

C

B

B×C A

)( CBA x

yz

This is called the “bac-cab” rule.

3

Example p282.1.

Problems 3.12.

)( Torque Frn

)()()( DCBADCBA

4

Read: Chapter 6:3Homework: 6.3.9,12,19.Due: April 25

5

April 18 Gradient

6.4 Differentiation of vectorsDerivative of a vector: . , kji

AkjiA

dt

dA

dt

dA

dt

dA

dt

dAAA zyx

zyx

Derivative of vector products:

dt

d

dt

d

dt

ddt

d

dt

d

dt

ddt

da

dt

daa

dt

d

BAB

ABA

BAB

ABA

AAA

)(

)(

)(

Derivative in polar coordinates:

jie

jie

cos sin

sin cos

r

eee

ee

eA

eeA

ejie

ejie

dt

dA

dt

dA

dt

dA

dt

dA

dt

dA

dt

dA

dt

dA

dt

dA

dt

d

AAdt

d

dt

d

dt

d

dt

ddt

d

dt

d

dt

d

dt

d

rrrr

rrr

rr

r

r

sincos

cossin

x

yree

r

A

6

6.6 Directional derivative; Gradient

Directional derivative: The changing rate of a field along a certain direction.

u

urr

c

zb

ya

xds

dz

zds

dy

yds

dx

xds

zyxd

cszz

bsyy

asxx

s

),,(

derivative lDirectiona

:Line

0

0

0

0

r0

rus

Gradient: kjizyx

Examples p292.1,3.

cos|| u

ds

d

1. The gradient is in the direction along which the field increases the fastest.

2. The gradient is perpendicular to the equipotential surface =constant.

u

u ds

d

0const, ds

d

7

Read: Chapter 6:4-6Homework:6.4.2,8;6.6.1,6,7,9.Due: April 25

8

zyx

kji

April 21 Divergence and curl

6.7 Some other expressions involving

The operator:

Vector function: kjiV ),,(),,(),,(),,( zyxVzyxVzyxVzyx zyx

z

V

y

V

x

V zyx

VDivergence:

Curl: kji

kji

V

y

V

x

V

x

V

z

V

z

V

y

V

VVVzyx

xyzxyz

zyx

Laplacian:2

2

2

2

2

22

zyx

Vector identities involving : p339.

VVV

3

1i i

ii

iii

i x

VV

xV

x

Examples p297.1,2.

9

Physical meaning of divergence:Let V be the flux density (particles across a unit area in a unit time).

dxdyVdxdzVdydzVdxdydz

dydzVdydzVVdxdydzx

V

dxdydzz

V

y

V

x

Vdxdydz

zyx

xxxdxxxx

zyx

V

V

00

V is the net rate of outflow flux per unit volume.

x0 x0+dx

Physical meaning of curl:Set the coordinate system so that z is along direction at the point (x0, y0, z0 ).V

dxVdyV

dxVVdyVV

dxdyy

V

x

Vdxdyd

xy

yxdyyxxydxxy

xyz

0000

)( VnV

V is the total circulation of V per unit area.

10

Read: Chapter 6: 7Homework: 6.7.6,7,9,13,18.Due: May 2

11

April 23 Line integrals

6.8 Line integrals

Line integral:

Circulation:

B

AdW lF

lF d A

B

Examples p300.1,2

Conservative field: A field is said to be conservative if does not depend on the path in the calculation.

B

AdW lF

Theorem: If F and its first partial derivatives are continuous in a simply connected region, then the following five statements are equivalent to each other.

1)

2) does not depend on the path.

3) is an exact differential.

4)

5)

.0 lF d

B

AdW lF

.0 F

dWd rF

.WF

12

00 :)1()5(

0

pathon dependnot does )(0

:)5()4()3()2()1(

lFF

kji

F

FrFrFlF

d

z

W

y

W

x

Wzyx

W

WddWdPWdP

O

Examples p304.3,5.

zxzyxWyxhzxzzxgyxzyfxzyx

yxhdzFzxgdyFzyfdxFW

zyxyx,x,,,

zyx

323232 ),(),(),(

),(),(),(or

),,()0,,()00()000(

Examples p306.6.

