=VS

dxdydzFdSnFGGGG

2013

Email: dsourlas@physics.upatras.gr

, 19 , , , . . , , . , .

, . , . .

1

1

11.111.2:21.331.441.541.661.771.881.9,.91.101220

23

232.1232.2252.3262.4262.5302.63239

41

413.1413.2.423.343

3.4443.5443.62463.7.483.8 F513.9573.10603.116468

IV71

I714.1714.2734.3774.4824.5.884.6894.79094

V97

975.1975.2985.3.995.41025.51065.61105.7.1115.81135.9114116

VI117

1176.11176.21196.31196.41226.51296.61336.7E135137

VII139

1397.11397.21397.3150153

VIII155

1558.11558.2.1558.31578.41618.5.1628.6'.1638.7.1658.8DS1668.9.1698.10.1748.11.1758.121778.131788.141818.15183185

IX189

GREEN,STOKESGAUSS1899.11899.1GREEN.1899.3GREEN1969.4STOKES1979.5STOKES2009.6GAUSS(2019.7GAUSS2039.8GAUSSSTOKES2049.9,207212

215

21510.121510.2,21510.3LAGRANGE22710.4232235

237

23711.1.23711.223711.323811.4,.24011.5GRAD,DIVCURL.24211.6242246

249

24912.124912.225112.325212.425312.5254

12.625512.725612.825812.926012.10CHRISTOFFEL26312.1126412.12265267

269

2691.2692.2723.2764.2785.280

B283

283288

289

2892891.()2902.()2913.2944.2965.2996.301

1305

2311

3319

4327

5337

6343

7357

8365

9385

10399

11413

12421

431

I 431432III434IV436V436VI438V439V440GREEN,STOKES,GAUSS441X442

444

I 444 446III 450IV 458V 461

VI 468V 479V 482 GREEN,STOKES,GAUSS487X 495

MAPLE500

531

531

1.1

, , , , , , , .. 6,28 , 22 0C, 220 Volts ... , . .. 400 300 , 700 700 . 400 300 , 700 , , (. 1.1.1). , . , .. , , , - , . , , , , " ".

- (1,

(1 , , , , . . .

2 2400 300 500+ =

400 300

300

. 1.1.1

2

, , v=OA (2, (. 1.1.2). - .. OXYZ, v=(vx ,vy ,vz ), (. 1.1.2).

. R3. v |v| (3 .

(1.1.1)

|v|=1 .

(2 .. , v,u,w, ..., .. . (3 ||v||, |v|, . , ||, R C, . |||| norm .

2 2 2x y zv v v= + +v

v, vG

A

. 1.1.2

vz

v

O

A

. 1.1.2

y

z

x

vx

vyO

v

3

1.2:

v w : O, , v w , - , v w, (. 1.2.1). r, , - v w. w v.

v w (vx ,vy ,vz ) (wx ,wy ,wz ) , v w, :

r=v+w=(vx+wx , vy+wy , vz+wz) (1.2.1)

.

1.3

v+v+v 3v . 3v 3 v v 3v v. , v , v v v. , v , v. =0, v=0 0 , ( , 0=(0,0,0) ).

O

x

z

y

A

w

v

. 1.2.1

rr

4

v v, .

. :

v-w=v+(-1)w (1.3.1)

v -w, w.

1.4

v w : v-w=0. v w ,

v=w (vx ,vy ,vz )=(wx ,wy ,wz ) vx = wx , vy = wy , vz = wz (1.4.1) , (), . , (.. )

, , : , . . , , . , .

1.5

v w, (4 vw , : (4 .. , (v,w)

5

vw=|v||w|cos (1.5.1) v w. (1.5.1) :

vw=wv (1.5.2) (1.5.1)

:

|w|cos== w v , :

vw= |v|( w v) vw= |w|( v w) : |v|=1 vw=

w v

vv=|v|2 (1..5.3) , ,

, :

(v+w)u=vu+wu (1.5.4) . v w vw=0, v w

vw=|v||w|cos=0 cos=0 =/2. OXYZ i, j, k OX, OY, OZ, v w :

v=vxi+vyj+vzk w=wxi+wyj+wzk (1.5.5)

vw (1.5.4) ij=ik=jk=0 ii=jj=kk=1, ( i, j, k, ), :

vw=vxwx+vywy+vzwz (1.5.6)

v

. 1.5.1

A

B

w

6

(1.5.1), (1.5.6). : (1.5.1) v w , (), v w , . . (1.5.6) , vx , vy , vz wx , wy , wz .

