10.1 differentiation of vector chapter 10 vector calculus

10.1 Differentiation of vector

Chapter 10 Vector Calculus

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)(

point someat continuous is )(

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able.differenti be may

vector basis the general in fixed, are ˆ ,ˆ ,ˆ s,coordinate Cartesian in :Note

ˆˆˆˆˆˆ)()(

ˆˆˆˆˆˆ)(

ˆ)(ˆ)(ˆ)()(

:physics In

2

2

2

2

2

2

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For two-dimensional plane polar coordinates

. and as changes direction

their but magnitudesconstant have ˆ and ˆ (2)

direction and magnitude inconstant are ˆ and ˆ (1)

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ˆˆsinˆcosˆˆcosˆsinˆ

ˆˆcosˆsinˆˆsinˆcosˆ

)2(ˆ)(ˆ

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ˆˆˆ

ˆ)()(

onaccelerati and velocity the find ,ˆ)()(vector position :Ex

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Differentiation of composite vector expressions


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dt

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02)(constant a is if *

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10.2 Integration of vector

)()()(

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2

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||||

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1 rate changing its 2/|| is area in change the

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2

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2

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dt

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dt

d

rrr

GM

dt

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10.3 Space curves

dudu

rd

du

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du

rd

du

rd

du

ds

du

rd

du

rd

du

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dzdydxrdrddskdzjdyidxurd

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2

1

is length arc The

)(

)()()()(ˆˆˆ)(

ˆ)(ˆ)(ˆ)()(

2

2222

Ex: A curve lying in the xy-plane is given by y=y(x), z=0. Evaluate the arc length along the curve between x=a and x=b.

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ˆˆ , 0

ˆˆ1|ˆ|

/1 curvature of radius The

|||ˆ

| torespect with

ˆvector tangent unit the of rate changing the : Curvature (1)

2

2

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torsion the of radius the 1

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Frenet-Serret

formula:n

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ˆ ˆˆˆ

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curvature of radius the : normal principal the :ˆ

trajectory the totangent unit the :ˆ particle the of speed the :

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trajectory a along travelling a of onaccelerati thethat Show :Ex2

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tv

nv

tdt

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dv ˆ

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2

2

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)(

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parameter some is ),( curve aFor

dt

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10.4 Vector functions of several arguments

j

n

j jn

n

n

jn

j j

jj

i

jn

j ji

n

niii

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adu

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12

21

1

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1

1

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1

1

2121

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)( scalars and of functionexplicit an is If (2)

......

),...,( of function a also is ,),.....,( If (1)

2

1

21

curve ofvector tangent the is (b)

curve ofvector tangent the is a)(

curves. coordinate called are and

is )( totangent vector The

))(),(()()( and )(

by drepresente be can surface

the on )( curve any ,parameter aFor

ˆ),(ˆˆ),(),( (4)

0),,( (3)

),( (2)

ˆ),(ˆ),(ˆ),(),( (1)

:is equation surface the coordinate Cartesian In

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10.5 Surface



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r

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u

rdv

v

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r

||||

area Total

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isvector normal a S, surface smooth the on Ppoint aFor

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vectors two the use can we t,independen linearly are and If


Ex: Find the element of area on the surface of a sphere of radius a,

and hence calculate the total surface area of the sphere.

2

0

2

0

222

2

1

4sinsinsin||

0cossinsinsin

sinsincoscoscos

ˆˆˆ

vector Normal

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is curve the totangent vector The (2)

ˆsinˆsincosˆcoscos

is curve the totangent vector The (1)

ˆcosˆsinsinˆcossin),(

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c

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10.7 Vector operator

vectortangent unit the is ˆ ˆ

curve the along length arc the , If

parameter a is ,)( If

)ˆˆˆ()ˆˆˆ(

to from in change The

ˆˆˆ grad),,( fieldscalar aFor

coordinate Cartesian in ˆˆˆ del

ttds

rd

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d

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du

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ddzz

dyy

dxx

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ky

jx

ird

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yj

xizyx

zk

yj

xi

Gradient of a scalar field


||)()0(ˆ||

||||cos||ˆ

is vector particular a in distance

the torespect with of rate changing The

max

ds

da

ds

da

ds

d

as

Ex: For a function at a point (1,2-1), find its rate of change with distance in the direction . At the same point, what is the greatest possible rate of change with distance and in which direction does it occur?

