Διανυσματικός Λογισμός-Λεοντάρης Γεώργιος
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Transcript of Διανυσματικός Λογισμός-Λεοντάρης Γεώργιος
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1 . .
pi 2005
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2 ,
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3
pi - pi pi pi pi . pi . , pi pi pi . pi pi - pi. , pi pi pipi - ij Kronecker ijk (LeviCivita). pipi pipi- pi pi pi . pi pi pi - pi pi . pi pi . - 3 4 pi pi , pi -. xii -. pi pipi pi . pipi pipi pi 5 6. - pi , .. - 7 , pi Stokes. pi - , 8 pi pi- 9 pi . pi pi pi pi , pi pi. pi- pi pi . , - pi pi pi , , .. - pi pi pi pi () pi pi pi.
, pi pi. pi pi - . pipi pi pi. pi pi pi pi pi. pipi pi
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4 pi, pipi pi pi pi pi.
- . pi - pi . pi pi . pi pi pi pi .
..., 29.01.2003
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Perieqmena
1 . 91.1 . . . . . . . . . . . . . . . 91.2 . . . . . . . . . . . . . . . . . 12
1.2.1 . . . . . . . . . . . 121.2.2 . . . . . . . . . . . . . . . . . . . . 141.2.3 pipi . . . . . . . . . . . . . . . . . . . 15
1.3 . . . . . . . . . . . . . . . 161.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.5 pi . . . . . . . . . . 23
1.5.1 pi . . . . . . . . . . . . . . . . 231.5.2 pi . . . . . . . . . . . . . . . 241.5.3 . . . . . . . . . . . . . . . . . 26
1.6 ij ijk . . . . . . . . . . . . . . . . . . . . . . 271.7 . . . . . . . . . . . . . . . . . . . . . . 29
1.7.1 . . . . . . . . . . . . . . . . . . . . . . . 301.7.2 . . . . . . . . . . . . . . . . 321.7.3 ijk pi. . 36
1.8 pi. . . . . 381.8.1 , . 381.8.2 ij , ijk . . . . . . . . . . . . . . 40
1.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2 472.1 . . . . . . . . . . . . . . 47
2.1.1 pi 482.2 pi . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.3 pi . . . . . . . . . . . . . . . . . . . . . 602.4 pi . . . . . . . . . . . . . . . . . . . . . . . 63
2.4.1 pipi . . . . . . . . . . . . . . . . . . . . . . . 672.5 . . . . . . . . 72
2.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . 732.6 pi pi. . . . . . . . . 752.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5
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6
3 pi, , . 833.1 pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.1.1 pi pi. . . . . . . . . . 853.1.2 pi, pipi pi. . . . . 87
3.2 . . . . . . . . . . . . . . . . . . . . . 883.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983.5 pi. . . . . . . . . . . 1023.6 pi . . . 103
3.6.1 . . . . . . . . . . . . . . . . 1043.6.2 . . . . . . . . . . . . . . . . . . 108
3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4 pi . 1154.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.2 pi . . . . . . . . . . . . . . . . . . . . . 120
4.2.1 pi (div). . . . . . . . . . . . . . . . . . . . . . . . 1204.2.2 (curl rot). . . . . . . . . . . . . . . . . . 1244.2.3 . . . . . . . . . . . . . . . . . . . . . . . 1264.2.4 pi (Laplacian). . . . . . . . . . . . . . . . . 132
4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5 pipi 1455.1 pipi . . . . . . . . . . . . . . . . . . . 1455.2 pi pi . . . . . . . . . . . . . . . . . . . . . . . 1515.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.3.1 . . . . . . . . . . . . . . . . . . 1565.3.2 . . . . . . . . . . . . . 160
5.4 . . . . . . . . . . . . . . . . . . . . . . 1635.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6 pi 1756.1 pi . . . . . . . . . . . . . . . . . . . . . . . 1756.2 pi . . . . . . . . . . . . . . . . . . . . 180
6.2.1 . . . . . . . . . . . . . . . . . . . 1806.2.2 pi pi pi. 185
6.3 pi . . . . . . . . . . . . . . . . . . . . . . . 1916.4 . . . . . . . . . . . . . . . . . 1936.5 pi pi. . . . 1966.6 pi . . . . . . . . . . . . . . . . . . . 1986.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
7 . 2057.1 pi . . . . . . . . . . . . . . . . . . . . 2057.2 Green. . . 2117.3 Green . . . . . . . . . . . . . . . . . . . . . . 2157.4 Stokes. . . . . . . . . . . . . . . . . . . . . . . 218
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7
7.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
8 pi 2338.1 pi , . . . . . 2338.2 . . . . . . . . . . . . . . . . . . . . . . . . . 2388.3 . . . . . . . . . . . . . . . . . . . . . . . . 243
8.3.1 Laplace . . 2488.4 Laplace: . . . . . . . . . . . . . . . . 2498.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
9 2599.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2599.2 . . . . . . . . . . . . . . . . . . . 2599.3 . . . . . . . . . . . . . . . . . . . . . . . 2659.4 N - . . . . . . . . . 2749.5 . . . . . . . . . . . . 2789.6 . . . . . . . . . . . . . . . . . . . . . . . 282
9.6.1 Christoffel . . . . . . . . . . . . . . . . 2869.6.2 . . . . . . . . . . . . . . . . . . . . . . . 287
9.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
293.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
.1.1 , pi Taylor . . . . . . . . . . . . 293.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 296.1.3 pi 5.31 . . . . . . . . . . . . . . 297.1.4 pi pi . . . . . . . . 298.1.5 pi . 300
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8
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Keflaio 1
Eisagwg sto
Dianusmatik Logism.
. pi - pi pipi. pipipi pi: Kronecker ij , Levi Civita ijk.
1.1 Snoyh Idiottwn twn Dianusmtwn
pi . pi pi . pi (. (1.1) ) M,N pi MN . MN pi, - MN , . pi -. pi pipi, MN NM . pi - pi pi . pi- .
, pi pi, 1. -, .
1
Sto keflaio twn tanustn ja meletsoume na diaforetik trpo orismo twn dianus-
mtwn.
9
-
10 1. .
M
N
.
.
MN
NM
1.1: MN - - - pi .
pi . pi, pi pi. (1.2) , .
B
A
A+BB
A
A
BA-
-
1.2: pi , pi .
,
~A (A1, A2, A3) (1.1)
pi A1, A2, A3 pi ~A .
pi () ~A, ~B pi
~A ~B = (A1 B1, A2 B2, A3 B3)
pipi pi ~A = (A1, A2, A3) (A1, A2, A3). ~A - | ~A| pi A. , .
-
1.1. 11
1.3: pi ~A .
pi . (1.3) i, j, k
i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) (1.2)
A1 = OP1, A2 = OP2 A3 = OP3. ,
~A = (A1, 0, 0) + (0, A2, 0) + (0, 0, A3)
= A1i+A2j +A3k (1.3)
pi pi - . pi - .1)
~A+ ~B = ~B + ~A (1.4)
2) pi
( ~A+ ~B) + ~C = ~A+ ( ~B + ~C) (1.5)
3) pi ~0
~A+~0 = ~A (1.6)
4) pi ~A = ~A~A+ ( ~A) = ~0 (1.7)
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12 1. .
5)pipi , R
( ~A) = () ~A (1.8)
6)
( ~A+ ~B) = ~A+ ~B (1.9)
(+ ) ~A = ~A+ ~A (1.10)
, pi pi pipi pi pi .
1.2 Eswterik Ginmeno Dianusmtwn
pi pi pipi -, pi . .
pi ~A ~B
~A ~B ~B ~A = A1B1 +A2B2 +A3B3 (1.11)
pi - . .
~A ~A = A21 +A22 +A23 (1.12)
pi A | ~A| =
~A ~A A21 +A
22 +A
23.
pipi pi . .., ~A ~A 0, ~A ~B = ~B ~A, ~A ( ~B + ~C) = ~A ~B + ~A ~C pi.
1.2.1 Idithtec tou Eswteriko Ginomnou
~A, ~B 1.4. ~A ~B pi
~A ~B = (A1 B1, A2 B2, A3 B3) (1.13)
~A ~B pi pipi
| ~A ~B|2 = (A1 B1)2 + (A2 B2)2 + (A3 B3)2= | ~A|2 + | ~B|2 2(A1B1 +A2B2 +A3B3) (1.14)
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1.2. 13
1.4: ~A, ~B, ~A ~B () pi | ~B|cos,| ~B| sin | ~A| | ~B|cos ().
pi pi ( ) pi
| ~A ~B|2 = | ~B|2 sin2 + (| ~A| | ~B|cos)2= | ~A|2 + | ~B|2 2| ~A|| ~B| cos (1.15)
(1.14) (1.15)
~A ~B = | ~A|| ~B| cos (1.16) pi (1.13) -
cos =3
i=1AiBi
| ~A|| ~B| (1.17)
. pi ~A, ~B. pi ~B pi ~A.
1.5: pi ~B|| ~B ~B ~A.
pi . pi ~B ~A pi . pi
~B|| = | ~B| cos ~A
| ~A|
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14 1. .
(1.17)
~B|| =~A ~BA2
~A (1.18)
pipi
~B = ~B ~B|| = ~B ~A ~BA2
~A (1.19)
1.2.2 Exswsh thc Eujeac
(x0, y0, z0) R3 , ~r0 = (x0, y0, z0) pi pi ~r0 pi pi ~A. ~r = (x, y, z) , ~r ~r0 pi ~A = (A1, A2, A3). ,
~r ~r0 = ~A (1.20)
pi pi pi pi , - (1.20)
x x0A1
=y y0A2
=z z0A3
(1.21)
y
x
y
z
r-r
r r0
0
e
A
1.6: ~r ~r0// ~A.
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1.2. 15
1.2.3 Exswsh tou Epipdou
~A, ~B O. ~r0 pipipi . , pipi ~r pi
~r ~r0 = ~A+ ~B (1.22)pi , . n pi pipi ~A, ~B , (. (1.7) ) pi pipi
(~r ~r0) n = 0 (1.23) pi ,
A
B
n^
1.7: pipi pi ~A, ~B. pi n.
n1x+ n2y + n3z = c, c = constant (1.24)
pi c = n1x0 + n2y0 + n3z0.. n1, n2, n3 Ai, Bi.. pi n pipi ,
n ~A = 0 3i=1 niAi = 0n ~B = 0 3i=1 niBi = 0
pi pi n1,
n1 = n2A2 + n3A3A1
=n2B2 + n3B3
B1(1.25)
pi
n2 = n3A3B1 A1B3A1B2 A2B1 (1.26)
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16 1. .
n1 = n3A2B3 A3B2A1B2 A2B1 (1.27)
n
n = 1(A2B3 A3B2, A3B1 A1B3, A1B2 A2B1) (1.28)
pi pi pi pi pi |n| = 1.
2 = (A1B2 A2B1)2 + (A3B1 A1B3)2 + (A2B3 A3B2)2= A21(B
22 +B
23) +A
22(B
21 +B
23) +A
23(B
22 +B
21)
2(A1B1A2B2 +A2B2A3B3 +A1B1A3B3)= A21( ~B
2 B21) +A22( ~B2 B22) +A23( ~B2 B23)2(A1B1A2B2 +A2B2A3B3 +A1B1A3B3)
= ~A2 ~B2 {(A1B1)2 + (A2B2)2 + (A3B3)2}2{(A1B1)(A2B2) + (A2B2)(A3B3) + (A1B1)(A3B3)}
= ~A2 ~B2 (A1B1 +A2B2 +A3B3)2= ~A2 ~B2 ~A2 ~B2 cos2 = ~A2 ~B2 sin2 (1.29)
(1.28) 6= 0, pi,
n =(A2B3 A3B2)i+ (A3B1 A1B3)j + (A1B2 A2B1)k
| ~A| | ~B| sin (1.30)
= 0, ~A// ~B. n pi pipi ~A, ~B.
1.3 Dianusmatik Exwterik Ginmeno
~A, ~B pipi pi pi . pi pipi pi | ~A| pi pi ~B pi ~A. - pipi pi ~A, ~B | ~A| pi ~B.
~A ~B = n| ~A|| ~B| sin (1.31)
pi n pipi ( ~A, ~B) .
-
1.3. 17
A
B
AxB
n^
1.8: pi .
ij
k
^^
^
xy
z
1.9:
pipi , pi .
~A ~B = ~B ~A (1.32) (1.31) i, j, k (x, y, z),
i j = j i = kj k = k j = ik i = i k = j
pi
i i = j j = k k = 0 pi (1.30) pi (1.31),
~A ~B = (A2B3 A3B2)i+ (A3B1 A1B3)j + (A1B2 A2B1)k (1.33) (1.33) pipi pi pi pi
~A ~B =
i j kA1 A2 A3B1 B2 B3
(1.34)
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18 1. .
pi (1.31), pi | ~A ~B| pipipi pi pi ~A, ~B.
.1.) pipi pi pi (1.10). pi ~ z. pipi pi pi O . v pi r pi pi r sin . pi
~v = ~ ~r (1.35) pi pi ~F pi ~r,~M = ~r ~F .
r sin q
r
q
w
v
O
1.10: pi r sin pi z, pi ~ pi z .