(1.5.1)

(1.5.7)

(1.5.7) , , .

: (1.5.1) (1.5.6) :

1) vv0 v vv=0 v=0 (1.5.8) 2) vw =wv v,w (1.5.9) 3) (v+u)w=(vw)+(uw) v, u, w , R (1.5.10) : , ( ), , .

1.6

v w, vw, , |v||w|sin, , v w

1 1

1

2 2 2 2 2 2

cos cos| || |

cos x x y y z z

x y z x y z

v w v w v w

v v v w w w

= = = + + = + + + +

v w v wv w v v w w

7

: , ( ). vw. vw v, w, vw , (.1.6.1). , .

:

vw=-wv (1.6.1) . vw=0, =0, v w . .

,

v(w+u)=vw+vu (1.6.2) v w :

vw=(vxi+vyj+vzk)(wxi+wyj+wz k) (vywz -vzwy)i+(vzwx-vxwz)j+(vxwy-vywx)k

(1.6.3)

(1.6.3) .

1.7

,

y z x yz xx y z

y z x yz xx y z

v v v vv vv v v

w w w ww ww w w

= + + =i j k

v w i j k

vw

wv v

w

8

. , . " ", ( ), " ", ( ), . :

v w

vw= (1.7.1) , , .

, (), w v, v.w=. , v* = v+u u w, u.w=0 (1.7.1). :

v*w=(v+u )w=vw+uw=+0= .

.

1.8

, . .

) v, w, u :

v(wu) (1.8.1) : v, w, u, , (. 1.8.1). |wu| , |wu|=|w||u|sin, w u . |v|cos. V :

V=|w||u|sin|v|cos=|wu||v|cos= v(wu)

9

>/2, v(wu) . :

V = |v(wu)| (1.8.2)

:

( ) x y zx y zx y z

v v vw w wu u u

=v w u (1.8.3)

) v, w, u :

v(wu) (1.8.4) :

v(wu)=(vu)w-(vw)u (1.8.5) ( :

a(bc)=b(ac)-c(ab) (1.8.6) back up).

1.9,.

, .

u

w

v

wu

. 1.8.1

10

, . . ' .

, , . rF F , r rF F . ' r . .

rF F . F. ' F . .

:

.. . .. . . u , |u|=. u ,

,

u1, u2 , u1, u2. :

. 1.9.1

2

3

1

u3

u1

u2

11

1 1 2, (. 1.9.1). u1 |u1|=1. 2 2 3 u2 |u2|=2. u1 u2 , 1 3 u1+u2. u3 13. u3 , u1 u2. 1=2=/2, u3 u1 u2.

. , .

- , , .

: (u1, 1), (u2, 2), (u3, 3) , 1, 2, 3 . :

2=1+12=1+(u11) (1.9.1) |u11|=1|1|=12=|12| u11 - 12 .

3=2+23=2+(u22) (1.9.2) (1.9.1) (1.9.2)

3=1+(u11)+u2[1+(u11)]= = 1+(u1+u2) 1+u2(u11) (1.9.3)

1, 2 , - :

3= 1+(u1+u2) 1 (1.9.4) (u3,3), :

3= 1+u3 1 (1.9.5) (1.9.4) (1.9.5) :

12

u3=u1+u2

.

.

1.10

1) ) .

) .

: ) A , (. 1). :

+= =+=-+=1/2(-)=1/2()

) , (. 2). 1 2 . :

2=+2=+1/2=

=+1/2(-)=1/2(+)

(1)

1=1/2=1/2(+)=

=1/2(+)=2 (2)

(1), (2) 1=AP2. 1 2

2) , , 2 1.

: , . =1/3, =1/3.

