E3
(x1x2x3) e1, e2, e3
X E3 x1 x2 x3
3

x = x1e1 + x2e2 + x3e3 = xiei
i=1
x = {x1 x2 x3} = {xi}i=1,2,3
X X
x, y
v = x - y
vi = xi - yi, i = 1, 2, 3.
3 3
V E3 V
3
V
3
E3 V
3
E3 × E3 - V
(a1, a2) E3 (a1, a2) " E3 × E3
3 3
a3 = a2 - a1 V a3 " V
3 3
V × V - R
3

a · b = aibi
i=1
a

"
|a| = a · a = a2 + a2 + a2
1 2 3
Õ a b
a · b
cos Õ = 0 d" Õ d" 
|a||b|
x
xi = x · ei , i = 1, 2, 3
3

x = (x · ei) ei
i=1
x = xi ei
x = (x · ei) ei
i

0 i = j

ei · ej =
1 i = j
ei · ej = ´ij
´ij
a · b = (ai ei) · (bj ej) = aibj ei · ej = aibj ´ij = aibi
(aiei) · (bjej)
 i, j ´ij j
aibj ´ij i
ai bj ´ij = aibi
ai bj ´ij = ajbj
3 3 3
V × V V
a × b = (ai ei) × (bj ej) = aibj ei × ej
e1 × e2 = e3, e2 × e3 = e1, e3 × e1 = e2,
e2 × e1 = -e3, e3 × e2 = -e1, e1 × e3 = -e2,
e1 × e1 = 0, e2 × e2 = 0, e3 × e3 = 0.
eijk
Å„Å‚ üÅ‚ Å„Å‚ üÅ‚
+1
òÅ‚ żł òÅ‚ żł
eijk = -1 i, j, k
ół þÅ‚ ół þÅ‚
0
ei × ej = eijkek
a × b = aibjeijkek = (a2b3 - a3b2) e1 + (a3b1 - a1b3) e2 + (a1b2 - a2b1) e3
a × b c
c = ckek = aibjeijkek
ck = aibjeijk = ekijaibj
em
ek · em = ´km k m
m k
eijkeijk = 6
eijkeijl = 2´kl
eijkeilm = ´jl´km - ´jm´kl
i, j, k ij i
1 2 3
1 2 3
k, l j, k, l, m
(x1x2x3)
(x2 x2 x2 )
1 2 3
(x2 x2 x2 ) e2 , i = 1, 2, 3 x
1 2 3 i
x = xiei = x2 e2
i i
 x2 x
i
ej e2
j

xi = ei · e2 x2 , x2 = (e2 · ej) xj i, j = 1, 2, 3
j j i i
ei, e2 ei · e2
i j
xi x2
j

ei · e2 = cos xi, x2 = cos x2 , xi = Qji
j j j
e2 · ej = cos (x2 , xj) = Qij
i i
Qij
Qij Qij = Qji

x2 = Qijxj, xi = Qjix2
i j
x

e2 = (e2 · ej) ej = Qijej, ei = ei · e2 e2 = Qjie2
i i j j j
1 2
xi = QjiQjkxk
QjiQjk = ´ik
x2 = QijQkjx2
i k
QijQkj = ´ik
3 × 3
3 3
V v%EÅ‚V
2
v = viei = vie2
i
2 2
vi = Qijvj, vi = Qjivj
v
3
V vi i = 1, 2, 3
Qij
îÅ‚ Å‚Å‚
12 9 4
-
25 25 5
ïÅ‚ śł
ïÅ‚ śł
3 4
ïÅ‚ śł
[Qij] = 0
ïÅ‚ 5 5 śł
ðÅ‚ ûÅ‚
16 12 3
-
25 25 5
29 4 3
v {0, 1, -1} {xi} {- , , - }
25 5 25
{x2 } (2.33)1
i
12 9 4 29
2
v1 = · 0 - · 1 - · 1 = -
25 25 5 25
3 4 4
2
v2 = · 0 + · 1 - 0 · 1 =
5 5 5
16 12 3 3
2
v3 = - · 0 + · 1 - · 1 = -
25 25 5 25
f = f (x1, x2, x3)

" " "
gradf = "f = e1 + e2 + e3 f
"x1 "x2 "x3
"f "f "xj "f
= = Qij
"x2 "xj "x2 "xj
i i
"xj
= Qij
"x2
i
"f/"x2 grad f
i
("f)2 ("f)j
i
("f)2 = Qij ("f)j
i
"f
1
a, b
a2 b2 = QikakQimbm = QikQimakbm = ´kmakbm = akbk
i i
a · b
A
32 = 9 Aij (x1x2x3) Aij
A2 = QikQjlAkl Aij = QkiQljA2
ij kl
Aij A2 A (x1x2x3) (x2 x2 x2 )
ij 1 2 3
A 3n A
ij . . . k

n
n
A2 = QipQjr . . . Qks A
pr...s
ij...k

n n
n = 3
A2 = QipQjrQksAprs
ijk
n = 4
A2 = QipQjrQksQltAprst
ijkl
Qij
A Aij
A
T
2
3
V T =
2
3 3 3 3 3
V " V V × V T a, b " V
2
a " b " T
2
3
V ei = 1, 2, 3
ei " ej T T A " T
2 2 2
A = Aijei " ej
Aij A ei " ej
A Aij
3
V
3
V
A = a " b = (aiei) " (bjej) = ai bj ei " ej
n n T n
n
3
V
3 3 3
T = V " V " . . . " V
n

n
ei
A n 3n T
n
A = A ei " ej " . . . " ek
ij...k

n
3n Aij...k A ei " ej " . . . " ek
Aij...k
A1, A2 m A = Ä…A1 + ²A2 m Ä…, ²
Aij...k
A1 ij...k A2 ij...k
A m e" 2 Aijk...l Aiik...l
m - 2
A2 = QipQjrQks . . . QltAprs...t,
ijk...l
A2 = QipQir Qks . . . QltAprs...t = Qks . . . Qlt ´prAprs...t = Qks . . . QltArrs...t
iik...l