13

Read: Chapter 6: 8Homework: 6.8.9,15,17.Due: May 2

14

April 25 Green’s theorem

6.9 Green’s theorem in the plane

Double integral in an area:

d

c Clr

x

xA

d

c

b

a Clu

y

yA

b

a

QdydyyxQyxQ

dxx

yxQdydxdy

x

yxQ

PdxdxyxPyxP

dyy

yxPdxdxdy

y

yxP

r

l

u

l

),(),(

),(),(

),(),(

),(),(

Green’s theorem in the plane:

CA

QdyPdxdxdyy

P

x

Q

A double integral over an area may be evaluated by a line integral along the boundary of the area, and vice versa.

Examples p311.1,2.

15

Examples p312.3: Divergence theorem in two dimension:

AA

C

C xy

A

yx

CA

yx

dsdxdy

dxdy

dyVdxVdxdyy

V

x

V

QdyPdxdxdyy

P

x

Q

PQVV

nVV

jiV

jijiV

)(

Examples p312.4: Stokes’ theorem in two dimension:

AA

C yx

A

xy

CA

yx

ddxdy

dyVdxVdxdyy

V

x

VQdyPdxdxdy

y

P

x

Q

QPVV

rVkV

jijiV

)(

16

Read: Chapter 6:9Homework: 6.9.2,3,8,10.Due: May 2

17

April 28 Gauss’ theorem

6.10 The divergence and the divergence theoremLetbe the density, v be the velocity of water. The water flow in a unit time through a unit area that is perpendicular to v is given by V=v, which is called flux density. The water flow rate through a surface with unit normal n is given by V·n.

Physical meaning of divergence:Let V be the flux density.

dxdyVdxdzVdydzVdxdydz

dydzVdydzVVdxdydzx

V

dxdydzz

V

y

V

x

Vdxdydz

zyx

xxxdxxxx

zyx

V

V

00

V is the net rate of outflow flux per unit volume.

x0 x0+dx

ddd

dnVV

1lim

0

Proof: For a differential cube,

Sum over all differential cubes, at all interior surfaces will cancel, only the contributions from the exterior surfaces remain.

18

Time rate of the increase of mass per unit volume: density source ,

Vt

Equation of continuity (when there is no sources): 0

t

V

The divergence theorem (Gauss’ theorem):(Over a simply connected region.)

dσd nVV

.

surfaces 6

0

0

0

0

0

0

dσd

dσ

dxdyVdxdzVdydzV

dxdydzz

V

y

V

x

Vdxdydz

dzz

zz

dyy

yy

dxx

xx

zyx

nVV

nV

V

dσnV

Examples p319.

A volume integral may be evaluated by a closed surface integral on its boundary, and vice versa.

19

Gauss’ law of electric field:

For a single charge,

For any charge distribution,

From the divergence theorem,

. ,

ddσqdσi

i nDnD

.4

22

qdrr

qdσ

nD

.

DDnD dddσ

20

Read: Chapter 6:10Homework: 6.10.3,6,7,9.Due: May 8

21

April 30 Stokes’ theorem

6.11 The curl and Stokes’ theorem

Physical meaning of curl:Set the coordinate system so that z is along at the point (x0, y0, z0 ).V

dxVdyV

dxVVdyVV

dxdyy

V

x

Vdxdyd

xy

yxdyyxxydxxy

xyz

0000

)( VnV

V is the total circulation of V per unit area.

ddd

drVnV

1lim)(

0

22

Stokes’ theorem:(Over a simply connected region. The surface does not need to be flat.)

Proof: Set the coordinate system so that x is along ndat an arbitrarily chosen point on the surface. Suppose the coordinates of that point is (x0, y0, z0 ).

Sum over all differential squares, at all interior lines will cancel, only the contributions from the exterior lines remain.

)( rVnV ddσ

sides 4

)(0

0

0

0rVVnV ddyVdzVdydz

z

V

y

Vdydzdσ

dzz

zy

dyy

yzyz

x

A surface integral may be evaluated by a closed line integral at its boundary, and vice versa.

rV d

Examples p328.1.

boundary. same thehave and if )()( 2121 σσσVσV dd Corollary:

23

Ampere’s law:

JHnHrH

nJrH

dd

dId

C

C

24

Read: Chapter 6:11Homework: 6.11.6,10,12,14.Due: May 8