A

B

. 1

1 2

. 2

13

= (1). = (2). :

=+=+ (1)

:

=+ (2)

(1) (2):

+=+ (3)

(3) - :

=1/2, =1/2, =-1/2+, =-1/2+ (4)

(4) (3) :

1/2+(-1/2+)=1/2+(-1/2+) (1/2--1/2)=(1/2--1/2)

B, BA , :

--1/2=0, --1/2=0 ==1/3. =1/3, =1/3

=1/3.

3) , /.

: . , , , , . :

(1)

: =-, =- (2)

(2) (1): (3)

= A B

= O OA OB O

14

(3) :

= (4)

(4) .

4) . .

: :

=1/2+1/2=1/2(+)

=1/2(+)

=1/2(+) (1)

=1/2(+)

+++=0

+=-(+) (2) (2) (1) :

=1/2(+)=-1/2(+)=-

=-1/2(+)=-1/2(+)=-=

(3) .

5) 20km/h. 20km/h . ; ;

: , 202km/h. 450 . ,

+ +

OB OA

(3)

20Km/h

20Km/h

20

20

45o

202

15

, . , 20 20.

6) 250miles/h . 80miles/h . ;

: v , v1 v2 . : v1=v2+v

:

(v2)2=(v1)2 +(v)2-2v1vcos(1350)=

=311.7miles/h

:

sin=0.567 =34.550.

, 311.7miles/h 34.550 .

7) , , ||=3 ||=2.

) .

) , ||=3 ||=2.

) .

2 10sin135 sin

v v=

20

20

20

20

1350

v1 v2

v

16

: ) ||=3, 3. ,

2. 1 5.

) ||=2. 2. (5, .

) .

1 5. 1 5.

8) r r .

cos2 +cos2+cos2=1.

: r=(x,y,z) , , , , . ,

cos=x/r (1)

r=|r| r. :

cos=y/r (2)

(5 . 8.3.

5

3

2

17

:

cos=z/r (3)

:

x2+y2+z2=r2 (4)

:

cos2+cos2+cos2=

=x2/r2+y2/r2+z2/r2=r2/r2=1

9) ) v(2,4,-5) , , . ) v(2,5,1) u(1,1,3).

: ) , , v , , . :

vi=|v||i|cos 2= cos cos= =0.298 =730

:

vj=|v||j|cos 4= cos cos= =0.596 =530

vk=|v||k|cos -5= cos cos= =-0.745 =1380

) v u v u. v u . u. :

u0=

:

4 16 25+ + 245

4 16 25+ + 445

4 16 25+ + 545

( )1 1,1,3| | 11

=uu

v

u

18

10) u1, u2 , , .

:

cos(-)=coscos+sinsin (1)

cos(+)=coscos-sinsin (2)

: 1 :

u1=cosi+sinj u2=cosi+sinj

u1 u2:

u1u2=|u1| |u2|cos(-) coscos+sinsin=cos(-)

2. :

u1=cosi-sinj

u2=cosi+sinj

u1 u2 : u1u2=|u1| |u2|cos(+) coscos-sinsin=cos(+) (2) (1) .

11) , . .

( ) ( ) ( )( ) ( ) ( )

0 01 1 3 12,5,1 , , 1,1,311 11 11 11

1 1 102 5 3 1,1,3 1,1,31111 11

= = = + + =

v u u

i

j

. 1

u1

u2

i

j

. 2

u2

19

, (6.

S1+ S2+ S3+ S4=0

S1, S2, S3, S4, .

: , a, b, ,

, ( 4).

:

S1= , S2= , S3= , S4=

:

S1+ S2+ S3+ S4= + + + =

=

.

(6 VIII

12

a b

12

a b 12

b c 12

c a ( ) ( )12

c a b a

12

a b 12

b c 12

c a ( ) ( )12

c a b a

[ ]12

+ + + + =a b b c c a c b c a a b a a 0

S1 S2

S3

S4

a c

b

b-a

c-a

20

1. :

(v+w )u =vu+wu 2. : |v+u| |v|+|u| 3. Cauchy-Schwarz: |vu||v||u| 4. :

S=1/2||=1/2sin , , .