2.) pi = 1/2
pi 2x = y = z. pi pi (2,3,5).pi. pi pi, ~ (1,2,2), pi
~ = 132
(i+ 2j + 2k
)(1.36)
pi
~v = ~r
= 132
i j k1 2 22 3 5
= 132
(4i j k
)(1.37)
-
1.4. 19
|~v| = 1.3.) .pi. pi ~r ~r0 pi ~A, ,
(~r ~r0) ~A = ~0 (1.38) , pi
(y y0)A3 = (z z0)A2(x x0)A3 = (z z0)A1(x x0)A2 = (y y0)A1
pi (1.21).
1.4 Askseic
1). n1, n2 pipi, cos(12) . sin(1 2).
1,2 pi n1, n2 x,
n1 = cos1i+ sin1jn2 = cos2i+ sin2j
n1 n2 = cos1 cos2 + sin1 sin2pi (1.16), pi
n1 n2 = |n1| |n2| cos(2 1)pi 2 1 . pi pipi
cos(1 2) = cos1 cos2 + sin1 sin2 (1.39) pi
n1 n2 = k|n1| |n2| sin(2 1)
n1 n2 =
i j kcos1 sin1 0cos2 sin2 0
= k(sin2 cos1 cos2 sin1),
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20 1. .
pi
sin(1 2) = sin1 cos2 cos1 sin2 (1.40)
2). ~A, ~B ~C pipi . ~C ~A, ~B.
. ~C = ~A + ~B. , ~A, ~B , pi ~B ~A = 0 ~B, ~A
~A ~C = ~A ( ~A+ ~B) ~A ~C = | ~A|2 (1.41)
~B ~C = ~B ( ~A+ ~B) ~B ~C = | ~B|2 (1.42)
,
~C =~A ~C| ~A|2
~A+~B ~C| ~B|2
~B
3). pi pi pi pi .. ~A, ~B pi pi. , pi pi - ~A+ ~B. ~C pi pi pi pi pi -.
~C =13( ~A+ ~B) (1.43)
1.11: ~C 1/3 .
pi ~C ~A+ ~B,
~C = s( ~A+ ~B) (1.44)
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1.4. 21
pi s . pipi, pi ~C pi pi ~A ~B
~C ~A = t(12~B ~A)
~C = ~A+ t(12~B ~A) (1.45)
(1.44,1.45)
s( ~A+ ~B) = ~A+ t(12~B ~A)
(s t2) ~B = (1 s t) ~A
pi ~A, ~B , pi ,
s t2= 1 s t = 0
pi pi pipi s = 1/3 t = 2/3.4). M,P,N O
OP =OM+
ON
1+ . , AB K,,M - pi B,A,AB, pi pi pi K,,M 1.
O
M
N
R
A
B
G
K
L
M
(a)
(b)
1.12:
. pi M,P,N , pi (pi () (1.12) )
ON OM = (OP ON)ON +
ON =
OM +
OP
OP =
OM +
ON
1 + (1.46)
, x, y, z ,BK = x
K,
= yA,
AM = z
MB. O -
, OA = ~a,
OB = ~b,
O = ~c,
OK = ~k,
O = ~l,
OM = ~m. ,
-
22 1. .
pi pi ,
~k =~b+ x~c(1 + x)
(1.47)
~l =~c+ y~a(1 + y)
(1.48)
~m =~a+ z~b(1 + z)
(1.49)
pi ~c pi pi = 1xy , s = 1y ,
( + s)~k + (1 s)~l1 +
=~a+ ~b1 +
(1.50)
(1.50) pi pi pi A,B, pi pi K , M . pi, (1.49) (1.50), pi z = 1xy , xyz = 1.
5). pipi x + y + z = 21 x 1 = y + 2 = 2z + 3.
f
epipedo x+y+z=21
e
u
q
e
i
a
x-1=
y+2=
2z+2
1.13: pi x1 = y+2 =2z + 3 pipi x+ y + z = 21.
. pipi pi pipi (1.23) (n1x + n2y + n3z = c), pi pipi (1, 1, 1),
n =13(i+ j + k) (1.51)
pi, pi
-
1.5. . 23
(1.21), pi -
~A = i+ j +12k (1.52)
pi ~A, n,
cos =n ~A|n|| ~A| =
533
pi pipi, pi ,
=pi
2 cos1 5
33
1.5 Ta Tripl Ginmena. Dianusmatikc Tauttht-
ec
pi ~A ~B ~C ~A ( ~B ~C) pipi pi .
1.5.1 To Mikt Tripl Ginmeno
pi ~A ( ~B ~C) pi pi pi
~A ( ~B ~C) =3i=1
Ai( ~B ~C)i (1.53)
= A1(B2C3 B3C2) +A2(B3C1 B1C3) +A3(B1C2 B2C1)(1.54)
pi pi
~A ( ~B ~C) =A1 A2 A3B1 B2 B3C1 C2 C3
(1.55) pi , pi pi ~E = ~B ~C pipi pi ~B, ~C. n , ~E = n| ~B| | ~C| sin,pi ~B, ~C , | ~E| pi pi pi ~B, ~C. h = ~A n ~A pi pipi ~B, ~C, pi
| ~A ( ~B ~C)| = Eh = V (1.56)
-
24 1. .
pipipi pi . pipi pi ~A pipi , n ~A = 0, pi pi . pipi pi , (1.55) ~A ~B, ~C. pi (1.53), (1.55)
1.14: pi pipipi, V = | ~A ( ~B ~C)|.
pi pi
~A ( ~B ~C) = ~C ( ~A ~B) = ~B (~C ~A) (1.57)
, pi pi pi .
. (1.15) ( - pi e) pi pi O. pi ~Fpi A pi pi
~M = (~r ~F e)e (1.58)
pi ~r ~F e : - pi pi O pi ~M = ~r ~F . ~M pi O. pi ~M pi pi ~M e ~r ~F e.pi ~M pi pi pi (1.58).
1.5.2 To Tripl Exwterik Ginmeno.
pi ~A ( ~B ~C). - ~B ~C ~B, ~C . ~A pipi ( ~B, ~C) (pi 1.16). pi ~B, ~C
~A ( ~B ~C) = ~B + ~C (1.59)
-
1.5. . 25
O r
FM
Mo
e^
S
1.15: pi pi pi ~r ~F e.
pi , pi pi. pipi (1.59) ~A, ,
~A ~B + ~A ~C = 0 (1.60)
pi pi = ~A ~C = ~A ~B, . (1.59)
~A ( ~B ~C) = {( ~A ~C) ~B ( ~A ~B)~C} (1.61)
pi . , , . pi, pi pipipi ~P , ~Q,
(~P ~Q)2 = |~P |2| ~Q|2 sin2 = |~P |2| ~Q|2(1 cos2 )= |~P |2| ~Q|2 (~P ~Q)2 (1.62)
pipi pi- (1.61). , pi (1.61) | ~A|| ~B||~C| {
A (B C)}2
= A2(B C)2 [A (B C)]2
= 1 cos2 [A (B C)]2
-
26 1. .
B
C
ABxC
Ax(BxC)
1.16: pi .
(1.61) ,
2{(A C)2 + (A B)2 2(A C)(A B)(B C)}= 2(cos2 + cos2 2 cos cos cos)
pipi
[A (B C)]2 = 1 cos2 2(cos2 + cos2 2 cos cos cos) (1.63)
pi (1.57) pi 2 = 1, = 1. pipi pi = +1.
1.5.3 Dianusmatikc Tautthtec
pipi pi pi pi .
~A ( ~B ~C) = ( ~A ~C) ~B ( ~A ~B) ~C (1.64)( ~A ~B) (~C ~D) = ~A (~C ~D) ~B ~B (~C ~D) ~A (1.65)( ~A ~B) (~C ~D) = ( ~A ~C) ( ~B ~D) ( ~A ~D) ( ~B ~C) (1.66)
(1.64) pi pi pipi pi pi pi- . ( pi pi .) pi-pi pi (1.64).
-
1.6. IJ IJK 27
1.6 Ta Smbola ij kai ijk
pi pi pipi Kronecker ij ijk. - pipi . i, j, k x1, x2, x3 ~A, A1, A2, A3.
~A = A1x1 +A2x2 +A3x3 Aixi (1.67)
(1.67), , pi pi pi pi -,
3i=1
Aixi Aixi (1.68)
Kronecker
ij ={
1 i = j0 i 6= j (1.69)
, .
pi - xi, i = 1, 2, 3
xi xj = ij (1.70), ij pi
~A ~B = (Aixi) (Bj xj)= AiBjij= AiBi
ijk =
+1 ijk = {123, 312, 231}1 ijk = {132, 321, 213}0 pi (1.71), +1 {123} , 1 {132} , pipi.
-
28 1. .
, , LeviCivita. ,
ijk = ikj = kij . (1.72) ijk
~B ~C = ijkxiBjCk (1.73)pi pi pi pi pi pi .
pi pi (1.71) pi
ijk = ei (ej ek). (1.74)
pi pi - pi
ijkmnk = imjn injm (1.75)
. pi pi pi
ijkpqr = ipjqkr + iqjrkp + irjpkqiqjpkr irjqkp ipjrkq (1.76)
pi pi (1.75) pi pipi .
. pi
ilmjlm = 2ij (1.77)ilmilm = 6 (1.78)
(1.75)
ilmjlm = ijll illj (1.79)pi pi pi pi- ,
ll 3l=1
ll = 3.
pi illj = ij .
ilmjlm = 3ij ij = 2ij
-
1.7. . 29
pi i = j (1.79). pi
ilmjlm = 3 3 3 = 6. pipi pi
(1.64). (1.73)
~A ( ~B ~C) = ijkxiAj( ~B ~C)k (1.80)pi xi
ijkAj( ~B ~C)k = ijkAj(kmnBmCn)= ijkmnkAjBmCn (kmn = mnk)= (imjn injm)AjBmCn= AnCnBi AmBmCi= ( ~A ~C)Bi ( ~A ~B)Ci
pi xi pi i,pipi (1.64).
. pi (1.53)
~A ( ~B ~C) = Ai( ~B ~C)i= ijkAiBjCk (1.81)= 1jkA1BjCk + 2jkA2BjCk + 3jkA3BjCk
pi pi pi :i = 1, 2, 3. pi j, k pi pi pi pi (1.71). , pi
1jkA1BjCk = 123A1B2C3 + 132A1B3C2 = A1B2C3 A1B3C2pi,
2jkA2BjCk = 213A2B1C3 231A2B3C1 = A2B1C3 +A2B3C13jkA3BjCk = 312A3B1C2 + 321A3B2C1 = A3B1C2 A3B2C1
, pi (1.54).
1.7
Peristrof twn Axnwn.
pi pi pi . pi . pi pipipi . pi , pi pipi ~r.
-
30 1. .
1.7.1 Do Diastseic.
, pi ~r = xi + yj. pi - pi (1.17). pi pi , (x, y) pi (x, y)
x = x cos + y sin (1.82)y = x sin + y cos (1.83)
x
x
yy
y
x
x
r
q
y
1.17: - pi .
, pi (A1(x, y), A2(x, y)) pi pi - pi
A1 = A1 cos +A2 sin (1.84)A2 = A1 sin +A2 cos . (1.85)
(1.84,1.85) (A1A2
)=
(cos sin sin cos
)(A1A2
)(1.86)
(1.86) - , - pi. pi 2 2
R() =(
cos sin sin cos
)(1.87)
RT () = R1() (1.88)RT ()R() =
(1 00 1
)(1.89)
-
1.7. . 31
, pi R1() = R() pi.
. pipi (1.18).
xx
y
y
x
x
y
y
q
-q
-q
1.18: .
. i, j.
pi. i (1, 0). pipi, (1.86) pi(
x
y
)=
(cos sin sin cos
)(10
)=
(cos sin
)(1.90)
(0, 1) pi,(x
y
)=
( sin cos
)(1.91)
, pi
i = i cos + j sin j = i sin + j cos (1.92)
1. pi pi pi , pi pi : ,
i = 11i+ 12jj = 21i+ 22j
-
32 1. .
ij pi . pi pi i,
i i = 11(i i) + 12(j i)= 11
, 11 = i i pi - i, i, 11 = cos . pipi .
2. ~r = xi+ yj,
~r = r(i cos + j sin ) (1.93)
pi r = |~r| pi pipi pipi ~r x. ~r
er = i cos + j sin (1.94)
. pi (xOy).
pi. pi pi pi. , pi (1.94). pi pi pi x, e ,
e er = 0e = i sin + j cos (1.95)
, pi .
1.7.2 Strofc se Treic Diastseic.
pi . pi
~r = xixi= x1x1 + x2x2 + x2x3 (1.96)
, pi ,
~r = x1x1 + x
2x2 + x
2x3 (1.97)
pi xi, - , xi xj = ij . pi (1.97)
xj ~r = xi(xj xi)= xiji = x
j (1.98)
-
1.7. . 33
1.19: 1j .
pi (1.96)
~r xj = xixi xj x1(xj x1) + x2(xj x2) + x3(xj x3) (1.99)
(1.98,1.99),
xj = x1(xj x1) + x2(xj x2) + x3(xj x3) (1.100)
pi x1x2x3
= (x1 x1) (x1 x2) (x1 x3)(x2 x1) (x2 x2) (x2 x3)
(x3 x1) (x3 x2) (x3 x3)
x1x2x3
(1.101) pi O,
O = (x1 x1) (x1 x2) (x1 x3)(x2 x1) (x2 x2) (x2 x3)
(x3 x1) (x3 x2) (x3 x3)
(1.102)pi, xi = ijxj , pi pi ij = x
i xj cos(xi, xj)
(pi (1.19).)
pi pi- .
