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v
e L L θ
v cos θ
v L
vL = v cos θ
θ v L
v (L ) = vL
T (L ) = T L
T L
T L
A v
v v
v
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v w
u = v + w w v
v w
v − w
v + ( − w )
v
w
v w
u = v + w
v
w
u = v + w
α
v αv |α |
v v α > 0 v α < 0 α = 0 αv = 0
V
v + 0 = 0 + v = v
v + ( − v ) = ( − v ) + v = 0
v + w = w + v
u + ( v + w ) = ( u + v ) + w
α (v + w ) = αv + αw
(α + β ) v = αv + β v
α (β v ) = ( αβ ) v
1 v = v
(V, + , R ,∗)
V +
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R
∗
v e v v v = v / v
v = vv
v
v = v1 e 1 + v2 e 2 + v3 e 3 ,
e 1 , e 2 , e 3 v1 , v2 , v3
v v1 e 1 , v2 e 2 , v3 e 3 v
X 1X 2
X 3
v
v1 e 1
v2 e 2
v3 e 3
v ≡v1
v2
v3
= ( v1 v2 v3 )T .
v
v = v21 + v2
2 + v23 .
v w
v · w = v w cosθ ,
θ v w
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v · w = w · v
u · (αv + β w ) = αu · v + β u · w
v · v ≥ 0 v · v = 0 ⇔ v = 0
v · w = ( vv ) · (ww ) = v (v · w ) = w (v · w ) = vwcos θ ,
v w w v
v
vw
v · w = cos θ ,
v
v1 = v · e 1 , v2 = v · e 2 , v3 = v · e 3 .
e 1 · e 1 = 1 , e 1 · e 2 = 0 , e 1 · e 3 = 0e 2 · e 1 = 0 , e 2 · e 2 = 1 , e 2 · e 3 = 0
e 3 · e 1 = 0 , e 3 · e 2 = 0 , e 3 · e 3 = 1 ,
∗
v · w = v1 w1 + v2 w2 + v3 w3 .
a b a · b = 0 a b
v
v = ( v · v )1/ 2 .
∗ v · w = ( v 1 e 1 + v 2 e 2 + v 3 e 3 ) · (w 1 e 1 + w 2 e 2 + w 3 e 3 ) = v 1 w 1 e 1 · e 1 + v 1 w 2 e 1 · e 2 + · · ·
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v · w = ( v1 e 1 + v2 e 2 + v3 e 3 ) · (w1 e 1 + w2 e 2 + w3 e 3 )= v1 w1 e 1 · e 1 + v1 w2 e 1 · e 2 + v1 w3 e 1 · e 3
+ v2 w1 e 2 · e 1 + v2 w2 e 2 · e 2 + v2 w3 e 2 · e 3
+ v3 w1 e 3 · e 1 + v3 w2 e 3 · e 2 + v3 w3 e 3 · e 3 ,
i vi i = 1 , 2, 3
v vi
vi v1 , v2 , v3
3 × 3 tr( A ) =
3
i=1 Aii = A 11 +
A22 + A33
tr( A ) = A ii
i i = 1 , 2, 3
Aii = A11 + A22 + A33 v v = vie i
i vi e i = v1 e 1 + v2 e 2 + v3 e 3
vi wiAmn vkwkAmn viwi Amn = Bmn
i 1, 2, 3
i m n
n
E n
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30
v
X i vi = v · e i v 31
A
A A i = A · e i
3 × 3 = 3 2
A
A = A 1 A 2 A 3
A i
A i =A1 iA2 iA3 i
A
A =A11 A12 A13
A21 A22 A23
A31 A32 A33
A i
v = vi e i
A = A ij e ie j A = Aij e i ⊗ e j
⊗
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v w A
Aij = vi w j .