´pr Arrs...t
 A
Aii = A11 + A22 + A33 Df tr A
=
A2 = Aii A
ii
Aij A Aij = Aji A
Aij = -Aji A
A " T
2
1 1
Aij = (Aij + Aji) + (Aij - Aji)
2 2

Aij
Aij
Aijk...l = Ajik...l (i, j)
Aijk...l = -Ajik...l (i, j)
A " T2 Aij ei " ej AT " T2
Aij
AT = Aijei " ej
A = AT
A = -AT.
A, B " T2
(A + B)T = AT + BT
(AT)T = A
A " T B " T C = A " B m + n C " T
m n m+n
A = A ei " ej . . . " er B = B ek " el " . . . " es
ij...r kl...s


n
m
C = A B ei " ej " . . . " er " ek " el " . . . " es
ij...r
kl...s

m+n
" a b C = a " b
a = aiei, b = biej,
C = aibjei " ej
" A " B = B " A

" C = a " b tr C = a · b
A " T B " T C m + n - 2
m n
A B AB
" A " B = A B ei " ej " . . . ep " ek " el . . . " er
ij...tp kl...r


n
m
" A p B k
C = Aij...tpBpl...r
ij...tl...r

m+n-2
A " T B " T m e" n C m - n
m n
A B A · B
" A " B
" n
C = A B
ij...s ijskl...r kl...r


n
m-n
n
m
A " T B " T C = A · B = AijkBjkei
3 2
m = n
A, B " T
m
A · B = A B
ij...k ij...k

m m
a, b " T
1
a · b = aibi
A b
Ab = A · b
A " T a, b " T
2 1
a · Ab = A · (a " b)
A a1, a2
a1 · A · a2 = a2 · A · a1
A, B " T2
(AB)T = BTAT
 A2 " T A " T
2 2
A2 = AA
A
A = Aijei " ej A2 = AijAjkei " ek
An " T (n - 1)
2
A " T
2
An = AAA . . . A

n
n A " T
2
An = AijAjmAmp . . . Ask ei " ek

n
1 = ´ijei " ej
A " T B " T
2 2
BA = AB = 1
A B = A-1
A-1Amjei " ej = ´ijei " ej
im
32 = 9
A-1
A-1Amj = ´ij
im
det Amj = 0

Amj det Amj A
det Amj = det A2 = det A
mj
A det A = 0

Q " T a, b " T
2 1
(Qa) · (Qb) = a · b
QQT = QTQ = 1
QT = Q-1
det(QTQ) = det 1 = 1 = det QT · det Q = (det Q)2
det Q = Ä…1
Q det Q = 1
R R a " T1 b " T1
a
b = Ra
A " T trA = 0
2
A " T A = 1  " R
2
1 1
A = 1trA + AD Aij = ´ijAkk + AD
ij
3 3
A
1
Aii = AD + 3Aii
ii
3
trAD = AD = 0
ii
F " T det F = 0

2
F = RU = VR
U V
R
F
v v2
2
vi = Fijvj
v2 = 0 v = 0 F
v2
2 2 T
|v2 |2 = vivi = FijFikvjvk = FjiFikvjvk
T T
Cjk = FjiFik, C = F F
"
Df
U = C
Df
R = F U-1
R
T
RTR = (F U-1)T(F U-1) = U-1F F U-1 = U-1CU-1 = 1
 a ai A Aij
bi = Aijaj
b2 = A2 a2 = QipQjrApr Qjsas = Qip ´rsApr as = Qip Apsas = Qijbj
i ij j

A2 2 a2 Aps bp
ij j
bi = Ajiaj
"Aij
A " T A = A(x)
2
"xj
"A2 "(QikQjlAkl) "xm
"Akl "Akl "Akl
b2 Df ij = = QikQjl Qjm = Qik´lm = Qik Df Qikbk
= =
i
"x2 "xm "x2 "xm "xm "xl
j j
"ai
a " T a = a(x) 2
1
"xj
"a2 "(Qik ak) "xm "ak Df
A2 Df i = = Qik Qjm = QikQjmAkm
=
ij
"x2 "xm "x2 "xm
j j
Q A
îÅ‚ " " Å‚Å‚
1/ 2 1/ 2 0
ïÅ‚ śł
ïÅ‚ śł
" "
ïÅ‚ śł
[Qij] = -1/ 2 1/ 2 0
ïÅ‚ śł
ðÅ‚ ûÅ‚
0 0 1
îÅ‚ Å‚Å‚
1 -1 0
ïÅ‚ śł
ïÅ‚ śł
ïÅ‚ śł
[Aij] = 0 0 0
ïÅ‚ śł
ðÅ‚ ûÅ‚
0 2 0
A Q
2 2
1 1
2
A11 = Q1kQ1lAkl = A11Q11Q11 + A12Q11Q12 + A32Q13Q12 + 0 = " · 1 + " · (-1) = 0
2 2