5. :

) :

) :

2=2+2-2cos , 2=2+2-2cos , 2=2+2-2cos

' , , .

6. :

) |v+u|2 -|v-u|2=4 v.u

) |v+u|2+|v-u|2=2|v|2+2|u|2

7. :

) v(uw) = u(vw)-w(vu) ) (vu)w = u(vw)-v(uw) ) (vu)(wr) = (vw)(u.r)-(vr)(uw)

8. Jacobi:

v(uw) + u(wv) + w(vu) = 0 9. v0 : vu=vw vu = vw

u=w. , uw. 10. a, b, c

. v a, b, c :

sin sin sin = =

21

11. n e1, e2, , en . ri ei (i=1,2, , n) . :

:

.

) .

) .

= + + v b c v a c v b av a b ca b c b a c c b a

n

i ii 1

e=

= p r

n

i ii 1

n n

i ii 1 i 1

e

e e

=

= =

= =

r

pR

n

ii 1

e 0=

n ii 1

e 0=

=

j i

k r

x

y

z

2.1

, , :

r=rxi+ryj+rzk

rx ,ry, rz , .. r=3i-2j+6k. , .. , . , .

, t, r :

r=x(t)i+y(t)j+z(t)k (2.1.1)

.

- :

r: IR AR3 r: tI r(t) (2.1.2) , , . x=x(t), y=y(t), z=z(t) - (2.1.1) .

24

: :

1. r(t) = Rcosti + Rsintj t[0,2) XY R.

, x=Rcost y=Rsint : x2+y2=R2.

2. r(t) = costi + sintj t[0,2) XY .

, x=cost y=sint : .

3. r(t) = costi + sintj + tk t[t1 ,t2]R OXYZ.

4. r(t) = coshti + sinhtj tR x2-y2=1. cosh2t-sinh2t=1 x=cosht, y=sinht . x=cosht>0, .

.

t .

:

2 2

2 2 1x y + =

25

1.

2.

3.

r(t) = costi + 2sintj + t/2k t[0 ,4] 4. ( )

r(t) = coshti + sinhtj tR

2.2

r=rx(t)i+ry(t)j+rz(t)k l=lxi+lyj+lzk t t0, :

( ) cos sin [0,2 ),t R t R t t = + r i ( ) cos sin [0,2 )t a t t t = + r i

26

, , (2.2.1)

:

1 :

(>0)((,t0)>0)[|t-t0|

27

rx(t0)i+ry(t0)j+rz(t0)k

:

r(t)= rx(t0)i+ry(t0)j+rz(t0)k (2.4.1)

.

2.4.1, , r(t).

: (x,y,z) - -, t, x,y,z t:

x=x(t), y=y(t), z=z(t).

: r(t) = x(t)i + y(t)j + z(t)k

:

:

1: (t)r(t), r(t)v(t) r(t)w(t)

0 00 0h 0 h 0

( ) ( )( ) ( )=lim lim y yx xr t h r tr t h r t

h h + + + +i j

0 0h 0

( ) ( )+lim z zr t h r th

+ =k

d tdtr( ) =

( ) ( ) ( ) ( ) ( )d t dx t dy t dz ttdt dt dt dt

= = + +rv i j k

( ) 2 2 2 22 2 2 2( ) ( ) ( ) ( ) ( )d t d t d x t d y t d z tt dt dt dt dt dt= = = + +v ra i j k

z

x

r(t0) r(t0+h)

r(t0) r(t0+h)-r(t0)

. 2.4.1

y

z

28

: . :

(2.4.2)

(2.4.2)

(2.4.2)

C r(t) dr/dt .

, .

1: r(t) , :

tI (2.4.3)

: . |r(t)|=r(t)=.

r2(t)=r.r=.

