−→ A =
n i=1 Ai
−→e
i
−→
−→
−→
−→
Ai
δ ij ij =
1 0
i=j
−→e .−→e i
i=j
j
i
qrs
Aklm ij
−→ e
= A1 e1 + A2 e2 + A3 e3 + ... + An en = (A1 , A2, A3 ,...,A n ) Bnp
= δ mn mn
T ijk ijk = T ikj ikj T ijk ijk =
−T
ikj ikj
y1 = a11 x1 + a12 x12 ∴
yk = aki xi
−→ −→ AB =A B 1
Aq 1
→y
k
= ak1x1 + ak2 x2 =
2
i=1 aki xi
⇒AB
B j
i
+ A2 B2 + A3 B3 = ABcos(θ ) jk
i
np
Ai + Bm = C qrs jk
pqr
jkpq r
Ai Bmn = C imn jkp
1kp
ikp
i=j
C im
1kp
2kp
3kp
N kp
C im = C 1m + C 2m + C 3m + ... + C N m
C 1m
n m
P (n, m) = n(n n! = m!(n m)!
−
−
− 1)(n − 2)...(n − m + 1)
C (n, m) =
1
e
e
ijk..l
= eijk...l =
1
0
ii δ ii ii = δ = N
aij δ ik ik = akj
eˆi
2
eˆi eˆ j =
⊗
3
⊗ eˆ = e
ijk eˆk
j
−
eˆk
−→ → → e ,− e ,− e 1
i = 1, 2, 3, ...N ...N
eˆk
0
e δ eijk eimn = δ jm δ kn δ jn δ km kn km 123...N e j1 j2 j3 ...jN = δ j1 j2 j3 ...jN
−
i
i
1
2
3
N
x = x (x ..., x ¯ , x¯ , x ¯ , ..., ¯ ) x¯i = x¯i (x1 , x2 , x3 ,...,x N )
x ∂x i = 0 J ( ) = =0 x ∂ x ¯ ¯i
| |
∂φ → ∂x
φ(x ¯i )
∇
∂φ ∂ x ¯ j ∂ x ¯ j ∂x i
=
i
∂ 2 φ ∂φ ∂ 2 x ∂ 2 φ ∂ x ¯ j ¯k ∂ x¯ j = + ∂x i ∂x m ∂ x ¯ j ∂x i ∂x m ∂ x ¯ j x¯k ∂x m ∂x i ∂φ ∂φ ˆ ˆ = = φ, i φ= n e e .gradφ i i i=1 ∂x i ∂x i ∂A i ∂A i . A = div A = N = = = Ai,i i=1 ∂x i ∂x i
−→ ∇ −→ −→ −→ ∇ × −→ A = B = eˆ .rot A = e A −→ −→ div F dv = F .n ˆ ds ↔ −→ ˆds = −→ F .dr ↔ e F (∇ × F ).n i
ijk
↔
v
k,j
s
s
)dxdy = − ∂F F dx + F dy ∂y −→ −→ −→ eˆ .∇ A = A ( B × ∇). A = e B A a a a | A| = e −→ −→ −→ −→ −→ A = A eˆ A = A .eˆ E (E , E , E ) −→ −→ −→ −→ E (E , E , E ) ⇒ eˆ = −→ E ⇒ | E | −→ −→ −→ E E = δ A = A E = A E −→ −→ −→ −→ E E = g = g E E = g = g g (
∂F 2 ∂x
ˆi ds k,j n
S ijk
C
F i,i dv =
v
ijk
1
C
j
i,k
1
2
2
1
3 i=1
i
i i
i
i
1
i i
i
i
j
ij
ji
i
2
3 i=1
i
ij
ji
i j
j
1
j
C
F i .nˆi ds
F i dxi
e3 jk F k,j dS =
C
F i dxi
i,jj
i1 i2 i3 ...iN
S
=
s
2
1i1
3i3
i
...aN iN
i
3
3
i
2i2
3 i=1
E i
−→−→
i
i
−→−→
Ai = A E 1
Ai = A E 1
g ij
ij
Ai = gik Ak
Ai = g ik Ak
Ai = Ai xi x2 (x1 , x2 , x3 )eˆ2 + x3 (x1 , x2 , x3 )eˆ3
−→ ∇ −→ −→ E j = gradx j =
dr =
T i =
dxi dt
¯ i = J w T
3 i=1
dxi eˆi =
x j
3 i=1
xi
2
1
j
∂ r dxi ∂ xi
2
3
1
1
¯i j dx T dx j
¯i = J w A j A
∂x j ∂ x ¯i
1
f = f (x1 , x2 , x3 ,...,x N )
→
N
3
→ −→
3
2
∂ x j
C : xi = xi (t) T (vectang) = i ¯ ¯i dx j dx dx C : x ¯i = x ¯i (x1 (t), x2 (t), x3 (t),...,x N (t)) = j dt d dt
¯(x¯1 , x¯2 , x¯3 , ..., x¯N ) A(x , x , x ,...,x ) = A 1
−r (x , x , x ) = x (x , x , x )eˆ + → −E → = ∂ −→r
⇒
¯ ∂ A ∂A ∂x j = j ¯i ¯i ∂ x ∂x ∂ x
∂A A j = ∂x j
N i i=1 T eˆi
¯ ∂ A ¯ Ai = ¯i ∂ x
lm...n = T ij...k E i E j ...E k E l E m ...E n qs
Arqp Br = C ps m
2
ds = dy dy
gij =
1 g
= m
= gij dxi dx j
m
gij =
h21
0
0 0
h22
0 0
0
h23
→
N i 2 i=1 (dy ) Adems m
: dy
m
∂y ∂y ∂x i ∂x j
∂y m j dx = ∂x j
∂y m ∂y m i j dx dx ∴ ds = ∂x i ∂x j 2
ds2 = h21 (dx1 )2 + h22 (dx2 )2 + h23 (dx3 )2
gij j
g ij gik = δ k
cof (gij ).
−→ A
Ai
Ai
mk nj A.nm g Aijk i.. = g
T pqrs = g pi g qj g rk g sm T ijkm
ds2 = gij dxi dx j
j
gij = δ i
⇒
V N g in = An Ai = Ai Ai = gin Ai An = A2 gij Ai A j = 1
g ij =
ijk =
√ ge
ijk
ijk =
√ 1g e
ijk
Ai = gij A j ; Ak = g jk A j p
T .jkm = g pi T ijkm
