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Chapter 8 向量分析 ( Vector Analysis). 物理量與符號. 物理量:. 1. 純量( scalar quantity): 有大小,無方向. 例如: 質量( mass), 溫度( temperature), 壓力( pressure), 能量( energy). 2. 向量( vector quantity): 有大小以及方向. 例如: 速度( velocity), 動量( momentum), 力矩( torque). 向量符號:. 1. 一般向量 : 長度. - PowerPoint PPT Presentation
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Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
物理量與符號物理量 :
1. 純量 (scalar quantity): 有大小 , 無方向
例如 : 質量 (mass), 溫度 (temperature), 壓力 (pressure), 能量 (energy)
2. 向量 (vector quantity): 有大小以及方向
例如 : 速度 (velocity), 動量 (momentum), 力矩 (torque)
向量符號 :
1. 一般向量 : 長度
2. 單位向量 : 長度 1
A
x
A
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
向量的基本運算1. 向量之相等 : 包括大小以及方向的相等 ,
2. 向量之反向 : 大小相等但方向相反 ,
3. 向量之合成 :
4. 向量之倍數 :
5. 向量之純量積 (scalar product): 功 (work) 的計算
6. 向量之向量積 (vector product): 力矩 (torque) 的計算
BA
A
BAC
An
SFW
Fr
向量之合成 :Commutative :
Associative :
ABBAC
)CB(AC)BA(
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
向量的座標表示法
xA
A
yA
zA
)z,y,x(AAAA
終點表示法 :
分量表示法 : zzyyxxAAAA
單位向量表示法 :
Au
AuAA
2
A
2
A
2
AzyxA
2
A
2
A
2
A
AAA
A zyx
zzyyxx
A
Au
x
y
z
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
座標軸的轉動 (Rotation of the Coordinate Axes)
X
Y
r
x
y
φ
X’
Y’
x’
y’φ
sinycosxx '
cosysinxy '
2222'2' ryxyx
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
座標軸的轉動 (Rotation of the Coordinate Axes)
sinycosxx ' cosysinxy '
1xx 2
xy
Let cosa11 sina
12
sina21
cosa22
212111
'
1xaxax 222121
'
2xaxax
The coefficient aij is the cosine of the angle between xi’ and xj
2
1jjij
'
ixax 2,1i
N dimensions
N
1jjij
'
iVaV N,...,2,1i
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
座標軸的轉動 (Rotation of the Coordinate Axes)
The coefficient aij is the cosine of the angle between xi’ and xj
2
1jjij
'
ixax 2,1i
j
'
i
ij x
xa
Using the inverse rotation :
yields
2
1i
'
iijjxax or
ij'
i
j ax
x
N
1jj'
i
jN
1jj
j
'
i'
iV
x
xV
x
xV
N
1jjij
'
iVaV
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
The orthogonality condition for the direction cosines aij:
jkikiji
aa or jkkijii
aa
k
j
k
'
i
'
i
j
i'
i
k
'
i
j
i x
x
x
x
x
x
x
x
x
x
The Kronecker delta is defined byjk
1jk for j = k
0jk for j k
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Vector and vector space
Vector : )x,x,x(x321
)y,y,y(y321
and
1. Vector equality : means xi = yi , i = 1,2,3.yx
zyx 2. Vector addition : means xi + yi = zi , i = 1,2,3.