. (A1, A2, A3) pi - ~A pi (1.101).
-
34 1. .
,
xi ~r = xi = (xi xj) (1.103)
(1.103) pi
O = (x1 x1) (x1 x2) (x1 x3)(x2 x1) (x2 x2) (x2 x3)
(x3 x1) (x3 x2) (x3 x3)
(1.104), piO pipi pi O
. , pi pipi 2 2, O O, O = OT . pi, pi (1.104) (1.102) pi O = O1,
O = O1 = OT (1.105)
pi - ( ) . (1.105)
OTO = OOT = I (1.106)
pi I pi pi 33. (1.106) pi pi pi pi .
. pi
pi x3 . y1 = (1,1, 0)/
2, y2 = (1, 1,2)/
6, y3 =
(1, 1, 1)/3, pi . , -
pi x1, x2, x3. . pi
yi pi , pi . pi,
y1 y1 = 12(1,1, 0) (1,1, 0) =12(1 + 1 + 0) = 1
y1 y2 = 12
16(1 1 + 0) = 0
y1 y3 = 12
13(1 1 + 0) = 0
pipi yi yj = ij yi .
pi pi pipi pi pi -. pi pi yi xj .
-
1.7. . 35
pi,
y1 x1 = 12(1,1, 0) (1, 0, 0) = 1
2(1.107)
y1 x2 = 12(1,1, 0) (0, 1, 0) = 1
2(1.108)
, pi
O = (y1 x1) (y1 x2) (y1 x3)(y2 x1) (y2 x2) (y2 x3)
(y3 x1) (y3 x2) (y3 x3)
pi
OT =
12
16
13
12
16
13
0 26
13
(1.109)
-
36 1. .
1.7.3
Efarmogc tou Sumblou ijk sthn 'Algebrapinkwn.
pi pi (1.54) pi (1.55)pi . pi (1.81), pi
D = ijkAiBjCk A1 A2 A3B1 B2 B3C1 C2 C3
(1.110) Ai = u1i, Bi = u2i, Ci = u3i, pi
U =
u11 u12 u13u21 u22 u23u31 u32 u33
(1.111) (1.110)
D = detU = ijku1iu2ju3k (1.112)
, pi Dlmn = ijkuliumjunk, pi l,m, n. pi {l mn} = {1 2 3} D, D, pipi pipi . pi Dlmn pi Dlmn = lmnD. pi lmn , pipi pi U uij pi
D = detU =16pqrupluqmurnlmn (1.113)
.
ijk detU = lmnuilujmukn (1.114)lmn detU = ijkuilujmukn (1.115)
pi pi .pi. pi pi pipi
pi . , pipi pi ijk pi ijk, ijkijk = 6 (1.113). pi . , pi pi pi(1.113) (1.114) pipi ijk
ijk detU =16ijkpqrupluqmurnlmn (1.116)
-
1.7. . 37
pi (1.76) ijkpqr pi pipi pi Kroneker, . ijkpqr = ipjqkr+ . , pipi (1.116). pi pi (1.76)
ipjqkrupluqmurnlmn = uilujmuknlmn (1.117)
, pi pipi (1.76)
iqjpkrupluqmurnlmn = ujluimuknlmn= ujmuiluknmln= uilujmuknlmn (1.118)
(pi l m mln = lmn.)pi (1.114). pi (1.115).
. pi U uij . pi pi (pi .15)
vpj =12D
pkljmnumkunl (1.119)
pi. pi uip pi vpj
uipvpj =12D
pkljmnuipumkunl
=12D
jmn (pkluipumkunl) (1.120)
(1.115) pi- pi imnD pi
uipvpj =12D
jmn (imnD)
=12jmnimn
=122ij = ij (1.121)
. Xi+ijkXjPk = Qi pi Xi, {i, j, k =1, 2, 3}.
pi. pi pipi , pi Q1, Q2, Q3.pi - pi. pi, i = 1, pi
-
38 1. .
X1 +X2P3 X3P2 = Q1 . , Xi - Xi = Xjij . , ( ij + ijkPk)Xj = Qi.pi pi Aij = ij + ijkPk pi AijXj = Qi. pi pi pi (A1) pi (1.119)
2D (A1)jk = jpqkabAapAbq= jpqkab(ap + aplPl)(bq + bqmPm)= 2jaqkaq + jaqkabbqmPm+ jpqkaqaplPl + jpqkabaplbqmPlPm (1.122)
pi D detU pi pi (1.113). pi- (1.122) (1.75) pi klijl = ijk. -pi ijk pi (1.122)
~X =1D(2 ~Q ~P ~Q+ (~P ~Q)~P ) (1.123)
1.8
Dianusmatikc Logismc me Hlektron-
ik Upologist.
pi pi pi. pi pi- pipi pi. pi pipi . pi- pi Mathematica pi pi Maple.
1.8.1 Eswterik kai Exwterik Ginmeno Dianus-
mtwn, Strofc.
pi - pi . , pi v1, v2,
v1:= {a1,b1,c1};, v2:= {a2,b2,c2};
pi pi
v1+v2
pi pi
{a1+a2,b1+b2,c1+c2}
-
1.8. .39
.
v1.v2
pi
{a1 a2, b1 b2, c1 c2}
Cross[v1,v2]
pi pi
{-b2 c1 + b1 c2, a2 c1 - a1 c2, -a2 b1 + a1 b2}
pi Mathematica, pi { } . pi, 2 2 pi
m =(
a bc d
)(1.124)
mathematica
m={{a,b},{c,d}}
( pi pi m (1.124), - MatrixForm[m]). pi ( v1)
m.v1
pi pi pi pi. pi
pi. pi, 2 2 pi R(q),(q =) pi pi . pi () q Mathematica
R[q ] :={{Cos[q],Sin[q]},{-Sin[q],Cos[q]}}
q, pi. pi pi pi , pi pi pi Mathematica :
R(0)
-
40 1. .
{{1,0},{0,1}}
1.8.2 Prxeic me ta Smbola ij, ijk
pipi pi pi - ij , ijk Mathematica. Kronecker KroneckerDelta[i,j]pi, pi pipi i pi ( v1 pi ), v1[[i]]. pi, pi -
Sum[KroneckerDelta[i,j]*v1[[i]]*v2[[j]],{i,3},{j,3}]
Sum[ ] pi pi i, j. pi -
. pi Signature[{1,2,3}] pi +1 1, 2, 3 1 pi, . pi eps[i ,j ,k ]:=Signature[{i,j,k}] pi pi, pi Sum[eps[1,j,k] v1[[j]] v2[[k]],{j,3},{k,3}]pi pi - comp[i ]:=Sum[eps[i,j,k] v1[[j]] v2[[k]],{j,3},{k,3}] pipi pi. pi, v3={a3,b3,c3} Sum[comp[i] v3[[i]],{i,1,3}]
-
1.9. 41
1.9 Askseic
1) 9 Newton, pi/4 y pi , pi pi pi (1,2) (5,4). pi .
2) pipi pi ~v. n pipi pi, n ~v pi pi .
3) n pi
~rc.m. =n
i=1mi~rini=1mi
(1.125)
~r , -
~A(~r) =ni=1
mi(~ri ~r) (1.126)
pi . 1) ~A(~rc.m.) =0. 2) pipi m1,m2,m3. ~A(~r2) = 0 .
4) n ~r1, ~r2, . . . ~rn. pi pi . pi pi pi pi 1/2 - pi . 1/3 pi pi pi pi 1/4 pi pi pi . pi pi;
5) x 13
=y 34
=z
5x 12
= 3 y = 2z
, pipi pi .
6) pi pi pipi x+ y + z = 21 x 1 = y + 2 = 2z + 3.
-
42 1. .
7) ~A, ~B( ~A ~B) ~A+ ( ~A ~B) ~A = ( ~A ~A) ~B
pi (1.18), (1.19).
8) ~A, ~B, | ~B| ~A+ | ~A| ~B | ~A| ~B| ~B| ~A .
9) (1.57).
10) pi (1.65,1.66). 11) ~A , pipi xi (xi ~A),
pi xi .
12) , pipi ixi = d,(i, d , xi ), ixi.
13) pi (1.55).
14) (1.15) pipi pi . pi ~F pi A pi pi A pi pi pi pipi .
15) (1.74). 16) (1.76)
(1.75).
17) pi pi (1.113) pi pi U .
18) ~A = ~B + ~C pi ~A ( ~B ~C) . pi 3 3 .
19) (~a~b) (~a~b) = a2b2 (~a ~b)2 (1.127)
20) pi
p2 =~L2
r2+(~r ~pr
)2pi ~L , ~L = ~r ~p.
-
1.9. 43
21)
~a =~b ~c~a ~b ~c ,
~b =~c ~a
~b ~c ~a , ~c =
~a~b~c ~a~b , (1.128)
~a =~b ~c
~a ~b ~c, ~b =
~c ~a~b ~c ~a
, ~c =~a ~b
~c ~a ~b.
22)
~a ~a = ~b ~b = ~c ~c = 1 (1.129)
~a ~b = ~a ~c = ~b ~c = ~b ~a = ~c ~a = ~c ~b = 0, pi .
23) ~A, ~B, ~C , Jacobi~A ( ~B ~C) + ~B ( ~C ~A) + ~C ( ~A ~B) = 0 (1.130)
24) ~v1 pi ~B pi pi
~B =04pi
q1~v1 ~rr3
. (1.131)
( Biot Savar). pi
~F2 =04pi
q1q2r3
~v2 (~v1 ~r).
~F1 pi q2 q1
~F1 =04pi
q1q2r3
~v1 (~v2 ~r)
Newton , ~F1 6=~F2. pipi pi pi- .
25) (1.8) pi-.
26) pi y1 =(1,1, 0)/2, y2 = (1, 1,2)/
6, y3 = (1, 1, 1)/
3, pi -
. , pi pi x1, x2, x3. pi pi .
-
44 1. .
27) pi (1.123). pi- .
28) pi Xi, Yi Xi + ijkYjPk = iYi + ijkXjPk = Mi
29) , pipi :
i) |~a~b|2 + (~a ~b)2ii) (~a~b) (~b ~c) (~c ~a)iii) (~a~b) (~b ~c) (~c ~a) (1.132)
pi ~a,~b,~c .pi.: i) |~a|2 |~b|2, ii) 0, iii) [(~a~b) ~c]2
30) ~R pi ~R ~A = 1, ~R ~A =~B, pi ~A, ~B . pi ~R ~A, ~B.pi.: . . ~A(~R ~A) pipi ~R = ( ~A ~B+ ~A)/A2.
31) ~A = Axi + Ay j. pipi (
i ( ~A i) + j ( ~A j))
12
(i ( ~A i) + j ( ~A j) + k ( ~A k)
) ~A = Axi+Ay j +Az k.
32) ~A ~A = ~B ~B = 4, ~A ~B = 0, ( ~A ~B) ~C = 0, ( ~A ~B) ~C = 8
pi ~C = Ck pi ~A, ~B, ~C.pi
~A ~C, |~C|, |~C ~B|
33) pi . i) pi- pi4 pipi. pi ,
-
1.9. 45
pi pi pi (pi4 ) pi pipi. . ii) pi 1 2 . pi 2 pi 1. iii) . (pi. 1200, ArcCos[ sin(1) sin(2)], ArcCos[ 1+sin 2 ].)
34) pi q = 1 pi~B pi ~F = ~v ~B. pipi pi pi
~v = i = ~F = 2k 4j,~v = j = ~F = 4i k,~v = k = ~F = j 2i
pi pi ~B.
35) aij , bjk 3 3 pi aijbjk = cik. (1.114) pipi DaDb = Dc.
36)
ijkyjak = bi cjyj = k (1.133)
yi.
-
46 1. .
-
Keflaio 2
Dianusmatikc
Sunartseic
2.1 Pargwgoc Dianusmatikc Sunrthsh-
c
. pi pi pi .
~F (t), pi pipi t t [ta, tb].
~F (t) = f1(t)i+ f2(t)j + f3(t)k (2.1)
pi fi(t), i = 1, 2, 3 t. ~F (t) t0 pi limtt0 ~F (t) = ~F (t0). pi pi
limt0
~F (t0 +t) ~F (t0)t
d~F
d t
t=t0
(2.2)
(2.1) (2.2)
d ~F (t)d t
=d f1d t
i+d f2d t
j +d f3d t
k.
~F (t), ~G(t) (t) ,
47
-
48 2.
d
d t
(~F (t) + ~G(t)
)=
d ~F (t)d t
+d ~G(t)d t
(2.3)
d
d t
((t) ~G(t)
)=
d(t)d t
~G(t) + (t)d ~G(t)d t
(2.4)
d
d t
(~F (t) ~G(t)
)=
d ~F (t)d t
~G(t) + ~F (t) d~G(t)d t
(2.5)
d
d t
(~F (t) ~G(t)
)=
d ~F (t)d t
~G(t) + ~F (t) d~G(t)d t
(2.6)
pi pipi pi- .
. , pi .
