1.1 vector algebra 1.2 differential calculus 1.3 integral calculus 1.4 curvilinear coordinate 1.5...
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1.1 Vector Algebra
1.2 Differential Calculus
1.3 Integral Calculus
1.4 Curvilinear Coordinate
1.5 The Dirac Delta Function
1.6 The Theory of Vector Fields
Chapter 1 Vector Analysis
1.1 Vector Algebra
1.1.1 Scalar , Vector , Tensor
1.1.2 Vector Operation 1.1.3 Triple Products
1.1.4 Vector Transform
1.1.1 Scalar , Vector , Tensor
Scalar:Vector:Function :
3A
A�
magnitude , 0 directionmagnitude , 1 directionmagnitude , 2 direction
Tensor
all quantities are tensor .
Scalar :
Vector :
function :
ˆ ˆ
i j ij
i je e
i j
1
0
1.1.1
i
n
iieAA ˆ
1
i
n
ii faf
1
iaan
ii
1
i
ie
base of number system
base of coordinates
if base of functions
Any component of the base is independent to rest of the base
orthogonal
that is ,
1.1.1
Tensor :
magnitude , 0 direction , 0 rank tensor scale
[ no dimension ]
magnitude , 1 direction , 1st rank tensor vector
[in N dimention space : N components ]
magnitude , 2 direction , 2nd rank tensor 2nd rank tensor
[in N dimention space : N2 component ]
magnitude , 3 direction , 3rd rank tensor 3rd rank tensor
[in N dimention space : N3 component]. . .
. . .
. . .
1.1.2 Vector operation
n
ii aaaa
1321
•Addition
642111 bac
Ex.jiA ˆ3ˆ2
jiB ˆ5ˆ4
BAjcicC
ˆˆ21
642111 bac 853222 bac
; ;
;
i i ic a b
n
iii
n
i
n
i
n
iiiiiii ecebaebeaBAC
11 1 1
ˆˆˆˆ
Vector addition=sum of components
i
1.1.2
ji
n
i
n
jji
n
jjj
n
iii eebaebeaBAc ˆˆˆˆ
1 111
cosn n n
i j ij i ii j i
a b a b A B
1 1 1
=
•Inner product
Ex.
jiA ˆ3ˆ2
jiB ˆ5ˆ4
;
jijiBAC ˆ5ˆ4ˆ3ˆ2
jjijjiii ˆ5ˆ3ˆ4ˆ3ˆ5ˆ2ˆ4ˆ2 231585342
sinBABAC
ijiikkijkjjkkji
kji
kji
kji
babaebabaebabae
bbb
aaa
eee
BAC ˆˆˆ
ˆˆˆ
n
kji
kjiijkkjiijk baebae
111
ˆˆ =
1.1.2
sinBABAC
Area of
•Cross product
kjiijk baeBA ˆ
ijk= 1 clockwise = -1 counterclockwise= 0
kji
kji ji kj ik or or
1
23 1-1
1.1.2
Define :
kjiijkkjijkiii cbacbaCBaCBA
ABC
ccc
bbb
aaa
321
321
321
(volume enclosed by vectors , ,and )
1.1.3 triple products
A
B
C
Figure 1.12
1.1.4 Vector transform
ˆ ˆx yA A x A y
ˆ ˆx yA A x A y
2
1
2221
1211
2
1
A
A
aa
aa
A
A
cossin
sincosR�
y
x
y
x
B
B
B
B
cossin
sincos
BRB�
y
x
y
x
A
A
A
A
cossin
sincos
ARA�
Vector transform jiji ARA
1.2 Differential calculus
1.2.1 Differential Calculus for Rotation
1.2.2 Ordinary Derivative
1.2.3 Gradient
1.2.4 Divergence
1.2.5 The Curl