v ⊗w = w ⊗ vu ⊗ (αv + β w ) = αu ⊗ v + β u ⊗w(αu + β v ) ⊗w = αu ⊗w + β v ⊗w(u ⊗ v ) · w = ( v · w ) uu · (v ⊗w ) = ( u · v ) w
Aij = e i · A · e j ,
e i e j
e 1 e 1 =1 0 00 0 00 0 0
e 1 e 2 =0 1 00 0 00 0 0
e 1 e 3 =0 0 10 0 00 0 0
e 2 e 1 =0 0 01 0 0
0 0 0
e 2 e 2 =0 0 00 1 0
0 0 0
e 2 e 3 =0 0 00 0 1
0 0 0
e 3 e 1 =0 0 00 0 01 0 0
e 3 e 2 =0 0 00 0 00 1 0
e 3 e 3 =0 0 00 0 00 0 1
A = Aij e i e j
n v 1 , . . . , v n
T = v 1 v 2 · · · v n
n th
w
T · w = ( v 1 v 2 · · · v n ) · w = ( v n · w ) (v 1 v 2 · · · v n − 1 )
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(v n · w ) v 1 · · · v n − 1 (n − 1) th
T
(n − 1)th
n th T
T = T i1 i2 ··· in e i1 e i2 · · · e in .
i1 · · · in ∈ 1, 2, 3 e i1 e i2 · · · e in nth
n 3n
m th A n th
B (m + n) th
C
C = A ⊗B = ( Ai1 ··· im e i1 · · · e im ) ⊗ (B j 1 ··· jn e j 1 · · · e jn )= Ai1 ··· im B j 1 ··· jn e i1 · · · e im e j 1 · · · e jn
= C k1 ··· km + n e k1 · · · e km + n .
n th
A + B = ( Ai1 ··· in + B i1 ··· in ) e i1 · · · e in ,
αA = ( αA i1 ··· in ) e i1 · · · e in ,
n
3n
n th
3n 3n
A = Ai1 ··· im e i1 · · · e im B = B j 1 ··· jn e j 1 · · · e jn
A · B = Ai1 ··· im e i1 · · · e im · B j 1 ··· jn e j 1 · · · e jn
= Ai1 ··· im − 1 kBkj 2 ··· jn e i1 e i2 · · · e im − 1 e j 2 e j 3 · · · e jn
(m + n − 2) th k
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A mth B
n th r r ≤ m, n C = Ar
B (m + n − 2r )
C = Ai1 ··· im − r k1 ··· kr Bk1 ··· kr j r +1 ··· jn e i1 · · · e im − r e j r · · · e jn .
r
A : B ≡ A2
B
σ = C : ε C
σ ε σ12 σ11
Ar
B =B r A r = 1 A B A B
I
v
I · v = v · I = v .
δ ij = 1 , i = j0 , i = j .
i, j ∈ 1, 2, 3
(δ ij ) =1 0 00 1 00 0 1
.
e i
e i · e j = δ ij .
u iδ ij = u j Aij ··· m ··· zδ mn = Aij ··· n ··· z .
δ ii = 3 .
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ijk =+1 , (i , j ,k ) (1, 2, 3)− 1 , (i , j ,k ) (1, 2, 3)0 , i = j j = k k = i
ijk = − jik = − ikj = − kji = kij = jki .
3 × 3 A
det( A ) = ijk A1 iA2 j A3 j .
ijk lmn = detδ il δ im δ inδ jl δ jm δ jnδ kl δ km δ kn
jkl jmn = δ
kmδ
ln− δ
knδ
lm,
ijk ijm = 2δ km .
v1 = ( v1 e 1 + v2 e 2 + v3 e 3 ) · e 1 = e 1 · e 1 v1 + e 1 · e 2 v2 + e 1 · e 3 v3
v2 = ( v1 e 1 + v2 e 2 + v3 e 3 ) · e 2 = e 2 · e 1 v1 + e 2 · e 2 v2 + e 2 · e 3 v3
v3 = ( v1 e 1 + v2 e 2 + v3 e 3 ) · e 3 = e 3 · e 1 v1 + e 3 · e 2 v2 + e 3 · e 3 v3 .
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v
X 1
X 2
X 3
e1
, e2
, e3
v = v1 e 1 + v2 e 2 + v3 e 3 .
v X 1 X 2 X 3
e 1 , e 2 , e 3 v1 , v2 , v3
v1 = v · e 1 , v2 = v · e 2 , v3 = v · e 3 .