3. Scalar multiplication : (with a real). )ax,ax,ax(xa321
4. Negative of a vector : )x,x,x(x)1(x321
5. Null vector : there exists a null vector )0,0,0(0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Scalar (Dot) Product
The projection of a vector onto a coordinate axis is a special case of the scalar product of and the coordinate unit vectors :
A
A
xAcosAAx
yAcosAAy
zAcosAAz
zAByABxAB)zByBxB(ABAzyxzyx
ABBAABABABABiiiiiizzyyxx
The scalar product is commutative :
z
x
y
A
B
θ
Definition :
cosABABBABAAB
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Distributive Law in the Scalar (Dot) Product
AAA)CB(AACABCABA)CB(A
A
C
B
CB
AC
AB
)CB(AA
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Normal vector
n
)yyxx(r
n
r
is a nonzero vector in the x-y plane 0rn
x
y
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Invariance of the scalar product under rotations
'
k
'
kk
'
z
'
z
'
y
'
y
'
x
'
xBABABABA
jzjj
izii
jyjj
iyii
jxjj
ixii
BaAaBaAaBaAa
jljilijil
'
k
'
kk
BaAaBA
(using the indices k and l to sum over x,y, and z)
iii
jiijji
jiljlijil
BABA
BA)aa(
iii
'
k
'
kk
BABA
take BAC
)BA()BA(CC
BA2BBAA
)BAC(2
1BA 222
invariant
invariant2AAA
Scalar quantity
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Vector (Cross) Product
Definition :
z)BABA(y)BABA(x)BABA(
BBB
AAA
zyx
BACxyyxzxxzyzzy
zyx
zyx
zCyCxCzyx
jkkjiBABAC i,j,k all different and with cyclic permutation of the indices i,j, and k
Magnitude of :C
2222 )BA(BA)BA()BA(CCC 22222222 sinBAcosBABA
Prove it!
sinABC
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Parallelogram representation of the vector product
sinABC BAC
x
y
θ
A
B
Bsinθ
ABBA
anticommutation
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
the vector product under rotations
'
j
'
k
'
k
'
j
'
iBABAC i,j, and k in cyclic order
mjmm
lkll
mkmm
ljll
BaAaBaAa mljmklkmjlm,l
BA)aaaa(
jkikiji
aa If i = 3, then j = 1, k =23312212211
aaaaa
3211232113aaaaa
3113222312aaaaa
l m
233131321233323113322133
'
3BAaBAaBAaBAaBAaBAaC
nn3n
333232131CaCaCaCa
N
1jjij
'
iVaV
C
is indeed a vector !
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Triple Scalar Product
)CBCB(A)CBCB(A)CBCB(A)CB(Axyyxzzxxzyyzzyx
)BA(C)AC(B
)CA(B)AB(C)BC(A
The dot and the cross may be interchanged :
CBA)CB(A
zyx
zyx
zyx
CCC
BBB
AAA
scalar
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Parallelepiped representation of triple scalar product
x
y
z
A
C
B
CB
)CB(A
Volume of parallelepiped defined by , , and A
B
C
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Construction of a reciprocal crystal lattice
Let , , and (not necessarily mutually perpendicular) represent the vectors that define a crystal lattice. The distance from one lattice point to another may be written as
a
b
c
cnbnanrcba
With these vectors we may form the reciprocal lattices :
cba
cba '
cba
acb '
cba
bac '
We see that is perpendicular to the plane containing and and has a magnitude proportional to .
'a
b
c
1a
1ccbbaa '''
0bcaccbabcaba ''''''
Fourier space
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Triple Vector Product
)CB(A
CB
x
y
z
)CB(A
C
A
B
CyBx)CB(A
CAyBAx0)]CB(A[A
CAzx BAzy
)BACCAB(z)CB(A
)BACCAB(
BAC-CAB rule
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 7 向量分析 (Vector Analysis)
Proof : z = 1 in )BACCAB(z)CB(A
Let us denote cosCB cosAC cosBA
2222 )]CB(A[)CB(A)]CB(A[ 2222 )BA(BA)BA(
22 )]CB(A[cos1 2222 )CB(CB)CB(
]CBCABA2)BA()CA[(z 222
)coscoscos2cos(cosz 222
)coscoscos2cos(coszcos1)]CB(A[ 22222
The volume is symmetric in αβ,γ z2 = 1 z = ± 1
For the special case y)yx(x 1z
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Gradient Suppose that φ(x,y,z) is a scalar point function which is independent of the rotation of the coordinate system.