yzyx 2kjia ˆ3ˆ2ˆ

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14

10)64(

14

1ˆ )ˆ3ˆ2ˆ(

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1

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max

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the in distance with field scalar)(or vector a of change of rate The

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point someat surface this tovector tangent a is ˆ (1)

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)ˆˆˆ()ˆˆˆ(

and 0(constant) ),,( If

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00

0,,

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000


Ex: Find the expression for the equations of the tangent plane and the line normal to the surface at the point P with the coordinates . Use the results to find the equations of the tangent plane and the line normal to the surface of the sphere at the point .

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2222 azyx ),0,0( a

z

yx

0n az

a

),0,0( a


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volumeunit per fluid of outflow of ratenet the is

fluid a in velocity local a is ),,( :Ex

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V

zyxV

a

z

a

y

a

x

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Scalar differential operator

zxyxz

zxy

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232

32

2

2

2

2

2

2

22

62

fieldscalar the of Laplacian the Find :Ex

of Laplacian the

Curl of a vector

]ˆˆ)(ˆ[2

)20(ˆ)22(ˆ)20(ˆ

ˆˆˆ

ˆˆˆ :Ex

alIrrotation 0 If

ˆˆˆ

ˆ)(ˆ)(ˆ)(

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222222

2222222

2222222

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2ˆ2

0

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. then ,ˆ andvector position the is If :Ex

k

xyzyx

kji

v

iyjxkzjyixkv

rvkr

Useful formulas:


baababbaba

aaa

baabba

aaa

abbaabbaba

baba

baba

)()()()()( (9)

)( (8)

)()()( (7)

)( (6)

)()()()()( (5)

)( (4)

)( (3)

)( (2)

)( (1)

10.8 Vector operator formula


kk

jji

iijkijkjji

i

ijjiji

ijkijkjji

ik

aa

aa

aaaa

aaa

)()(

)(

][)()]([

)(that Show :Ex

,,

,,

Useful special cases:

rdr

d

r

kzjyix

dr

d

kz

r

dr

dj

y

r

dr

di

x

r

dr

dk

zj

yi

xr

rdr

dr

zyxrrkzjyixr

ˆ)ˆˆˆ(

ˆˆˆˆˆˆ)(

ˆ)( 1)(

)(|| vector position a is ˆˆˆ 2/1222


dr

d

r

zyx

dr

d

r

z

z

r

dr

d

r

y

y

r

dr

d

r

x

x

r

dr

d

r

z

dr

d

zr

y

dr

d

yr

x

dr

d

xr

dr

ddr

rd

dr

rd

rr

dr

dr

dr

dr

dr

drr

rrr

zyx

rr

r

x

rzyxx

x

zyx

z

zyx

y

yzyx

x

xr

dr

rdrrr

dr

dr

z

z

y

y

x

x

rrrr

2

2

2

222

2

2

2

2

2

2

2

2

2

22

3

222

3

22/1222

222222222

)(

)()()(

)()()(ˆ)( :term 2nd thefor

)()(2ˆ)()ˆ()ˆ())(()( (4)

213

r

3ˆ

similar are termsanother 1

])([ :term1st thefor

)()()(ˆ (3)

)()(3ˆ)(

])([ (2)


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r

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r

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r

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d

r

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d

r

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x

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x

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x

x

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y

y

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ˆ)ˆˆˆ(1ˆˆˆˆˆˆ (6)

0)(ˆ)(ˆ)(ˆ

for

)(ˆ)(ˆ)(ˆ

ˆˆˆ

(b)

0)(ˆ)(ˆ)(ˆ

ˆˆˆ

(a)