. ~R(t) |~R(t)| = c pi c ., ~R(t)2 = |~R(t)|2 = c. pi pi
d
d t~R(t)2 =
d
d t|~R(t)|2
~R(t) dd t
~R(t) = |~R(t)| dd t|~R(t)| = 0 (2.7)
pipi pi |~R(t)| . ,pi pi (2.7) pipi ~R(t) pi dd t
~R(t).
2.1.1 Paradegmata Dianusmatikn Sunartsewn ap
th Fusik
1. pi- pi
~r(t) = r0(i cos(t) + j sin(t)) (2.8)
pi pi pi pi pipipi pi ~r(t).
~v(t) d~r(t)d t
= r0(i sin(t) + j cos(t)) (2.9)
, |~r(t)| = const, pi ~r(t) ~v(t) = 0.
-
2.1. 49
y
x
~r(t)
= t
. , = t ~r(t).
pi ~r(t) pi-
~(t) = r02(i cos(t) + j sin(t))= 2~r(t) (2.10)
pi ( ) pi .
2. pi ~r(t) = x1 cos(t) + x2 sin(t)
pi , , , . , , , pi ~L. pi pi pipi pi .
. pi , x1 = cost, x2 = sint, (x1
)2+(x2
)2= 1 (2.11)
pi T = 2pi/. pipi pi pi pi pi,
~v(t) = (x1 sint+ x2 cost)~(t) = 2(x1 cost x2 sint) (2.12)
~L = m~r ~v
~L = m
x1 x2 x3
cost sint 0 sint cost 0
= x3m (2.13) pi pi pi |~r(t)| = (2 cos2 t+2 sin2 t)1/2. - ~v ~r
-
50 2.
.
~r ~v = 12(2 2) sin(2t)
= ( ), sin 2t = 0, n tn =npi2 , tn =
npi2 , n = 0, 1, 2, . . . .
3. Newton, ,
d
d t
(~r d~r
d t
)(2.14)
pi
. pi (2.14) (2.6),
d
d t
(~r d~r
d t
)=
d~r
d t
0
d~rd t
+ ~r d2 ~r
d t2
= ~r d2 ~r
d t2(2.15)
Newton F = md2 ~rd t2 (2.15),
pi
~M = ~r ~F=
d
d t
(~r md~r
d t
)(2.16)
~p = md~rd t , ~L = ~r ~p,
~M =d
d t(~r ~p) d
~L
d t(2.17)
pi pi pi , .
4. pi pi ( ~E) ( ~B) pi. .
. pi pi ~E ~B, pi
md~v
d t= q( ~E + ~v ~B)
-
2.2. . 51
pi ~v = d~rd t . ,pipi pipi
m~v d~vd t
= q~v ~E + q~v (~v ~B)
pi ~v (~v ~B) = 0, pi pi ,
12md~v2
d t= q~v ~E
12m~v2 C0 = q
d~r
d t ~Ed t
= q
~E d~r (2.18)
pi C0, . pi ~B , Lorentz pipi ~v, ~B ( pi). pi pi, pi pi
12m~v2 = C0
pi . pi ~v .
2.2 Kamplec Qrou.
~r(t). pipi , pi, t . t, ~r(t)pi pi pi pi . t [ta, tb], ~r(ta), ~r(tb) pi. ~r(t)
~T =d~r(t)d td~r(t)d t (2.19)
-
52 2.
~T pi pi. -pi s pi pi ~r(t) pi -. ~r(t), ~r(t)+~r(t), - pi , t 0 pi limt0~r/t pi pi.
7
>
*
*
s(t) ~r
~r
~r/t
~r +~r
. pi s(t) pi limt0~r/t . pi pipi . pi
,Q0, Q1, . . . Qi, Qi+1, . . . QN
~r0, . . . ~ri, ~ri+1, . . . ~rN .
, pi pi pi
s =N1i=0
|~ri+1 ~ri|
N ,
s(t) = limN
Ni=0
~ri+1 ~ritt t
t0
d~rd t d t (2.20)
pi pipi, pi pipi (x1, x2), r2 = x21 + x
22 pi
s = tbta
d t
{(d x1d t
)2+(d x2d t
)2}1/2
= ba
d x1
1 +
(d x2d x1
)2(2.21)
pi s. pipi ,
-
2.2. . 53
pi (2.20) pi t = t(s) ~r(t) ~r(s).
pi pi pipi pi - pi . pipi z pi pi :
x = R cos ty = R sin t (2.22)z = t
,
~r = i R cos t+ j R sin t+ k t (2.23)
pi t = 2pi, pi pi pipi x, y, pipi pi z pi
= 2pi (2.24)
. pi pi pipi(x, y)
tan =
2pi R=
R(2.25)
pi pi
s0 = 2pi0
d~rd t d t
= 2piR2 + 2
=2pi Rcos
(2.26)
pi t = ( rad) s = Rcos (rad).
1. pi pi - ~T = d~r/d s.
. ~T = d~rd t /|d~rd t |. (2.20) d
d ts(t) =
d
d t
t d~rd t d t d~rd t
(2.27) ~T
~T =d~rd td sd t
d~rd s
(2.28)
-
54 2.
2. pipi pi ~r(t) = 2 cos t i+2 sin t j pi . pi pi.
.
s(t) = t0
d~rd t d t
= t0
2
sin2 t + cos2 t d t = 2 t
pi t = s/2. ~r(t)
~r(s) = 2 (coss
2i+ sin
s
2j) (2.29)
pi pi pi
~T =d~r
d s= sin s
2i+ cos
s
2j
pi pi pi pi. 3. pi pi
~r(t) = cos t i+ sin t j + 2 t k (2.30)
pi = pi/2. t0 = 1sec pi pi pi pi . t > 1 t = 2sec.
. pi pi . pi pi ~r (t),
~r (t) = ( sin t i+ cos t j + 2 k) (2.31)
t = t0 = 1,
~r (t = 1) r0 = j + pi k~v (t = 1) =
pi
2
(i+ 2k
) t > 1 pi ~R(t) = ~r0 + ~v0(t t0),
~R(t) = pi2(t 1) i+ j + pi t k (2.32)
4. pipi pi pi
x(t) = e2pi3 t cos
pi
2t, y(t) = e
2pi3 t sin
pi
2t, (2.33)
-
2.2. . 55
t = [0, 1]. pipi .
. pi pi
d x
d t= pi
6e
2pi3 t (4 cos
pi
2t+ 3 sin
pi
2t)
d y
d t= +
pi
6e
2pi3 t (3 cos
pi
2t 4 sin pi
2t)
pi
d~rdt =
[dxdt2 + dydt
2] 12
=5pi6e
2pi3 t (2.34)
s = 10
d~rdt d t = 5pi6
10
e2pi3 td t =
54
(1 e 2pi3
) pipi pi pi
t
s = t0
d~rdt d t = 5pi6
t0
e2pi3 td t =
54
(1 e 2pi3 t
)pi pi t
t = 32pi
ln(1 45s) (2.35)
x(s) = (1 45s) cos
(34ln(1 4
5s))
y(s) = (1 45s) sin
(34ln(1 4
5s))
. 5 q ~v - pi ~B ~v 6= 0, pi/2, pi pi~B. pi` pi .
x3 pi pi
~B. , pi Lorentz
~F = q~v ~B = nqBv sin (2.36)
-
56 2.
pi n ~v, ~B. pi, pi pi ~B. pi ~B, v = v sin , P = mv pi ~B, pi
F = m = |q|vB sin mv
2r
= |q|vB
pi pi pipi
r =mv|q|B =
P sin |q|B (2.37)
pi P = m|~v|. pi pi- = 2pi/T pi T pi . s = 2pir = vT , T = 2pir/v (2.37),
T =2pim|q|B , =
|q|Bm
. (2.38)
pi, pipi (x1, x2),
~r(t) =P sin |q|B
{x1 cos
[ |q|Bm
t
]+ x2 sin
[ |q|Bm
t
]} ~B v = v cos pi ~B, s = vt = Ptm cos , pi
~r(t) =P sin |q|B
{x1 cos
[ |q|Bm
t
]+ x2 sin
[ |q|Bm
t
]}+ x3
P cos m
t (2.39)
pi. pi , pi pi ~r(t).
d~r
d t=
P sin m
{x1 sin
[ |q|Bm
t
]+ x2 cos
[ |q|Bm
t
]}+ x3
P cos m
(2.40)
d~rd t = Pm (2.41)
, pi T = 2pi/ = 2pim/|q|B,
sT = T0
d~rd t d t = Pm (T 0) = 2piP|q|B
-
2.2. . 57
2.1: pi pi ( q < 0) pi ~B pi x3
, pi pi pi pi ~B,
= vT =P cos m
2pim|q|B =
2piP|q|B cos
pi pi
so = sT`
=
`
cos (2.42)
pi t = sov =`
v cos . 6. pi pi ~r1 =
(1, 0, 0), ~r2 = (0, 1, 0), ~r3 = (0, 0, 1).. pipi pi .
~r0 pipi, ~r pi pipi (~r ~r0) n = 0, pi n = n1i + n2j + n3k . pi
(~r1 ~r2) n = 0 n1 n2 = 0(~r2 ~r3) n = 0 n2 n3 = 0
pi pi pipi pipi n1 = n2 = n3. |n| = 1 pi 3n21 = 1, pi n1 = n2 = n3 = 13 . pi pi- () pipi. pi
n =i+ j + k
3
~r = ~r0 + a(e1 cos+ e2 sin) (2.43)
-
58 2.
pi a , ~r0 pi pi e1,2 pi pipi, e1 e2 = 0. pi - e1 = 1i+2j. pi pipi pipi ( pi pi n), e1 n = 0, pi pi 1n1 + 2n2 = 0, 1 = 2 pi 21 + 22 = 1,pipi
e1 =i j
2(2.44)
e2 e1, n. e2 = n e1, pi
e2 =16(i+ j 2k) (2.45)
pi pi ~r0 =
n
e
e^
^
^
1
2|r-r|
2 0
r0
x y
z
2.2: pi pi (1, 0, 0), (0, 1, 0), (0, 0, 1).
x0i + y0j + z0k. pipi pi pi ~ri = ~r1,2,3. pi ~ri ~r0 pipi, pi (~ri ~r0) n = 0, i = 1, 2, 3. pi pi x0+ y0+ z0 = 1. pipi, pipi |~ri ~r0| = a, pi,
(x0 1)2 + y20 + z20 = x20 + (y0 1)2 + z20 = x20 + y20 + (z0 1)2 (2.46)
x0 = y0 = z0 = 13 .
(x0 1)2 + y20 + z20 =
63 .
-
2.2. . 59
~r =13
(i+ j + k
)+63
(i j
2cos+
16(i+ j 2k) sin)
)(2.47)
7. Skier pi pi pi pi z pi pi . pi pi pi ( |r| = 1) pi z. pipi pi pi pi pi pipi (xy) pi pi pi x.
(t), h(t) pi pi
~r(t) = (t)(cos t i+ sin t j) + h(t) k. (2.48)
pi pi pi , |~r(t)| = 1, pi
(t)2 + h(t)2 = 1 (2.49)
pi
~r(t) =1 h(t)2 (cos t i+ sin t j) + h(t) k. (2.50)
pipi h(t0 = 0) = 0 h(tf = 2pi) = 1, (2.50) x pi z = 1. pi
~r (t) = ( cos t sin t) i+ ( sin t+ cos t) j + h k (2.51) pi pipi (xy) pi pi z, pi
~r k = const, h(t) = const h(t) = t+ (2.52) pi pi z (. . (2.3), tf = 2pi, pi = 12pi , x, pipi = 0.
pi pi (2.48)
|~r(t)| =2 + 2 + h2 (2.53)
pipi , h(t) pi pi-pi
s(tf ) = tf0
16pi4 + t4 + 4pi2(1 2t2)
4pi2(4pi2 t2) d t (2.54)
tf = 2pi pi s(2pi) 5.42.
-
60 2.
-0.5
0
0.5
1
-0.5
0
0.5
1
0
0.25
0.5
0.75
1
-0.5
0
0.5
1
2.3: pi pi pi pi z.
2.3
Kampulthta kai Stryh
pi pipi pi s, pi pi ~T pi
~T (s) =d~r
d s. (2.55)
pi |~T | = 1 pi ~T d ~Td s = 0, pi- ~T pi . ~N ,
d ~T
d s= k ~N, | ~N | = 1. (2.56)
pi k 1 pi pi. ~B = ~T ~N , ~N, ~B, ~T pi pi pi ~r(t). ~B
d ~B
ds=
d~T
ds ~N + ~T d
~N
ds(2.57)
pi (2.56) d~Tds ~N = k ~N ~N = 0. pi, d
~Nds ~N = 0,
d~Nds pipi pi
~B, ~T . ,
1
Prgmati,
~N = d~Tds/| d~Tds| = d~T
dskai k = 1
= | d~T
ds| 0.
-
2.3. 61
pi pi,
~T d~N
ds ~T ~B ~N
pi (2.57)
d ~B
d s= ~N (2.58)
pi pi. pipi
() > 0 d ~Bd s pi ~N . pi ~N = ~B ~T pi
d ~N
d s=
d
d s( ~B ~T ) = d
~B
ds ~T + ~B d
~T
ds
= ~N ~T + k ~B ~N pi ~N ~T = ~B ~B ~N = ~T ,
d ~N
ds= k~T + ~B (2.59)
(2.56,2.58,2.59) pi Frenet. pi Frenet
Darboux, ~D = ~T+k ~B: ,
~D ~N = ( ~T + k ~B) ~N = ~T ~N + k ~B ~N = ~B k~Tpi (2.59). , ~D ~B, ~D ~T pipi.
d~Q
ds= ~D ~Q; ~D = ~T + k ~B, ~Q = { ~N, ~B, ~T} (2.60)
. pi Frenet pi pipi pi, k = 0.