X 1X 2
X 3
v
X 1
X 2
X 3
v1
v2
v3
=e 1 · e 1 e 1 · e 2 e 1 · e 3
e 2 · e 1 e 2 · e 2 e 2 · e 3
e 3 · e 1 e 3 · e 2 e 3 · e 3
v1
v2
v3
,
v j = e j · e i vi .
Q = e j · e i
v = Qv .
Q
Q T Q = I Q − 1 = Q T
n th A X 1 X 2 X 3
A = A i1 ··· in e i1 · · · e in ,
X 1 X 2 X 3
A j1 ··· jn = An
e j 1 · · · e jn
= Ai1 ··· in e i1 · · · e in
ne j 1 · · · e jn
= e i1 · e j 1 · · · e in · e jn Ai1 ··· in ,
A
A = QAQ T , Q = e j · e i .
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θ X 1
X 1 X 2 X 3
A
(Qij ) =cosϕ sinϕ 0
− sin ϕ cosϕ 00 0 1
.
v
v = v1 v2 v3T
v1 , v2 , v3 ∈ R
X 1 v
v = v1 0 0 T
A ψ
A λ
A · ψ = λψ
λ
ψ
ψ
λ
X 1 ψ
A · e 1 = A11 e 1 + A21 e 2 + A31 e 3
A · e 1 = λe 1
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A11 = λ , A21 = 0 , A31 = 0 .
R 3
σ
σ =σ11 0 00 σ22 00 0 σ33
,
n
A n
ψ 1 , · · · , ψ n n
λ 1 , · · · , λn A
A = ΨΛΨ − 1
Ψ
n × n ψ Ψ = ( ψ 1 , · · · , ψ n )
Λ
n × n ith λ i
A = ΨΛ ΨT
Ψ
Λ
A
Λ =n
i=1
λ i n in i .
(A − λ I ) ψ = 0 .
det( A − λ I ) = 0
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A
A11 − λ A12 A13
A21 A22 − λ A23
A31 A32 A33 − λ= 0 ,
λ 3 − (A11 + A22 + A33 ) λ 2 + A22 A23
A32 A33+ A11 A13
A31 A33+ A11 A12
A21 A22λ −
A11 A12 A13
A21 A22 A23
A31 A32 A33
= 0
v
v = ( vi vi)1/ 2
f (v1 , v2 , v3 )
A
λ, λ 2 λ 3
A
IA = A11 + A22 + A33
IIA = A22 A23
A32 A33+ A11 A13
A31 A33+ A11 A12
A21 A22
IIIA =A11 A12 A13
A21 A22 A23
A31 A32 A33
I, II III A
IA = tr( A ) = λ 1 + λ 2 + λ 3
IIA = 12
(tr( A )) 2 − tr A 2 = λ 1 λ 2 + λ 2 λ 3 + λ 3 λ 1
IIIA = det( A ) = λ 1 λ 2 λ 3 .
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φ(A )
φ
A
φ = φ(IA , IIA , IIIA ) .
A A T A A T A
λ 2i A T A λi > 0 A
A A
v w
u = v × w
θ
v w u = v × w u v w
(v , w , u )
u
u = vwsin θ 0 ≤ θ ≤ π .
v × w = − w × v
u × (αv + β w ) = αu × v + β u × w
e 1 × e 1 = 0 , e 1 × e 2 = e 3 , e 1 × e 3 = − e 2
e 2 × e 1 = − e 3 , e 2 × e 2 = 0 , e 2 × e 3 = e 1
e 3 × e 1 = e 2 , e 3 × e 2 = − e 1 , e 3 × e 3 = 0 ,
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v × w = e 1 e2 e 3v1 v2 v3
w1 w2 w3
,
v × w = ijk v j wke i [v × w ]i = ijk v j wk .
v
w
w s
i n θ
θ
v × w
v w
A v w
A = 12
v × w .