)x,x,x()x,x,x(321
'
3
'
2
'
1
'
jj
ijj
'
i
j
j
'
i
321
'
i
'
3
'
2
'
1
'
xa
x
x
xx
)x,x,x(
x
)x,x,x(
N
1jj'
i
j'
iV
x
xV
We construct a vector with components : j
x
zz
yy
xx
or
zz
yy
xx
A vector differential operator
)potential(force
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
is a vector having the direction of the maximum space rate of change of φ.
Chapter 8 向量分析 (Vector Analysis)
A Geometrical Interpretation
P Qdr
z
x
y
y
z
x
φ(x,y,z)= C
φ= C2 > C1
φ= C1P
Q
zz
yy
xx
dzzdyydxxrd
0ddzz
dyy
dxx
rd
is perpendicular to rd
rd)(CCCd12
For a given dφ, is a minimum when it is chosen parallel to (cosθ = 1).
rd
For a given , is a maximum when is chosen parallel to .
rd d rd
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Exercise : 試求曲面 上一點 (2,2,8) 之切面與法線方程式 (88 台大造船 )
)yx(8z 222
Solution : 取 , 而曲面在 之法向量 為 :k8j2i2r0
0r
N
kz2jy16ix16)zy8x8( 222
kj2i2k16j32i32N
根據直線與平面之點向式 :
切面 :
法線 :
0zy2x20)8z()2y(2)2x(2N)rr(0
1
8z
2
2y
2
2x0N)rr(
0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Example : Calculate the gradient of f(r) = )zyx(f 222
z
)r(fz
y
)r(fy
x
)r(fx)r(f
x
r
dr
)r(df
x
)r(f
r
x
)zyx(
x
x
)zyx(
x
r2/1222
2/1222
r
x
dr
)r(df
x
)r(f
r
y
dr
)r(df
y
)r(f
r
z
dr
)r(df
z
)r(f
dr
dfr
dr
df
r
r
dr
df
r
1)zzyyxx()r(f
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Divergence
x
y
)t(r
)tt(r
r
vt
rlim
0t
Differentiating a vector function
vt
)t(r)tt(r
dt
rdlim
0t
: vector
: differential property
z
V
y
V
x
VV zyx
Scalar
Vector
z
VfV
z
f
y
VfV
y
f
x
VfV
x
f)fV(
z)fV(
y)fV(
x)Vf( z
z
y
y
x
xzyx
VfV)f()Vf(
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Example : Calculate the divergence of f(r) r
)]r(zf[z
)]r(yf[y
)]r(xf[x
))r(fr(
dr
)r(df
r
z
dr
)r(df
r
y
dr
)r(df
r
x)r(f3
222
dr
)r(dfr)r(f3
if 1nr)r(f
1n2n1n1n r)2n(r)1n(rr3)rr(
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
A Physical Interpretationz
x
y
G
C
A
E
B
F
H
D
dy
dz
dx
Consider )v( )z,y,x(v
: the velocity of a compressible fluid
)z,y,x( : the density of a compressible fluid
The rate of flow in (EFGH) = dydzv0xx
The rate of flow out (ABCD) = dydzvdxxx
dydz]dx)v(x
v[0xxx
Expand in a Maclaurin seriesNet rate of flow out|x = dxdydz)v(
x x
Net rate of flow out = dxdydz))v((dxdydz)]v(z
)v(y
)v(x
[zyx
)v( : the net flow of the compressible fluid out of the volume element dxdydz per unit
volume per unit time. (divergence)
The continuity equation : 0)v(t
ρ(x,y,z,t)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Exercise : For a particle moving in a circular orbit tsinrytcosrxr
(a) Evaluate (b) Show that (The radius r and the angular velocity are constant)
rr
0rr 2
tsinrytcosrxr tcosrytsinrxr
rtsinrytcosrxr 222
(a)
)tcosrytsinrx()tsinrytcosrx(rr
22222 rz)tsinrtcosr(z
(b) tsinrytcosrxr
0rr 2 Proof !