. pi pi d~Tds = 0 ~N ~0, ~T ,
, ~T = ~c. pi pi d~rds = ~T = ~c, ~r = ~cs+ ~d, , . ~N , pipi , pi pipi pipi pi. ~N ~T . ~B = ~T ~N pi , d~Tds = ~0, = 0.
. pi Frenet, pi ~r, ~r,
...~r pi (~T , ~N, ~B). -
pi pi ~r, ~r,
...~r .
-
62 2.
r(t)
O
N
B
T
s(t)
dNds
2.4: pi s(t) ~T , ~B, ~N .
. pi pi ,
~r d~rdt
=ds
dt
d~r
ds= s ~T
~r = s ~T (2.61)
(2.61) pi (~T , ~N, ~B) pi ~T .
pi pi ~T pi
~T =d~T
ds
ds
dt=
d~Tdsd~Tds
d~Tds dsdt = ~Nks (2.62)
~r d2~r
dt2=
d
dt(s ~T ) = s ~T + s ~T = s ~T + ks2 ~N (2.63)
-
2.4. 63
pi pipi pi ~T , ~N .
~r = t ~T + n ~N (2.64)
pi t = d2sdt2 pi , n = k
(dsdt
)2
pi. pi pi, pi pi
pi
...~r =
d
dt(s ~T + ks2 ~N)
= (...s k2s3)~T + (ks2 + 3ssk) ~N + k s3 ~B (2.65)
(2.61,2.63), ~r ~r = ks3 ~B,
k =|~r ~r|s3
=|~r ~r||~r|3
(2.66)
=~r ~r
...~r
|~r ...~r |2
(2.67)
2.4
Kntro Kampulthtac
pi s(t). pi pi pi pi pi pi pi . s pi pi pipi ,
s = (2.68)
~T1, ~T2 pi s,
T |~T1 ~T2| = |~T | = s = T
lims0
~Ts =
d~Tds = 1 = k (2.69)
, pi pi pi. ~rc pi ~r
~rc = ~r + ~N (2.70)
. pi ~r(t) = 0. pi pipi k =.
-
64 2.
Dq
S(t)
NT
T
DT
2
1
T T1 2
Dq
r(t)
(t)rc
2.5: pi pi s(t).
. pi pi Frenet d~Bds = 0, ~B
. ~T ~N = ~B pi , ~N, ~T pipi pi pi pipi. k =,pi pi Frenet
k~T = d~N
ds
kd~rds
=d ~N
ds
k ~r + ~c1 = ~N + ~c2pi ~c1,2 . k ~r0 = ~c2~c1 k = 1/, , ~r ~r0 = ~N pi .
. Darboux
~D = ~N d~N
ds
. pi (2.60) d~Nds = ~D ~N
~N d~N
ds= ~N ( ~D ~N)
(1.64) pi ~D pipi (~T , ~B) (, ~N),
~N ( ~D ~N) = ( ~N ~N) ~D ( ~N ~D) ~N = ~D (2.71)
-
2.4. 65
. ~D = f(t)e, pi e
, ~D ~D = 0.. , ~D = f(t)e+ f(t) e = f(t)e,
~D ~D = f(t)f(t)e e = 0 (2.72)
. pi ~r(s) e ~T (s) = cos, pi e . -
~N , d~Nds ,
~D. pi pi ~D d~Dds . k = .
. pi pi ~r(t) e . e ~T (t) = cos,
e d~T
ds= 0 e ~N = 0
~N e.
, e d ~Nds = 0 pi d~Nds pi
e. pi pi ~D = ~N d ~Nds , Darboux pi e. pi ~D d~Dds , pi
d ~D
ds=
d
ds
( ~T + k ~B
)=
d
ds~T +
d~T
ds+dk
ds~B + k
d ~B
ds
pi Frenet
d ~D
ds=
d
ds~T +
dk
ds~B
~D d~D
ds= ( ~T + k ~B)
(d
ds~T +
dk
ds~B
)=
(dk
ds kd
ds
)~N
~D pi pi ,
~D d~D
ds= 0
dk
ds kd
ds= 0,
-
66 2.
pi pi pipi
dk
ds= k
d
ds
lnk = ln + ck
= ec
pi pi k .
pi ~N , , pi pi . pi e ~T =
e d~rds
= 0 e ~r = s cos+ (2.73) e x3,
x3 = s cos+ (2.74)
~r(s) pipi x1, x2,
~r(s) = ~r0 + x3(s cos+ ) (2.75)
pi pi pi, pipi pi s(t). pi - C pipi pi (evolute). (2.70)
d~rcds
=d~r
ds+
d ~N
ds+ ~N
d
ds(2.76)
pi pi Frenet pipi pi, d~rds+d ~Nds =
0,
d~rcds
= ~Nd
ds(2.77)
` pi C, d~rcd` = ~Tc, |~Tc| = 1d~rcds
=d`
ds
d~rcd`
=d`
ds~Tc =
d
ds~N (2.78)
, pi ~Tc pi C, pipi ~N pi s(t).
d`
ds=
d
ds(2.79)
pi pi
` = + constant (2.80)
` = , ` C pi s(t).
-
2.4. 67
T
TC
N
O
S(t)
C
rC
r
r
2.6: pi pi C pi pipi pi s(t).
2.4.1 Eppedh Knhsh
pi pi pi pi - pipi. pipi pi pi pi ,pi pi pi pi pi .
pipi pipi (x, y) pi pi x = x(t), y = y(t). , pipi pi , x, y, pi pi - r, .
x = r cos (2.81)y = r sin (2.82)
r = (x2 + y2)12 (2.83)
= sin1y
r= cos1
x
r(2.84)
pi r = r(t), = (t) pipi pi pi pi pi - . pi [pi, pi]. ,er, e , pi ~r(t) (pi 2.7).pi ~r(t) = rer
-
68 2.
r(t)
s(t)
y
x
e
e
q
^
^
2.7: pipi .
,
er = i cos + j sin (2.85)
pi , pi e (pi- )
e = i sin + j cos (2.86) er, e i, j , .
derd
= i sin + j cos e (2.87)
ded
= i cos + j sin er (2.88)
pi , pi pi t, ~v(t) = d~vdt ,
d~r(t)dt
=d{r(t)er(t)}
dt
=dr(t)dt
er(t) + r(t)der(t)dt
(2.87), der(t)dt =ddt e,
~v(t) =dr
dter + r
d
dte (2.89)
(2.89) pi
~(t) =
[d2r
dt2 r
(d
dt
)2]er +
[rd2
dt2+ 2
dr
dt
d
dt
]e (2.90)
-
2.4. 69
, pi ,~(t) = r er + e. r, pi d
2rdt2
(ddt
)2pi pipi r =
constant pi. r d2dt2
pi, 2drdtddt pi pipi
pi Coriolis pi pi, . Newton,
~F = Fr er + F e (2.91)
Fr = md2r
dt2mr
(d
dt
)2(2.92)
F = mrd2
dt2+ 2m
dr
dt
d
dt(2.93)
pi (2.93) r
rF =d
dt
(mr2
d
dt
)(2.94)
pi pi pi ~r ~F , pi mr2 ddt
~L. pi (2.94)pi pi .
pi , pipi pi, F = 0. ,
mr2d
dt= c
pi pi pi . pi pi pi Kepler pi pi - . ,
2.8: . pi pi .
-
70 2.
pi
d ~A
dt=
12~r d~r
dt(2.95)
pi (2.8).
dA
dt=
12r2d
d t(2.96)
, dAdt L pi .
. pipi
r =1
1 + 2 cos ,d
dt=
1r2
(2.97)
pi pi .. pi (2.89) ~v(t) = drdt er + r
ddt e.
(2.97)
~v = 2 sin er +er
, pi pi (2.90), pi
r = 1r2, = 0
pi pi F = 0, . pi .
. pi m pi pi ( M > m).
.
~F = GMmr2
er
Newton,
~F = md2 ~r
d t2
pi pi
~r d2 ~r
d t2=
d
d t
(~r d~r
d t
)= GM
r2~r er = 0
pi
~r d~rd t
= o.
-
2.4. 71
pipi pi pi pi
d ~A
d t=
12~r d~r
d t= o.
, pi pi pipi pi ~r ~v = d~rdt pi, pipi. pi, pi ~r = rer, ~v = re
d ~A
d t=
12r2ez
pi ez = er e pipi .
d2 ~r
d t2 d
~A
d t= GM
r2er 12r
2ez =12GMe.
pi d~A
d t , pi -,
d
dt
(d~r
dt d
~A
dt
)=
12GMe.
d~r
dt d
~A
dt=
12GM
e d t =
12GM(er + ~)
pi ~ pi pipi pi . pi-
~r (~v d~A
dt) =
12GM(~r er + ~r ~).
pi pi pi
~r (~v d~A
dt) = (~r ~v) d
~A
dt 2d
~A
dt d
~A
dt= 2A2 (2.98)
pi ~ , pi 2 t = 0, pi ~r ~ = r cos , ~r er = r. (2.98)
r =4A2
GM1
1 + cos (2.99)
< 1 pi - pi . > 1, m pi .2
Se kje perptwsh, mporome na gryoume ~r = r cos(+ ), pou h gwna twn ~r kai~ gia t = 0.
-
72 2.
2.5
Dianusmatikc Sunartseic Polln Metabl-
htn
pi pi pi pi pi pi . pi pi . .
pi pi (u,w),
~F (u,w) = f1(u,w)i+ f2(u,w)j + f2(u,w)k (2.100)
pi fi(u,w) . pi pipi pi , ~F (u,w) pi.
~Fu u
~F (u,w) = limu0
~F (u+u,w) ~F (u,w)u
(2.101)
, pi pi - u, pi pi . pi w. pi , (2.101)
u~F (u,w) =
f1u
x1 +f2u
x2 +f3u
x3 (2.102)
pi pi pi.,
2
uw~F (u,w) =
2f1(u,w)uw
x1 +2f2(u,w)uw
x2 +2f3(u,w)uw
x3(2.103)
pi,
2fi(u,w)uw
=2fi(u,w)wu
, 2
uw~F (u,w) =
2
wu~F (u,w)
pi pi pi u1, u2, . . . un pi pi pi t, uj = uj(t). - pi pi pi pi t. -
~F (u1(t), u2(t), . . . , un(t), t) =3i=1
fi(u1(t), u2(t), . . . , un(t), t)xi
-
2.5. 73
,
~F (uj(t), t) = fi(uj , t)xi, j = 1, 2 . . . , n; i = 1, 2, 3. (2.104)
pi (2.104) pi t pi
d~F
dt=
nj=1
~F
uj
dujdt
+ ~F
t(2.105)
pi
~F
uj=
3i=1
xifiuj
(2.106)
pi pipi pi , pi fi, .
2.5.1 Paradegmata.
. Q, q pi pi pir = |~r| =
x2 + y2 + z2 pi
pi
~F (~r) = Qq
r2er
pi er ~r,er = ~rr . , pi pi
~F (~r) = Qq
r3~r
~F (~r) pi pi r ,
|~F (~r)| F (r) = Qqr2
F (r) F (~r).
pipi pipi pi pi pi N qi pi ~ri, i = 1, . . . N . pi Q
~F (~r) =Ni=1
Qqi
|~r ~ri|3 (~r ~ri) (2.107)
-
74 2.
, pi pi
~E(~r) =Ni=1
qi
|~r ~ri|3 (~r ~ri)
6r
r
r
1r
~ri ~r~r1
~r
~ri
~r2
. .
( pi ). pi pi pi pi pi Newton. , pi pipi pi pi pipi . , pi pi (6.13) pi pi. , pi pi pi pi pi , . pipi , pi , pi pi pi . pi pi pi . E,M,S
1
2
3
2.9: pi E,M,S mE ,mM ,M.