1.2.6 Product Rules
1.2.7 Second Derivatives
yyxx BABABABA cos
yyxx BABABABA cos
yxx ˆsinˆcosˆ yxy ˆcosˆsinˆ
y
x
y
y
x
y
y
x
x
x
y
x
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
yxy
xy
x
xxx
ˆsinˆcos
ˆ
ˆˆ
ˆ
ˆˆˆ
yxy
yy
x
yxy
ˆcosˆsin
ˆ
ˆˆ
ˆ
ˆˆˆ
1.2.1 Differential Calculus for Rotation
1.2.2 Ordinary Derivative
dxdx
dfdf )(
Geometrical Interpretation: The derivative is the slope of the graph of f versus x dxdf
ld
=
ldfldfee yyxx
)ˆˆ(
Ex. f =xy f=xy2
yx
f
x
y
f
;
2yx
f
; xyy
f2
1.2.3 Gradient
For a function ),( yxf
ye
xe yx
ˆˆ
Define i
ey
ex
e iyx
ˆˆˆ
f
: gradient of f
ld
dffldfldfdf cos
ˆ ˆ ˆ ˆ( ) ( )df df f f f fx y x ydx dy x y x ydf dx dy dx dy e e dxe dye
1.2.4 Divergence
y
V
x
Viiii
yxvvv
iiijjijijijjii vveevveev
ˆˆˆ)ˆ(
[a scalars]
dx
xVdxxV
xdxx
xVdxxV
x
Vx
)(
>0 : blow out
<0 : blow in
dyy y xV dxxV
dxx x
1.2.5 The Curl
ˆ ˆ
ˆ( )y x
ijk i j k ijk i j k
V Vx y
v v e e v
z
)(ˆ)(ˆ)(ˆ
ˆˆˆ
y
v
x
vk
x
v
z
vj
z
v
y
vi
vvvzyx
kji
v xyzxyz
zyx
0x
Vy
0y
Vx
0)(ˆ
y
V
x
VxyzV
1.2.6 Product Rules
The similar relations between calculation of derivatives and vector derivatives (1)Sum rules
dx
dfkkf
dx
d)(
)()(
)()(
)(
AkAk
AkAk
fkkf
(2)The rule for multiplication by a constant
( )d df dg
f gdx dx dx
( )
( ) ( ) ( )
( ) ( ) ( )
f g f g
A B A B
A B A B
1.2.6 (2)
(produce of two scalar functions) (dot product of two vectors) BA
gf
dx
dfg
dx
dgffg
dx
d)(
(3) Product rules there are six product rules:two each for gradient, divergence and curl. Product rule for divergence:
ABBA
ABBABA
)()(
)()()(
fggffg )(
two product rules for gradient
1.2.6 (3)
)()()(
)()()(
BAABBA
fAAffA
two product rules for divergences
Af
(scalar times vector)(cross product of two vector functions))( BA
two product rules for curls
( ) ( ) ( )
( ) ( ) ( ) + ( ) ( )
fA f A A f
A B B A A B A B B A
1.2.6 (4)
(4)The quotient rule
2
2
2
)()()(
)()()(
)(
g
gAAg
g
A
g
gAAg
g
A
g
gffg
g
f
2)(g
dx
dgf
dx
dfg
g
f
dx
d
The quotient rule for derivative:
The quotient rules for gradient,divergence ,and curl
1.2.7 Second Derivatives
1. Divergence of gradient:
TTTee
TeeTeT
ijiijjiji
jjiijj
2)ˆˆ(
)ˆ()ˆ()ˆ()(
jjii eVVeV ˆ)()ˆ( 222
TTTT zyx2222
so
2 is called Laplacian,T is a scalar; is a vectorV
iie ˆ a derivative vector
(inner product of same vector; ) A A A 2
1.2.7 (2)
0
)(ˆ)(ˆ)(ˆ
)(ˆ
)(ˆ)(ˆ
ˆ
)ˆ()ˆ()(
122133113223321
12321213123
1323131213223132321231
TTeTTeTTe
TTe
TTeTTe
Te
eTeT
jikkij
jjii