0])([ (5)


],[ if 0

],[ if )()()(

for 0)(

function delta Dirac is )(

)(4)1

()1

()ˆ

( (9)

ˆ)

1( (8)

3 (7)

0

000

00

22

2

bax

baxxfdxxxxf

xxxx

r

rrrr

rr

r

r

r

b

a

Combinations of grad, div, and curl


solenoidal is 0)( (4)

alirrotation is 0)( (3)

0///

///

)(

0)( (2)

0)(ˆ)(ˆ)(ˆ

///

///

ˆˆˆ

)(

0)( (1)

fieldvector : fieldscalar :

222222

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aaa

aaa

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zyx

a

a

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a

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0)()()( (8)

them. onact can so ,

constantnot are vectorsunit the s,coordinate spherical and lcylindrica In

them. onact not does

operator the so constant, arevector unit the s,coordinate Cartesian In :

)ˆˆˆ)((

)()( (7)

)(ˆ)(ˆ)(ˆ

))(ˆˆˆ()( (6)

)( (5)

2

2

2

2

2

2

2

22

2

2

2222

2

2222

2

2

2

2

2

2

2

22

Note

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a

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a

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a

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a

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10.9 Cylindrical and spherical polar coordinates


A. Cylindrical polar coordinates

zz

zz

edzedededzeded

dzz

rd

rd

rrd

kekz

re

jiejir

e

jiejir

e

kzjir

z

zzyx

ˆˆˆ

ˆˆ ˆ

ˆcosˆsinˆ ˆcosˆsin

ˆsinˆcosˆ ˆsinˆcos

ˆˆsinˆcos:Ppoint for

20 0

sin cos

ly.respective ,ˆ and ,ˆ ,ˆ directions along

1 and , ,1 are factors Scale

is ˆ (3) is ˆ (2) is ˆ (1)

along distance the of change The

),,( to ),,( from Position

z

z

z

eee

hhh

dzedede

dzzddz


z

aaaa

ez

ee

eaeaeaaz

dzddedzededdV

dddA

dzddrdrddsds

z

z

zz

z

1)(

1 (2)

ˆˆ1

ˆ )1(

ˆˆˆ fieldvector a and ),,,( fieldscalar aFor

|)ˆˆ(ˆ|

iselement volume The

is plane y-x in area The

])()()[()()( 2/122222/12


.vector intoput are above The

sin cos

ˆˆ ˆcosˆsinˆ ˆsinˆcosˆ

ˆˆ ˆcosˆsinˆ ˆsinˆcosˆ

scoordinate lcylindrica in expressed ˆˆˆ :Ex

1)(

1 (4)

ˆˆˆ

1 (3)

2

2

2

2

2

22

a

zzyx

ek eejeei

kejiejie

kxzjyiyza

z

aaaz

eee

a

z

z

z

z


B. Spherical polar coordinates

jieee

kjieee

kjiee

reree

jrirr

e

kjrirr

e

kjir

re

krjrirr

r

rzryrx

rr

r

r

ˆcosˆsin||/ˆ

ˆsinˆsincosˆcoscos||/ˆ

ˆcosˆsinsinˆcossinˆ

sin|| || 1||

ˆcossinˆsinsin

ˆsinˆsincosˆcoscos

ˆcosˆsinsinˆcossin

ˆcosˆsinsinˆcossin

20 0 0

cos sinsin cossin

ddrdredrerdedrdV

ddaeadedadA

drdrdr

rdrddsd

edrerdedr

edededr

dr

dr

drr

rrd

r

r

r

sin|)ˆsinˆ(ˆ|

:element volume The

sin|ˆˆsin|

a radius a with sphere a of surface the onelement area The

)(sin)()(

)(s

:ntdisplaceme of Magnitude

ˆsinˆˆ

:ntDisplaceme

2

2

222222

2



2

2

2222

22

2

22

sin

1)(sin

sin

1)(

1 (4)

sin

ˆˆˆ

sin

1 (3)

sin

1)(sin

sin

1)(

1 (2)