-
2.6. . 75
, ~r1, ~r2, ~r3 , (pi ) pi Newton
mE~r1 = GmEmM|~r2 ~r1|3 (~r2 ~r1) +G
mEM|~r3 ~r1|3 (~r3 ~r1) (2.108)
mM ~r2 = GmMmE|~r1 ~r2|3 (~r1 ~r2) +G
mMM|~r3 ~r2|3 (~r3 ~r2) (2.109)
M~r3 = GMmE|~r1 ~r3|3 (~r1 ~r3) +G
MmM|~r2 ~r3|3 (~r2 ~r3) (2.110)
pi mE ,mM ,M G pi . - ~r = ~r2~r1 ~R = ~r3~r1, pi r = |~r|, R = |~R| = |~r3~r2| pi . , , pi
~r1 = GmMr3
~r +GMR3
~R (2.111)
~r2 = GmEr3
~r +GM3
(~R ~r) (2.112)
pi
~r = G(mE +mM ) ~rr3GM
(~R
R3
~R ~r3
)(2.113)
pi, pi pi
~F GM((
13 1R3
)~R ~r
3
)(2.114)
2.6
Grfhma kampuln me Hlektronik Up-
ologist.
pi - pi pi. - y = f(x) pi . , pipipi pipi, pi pi, pi pi pi pi. pipi pi pi pi. pi, pi (2.1) pi Mathematica. - pi pi pi pi . pi pipi
-
76 2.
pipi pi pi pi . pi pi
~r(t) = r0(t)(i sin(t) + j cos(4t)
), r0(t) = et/10 (2.115)
pi Mathematica pi pi , pi pi ( = 1) pi
ParametricPlot[{Exp[-t/10]*Sin[t], Exp[-t/10]*Cos[4*t]}, {t, 0, 30}]
pi pi pi pi . pi pi -
-0.6 -0.4 -0.2 0.2 0.4 0.6 0.8
-0.75
-0.5
-0.25
0.25
0.5
0.75
1
2.10: pi pi (2.115) piMathematica.
pi , .,
~r(t) = r0(t)(i sin(t+ pi/2) + j cos(4t+ pi/2)
), r0(t) = et/10 (2.116)
pi pi = 1, pipi
ParametricPlot[{Exp[-t/10]*Sin[t+Pi/2], Exp[-t/10]*Cos[4*t+Pi/2]}, {t, 0, 30}]
pi pi . pi pi,
, pipi . pi, pi pi
~r(t) = 2(sin(t)i+ cos(t)j) + 3tk (2.117)
[0, 4pi]
ParametricPlot3D[ {2*Sin[t],2*Cos[t], 3*t},{t,0, 4 Pi}]
pi pi (2.1)
-
2.6. . 77
-0.75 -0.5 -0.25 0.25 0.5 0.75 1
-0.75
-0.5
-0.25
0.25
0.5
0.75
1
2.11: pi pi (2.116) piMathematica.
pi pi pi r(t) = 1/(1 cos t)
~r(t) = (r(t) cos(t), r(t) sin(t)) (2.118)
pi < 1 pi , pi -. pi . pi = 0.7, 0.5, 0.3, 0.1 pi
r[t ,e ] := 1/(1-e Cos[t])lista={r[t,#] Cos[t], r[t,#] Sin[t]}&/@ {0.7,0.5,0.3,0.1}ParametricPlot[Release[lista],{t,0,2 Pi}, AspectRatio >Automatic] pi pi pipi (2.12).
-1 1 2 3
-1
-0.5
0.5
1
2.12: pi pi (2.118) piMathematica.
-
78 2.
2.7 Askseic
2.1) pi (2.2), pi (2.3-2.6).
2.2) pi pi pi
x1 = a cos(t), x2 = b sin(t), x3 =a2 b22
t
2.3) . pi .
2.4) pi :
x = et, y = 2 cos(0t), z = 2 sin(0t)
pi 0 = 3. , pi pi .
2.5) m q piB. pi ~F = q~v ~B, pi |~v| = .
2.6) pi pi
x = cos( t), y = sin ( t), z = b t
2.13: pi pi 2.6.
2.7) pi pipi pi- pi pi pi
~c(t) = cos3 t i+ sin3 tj (2.119)
(pi 2.13). pi pi-. (pi. v = 3 sin t cos t, stot. = 6)
-
2.7. 79
2.8) pi~c(t) = (t sin t) i+ (1 cos t) j (2.120)
2.9) (2.14) pi h = 22.5m. pi pipi 1/3 pi R = 150pi m pi pi pipi . . .(pi. 102.5m)
2.14: pi 2.8 pi pi. pi .
2.10) pi pi 7 2.2 pipipi pi pi (t) = tanh t. , pi h(t) pi pi pi pi pi z. (pi. |~r| = 1 pi).
2.11) pi pipi h(t) = sin t4 . pi pi pi-
. (pi. |~r| = 149 + 8 cos2 t2 , s = 4.41).
2.12) pi h = z0 pi pi pi pi pi
x = a cos, y = a sin, z = b, > 0
z pipi . - . pi h = z0 h = z1. pi pi - (z = 0). pi. dsdt =
2g(z0 z1),
s =a2 + b2 z0z1b , t0 = (a
2 + b2)/b2z0/g.
-
80 2.
2.13) pi pi d~dt =r er + e
r =d3r
dt3 3dr
dt
(d
dt
)2 3r d
dt
d2
dt2
= 3d2r
dt2d
dt+ 3
dr
dt
d2
dt2+ r
d3
dt3 r
(d
d t
)3
2.14) ~F , pi pi ~B q. , ~v , ~F = q~v ~B. ~v, ~B, ~F pi pi pi .
2.15) ~T , ~N, ~B, ~D, k, ,~r(t) = r0(costi+ sintj)
2.16) pi d~rds ~T , ~T ~N , ~T ~N ~B,~T ~Tds ,
d~rdt ~T , d
~Nds ~B, d
2~rds2 ~T .
2.17) ~r(t) pi pi pi , ~r = ~N 1 dds ~B. pi pi + dds
(1
dd s
)= 0.
2.18) pi (2.65). 2.19) pi y = y(x), z = 0, pi-
= (1 + y2)3/2/|y|. 2.20) pi ~r = cos i+ sin j,
pi = 1 (2 sin2 + 2 cos2 )3/2.
2.21) m = 1 pipi ,
r(t) = 1 + t, (t) =pi
1 + t, t 0
pi t = 1. Fr, F . pi pi t = 1, t = 2.
2.22) pi- pi.
2.23) pipi pi r = t = t. pi. pi. ~v = er + ( t) te.
-
2.7. 81
2.24) pi , ( = t), , pi pi pi r(t) = t2. pi pi.
2.25) ~v pi ~a pi pi = v3/|~v ~a|.
2.26) pi ~F = cos (yx)i+ (2xy x2)j + (3x+ 2y)k.
2.27) G = ex(c siny i + d cosy j), (, c, d , ) pi- pi
2 ~Gx2 +
2 ~Gy2 .
2.28) u(x, t) = 3c sech2[
c2 (x ct)
], ut+uxxx+uux = 0,
pi ut pi, pi. pi u(x, t) c. Korteweg deV ries(KdV ) pi .(sech = 1cosh )
2.29) pi pi pi pi (2.118) pi e > 1 pi pi .
2.30) r = A1+ cos . pi pi pi pipi = 0 () < 1 (), = 1 (pi), > 1 (pi) (pi 2.116).
2.31) pi pi
x = et/10 cos t, y = et/10 sin t, z = t/4 pi t = [0, 4pi].
2.15: pi 2.30.
-
82 2.
-
Keflaio 3
Epifneiec, Kateujuntik
Pargwgoc, Klsh.
3.1 Epifneiec
pi pi pipi ~r(t), - pi xi, i = 1, 2, 3 pi pi t, xi = xi(t). pi pi. , - pi pi .pi pi pi -pi pi pipi pi . pi, pi (x1, x2) = ( cos , sin ), [0, pi] pi 1
x2 = f(x1) = (2 x21)1/2 (3.1)
, f(x1) pi pi R R, pi pipi pi R2. , pi. pi,
x3 = g(x1, x2) = (2 x21 x22)1/2 (3.2)
pi pi pi x3. pi R2 R pi R3., pi pi pi pi
1
Anafroume ed wc pardeigma thn aplosterh perptwsh eppedhc kamplhc. Ja
parousisoume th genkeush se mh eppedec kamplec sth sunqeia auto tou kefalaou.
83
-
84 3. , , .
pi .
(x1, x2, x3) . (x1, x2, x3), pipi R3 pi R. pi . pi, pi (x1, x2, x3) (x1, x2, x3)
(x1, x2, x3) = (x1, x2, x
3) (3.3)
pi, T (x1, x2, x3) , - pi pi T = T (x1, x2, x3). pi , T , , T = T (x1, x2, x3, t). pi - (x1, x2, x3); -pi pi R3 R pi pi R4. pi pi - ( R3) pi -pipi R4. pi. , pi pi - T = T (x1, x2, x3), pi pi c. , pi pi pi R3 pi pi . , pi, ...
, ~r = (x1, x2, x3) pi pi
(x1, x2, x3) = c (3.4)
pi c pi , pi. pi pi pi, pi pi . n1x1 + n2x2 + n2x3 = c, pipi. pi pipi pi.pi (3.4) pi pi , (pi.. x3)pi
x3 f(x1, x2) (3.5) pi, x21+x
22+x
23 = c pi pi
x3, (3.2). pi pi pi pi
f(x1, x2) =x21 x22x21 + x
22
sin(x21x22) (3.6)
-
3.1. 85
3.1: pi f(x1, x2) =x21x22x21+x
22sin(x21x
22).
pi pi Mathematica. pi (3.5) pi
pi x1, x2
~r(x1, x2) = x1x1 + x2x2 + f(x1, x2)x3. (3.7)
pi (3.1), pi f(x1, x2) pi (3.6).
~r(x1, x2) pi , ( x2 = c, pi c ), (3.7)
~r(x1, c) = x1x1 + cx2 + f(x1, c)x3. (3.8)
pi 2 - pi pi. pi pipi pi pipi pi pi pi pi (3.7). (3.1) pi (x1 = c, x2 = c) pi pi. pi c. , pi pi pi x1 x2 = 1 ...
3.1.1 Efaptmena Diansmata se Epifneia.
pi pi pi - x1, x2. , - pi pi pi pi . p, q xi - xi = xi(p, q), pi
~r(p, q) = x1(p, q)x1 + x2(p, q)x2 + x3(p, q)x3 (3.9)
-
86 3. , , .
pipi p = c q = c pi pi pipi pi, ~r(p = c, q) ~r(p, q = c) . (3.2)pi pi pi ~r(c1, q) ~r(p, c2). ~r(p, q) , pipi pi (p, q) pi , pi pi - pi. pi . ~rp ~rq 6= ~0 (pipi), pi pi . pi pipi , . - (3.3) (x1, x2) = (0, 0).
3.2: pi pi pi ~r(const, q) ~r(p, const) pi .
3.3: pi f(x1, x2) = Exp[(10x1)2+(10x2)
2]x21+x
22
(x1, x2) = (0, 0). , pi pi.
pi (p, q) pi. pi
~rp =~r
p
q=const
(3.10)
~rq =~r
q
p=const
(3.11)
-
3.1. 87
pi pi pi ~r(p, const) ~r(const, q). pipi pi pi pi. ~rp ~rq pi pi pi pipi pi .
n =~rp ~rq|~rp ~rq| (3.12)
. (3.12) pipi pi (p, q) (x1, x2). (3.7) pi x1, x2,
~rx1 = x1 + x3f(x1, x2 = c)
x1(3.13)
~rx2 = x2 + x3f(x1 = c, x2)
x2(3.14)
(3.13,3.14), pi pi pi
~rx1 ~rx2 = x1f
x1 x2 f
x2+ x3 (3.15)
pipi - .
. pipi (x1, x2),~r = x1x1 + x2x2. ~rx1 = x1 ~rx2 = x2. n = x3. d~S = ~rx1 ~rx2dx1dx2, pipi (dx1, dx2), dS = dx1dx2.
3.1.2 Isodunamikc Epifneiec, Omoeppedec Kam-
plec.
pi (~r). = c pi pi pi . pi- pi pi - pi pi . - (3.4) pi pi (x, y, x) =z sin(xy2), = 0 = 2.
pi z = f(x, y) pipi () pi f(x, y) = c. pi, z = x2+ y2
pi pipi pi pi x2 + y2 = c. (3.5) pi pi z = h(x2+2y2)ex
24y2 , (pi h ) pi pi. pi (3.6).
-
88 3. , , .
22.5
33.5
4 1
1.25
1.5
1.75
2
-10123
22.5
33.5
4
3.4: pi (x, y, x) = z sin(xy2).
3.2 Kateujuntik Pargwgoc
pi , (x, y, z) , pi pi (x, y, z) = cpi c , pi pi. pi , pi pi.pi pi, pi - (x, y, z) pi (x0, y0, z0). pi pipi pi . pipi , pi (x, y, z) . pi pi ~T . pi , pi |~T | = 1. pi pi pi ~T pi ~r = ~r0+ s ~T . pi ~T pi
dds
~r0
lims0
(~r0 + s~T ) (~r0)s
(3.16)
pipi pi ~T pi x, pi pi (3.16) pi pipi pi /x. . pipi 3.16, pi
dds
=xi
dxids
-
3.2. 89
-2 -1 0 1 2
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
3.5: pipi pi z = h(x2 + 2y2)ex24y2 .
pi pi
xi
dxids
(xi xi
) (xj dxjds
) (3.17)
pi pi x1,2,3 - x, y, z .
(3.17), pi pi - pi pi pi pi (x1, x2, x3) - pi , grad
~ = xi xi
(3.18)
pi pi pi , pi d~rds = ~T ., (3.17) ,
dds
= ~ ~T (3.19)
pipi pi pipi pi pi pi ~r0.
pi pi pi ~r0. pi pipi pi pipi pi ~c(t) ~r0.
-
90 3. , , .
-1
0
1-1
0
1
0
2
4
6
8
-1
0
1
3.6: z = (x2 + 2y2)ex24y2 .
(3.16-3.19) pi , pi ~T pi pi ~c(t) ~r0. pipi, pi , , pipi- grad ~. pi pi pi pi . pi ~T pi pi pi, (3.17)
dds
= |~||~T | cos = |~| cos (3.20)
|~T | = 1. pi (3.20) . 1: grad pi
(~r).pi (3.20) pipi ~T pi-
grad, cos = 1 pi .
pipi pi pi s. pi pi pi pi pi pi. pi - pi ~s(t), pi t pi - pi .
dd t
=d s
d t
dd s
= |~v|~ ~T (3.21)
-
3.2. 91
3.7: ~r = ~r0+s~T pi .