A v
A × v = jmn Aim vn e ie j v × A = imn vm Anj e i e j .
u v w
u · (v × w ) ,
u
v
w
u · (v × w ) =u1 u2 u3
v1 v2 v3
w1 w2 w3
,
u · (v × w ) = ijk u i v j wk .
u v w
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u × (v × w ) = ( u · w ) v − (u · v ) w
(u × v ) × w = ( u · w ) v − (v · w ) u
(u × v ) · (w × x ) = ( u · w ) (v · x ) − (u · x ) (v · w )
(u × v ) × (w × x ) = ( u · (v × x )) w − (u · (v × w )) x
= ( u · (w × x )) v − (v · (w × x )) u
T = I + vw v wv · w = 0 T 2 , T 3 , · · · , T n eT eT =
∞n =0
1
n ! T n
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A (u) A u
A : R −→ T
u −→ A (u)
T T =R 3 × 3
T = R 3 T = R 3 × 3 × 3 × 3
O X 1
X 2
x ( t )
x (t) t
x
A (x ) A x
A : R 3 −→ T
x −→ A (x ) .
R 3
T (x )
x
ε (x )
ε x
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A (u)
dAdu
= lim∆ u→ 0
A (u + ∆ u) − A (u)∆ u
.
A (u) = Ai (u) e i
dAdu
= dAi
du e i =
dA1
du e 1 +
dA2
du e 2 +
dA3
du e 3 .
ei
A (u) = Aij (u) e ie j
A (u) , B (u) , C (u) a (u)
1. d
du (A · B ) = A ·
dBdu
+ dAdu
· B
2. d
du (A × B ) = A ×
dBdu
+ dAdu
× B
3. d
du [A · (B × C )] =
dA
du · (B × C ) + A ·
dB
du × C + A · B ×
dC
du
4. a · dadu
= adadu
5. a · dadu
= 0 a = const
4th 5th
A (x ) dA
dx dx = dx i e i dx
A (x ) = A (x1 , x 2 , x 3 )
∂ A∂x 1
= lim∆ x1 → 0
A (x1 + ∆ x1 , x 2 , x 3 ) − A (x1 , x 2 , x 3 )∆ x1
∂ A∂x 2
= lim∆ x2 → 0
A (x1 , x 2 + ∆ x2 , x 3 ) − A (x1 , x 2 , x 3 )∆ x2
∂ A∂x 3
= lim∆ x3 → 0
A (x1 , x 2 , x 3 + ∆ x3 ) − A (x1 , x 2 , x 3 )∆ x3
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∂ A∂x i= lim
∆ x i → 0A (x + ∆ x i e i) − A (x )∆ x i
.
dx
dA (x1 , x 2 , x 3 ) = ∂ A∂x i
dx i = ∂ A∂x 1
dx 1 + ∂ A∂x 2
dx 3 + ∂ A∂x 3
dx 3 ,
dA = grad( A ) · dx grad( A ) = ∂ A∂x i
e i .
grad( A )
A
grad( A ) = ∂ A∂x i
⊗ e i .
∇ ≡ e i∂
∂x i= e 1
∂ ∂x 1
+ e 2∂
∂x 2+ e 3
∂ ∂x 3
.
∗
grad( A ) = A ∇ = A ⊗∇ .
A (u) B (u) A (u) = d/du B (u)
A
A (u) du = B (u) + C
C = const A [a, b]
b
aA (u) du = B (b) − B (a) .
Aij (u) du = B ij (u) + C ij ,
Aij
∗ ⊗
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X 1X 2
X 3
C
P 1
P 2
x n
C P 1 (a1 , a 2 , a 3 ) P 2 (b1 , b2 , b3 )
N x 1 , . . . , x N − 1
A (x ) C
C A · dx =
P 2
P 1A · dx = lim
N →∞
N
i=1
A (x i ) · ∆ x i
∆ x i = x i − x i− 1 N → ∞
maxi
∆ x i → 0 .
C A · dx = C
(A1 dx 1 + A2 dx 2 + A3 dx 3 ) .