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Curl Definition :
zyx
xyzxyz
VVVzyx
zyx
)Vy
Vx
(z)Vx
Vz
(y)Vz
Vy
(xV
)Vz
f
z
VfV
y
f
y
Vf()]fV(
z)fV(
y[)Vf(
y
y
z
z
yzx
xxV)f(Vf
V)f(Vf)Vf(
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
A Physical Interpretation
x
y
x0+dx, y0
x0+dx, y0+dyx0, y0+dy
x0, y0 1
2
3
4
Circulation around a differential loop
4
yy3
xx2
yy1
xx1234d)y,x(Vd)y,x(Vd)y,x(Vd)y,x(Vncirculatio
dxdy)y
V
x
V(
)dy)(y,x(V)dx](dyy
V)y,x(V[
dy]dxx
V)y,x(V[dx)y,x(V
xy
00y
x
00x
y
00y00x
circulation per unit areaz
V
Vorticity ( 渦度向量 ):V
0V
V
is labeled irrotational
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
A Physical Interpretation
0V
V
is labeled irrotational
(the gravitational and electrostatic forces)
32 r
rC
r
rCV
21mGmC Newton’s law of universal gravitation
0
21
4
qqC
Coulomb’s law of electrostatics
V)f(Vf)Vf(
Calculate: ))r(fr(
r)]r(f[r)r(f))r(fr(
0
zyxzyx
zyx
r
)dr
df(r)r(f
0rr)dr
df(
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Example : verify that
)B(A)A(BB)A(A)B()BA(
Gradient of a Dot Product
BACCAB)CB(A BAC-CAB rule
B)A()BA()B(A
A)B()BA()A(B
)B(A)A(BB)A(A)B()BA(
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Successive Applications of
The divergence of the gradient : the Laplacian of
)z
zy
yx
x()z
zy
yx
x(
2
2
2
2
2
2
zyx
When φ is the electrostatic potential 0
Laplace’s equation of electrostatics
2
in the European literature
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Example : Calculate )r(g
Example :dr
)r(dgr)r(g )
dr
dgr()r(g
Example :dr
)r(dfr)r(f3))r(fr(
replacingdr
)r(dg
r
1)r(f
dr
)dr
)r(dgr1
(dr
dr
)r(dg
r
3)
dr
dgr()r(g
2
2
2
2
2 dr
)r(gd
dr
)r(dg
r
2]
dr
)r(gd
r
1
dr
)r(dg
r
1[r
dr
)r(dg
r
3
If nr)r(g 2n2n2n
2
n2n
n r)1n(nr)1n(nnr2dr
rd
dr
dr
r
2)r(
ttancons)r(g n = 0 0)r(g
r
1)r(g n = -1 0)r(g
A consequence of physics
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Successive Applications of : The curl of the gradient
zyx
zyx
zyx
0)xyyx
(z)zxxz
(y)yzzy
(x222222
All gradients are irrotational A mathematical identity !
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Successive Applications of : V
The divergence of a curl
zyxVVVzyx
zyx
V
)y
V
x
V(
z)
z
V
x
V(
y)
z
V
y
V(
xxyxzyz
0zy
V
zx
V
zy
V
yx
V
zx
V
yx
Vx
2
y
2
x
2
z
2
y
2
z
2
All curls are solenoidal
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Successive Applications of :
VV)V(
BACCAB)CB(A
BAC-CAB rule
Example : Maxwell’s equation (in vacuum)
0B
0E
t
EB
00
t
BE
t
BB
t
)E(t
E2
2
00
EEE
2
2
00 t
EE
The electromagnetic vector wave equation
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 8 向量分析 (Vector Analysis)
Successive Applications of :
Exercise : 試證明 P)a(a)P(P)a(a)P()Pa(
(74 台大材料 , 清華材料 )
)Pa()Pa()Pa(Pa
P)a(a)P(P)a(a)P(PPaa
BACCAB)CB(A
P)a(a)P(P)a(a)P(