'
'
'
3.8: pi = c pi pi pi. pi pi ~T pi.
pi ~v pi ~s(t).. pi
(x, y, z) =02(x2 + y2 z) (3.22)
i) pi pipi pi . ii) |~v| = 5m/s, 4i + 4j 2k,pi pi (, , 2); (0/ = 1/m.)
. pi , pipi pi .
= 0a
(2x
ai+ 2
y
aj k
)
-
92 3. , , .
pi dd s = ~T ,
3.9: 4i+ 4j 2k pi pi-.
pi ~T ,
~T =13
(2i+ 2j k
) (a, a, 2a) = 2i + 2j k,
dd t
=dd s
d s
d t
= |~v|(~T )= v
03a
(4x
a+ 4
y
a+ 1)
(a,a,2a)= 15./s
pi pipi pi pi pi pi pi pi ( 3.10). pi pipi. ~r0 (~r0) = k. pi (~r) = k pi pi pi. pi pipi . = k pipi (3.18) ~. pi pi pi pi pipi pipi = k
dd s
~c(t){=k}
= 0
(3.23)
-
3.2. 93
3.10: pi pipi pi. pi pi .
, pi pi pi, 2 pi pi pi pi = k,
~ ~T = 0 (3.24)
pi grad pi pi pipipi. ~r0 pi, pipi pi pi pi , pi
~(r)~r=~r0
(~r ~r0) = 0 (3.25)
pi pipi . , f(x, y) , f(x, y) = k pi ~f(x, y) (x, y) pi. pi (x0, y0) pi pi
~f(x, y)(x0,y0)
(x x0, y y0) = 0 (3.26)
pi -pi pi . f f pi . , pi h = f(x, y) (pi , pi) pif(x, y) = k pipi pi ( h). ~f(x, y) pi pi , pi .
-
94 3. , , .
. pi pi . h pi pi
z = f(x, y) = h ex2(2+x)y2 (3.27)
pi pi (1, 4, he49).. pi pi ,
pi pi . piz = constant. pi, f(x, y) pi z pi. pi, pi pi f(x, y),
f(x, y) = h ex2(2+x)y2{(2x+ y2)i+ 2y(x+ 2)j
}(3.28)
(3.11) pi pi z =
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
3.11: f(x, y) pi .
f(x, y). pi (3.28) pipi (x, y), , pi pi . (x, y) = (1, 4), pipi (3.28).
n|1,4 = 15(3i+ 4j)
. pi r (r) = r , pi . pi.
-
3.3. 95
.
= (i
x
1r+ j
y
1r+ k
z
1r
)(3.29)
x
1r= 1
r2
xr = x
r3
pi pi. ,
= r3
(ix+ jy + kz
) ~r
r3
~r er = ~r/r,
= r2er ~E
~E Eer = (3.30) = V1, V2, . . . , pi
( q > 0, pi , pi pi ). pi (3.19) pi -pi s = r, , pi ~T = er.
dds
= ~T ddr
= er
(3.30)
ddr
= Er (3.31)
pi (3.30). .
3.3 Akrtata
pi pi pi pi pi pi . - (3.6) pi, pi pi -. pi pi pi pi -
-
96 3. , , .
pi- : , . . (, ) pi pi .
( ) pipi - f = f(x) . pi - pi pi pi pi. f (x0) > 0 pi , f (x0) 0 (3.37)
2f
x2
(x0,y0)
2f
y2
(x0,y0)
[2f
xy
(x0,y0)
]2> 0 (3.38)
( pi .) pi , (3.37) pipi pi < 0, pi . , pipi . < 0, . = 0, pi pi pipi pi pipi.
3.4 Efarmogc
Q = Q(x, y, z) , pi-
dQ
ds=
Q
xcos1 +
Q
ycos2 +
Q
zcos3 (3.39)
cos2 1 + cos2 2 + cos2 3 = 1 (3.40)
pi. pi (3.39)
dQ
ds=
(Q
xi+
Q
yj +
Q
zk
) (cos1i+ cos2j + cos3k) (3.41)
pi pi , Q, pi ~T . (3.40) , |~T | = 1.
dQ
ds= Q ~T = |Q| cos
pi cos pi ~T . pi cos = 1
dQ
d s
max
= |Q|
-
3.4. 99
pipi pi pi piz = f(x, y) = x3 + y3 6xy, (1, 2,3) .
pi. pi pi
f
x= 3x2 6y
f
y= 3y2 6x
pi pipi pi ,
~rx ~ry = (6y 3x2)i+ (6x 3y2)j + k (3.42) ~r0 = (1, 2,3),
n = (9i 6j + k)/118
pi pipi pipi pi (~r~r0) n = 0, pi 9x 6y + z + 6 = 0.
= (~r, t) pi pi . P ~v + t .
pi. pi t - P ~r(t) pi t, P ~r + ~r (pi 3.13).
3.13: .
PP
limt0
t= lim
t0(~r + ~r, t+ t) (~r, t)
t
-
100 3. , , .
d
dt=
xi
dxidt
+
t
=(xjdxjdt
)(xi
xi
)+
t(3.43)
pi pi ,
d
dt= ~v +
t(3.44)
pi ddt pipi pi, pi pi . pi t .pi, pi ~v pi .
. Q(~r, t)
dQ
dt=
d~r
dt Q+ Q
t(3.45)
dQ = d~r Q+ Qt
d t (3.46)
, pi (3.44). , pipi
d~v
dt= ~v ~v + ~v
t(3.47)
pi, pi t ,
d
dt= ~v +
t(3.48)
pi pi pipi pi , Q = Q(u), u = u(~r).
Q = Qxi
xi
Qxi =dQdu
uxi
,
Q = dQduu(~r) (3.49)
-
3.4. 101
pi Q = |~r|n, n 6= 0.pi. (3.49), u = |~r| r
rn = drn
drr
= (nrn1) (r
xixi)
= nrn1~r
r nrn2~r (3.50)
z = (x2 + 2y2)ex
24y2 (3.51)
pi pi (3.6). pi 3.3, pipi pi pi pi (3.36-3.38). pipi
z(x, y)x
= 2x (1 + x2 + 2 y2)ex24y2 = 0z(x, y)y
= 4 y (1 + 2x2 + 4 y2)ex24y2 = 0
pi pi
(x, y) (0, 0) (3.52)(x, y) (1, 0) (3.53)(x, y) (0,1
2) (3.54)
pi . pi pi pi .
2z(x, y)x2
= (2 + 4x4 4y2 + 2x2(4y2 5))ex24y2
2z(x, y)x2
= (4 80y2 + 128y4 + 8x2(8y2 1))ex24y2
2z(x, y)xy
= 8xy(3 + 2x2 + 4y2)ex24y2
pi (0,0): pi 2z
x2 (0, 0) = 2 > 0,2zy2 (0, 0) = 4 > 0
2zxy (0, 0) = 0. pi =
2 4 02 = 8 > 0. (0,0), pi.
(1, 0) 2zx2 (1, 0) = 2z
y2 (1, 0) = 4/e < 0, 2z
xy (1, 0) =0 = 16/e2 > 0 pi .
(0,1/2) = 8/e2 < 0 pi .
-
102 3. , , .
3.5
Grafmata me ton Hlektronik Upol-
ogist.
pi (pi Mathematica, Maple ..)pi pi pi pi - pi pi pi. (3.4) pi (x, y, x) =z sin[xy2], = 0 = 2. pi piMathematica:
p1=Plot3D[Sin[x y2],{x,-2,2},{y,-2,2}]p2=Plot3D[Sin[x y2]+2,{x,-2,2},{y,-2,2}]Show[p1,p2]
pi = 0 - x [2, 2], y [2, 2]. , = 2 pi x, y. , pi pi pi .
, pi (pi 3.27) pi piMathematica. f(x, y).
f[x ,y ]:=Exp[-x2-(x+2) y2]
pi ( ) pi x, y -.
Plot3D[ f[x,y], {x, -1.2, 1.2}, {y, -1.2, 1.2}, PlotPoints > 50]
pi pipi (3.14). option PlotPoints > 50 -
. ,
p1 = ContourPlot[ f[x,y], {x, -1, 1}, {y, -1, 1}, ContourShading > False,PlotPoints > 30, Contours > 4]
p1 pi pi pi pi . pi pi pi pipi x, y pi . pipipi Mathematica , pi pi pi pi. pipi, pipi - pi pi . pi pipi
-
3.6. 103
-1
0
1 -1
-0.5
0
0.5
1
0
5
10
-1
0
1
3.14: pi f(x, y).
-
104 3. , , .
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1
3.15: pi f(x, y) .
pi pi - pi pipi (, pi) . pi , pi . pi pi pi ( pi) .
3.6.1 Kulindrikc Suntetagmnec
pi -, pi pipi . x1, x2 x1, x2 .
~r = x1x1 + x2x2 (3.55)
, pi ~r pi pi r, , er e pi (3.16). 2 x1 = r cos, x2 = r sin , ~r = rer.
pi pi pi -.
r = |~r| =x21 + x
22 (3.56)
cos =x1
x21 + x22
(3.57)
-
3.6. 105
x
x1
2
^
^
e
e
f
f
r^
^
x2
x1
3.16: pi . er e, r, .
(ere
)=
(cos sin sin cos
)(x1x2
)(3.58)
pi pi pi . ,pi pipi x3, pipi (x1Ox2), r , pi
=x21 + x
22 (3.59)
x1 = cos, x2 = sin, x3 = x3 (3.60)
0 2pi, 0
-
106 3. , , .
x
f=staqero
r=staqero
=staqero
f
r
1 x2
x3
x3
e
er
^
x3
^
e^f
3.17: . pi =., =., x3 = . - e, e, e3.
x2 =cos ,
grad =1( sin i+ cos j)
|grad | = 1 .
pi e3. pipi : pipi , . pi = . grad =.,x3 = . grad grad x3 . pi ( 3.16)3,
e =grad
|grad | , e =grad
|grad | , ex3 =grad x3|grad x3| , (3.61)
pi pi . e pi = pi , e pi pi pi pi e e = 0. pipi e, e pi x3, e3.
3
Upenjumzoume ti , enai sunartseic twn x1, x2, bl. (3.56,3.57).
-
3.6. 107
pi pipi (3.61). pi -, ,
dds
= |grad| (3.62)
pi s pi . x3 ds = dx3 , |grad x3| = 1,
e3 = grad x3 (3.63)
, |grad | = dds = dd = 1,
e = grad (3.64)
pi , ds = d ( 3.18.) |grad | = dds =
e
e
O
ds
^
^
r
r
f
3.18: pi |grad | = 1 , e = grad de/d = e.
1 pi pi (3.61)
e =grad
|grad | = grad (3.65)
e, e, e3, pi . pi
.
~r = x1x1 + x2x2 + x3x3= (
x1x1 +
x2x2) + x3x3
= e + x3e3 (3.66)
d~r pipi pipi e pi . pi (3.66)
d~r = d(e) + d(x3e3)
-
108 3. , , .
de = ed (pi (3.18) ) :
d~r = ed + e d+ e3 d x3 (3.67)
ds = |d~r| = (d2 + 2d2 + dx23)1/2 (3.68) pi
. pi f(, , x3) ., f = gradf pi e, e, e3
grad f = f = (e f)e + (e f)e + (e3 f)e3 (3.69) ei f pi f pi ei, 3.2, pi pi f .
e f = f s
,x3=
f
e f = f s
,x3=
1
f
(3.70)
e3 f = f s
,=
fx3
f pi (3.69) (3.70),
grad f = f
e +
1
f
e +
f
x3e3 (3.71)
pi . pi -.
3.6.2 Sfairikc Suntetagmnec
pi pi pi. , pi r = |~r| pi , x3 ~r pi x1 pi ~r pipi x1Ox2. r = . pi r. =. pipipi pi x3 pi pipi
-
3.6. 109
x1Ox2. = . pi pi x3. r [0,), [0, 2pi] , [0, pi]. pi pi . :
x1 = r sin cos, 0 r
-
110 3. , , .
3
f
x
2x
1
r
x
q
D
D
D
r
f
q
3.20: er, e e, r,, , , . pi .
ds = rd pi grad = dds =1r , pi
e = r grad (3.77)
, ds = r sin d grad = d ds =1
r sin
e = r sin grad (3.78)
pipi . pi pi
~r = rer (3.79)
(pi 3.20)
d~r = er d r + r d er, (3.80)
pi der = ed + e sin d. :
d~r = er d r + e r d + er sin d (3.81)
d s = (d~r d~r)1/2 = (d r2 + r2 d 2 + r2 sin2 d 2)1/2 (3.82)
~F pi -
-
3.6. 111
3
f
x
2x
1
r
x
q
dr
r+drdq
df
3.21: d~r pi er, e, e. d~r = er d r + e r d + e r sin d .
~F (r, , ) = Fr er + F e + Fe (3.83)
pi pi pi
= r
er +1r
e +1
r sin
e (3.84)
-
112 3. , , .
3.7 Askseic
3.1) pi (3.6) pi ~rx ~ry. pi pi n (x, y) =(0, 0).
3.2) pi ~rx ~ry pipi (3.3). pi (x, y) (0, 0). pi pi n . pi pi pi (, ) pi .