1. P 2
P 1A · dx = −
P 1
P 2A · dx
2. P 2
P 1A · dx =
P 3
P 1A · dx +
P 2
P 3A · dx P 3 P 1 P 2
x (s)
x1 = x1 (s) , x2 = x2 (s) , x3 = x3 (s) .
s
x 2
x 1
A · dx = s2
s1
A · dxds
ds .
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P 1 P 2
C A (x )
C A (x )
P 1 P 2
P 2 x
φ (x ) = x
P 1A · dx .
φ (x )
dφ = A · dx ,
A = grad( φ ) = φ∇ .
C A · dx
φ (x ) A = φ∇
φ (x ) A = φ∇
C
C A · dx = 0 ,
C C
X 2
X 3
X 1
n i∆ S
i
S
S
N ∆ S i i = 1 , . . . , N x i n i
A (x ) Ω ⊇ S
A S
S A · n dS = lim
N →∞
N
i=1
A (x i) · n i ∆ S i
N → ∞ : maxi
∆ S i → 0 .
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A S
S A × n dS = lim
N →∞
N
i=1
A (x i) × n i ∆ S i
A
A · n A A × n
S
S X 1 X 2 S
S A · n dS =
S
A · n dx1 dx 2
n · e 3.
dS dx1 dx 2
dx 1 dx 2 = ( n dS ) · e 3 = ( n · e 3 ) dS ,
n dS e 3 dx1 dx 2
v =1/r 2 e r
X 2
X 3
X 1
ndS
S
S
†
S
n Ω V
A (x ) S
Ω
D = 1V S
A · n dS C = 1V S
A × n dS ,
†
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V Ω V → 0
A
div( A ) = limV → 0
1V S
A · n dS
curl( A ) = limV → 0
1V S
A × n dS .
div( A ) = A · ∇
curl( A ) = A × ∇
u (x )
div( u ) = u · ∇ = u i∂ i curl(u ) = u × ∇ = kij u i ∂ j .
grad( A ) = A ∇ , div(A ) = A · ∇ , curl(A ) = A × ∇
φ = dx · ∇ φ limV → 0
1V S
n · A dS = ∇ · A limV → 0
1V S
n × A dS = ∇ × A
∇ × (∇ × v ) v
A
A = φ∇
A
div( A ) = div(grad( φ )) = φ∇ · ∇
∆ ≡ ∇ · ∇ = ∂ k∂ k .
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∆ A ∇2 A
∇4 A = ∆∆ A = ∆ (∆ A )
∆∆ φ = ∂ 4 φ∂x 4
1+
∂ 4 φ∂x 4
2+
∂ 4 φ∂x 4
3+ 2
∂ 4 φ∂x 2
1 ∂x 22
+ 2 ∂ 4 φ∂x 2
2 ∂x 23
+ 2 ∂ 4 φ∂x 2
3 ∂x 21
.
S Ω
V n A (x )
V
A · ∇ dV = S
A · n dS .
S
C
C A · dx = S
(A × ∇ ) · n dS
n
X 2
X 3
X 1
C
ndS
S
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v w A B α β
∇ (A + B ) = ∇A + ∇B
∇ · (A + B ) = ∇ · A + ∇ · B
∇ × (A + B ) = ∇ × A + ∇ × B
∇ · (αA ) = ( ∇α ) · A + α(∇ · A )
∇ × (αA ) = ( ∇α) × A + α(∇ × A )
∇ × (∇ × A ) = ∇ (∇ · A ) − ∇ 2 A
∇ · (v × w ) = ( ∇ × v ) · w + v · (∇ × w )
∇ × (v × w ) = ( w · ∇ )v − (∇ · v )w − (v · ∇ )w + ( ∇ · w )v
∇ (v · w ) = ( w · ∇ )v + ( v · ∇ )w + w × (∇ × v ) + v × (∇ × w )
A
∇ × (∇A ) = 0 (A∇ ) × ∇ = 0
∇ · (∇ × A ) = 0 (A × ∇ ) · ∇ = 0
A
S (A × ∇ ) · n dS = V
(A × ∇ ) · ∇ dV = 0
C (A∇ ) · dx = S
(A∇ ) × ∇ · n dS = 0 .