ˆsin

1ˆ

1ˆ (1)

ˆˆˆ fieldvector a ,),,( fieldscalar A

rrrr

rr

arraar

eee

ra

a

ra

rar

rra

er

er

er

eaeaeaar

r

r

r

r

rr


10.10 General curvilinear coordinate

|| || || :factors Scale

1

ˆ 1

ˆ 1

ˆ :vectorsUnit

|| || ||

Pat curve- totangent vector a



Ppoint at vector position the is ),,(

),,( ),,( ),,(

),,( ),,( ),,(

33

22

11

333

222

111

332211

33

3

22

2

11

1

321

332211

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rh

u

rh

u

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r

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u

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333222111

332211

33

22

11

1

ˆˆˆ

sin 1 :coordinate Spherical

1 1 :coordinate lCylindrica

is changing ofelement distance The

eduheduheduh

edueduedu

duu

rdu

u

rdu

u

rrd

rhrhh

hhh

duhdu

r

z

ii

321321333222111

332211

23

23

22

22

21

21

2

|)ˆˆ(ˆ|

|)(| iselement volume The

3,2,1for ˆ)(

vector the by defined pedparallelpi malinfinitesi The

)()()()(

is length arc ofelement the lar,perpendicu

mutually are ˆ s,coordinater curvilinea orthogonalFor

dududuhhheduheduheduh

edueduedudV

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r

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3322113

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21

1

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333

33

222

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111

11

:vectors of sets Two

surface the to normal ||

ˆ


ˆ


ˆ

:vectorsunit three usefulAnother

uuuu

r

u

r

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r

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u

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re

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u

ii

ii


vectors of system reciprocal are }{ and }{ iie

otherwise 0

if 1

)ˆˆˆ()ˆˆˆ(

jie

u

u

z

u

u

z

y

u

u

y

x

u

u

x

kz

uj

y

ui

x

uk

u

zj

u

yi

u

x

uu

re

ji

i

jj

i

j

i

j

i

jjj

iii

ji

ji


Gradient

333

222

111

332211

33

22

11

ˆ1

ˆ1

ˆ1

)ˆˆˆ(

euh

euh

euh

kduhj duhiduhrd

duu

duu

duu

d

Divergence

)]()()([1

ˆˆˆ

3213

2132

1321321

332211

ahhu

ahhu

ahhuhhh

a

eaeaeaa

Proof:

)()()(

)()ˆ(

ˆˆˆ

3232132321

3232111

3322321

uuhhauuhha

uuhhaea

uhuheee


)(1

)ˆ( and )(1

)ˆ(

)ˆ( and )ˆ( for same the

)(1

ˆ)()

ˆˆ()()ˆ(

0)(0)( 10.43 eq. From

3213321

332132321

22

3322

3211321

32

1321

3

3

2

232111

32

ahhuhhh

eaahhuhhh

ea

eaea

hhauhhh

hh

ehha

h

e

h

ehhaea

uu

Laplacian

)]()()([1

ˆ1

ˆ1

ˆ1

33

21

322

13

211

32

1321

2

333

222

111

uh

hh

uuh

hh

uuh

hh

uhhha

euh

euh

euh

a


Curl

332211

321

332211

321332211

ˆˆˆ1

ˆˆˆ

ahahahuuu

eheheh

hhhaeaeaeaa

Proof:

)ˆ( and )ˆ(for same the

)(ˆ

)(ˆ

ˆ]ˆ)(

1ˆ)(

1ˆ)(

1[

ˆ)(

)()()ˆ(

3322

11221

311

313

2

1

1311

33211

22111

11

1

111

11111111111

eaea

hauhh

eha

uhh

e

h

eeha

uheha

uheha

uh

h

eha

uhauhauhaea

10.1 differentiation of vector chapter 10 vector calculus

Documents

vector calculus slide

integration of vector

vector calculus ex

vector calculus divergence

vector functions

vector calculus curl

scalar field slide

surface chapter