3.3) pi pi pi - pi ~T pi -
f(x, y) = logx2 + y2, (x0, y0) = (1, 0), ~T = (2i+ j)/
5
Q(x, y, z) = ex + yz, (x0, y0, z0) = (1, 1, 1), ~T = (1,1, 1)/3
(pi. 2/5, e/
3.)
3.4) pi pi (x, y, z) = 2x+y+z2 i+ j+ k . (pi.3) 3.5) pi
z2 ex cos y, log(x2 + y2 + z2),xyz
x2 + y2 + z2
3.6) pipi pi pi piz = cos x sin y (0, pi/2, 0).
3.7) R = |~R| pi ~r = xi + yj + zk pi ~A = ai+ bj + ck. |~R| ~R.
3.8) A B P pi. AP , BP pi P . (pi AP +BP = .)
3.9) , x, y, z, pi () ().
3.10)pi pi T (x, y, z) = ex2y22z2pi pi pi . - pi pipi . pi pi pi v = e8m/s pi n = 12 (i+ j2k); pi pi ;
3.11) pi pi z = hx2 y2/3. pi
-
3.7. 113
3.22: pi pi pi pi 3.8.
(x, y) = (1, 1). pi pi pi , pi tan =0.04.
3.12) pix21a21
+x22a22
+x23a23
= 1,x21b21
+x22b22
+x23b23
= 1. (3.85)
a21b22 = a
22b21, pi pipi (x1, x2).
3.13) F (r, ) = f(r cos , r sin ) f(x, y) = x/(x2 + y2).
F
r= cos
f
x+ sin
f
y
2F
r2= cos2
2f
x2+ sin2
2f
y2+ 2 cos sin
2f
xy
|f |2 =(F
r
)2+
1r2
(F
)2(3.86)
3.14) m ~r(t) Newton, pi ~F = , pi (~r(t)) pi pi . E = 12m~v2+(~r) ( ). pi pi pi, .
(pi. dEdt = m~v d~vdt +(~v (~r)+ t ) ~F = m~.) 3.15) V (~r) = V0 2e~r2/2 , ~r = xi +
yj. pi pi V0, , . pi V (~r) .
-
114 3. , , .
3.16) pi pipi (x0, y0) piz = f(x, y) pi
z = f(x0, y0) +f
x
(x0,y0)
(x x0) + fy
(x0,y0)
(y y0) (3.87)
(. (3.87) pipi- z = ax+ by + c pi pi .)
3.17) pi ij (3.35) pipi (3.34).
3.18) pi ( Hessian)
(x1, x2)(
a bc d
)(x1x2
)(3.88)
pi pi xixj . pi pi pi .(pi. pi a2 (x1 +
bax2)
2 + (c b2a )x22 pi...) 3.19) f(x, y) = 1/(xy). pi pi pi . ( pi pi d =
x2 + y2 + z2 . pi. (1,1,1),
(1,-1,-1), (-1,1,-1)). 3.20) cos x+sin y
(x, y), [0, 2pi] [0, 2pi]. 3.21) pi ,
xy + 1x +8y . x
3 + y2 6xy + 6xq + 3y.
-
Keflaio 4
Apklish kai
Strobilismc.
4.1 Grammc Roc.
pi pi . pi pi - (x, y, z) = c, pi pi. pipi, pi pi - ,pi . pi pi pi , . ( pi - pi pi pi .) pi pi pipi -
~F (x, y, z) = Fxi+ Fy j + Fz k (4.1)
pi Fx = Fx(x, y, z), Fy = Fy(x, y, z) Fz = Fz(x, y, z). , pipi , pi ~F (x1, x2, x3) = Fixi, Fi = Fi(x1, x2, x3)pi i i = 1, 2, 3.
pi pi (4.1);pi pipi pi (4.1) ( pi R3 R3), pi R3 . (4.1) pi pipi pi.
pi pi pipi pipi
115
-
116 4. .
4.1: pi pi .
pi, . , pipi - . pi pi pi pi pi- . ( (4.2) pi pi .) - pi pi . pi, pi pi pi pi. pi , pi pi ~r = ~c(t) pi pi .
~A pi, pi ~c(t) pi
d~c
d t= ~A(~c(t)). (4.2)
pi . (4.2) . ~c(t) = x(t)i+ y(t)j + z(t)k,
d x
d t= Ax
d y
d t= Ay
d z
d t= Az
-
4.1. . 117
pi pi
d x
Ax=
d y
Ay=
d z
Az(4.3)
. pi pi d~cd t ~A; (4.3).
4.2: pi pi .
pi ~c(t) - pi. pi pi pi , pipi ~r0 = ~c(t0). pi pi .
pipi pi pi pi pi , pi pi.
. pi pi pi pi , pipi pi pipi t. pipi pi pi pipi pi pi . ~v pi , ~vt = 0. pi pi t.
. 1. pi pi pi ~c(t) =
(e2t, ln|t|, 1/t), t 6= 0, pi ~F = (2x, z,z2).pi. pi (4.3).
pi
d
dt~c(t) = (2e2t,
1t, 1
t2)
-
118 4. .
2e2t = 2x, 1t = z, 1t2 = z2, (4.2) pi. 2. pi
~v = xi yj pipi pi .pi. pipi (x, y) pi
(4.3)
dx
vx=
dy
vy
dx
x=
dy
y xy = c
pi (4.3)
4.3: pi pi pi 2.
. pi pit x = x0et, y = y0et. ~c(t) = ix0et+ jy0et. x0, y0 pi pi . pi pi et = x/x0, et = y/y0, pipi pi xy = x0y0 = c. pi pi pi pi (1, 1) x0 = y0 = 1 t = 0, pi ~c1,1(t) = iet + jet pi xy = 1.
3. pi ~v = yi+ xj pipi pi .
pi. , pi (4.2)
dx
vx=
dy
vy
dx
y =dy
x x2 + y2 = c
-
4.1. . 119
pi pi . pi pipi, pi- pi pi (r =(x2 + y2)1/2, = sin1 x/r)
~v = r(yri+
x
rj) = ru
. 4. pi pi pi
~v =1r(a e + b e) (4.4)
pi e = cos i+ sin j e = sin i+ cos j, - pi . pipi .
pi, pi
d r
vr=
r d
v
pi vr = ar , v = br . ,d r
r= a
bd
pi
r = c eab (4.5)
,
vr =d r
d tv = r d d t (4.6)
pi,
r2 = r20 2a t (4.7)(pi pi r(t = 0) = r0), ,
= b2a
ln |r20 2at|+ d (4.8)
pi d pi pi pi pi . pi- r,
0 = baln
r0r
(4.9)
pi (4.5)
-
120 4. .
4.2 Apklish kai Strobilismc.
pi - pi . pi pi. pi pi pipi - pi pi (div) (rot curl). , .
4.2.1 Apklish (div).
pi (divergence) pi pi pi pi pi (pi (4.4) ). , pi pipi pi pi pi - .
4.4: pi.
pi ~F (x, y, z), pi
div ~F ~ ~F=
Fxx
+Fyy
+Fzz
(4.10)
pi pipi , pi pi pi pi ~F (x, y, z). pi pi .
-
4.2. . 121
pi pi pi . pi (x, y, z) pi ~v(x, y, z). pi pi pi, pi S ~h = ~vt. pipi pi pi m = (~vt) nS. pi pi S
mt
= ~v nS (4.11)
pi ~J = ~v, (4.11)
mt
= ~J nS.
pi -
DS
nn^
v
D
t
4.5: S t.
pi pipipi x,y,z pi ~J pi pipipi. x pi I, II( 4.6)(
~J nI + ~J nII)S = (Jx(x+x, y, z) Jx(x, y, z) )yz
=Jx(x+x, y, z) Jx(x, y, z)
xxyz
Jxx
V (4.12)
pi V = xyz pipipi. pi pi pipipi ,
-
122 4. .
~J ~S|pi. (Jxx
+Jyy
+Jzz
)V
m/t pi pi pi. pi pi ~J pi pipipi. V , ,
mVt
= ~ ~J (4.13)
m/(Vt) t () pi,
~ ~J + t
= 0 (4.14)
(4.14) .
D x
D y
D z
(X,y,z)
nnI
II
^ ^
J(x,y,z)
4.6: pi pi pipipi.
. 1. pi,
pi pi pi ~v(x, y) pi P pi pi (4.7), (4.8).
pi. pi (4.7) pi pi j, pi ~v = vx(x, y)i. y pi pi pi pi x. pi, vx pi x, vx = vx(y). pi y = y0 pi i, pi y > y0. vx = vx(y, y0) vx < 0 y < y0 vx > 0 y > y0. pi div~v = vx(y,y0)x = 0. pipi ~v = vxx > 0. pi, pi pi
-
4.2. . 123
. P . P
4.7: pi pi 1
. P.P
4.8: pi pi 1
-
124 4. .
(4.8) ~v = vyy > 0 ~v = 0.
2. pi pi ~E = ~rr3 . pi ;
pi. pi pi pi pi pi. pi pi pi pi pi pi-pi . pi ~E - ~A = ~r = 1r3 . pi, ~A,
( ~A) = xi
(Ai)
=
xiAi +Ai
xi
( ~A) = ~A+ ~A (4.15)
pi pi pi
(u(~r)) = dduu(~r) (4.16)
pipi ,
(1r3~r
)=
1r3 ~r + ~r 1
r3
=1r3
3 + ~r (3 ~r
r5
)= 0, r 6= 0
pi r = 0. pi ~E pi, ~r = 0 (pi ) pi . pi pi pi pi pipi.
4.2.2 Strobilismc (curl rot).
pi pi - pi . ~F (x, y, z) =F1i + F2j + F3k , curl ~F rot ~F
curl ~F = ~F=
(F3y
F2z
)i+(F1z
F3x
)j +
(F2x
F1y
)k(4.17)
-
4.2. . 125
pi, (4.17) pi pi pi , pi x . . . Fi,
~ ~F =
i j kx
y
z
F1 F2 F3
(4.18) 1. pi pi
~v = (yi xj)/(x2 + y2)..
~ ~v =
i j kx
y
z
yx2+y2 xx2+y2 0
= 0 i+ 0 j
{
x
(x
x2 + y2
)+
y
(y
x2 + y2
)}k
= (
y2 x2(x2 + y2)2
+x2 y2
(x2 + y2)2
)k = 0 (4.19)
2. pi ~v = yi xj.
~ ~v =
i j kx
y
z
y x 0
= 0 i+ 0 j 2 k = 2 k (4.20)
3. pi
~u = z2yi+ z2xj . pi ~ ~v = ~u ~,~v.
pi. ~v :
d x
z2 y= d y
z2 x x2 + y2 = c (4.21)
. pi, pi ~u = ~r, pipi ~ pi ~ = z2k. pi, ~ ,
d z
z2= cdt 1
z= b c t
-
126 4. .
dx = dy = 0. ~u
~v = ~u= 2z(x i y j + z k)
, pi ~u :
d x
x=
d y
y= d z
z y = c1 x, x z = c2, y z = c3
(4.9) ~v.
01
23
4 0
1
2
3
4
1
2
3
01
23
4
4.9: pi ~v
4.2.3 Fusik Ermhnea.
pi pi . (4.10) pi ~J = ~v. , pi pi pi z. pi - z. pi pi , pi pi pi pi
e = sin i+ cos j (4.22)
pi pi
~J(~r) e (4.23)
-
4.2. . 127
pi v , r , ( (4.10) ) pi 1
vx(x+x, y +y, z) vx(x, y, z) + vxx
x+vxy
y (4.24)
vy(x+x, y +y, z) vy(x, y, z) + vyx
x+vyy
y (4.25)
x,y r ,
4.10: .
x = r cos , y = r sin . (4.23) pi
v = ~v e= sin (vx + vx
xr cos +
vxy
r sin )
+ cos (vy +vyx
r cos +vyy
r sin )
pi pi
v = sin vx + cos vy +(vyy
vxx
)r sin cos
+vyx
r cos2 vxy
r sin2 (4.26)
1
Qrhsimopoiome ed thn anptuxh tou tpou tou Taylor,
f(x) =n=0
(x x0)nn!
dn
dxnf(x0).
Anaptssoume gia tic do metablhtc x, y kai kratome mno touc grammikoc rouc.
-
128 4. .
v pi pi ,
v = 12pi 2pi0
~v ed (4.27)
pi (4.26) 2pi0
sin d = 2pi0
cos d = 2pi0
sin cos d = 0.
, pi (4.26), pi
v = 12pi 2pi0
(vyx
r cos2 vxy
r sin2 )d
=r
2pi
(vyx
2pi0
cos2 d vxy
2pi0
sin2 d )
=r
2
(vyx
vxy
)(4.28)
pi z pi,
z =vr
=12
(vyx
vxy
)(4.29)
pi pi , x, y. pi pi pipi x y z, pi ( pi1/2)
~ = xi+ y j + z k
=(vzy
vyz
)i+(vxz
vzx
)j +
(vyx
vxy
)k (4.30)
pi ~v.
~ = ~v (4.31). pi pipi
~v = jv0ey2
2 (4.32)
~u = ju0ex2
2 (4.33)
. pi pipi (4.11). pi pi pipi
-
4.2. . 129
~v = 0 ~u = k 2u0
2x e
x2
2
X
Y
Ftervt