∇ · A × ∇ = 0 ∇ × A · ∇ = 0
φ ψ
V [φ (ψ∆) + ( φ∇ ) · (ψ∇ )] dV = S
φ (ψ∇ ) · n dS
V [φ (ψ∆) − (φ∆) ψ] dV = S
[φ (ψ∇ ) − (φ∇ ) ψ] · n dS .
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A φ
V A × ∇ dV = S
A × n dS C φ dx = S
(φ∇ ) × n dS .
V ∇ × A dV = S
n × A dS C φ dx = S
n × (∇φ) dS .
e 2
e 3
e 1
u2
u3
u1 u3 = c3
u 1 = c 1
u 2 =
c 2
P
(x1 , x 2 , x 3 ) (u1 , u 2 , u 3 )
x = x (u ) u = u (x )
det∂ (u1 , u 2 , u 3 )∂ (x1 , x 2 , x 3 )
=
∂u 1∂x 1
∂u 1∂x 2
∂u 1∂x 3
∂u 2∂x 1
∂u 2∂x 2
∂u 2∂x 3
∂u 3∂x 1 ∂u 3∂x 2 ∂u 3∂x 3
= 0
f (x ) f : R m → R n
m × n J = ∂f i∂x j
i = 1 , . . . , m j = 1 , . . . , n
P
u1 = const , u2 = const , u3 = const
P
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u i
r
∂ r /∂u i
ui
e 1 = ∂ r /∂u 1
∂ r /∂u 1, e 2 =
∂ r /∂u 2
∂ r /∂u 2, e 3 =
∂ r /∂u 3
∂ r /∂u 3
h i = ∂ r /∂u i
∂ r /∂u 1 = h1 e 1 , ∂ r /∂u 2 = h2 e 2 , ∂ r /∂u 3 = h3 e 3
e 1 , e 2 e 3 (u1 , u 2 , u 3 )
h i
e i
r
r (u1 , u 2 , u 3 )
dr = ∂ r∂u 1
du 1 + ∂ r∂u 2
du 2 + ∂ r∂u 3
du 3 = h1 du 1 e 1 + h2 du 2 e 2 + h3 du 3 e 3 .
dr 1 = h1 du 1 , dr 2 = h2 du 2 , dr 3 = h3 du 3
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ds
(ds)2 = dr · dr = h21 (du 1 )2 + h2
2 (du 2 )2 + h23 (du 3 )2 .
h i = 1
dr = dx1 e 1 + dx 2 e 2 + dx 3 e 3 (ds)2 = ( dx 1 )2 + ( dx 2 )2 + ( dx 3 )2 .
dS = dS n ,
dS = ( dS · e 1 ) e 1 + ( dS · e 2 ) e 2 + ( dS · e 3 ) e 3
dS · e i dS e i
dS 1 e 1 = dr 2 e 2 × dr 3 e 3 = h2 h3 du 2 du 3 e 1
dS 2 e 2 = dr 3 e 3 × dr 1 e 1 = h3 h1 du 3 du 1 e 2
dS 3 e 3 = dr 1 e 1 × dr 2 e 2 = h1 h2 du 1 du 2 e 3
dS = ( h2 h3 du 2 du 3 ) e 1 + ( h3 h1 du 3 du 1 ) e 2 + ( h1 h2 du 1 du 2 ) e 3
dS = dx2 dx 3 e 1 + dx 3 dx 1 e 2 + dx 1 dx 2 e 3 .
dV dr 1 e 1
dr 2 e 2 dr 3 e 3
dV = ( dr 1 e 1 ) · (dr 2 e 2 ) × (dr 3 e 3 ) = h1 h2 h3 du 1 du 2 du 3 .
