Διανυσματικός Λογισμός-Λεοντάρης Γεώργιος

1 . .

pi 2005

3

pi - pi pi pi pi . pi . , pi pi pi . pi pi - pi. , pi pi pipi - ij Kronecker ijk (LeviCivita). pipi pipipi 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. pipi 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

4 pi, pipi pi pi pi pi.

- . pi - pi . pi pi . pi pi pi pi .

..., 29.01.2003

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

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

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

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)

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)

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|

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.

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)

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)

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),

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)

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) pipi 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 pipipi pi. pi pipi . pipi 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. pipi

~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


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


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 pipi 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) pipi.

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 , ) pipi

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.

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 . pipi 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).

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, pipi 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

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