2. Curl of gradient :
a
(cross product of same vector, )0AA
1.2.7 (3)
)()(2 VVV
)( V
3. gradient of divergence:
4. divergence of curl
3
1,
ˆ ˆ( ) ( ) ( )
( ) 0
l l kij k i j kij lk l i j
ijk k i j k i i k jj cw
i j k
V e e V V
V V
(similar to ) 0)( VAA
1.2.7 (4)
2
2
2
2
ˆ( ) ( )
ˆ
ˆ ( )
ˆ ˆ
ˆ ˆ( ) ( )
ˆ ˆ( ) ( )
( )
ijk i j k
l lmk m ijk i j
l li mj lj mi m i j
i j i j j i j
i i j j i j j
i i j j
V V e
e V
e V
e V e V
e V V e
e V V e
V V
VVV 2)()(
5.Curl of curl
Poof:
(similar to ) )( VAA
1.3 Integral Calculus
1.3.1 Line, Surface, and Volume Integrals
1.3.2 The Fundamental Theorem of Calculus
1.3.3 The Fundamental Theorem of Gradients
1.3.4 The Fundamental Theorem of Divergences
1.3.5 The Fundamental Theorem for Curls
1.3.6 Relations Among the Fundamental Theorems
1.3.7 Integration by parts
1.3.1 Line,Surface,and Volume Integrals
(a) Line integrals
(b)Surface Integrals
b
aPldV
P:the path(e.g.(1) or (2) )
s
adV
S:the surface of integral
1.3.1 (2)
(c)Volume Integral
VTd dzdydxd
dVzdVydVxdzVyVxVdV zyxzyx ˆˆˆ)ˆˆˆ(
Example1.8
?2 TdzyxT
8
3)
12
1)(9(
2
1)1(
2
1
}][{
1
0
23
0
2
1
0
1
0
3
0
2
dyyydzz
dzdydxxyzdTy
Solution:
Suppose f(x) is a function of one variable. The fundamental theorem of calculus:
)()()(
)()()(
afbfdxxF
afbfdxdx
df
b
a
b
a
dx
dfxF )(
1.3.2 The Fundamental Theorem of Calculus
Figure 1.25
Suppose we have a scalar function of three variables T(x,y,z) We start at point ,and make the journey to point
(1)path independent(2) ,a closed loop(a=b)
),,( zyx aaa ),,( zyx bbb
0)( dlT
A line is bounded by two points
1.3.3 The Fundamental Theorem of Gradients
Figure 1.26
( )
( ) ( ) ( )b
a
dT T dl
T dl T b T a
1.3.4 The Fundamental Theorem of Divergences
ss
kj
s
ikjii
v
i
volumn
adVidaiVddVdddVdV ˆ)ˆ()(
Proof:
is called the flux of through the surface.V
( )volumn surface
V d V da
surface
V da
1.3.4 (2)
Example1.102 2ˆ ˆ ˆ(2 ) (2 )
sV y x xy z y yz z V d Vda
Solution:
2
dV
)(2 yxV
1
0
1
0
1
0)(2)(2 dzdydxyxdyx
ydxyx 1
0 2
1)( 1)(,
1
0 2
1 dyy
1
01 1dz
3
11
0
1
0
2 dzdyyadV
3
11
0
1
0
2 dzdyyadV
3
4)2(
1
0
1
0
2 dxdzzxadV
1
0
1
0
2
3
1dxdzzadV
121
0
1
0 ydxdyadV
1
0
1
000dxdyadV
2013
1
3
4
3
1
3
1 s adV
(2)
(1)
1.3.5 The Fundamental Theorem for Curls
line
k
line
kkjk
s
jikji
s
ijk
surface
ldVdVddVdaeVeadV
)ˆ()ˆ()(
Proof:
adVsurface
)(
0)( adVsurface
(1) dependents only on the boundary line, not on the particular surface used.
(2) for any closed surface.