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(u1 , u 2 , u 3 ) (q 1 , q 2 , q 3 )
du i ∂ r /∂u i hi
u (q )
du i = ∂u i
∂q jdq j
∂ r∂u i
= ∂ r∂q j
∂q j∂u i
∂ r∂u i
= j
∂ r∂q j
∂q j∂u i
21 / 2
.
dA q =∂ (q 1 , q 2 , q 3 )∂ (u1 , u 2 , u 3 )
∂ (q 1 , q 2 , q 3 )∂ (u1 , u 2 , u 3 )
− T · dA u
dV q =∂ (q 1 , q 2 , q 3 )∂ (u1 , u 2 , u 3 ) dV u
φ a = a1 e 1 + a 2 e 2 + a3 e 3
∇φ = 1h1
∂φ∂u 1
+ 1h2
∂φ∂u 2
+ 1h3
∂φ∂u 3
∇ · a = 1h1 h2 h3
∂ ∂u 1
(h2 h3 a1 ) + ∂ ∂u 2
(h3 h1 a 2 ) + ∂ ∂u 3
(h1 h2 a 3 )
∇ × a = 1
h1 h2 h3
h1 e 1 h2 e 2 h3 e 3
∂/∂u 1 ∂/∂u 2 ∂/∂u 3
h1 a1 h2 a 2 h3 a 3
∆ φ = 1
h1 h2 h3
∂ ∂u 1
h2 h3
h1
∂φ∂u 1
+ ∂ ∂u 2
h3 h1
h2
∂φ∂u 2
+ ∂ ∂u 3
h1 h2
h3
∂φ∂u 3
.
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X 2
X 3
X 1
x
rϕ
z = x3
x1
x2
(r,ϕ ,z )
x1 = r cosϕ , x2 = r sin ϕ , x3 = z
r = x21 + x2
2 , ϕ = arctan( x2 /x 1 ) , z = x3
hr = 1 hϕ = r h z = 1 ,
dr = dr e r + r dϕ eϕ + dz e z
dS = r dϕdz e r + drdz eϕ + rdrdϕ e z
dV = r dr dϕdz .
∇ ≡ e r∂ ∂r
+ eϕ1r
∂ ∂ϕ
+ e z∂
∂z
∇ · a = 1r
∂ ∂r
(ra r ) + 1r
∂a ϕ∂ϕ
+ ∂az
∂z
∆ φ = 1r
∂ ∂r
r∂φ∂r
+ 1r 2
∂ 2 φ∂ϕ 2 +
∂ 2 φ∂z 2 =
∂ 2 φ∂r 2 +
1r
∂φ∂r
+ 1r 2
∂ 2 φ∂ϕ 2 +
∂ 2 φ∂z 2
∇ × a =1r
∂a z
∂ϕ −
∂aϕ∂z
e r +∂a r
∂z −
∂az
∂reϕ +
1r
∂ ∂r
(ra ϕ) − ∂a r
∂ϕe z
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X 2
X 3
X 1
r
ϕ
θ
x3
x1
x2
(r,θ,φ )
x1 = r sin θ cos φ , x2 = r sin θ sin φ , x3 = r cos θ
r = x21 + x2
2 + x23 , θ = arccos( x3 /r ) , φ = arctan( x2 /x 1 ) .
h r = 1 hθ = r hφ = r sin θ ,
dr = dr e r + r dθ e θ + r sin θ dφ e φ
dS = r 2 sin θdθdφ e r + r sin θdrdφ e θ + rdrdθ e φ
dV = r 2 sin θdrdθdφ.
∇ ≡ e r∂ ∂r
+ e θ1r
∂ ∂θ
+ e φ1
r sin θ∂
∂φ
∇ · a = 1r 2
∂ ∂r
r 2 a r + 1r sin θ
∂ ∂θ
(sin θaθ) + 1r sin θ
∂a φ
∂φ
∆ ξ = ∂ 2 ξ ∂r 2 +
2r
∂ξ ∂r
+ cosθr 2 sin θ
∂ξ ∂θ
+ 1r 2
∂ 2 ξ ∂θ 2 +
1r 2 sin2 θ
∂ 2 ξ ∂φ 2
∇ × a = 1r sin θ
∂ ∂θ
(sin θaφ) − ∂aθ
∂φe r +
1r
1sin θ
∂a r
∂φ −
∂ ∂r
(ra φ) e θ
+1r
∂ ∂r
(ra θ) − ∂a r
∂θe φ
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