: . : r2(t) =rr

r2(t)=. r(t)=|r(t)|=. 2: r(t) , :

[ ]d d (t) d (t)(t) (t) (t) (t)dt dt dt

= + rr r

[ ]ddt

t t d tdt

t t d tdt

r v r v r v( ) ( ) ( ) ( ) ( ) ( ) = +

[ ]ddt

t t d tdt

t t d tdt

r v r v r v( ) ( ) ( ) ( ) ( ) ( ) = +

( )( ) 0d ttdt

=rr

[ ]( ) ( ) 0d t tdt

=r r ( ) ( )( ) ( ) 0d t d tt tdt dt

+ =r rr r

( )2 ( ) 0d ttdt

=rr ( )( ) 0d ttdt

=rr

( )( ) 0d ttdt

=rr

2 ( )( ) 2 ( ) 0d d tr t tdt dt

= = rr

29

tI (2.4.4)

: : :

r0(t)=

|r0(t)|=1 , r0(t) r0(t)=0. : r(t)=r(t)r0(t) r(t)=r(t)r0(t)+r(t)r0(t)=r(t)r0(t)

r(t)r(t)=r(t)r(t)r0(t)= =r(t)r(t) r(t)r(t)=0

r(t)r(t)=0

: r(t) =0. :

=

(2.4.5)

r2(t)=r(t)r(t) 2r(t)r(t)=2r(t)r(t) r(t)r(t)=r(t)r(t) (2.4.6) (2.4.5) (2.4.6) :

=

r(t)r(t)=0 r0(t)=.

r(t) .

: r v. :

( )( ) d ttdt

=rr 0

( ) ( )| ( ) | ( )

t tt r t

=r rr

( ) ( )( ) ( )t r t

r t r t=r

( )d tdtr

0 ( ) ( )( )

d t d tdt dt r t

=r r 2( ) 1( ) ( )( ) ( )r t t tr t r t

+ =r r2

3

( ) ( ) ( ) ( ) ( )( )

r t r t t r t tr t

+= r r

[ ] [ ]03

( ) ( ) ( ) ( ) ( ) ( )( )( )

t t t t t td tdt r t

+ = r r r r r rr

{ } { }31 ( ) ( ) ( ) ( ) ( ) ( )( ) t t t t t tr t = r r r r r r{ }31 ( ) ( ) ( )( ) t t tr t = r r r

0 ( )d tdt

=r 0

30

r2=rr=, v2=vv= , (2.4.3), :

2 r =0 rv=0 (2.4.7)

v=0 (2.4.8)

rv=0 :

vv+r=0 r=-v2 (2.4.9)

(2.4.7) v r. (2.4.8) v . r r 0 180 . (2.4.9) :

r=|r|||cos=-v2 cos

31

(2.5.3)

c .

1:

r(t)=(t2+1)i+2j-t3k.

: ( )( )2 3(t) (t)dt t 1 2 t dt= = + + = u r i j k ( )3 41 2 3t tt c 2t c c3 4

= + + + + = i j k

= ( )3 4t tt 2t3 4

+ + + i j k c c=c1i+c2j+c3k.

1: , . :

(2.5.4)

, , , :

(2.5.5)

2: :

:

( ) ( )t t dt= +u r c

( ) ( ) ( ) ( ) ( )t t dt x t dt y t dt z t dt= + = + + + u r c i j z c

[ ]2 21 1

( ) ( ) ( ) ( )t t

t tt dt x t y t z t dt= + + r i j k

( )( )3 2 32

1 2t t dt+ + i j k

( )( ) ( ) 33 3 42 32 2

1 2 23 4t tt t dt t t

+ + = + + = i j k i j k

[ ] ( )8 81 22 659 3 2 6 4 4 23 4 3 4

= + + + = + i j k i j

32

, :

(2.5.6)

(2.5.7)

3: :

) )

r(t)=t2i+(t+1)j-t3k, v(t)=2i-tj+t2k

: ) r(t)v(t)=2t2-t2-t-t5=-t5=t2-t.

=

) r(t)v(t)=(t3+t2-t4)i+(-2t2-t4)j+(-t-2t-2)k.

2.6

1) a(t)=t2i+sintj-t3k. t. t=0 .