A surface is enclosed by a closed line
Figure 1.31
( )surface boundary
line
V da V dl
1.3.5 (2)
Examples 1.11
( )s p
V da V dl
zyzyyxzV ˆ)4(ˆ)32( 22
xdzdyadandzzxxzV ˆˆ2ˆ)24( 2
3
44)(
1
0
1
0
2 dzdyzadVs
,13,3,001
0
22 dyyldVdyyldVzx
,4,4,103
41
0
22 dzzldVdzzldVyx
,13,3,100
1
22 dyyldVdyyldVzx
,00,0,000
1 dzldVldVyx
3
4
3
4011 dlV
Solution:(1)
(2)
1.3.6 Relations Among the Fundamental Theorems
(1)Gradient :
lineboundarysurface
ldVadV
)(
•combine (1)and (3)
•combine (3)and(2)
0)(0)]([
0)(
TadT
ldT
surface
line
(2)Divergence :
(3)Curl :
( ) ( ) ( )b
aT dl T b T a
( )volumn surface
V d V da
( ) 0
[ ( )] 0 ( ) 0
surface
volumn
V da
V d V
1.3.7 Integration by parts
b
a
b
a
b
a
b
a
b
a
b
a
b
a
fgdxdx
dfgdx
dx
dgfor
dxdx
dfgdx
dx
dgffgdxfg
dx
ddx
dfg
dx
dgffg
dx
d
)()(
,)()()(
)()()(
adAfdfAdAfor
adAfdfAdAfdAf
fAAfAf
s
)()(
,)()()(
)()()(
1.4 Curvilinear Coordinates
1.4.1 General Coordinates
1.4.2 Gradient
1.4.3 Divergence
1.4.4 Curl
1.4 Curvilinear Coordinates
Spherical Polar Coordinate and Cylindrical Coordinate),,( r
cossinsincossin rzryrx
ˆsinˆˆ
ˆˆˆˆˆˆ
ˆˆˆˆˆˆ
drdrrdr
dldlrdlkdzjdyidxld
AArAkAjAiAA
r
rzyx
),,( zr
zdzdrrdrdl
zAArAA
zzryrx
zr
ˆˆˆ
ˆˆˆ
sincos
Figure 1.36 Figure 1.42
1.4.1 General Coordinates
Cartesian coordinate 1,1,1,, ii hzyxq
iii
zyx
zyx
edqh
edqhedqhedqh
edzedyedxld
ˆ
ˆˆˆ
ˆ1ˆ1ˆ1
332211
kjjiiijk
z
edqhdqhad
edqhdqhedydxad
ˆ
ˆˆ 32211
321321
332211
dqdqdqhhh
dqhdqhdqhdzdydxd
ze
ze
ye
xe
1ad
dxdy
dz
1.4.1 (2)
Cylindrical coordinate 1,,1,, rhzrq ii
)( rs
ˆˆ ˆˆ
ˆ
s z
i i i
dl ds dl sd dl dz
dl dr r r d dz z h dq e
da rd dz r
321321 dqdqdqhhh
dzhdhdrh
dzrddr
dldldld
zr
zr
1.4.1 (3)
Spherical coordinate sin,,1,, rrhrq ii sin
ˆ ˆˆ sin
ˆ ˆsin
ˆ ˆ
ˆ ˆ
sin
r
r
r
r
dl dr dl rd dl r d
dl drr rd r d
da dl dl r r d d r
da dl dl r dr d
da dl dl r dr d
d dl dl dl r dr d d
21
2
3
2
(C) Figure 1.38
(a)
(b)
Figure 1.39
1.4.1 (4)
i
iii edqhld ˆ
321321
ˆ
dqdqdqhhhd
edqhdqhad kjjiiijk
In summary:
Gradient
iii
ii
iii
iiiii
q
T
he
dqh
dqqTeT
edqhTdqq
T
ldTdT
1ˆ
)(ˆ
)ˆ()(
)(
1.4.2 Gradient
So for
sinˆˆˆ),,,(
ˆˆˆ),,,(
r
Te
r
Te
r
TeTr
z
Te
r
Te
r
TeTzr
r
zr
ˆ ˆ ˆ( , , ),T T T
x y z T x y zx y z
for
for
( )V s
V d V da
1.4.3 Divergence
332211
221131133233221
dqhdqhdqhdqhdqhVdqhdqhVdqhdqhV
V
( ) i iV h dq V da jjiik
kijk qdhqdhV
221131133233221
332211332211 )()(
dqhdqhVdqhdqhVdqhdqhV
AVAVAVdqhdqhdqhV
])()()([
3
3
2
2
1
1321 2133123211
A
q
A
q
A
qhhh hhVhhVhhV
( )d dd d s
V d V da
1.4.3 (2)
( )k kkl l
l
V V dah q
( )k i i j j ijkkl l
l
V h q h qh q
( )V h q h q V h q h q V h q h qh h h q q q
1 2 2 3 3 2 3 3 1 1 3 1 1 2 21 2 3 1 2 3