: :

2

1

( ) ( )t

tt t dt r v

2

1( ) ( )

t

tt t dt r v

3

0( ) ( )t t dt r v 20 ( ) ( )t t dt r v

3

0( ) ( )t t dt r v ( )

36 3 23 5 2

00

-t +t -t 1176 3 2t t tdt = + =

( ) ( ) ( )2 2 4 3 2 4 3 30 0

( ) ( ) 2 2 2t t dt t t t t t t t dt = + + + + = r v i j k25 4 3 5 4 4

2

0

4 722 125 4 3 5 2 4 15 5t t t t t t t t

= + + + + = i j k i j k

ddt

= va

33

c1 , :

t=0 v(0)=0 -j+c1=0 c1=j.

r(t) :

r(0)=0 : c2=0.

2) . ( r ).

:

: (1)

: . (2)

. f(r)>0 f(r)

34

(3) r:

=c

m , :

f(r)0 a(t) r(t) .

f(r)

35

:

, :

v .

.

r : hr=0

h, r . h , h.

m :

m . . :

, .

Kepler .

1t 2 t

= rr

01 1lim2 2t

dE ddt t dt

= = = rr r v

12

= h r v

3

mMGr

= F r

2 2

2 3 2 3

d mM d Mm G Gdt r dt r

= = r rr r. 2.6.2

36

tAB t, tAB=t , :

()=(BAO)

4) , :

(1)

= (2)

. .

: r0 , r,

r=r r0 (3)

(3): (4)

:

L=2mh=mrv= (5)

(1) (5): 2

02 2

d d MGdt dt r

= = r vL L r L

(6)

(5) (6):

(7)

2

2 3

d MGdt r

= r r

1 d2 dt

=rr h

00

dd drrdt dt dt

= = +rr v r

20 00 0 0

d ddrmr r mrdt dt dt

+ = r rr r r

2

02 2

d d MmGdt dt r

= = r vL L r L

( )

2 00 02

0 0 00 0 0 0

dd MmG rdt r dt

d d dGmM GmMdt dt dt

= = = =

rv L r r

r r rr r r r

37

( : v(uw) = u(vw)-w(vu) 1). L , :

(8)

(7), (8) :

(9)

(9):

vL=GmMr0+c (10) c . (10) r :

r(vL)=GmMrr0+rc=GmMr+rr0c= GmMr+rc cos (11) r0 c.

r(vL)=(rv)L=(L/m)L=L2/m (11) :

L2=GmMr+rc cos (12)

(13)

:

(14)

b e . (13)

.

e=1 e>1 . , , e

38

5)

r=costi+sintj , , , .

: : h=1/2rv. : v=dr/dt=-sinti+costj . :

h=1/2rv=1/2(costi+sintj)(-sinti+costj)=1/2k.

6) , : x=cost, y=sint, z=0 0t2 . .

: : r=r(t), t1tt2 : ) dr/dt t1tt2 ) tatb r(ta)r(tb) ( ) ) dr/dt0 t, t1tt2 :

r(t)=costi+sintj dr/dt=-sinti+costj dr/dt . ) ). 0t2 dr/dt0. . r(0)=r(2). r(0)=i r(2)=i. .

, t , :

dr/dt=-sinti+costj

, ( ).

39

1. :

2. r(t) t r(t), ( , r(t) t ), :

) (x-5)2+(y-3)2=9 (7

) 4x2+9y2=36

) y=x2 '

) y=x3 '

3. :

r1(t)=(et-1)i+2sintj+ln(t+1)k r2(u)=(u+1)i+(u2-1)j+(u3+1)k

. .

4. r(t) [0,2], r(0)=i

r(t) :

:

) (7 . .

[ ] ( ) ( )( ) ( ) ( ) ( )d d t d tt t t tdt dt dt

= + rr r

[ ] ( ) ( )( ) ( ) ( ) ( )d d t d tt t t tdt dt dt

= + r vr v v r

[ ] ( ) ( )( ) ( ) ( ) ( )d d t d tt t t tdt dt dt

= + r vr v v r

2 2

2 2 1x y + =

40

)

)

)

5. x(t)=e-t, y(t)=2cos3t, z(t)=2sin3t, t . ) t. ) t=0.

6. x(t)=2t2, y(t)=t2-4t, z(t)=3t-5, t . u=i-3j+2k t=1.