V h h V h h V h hh h h q q q
1 2 3 2 3 1 3 1 2
1 2 3 1 2 3
1
zV
y
V
xV zyxV
zyxqi ,,
,,rqi
zrqi ,,
)](
)sin()sin([ 2
sin1
2
rV
rVrVV rrr
)()1()(1 rVVrVV zzrrr
( ) ˆijkV h q h q e V h qi i j j j j jk j
kqqijkhhh eV khjVjhjqjikjiˆ)]([1
kekhi
jj
kji q
hV
ijkhhh ˆ)(1
)()()( ldVadVldVadVs da
dadd
c
1.4.4 Curl
zyx ,,
zr ,,
,,r
ˆ])([
ˆ)]([
ˆ])(sin[
1
sin11
sin1
r
r
vrr
rv
r
vr
rv
rv
rvv
zrv
rv
r
zrz
vrr
rv
zv
z
vvr
ˆ])([
ˆ)(ˆ)(
1
1
z
yxv
yv
x
v
xv
zv
z
v
yv
xy
zxyz
ˆ)(
ˆ)(ˆ)(
1.4.4 (2)
1.5 The Direc Delta Function
1.5.2 Some Properties of the Delta Function
1.5.3 The Three-Dimensional Delta Function
1.5.4 The Divergence of 2/ˆ rr
1.5.1 The Definition of the Delta Function
The definition of Delta function ：0)( x 0x
)(x 0x
1)( dxx
1
1
)(xf1
212
1
1
)(xg
1.5.1 The Definition of the Delta Function
)()(lim0
xxf
)()(lim
0xxg
Figure 1.46
dyayayday )(1)()(
0)()(
ayxayx
ay
)()( ayxayx
ay
1.5.1 (2)
• Definition with shifted variable.
1.5.2 Some Properties of the Delta Function
• )()( 1 ykyk
dykykkydkydxx )()()()(1
1)()(1
dykykdyy
• ( ) ( ) (0) ( ) (0) ( ) (0)f x x dx f x dx f x dx f
• ( ) '( ) ( ) ( ) '( ) ( ) '(0)f x x dx f x x f x x dx f
0
•
•
)()()()( axafaxxf
)()()( afdxaxxf
• 1| |( ) | | ( ) ( ) ( )ky k k y k y y
1.5.3 The Three-Dimensional Delta Function
)()()()(3 zyxr
zyx ezeyexr ˆˆˆ
3( )
( ) ( ) ( )
1
all spacer d
x y z dxdydz
3( ) ( ) ( )all space
f r r a d f a
1.5.4 The Divergence of 2/ˆ rr
1 ( )rrrV Vr
22
2ˆV rr
)0( r
21 (1) 0rr
V s
adVdV
2
21( ) ( sin ) 4
s RR d d
Fig. 1.44
1.5.4 (2)
14 1
VV d
314 ( ) ( ) ( ) ( )V r x y z
231 ˆ
4 ( ) ( )rr
r
23ˆ( ) 4 ( )r
rr
23ˆ( ) 4 ( )r
rr
0rrr
22 3ˆ1 1( ) ( ) 4 ( )r
rr r r
1.6 Theory of Vector Fields
1.6.1 The Helmholtz Throrem
1.6.2 Potentials
CF
DF
AVF
requirement：
0 C 2
0limr
Cr
02
0limr
Dr
0, ,
( ')'
14 'D r
V r rV d
( ')'
14 'C r
V r rA d
2 ( ')14 '
( ) 'VD rr r
F V A d
3 ( ) ( )' ' 'V
D Dr r r d
0
1.6.1 The Helmholtz Theorem
Helmholtz theorem:
Proof : Assume
)()( AVF
CAA
)(20
2 214 '
( ')| '|V
A d CC rr r
'''
'1)(
rriirr
ceiirr
rC ce ii
' 'ˆ ˆi i i i r r r r
c e e c
1 1
V rrVrr
rC drCdA '1'')(''
'
)(4
0)( '1'''1'' V rrrrsdrCadC
)( ''1''1
rrrrd
1.6.1 (2)
'312 4' rrrr
)()()( 00 xfdxxfxx0,0, ' crcrs
s r
c
srdr 2lim rc 1
ss rrrdrc ln11
2
031 r
c
nr
r
c
rc
1
2 0
When n >20
r
1.6.1 (3)
Curl-less fields :
( ) ( )s
l
F da
F dl
3 0
0
( ) ( ) ( )b
aF dl b a 4
1.6.2 Potentials
( ) F 1 0
0 .F F V V is a scalar potential
( )F V 2
Divergence-less fields :
(1) 0F
AF
)2(
( )sF da 3 0
is independent of surface
1.6.2 (2)
.0 potentialvectoraisAAFF
(4)sF da
top related