7. C x=x(s), y=y(s), z=z(s), s C C. r o C, dr/ds C.

8. ) x(t)=t2+1, y(t)=4t-3, z(t)=2t2-6t. ) t=2.

9. :

r(t)=2costi+sintk, v(t)=

:

) , ) , )

10. :

r(2)=3i+1j+1k,

r(4)=6i+7j+3k,

t 31e tt

+ i j k

/ 2

0( )t dt

r / 20 ( )t dt v / 20 ( ) ( )t t dt r v4

2

2

2

( )( ) d tt dtdt

rr

2 4

( ) ( ),t t

d t d tdt dt= =

= = + +r r0 i j k

3.1

. .. ' . ' ' . (x,y,z) r=xi+yj+zk , :

=(x,y,z)=T(r)

f :

f: ARn BR f: (x1 ,x2 ,...,xn )A f(x1 ,x2 ,...,xn)B (3.1.1) n .

: z=f(x,y), R3, ..

R. : w=f(x,y,z), R4 . f(x1,x2,...,xn) n ' n+1-.

2 2 2z R x y=

42

(x,y,z), (8. .

3.2.

f=f(r) , r r0 , :

(>0)(>0)[|r-r0|

43

f (0,0), . 2.

3.3

f=f(r) r0 :

(3.3.1)

, . ..

) R3

(0,0,0)

) R2

y=x

) R2

y=x2

) R3

x+y+z=0.

)

tan-1

x+y=0. R3 s x+y=0.

0 0lim ( ) ( )f f =r r r r

( ) 22 2 2xy yzf x y z= + +r

( ) 4xfx y

= r

( ) 2 22x yzf x y+= r

( ) 5x yfx y z

= + +r

( ) 21tan xzfx y

= + r

2xzx y+

44

3.4

f=f(x,y), ( , , ). f (x0, y0) x :

0 0 0 0

lim ( , ) ( , )x x f x y f x y = (3.4.1) f (x0, y0) y :

0 0 0 0

lim ( , ) ( , )y y f x y f x y = (3.4.2)

. , . .

. 2 22 ( , ) (0,0)

( , )0 ( , )=(0,0)

xy x yx yf x y

x y

+=

f(x,0)=0 x f(0,y)=0 y

(0,0) f x y. , (0,0), (0,0) y=x :

( ) (0,0)lim f ( ) 1 0x,y y=x

x,y =

3.5

f=f(x,y,z). :

0lim ( ,0) 0 (0,0)x f x f = =0lim (0, ) 0 (0,0)y f y f = =

2

2 2

2( , ) 1y xxf x y

x x== =+

45

) x f :

0( , , ) ( , , ) ( , , )lim ( , , )

xh xf x h y z f x y z f x y zf x y z

h

+ = =

(3.5.1)

) y f :

0

( , , ) ( , , ) ( , , )lim ( , , )yh y

f x y h z f x y z f x y zf x y zh

+ = =

(3.5.2)

) z f :

0

( , , ) ( , , ) ( , , )lim ( , , )zh z

f x y z h f x y z f x y zf x y zh

+ = =

(3.5.3)

, .

. f=f(x,y)=exy +ln(x2+y). :

( ) 2, 2xyf x y xyex x y

= + + ( )

2

, 1xyf x y xey x y

= + +

2 2(2,1) 4 44 1 5

f e ex

= + = ++

f x (2,1).

2 2(2,1) 1 12 2

4 1 5f e e

y

= + = ++ f y (2,1)

. , , f(x), . , , , .. :

: (x,y)(0,0)

2 2

2 ( , ) (0,0)( , )

0 ( , ) (0,0)

xy x yx yf x y

x y

+= =( )

( )2 2

22 2

2( , ) y y xf x yx x y

=+

46

(x,y)(0,0)

: f(x,0)=0 f(0,y)=0,

(x,y) (0,0). (0,0). :

f (x0+h,y0).

f (x0 ,y0+k).

f (x0,y0) f (x0+h,y0+k).

f , f f .

: , .

: .

3.62

f=f(x,y).

x y,

:

( )( )

2 2

22 2

2( , ) x x yf x yy x y

=+

0

( ,0) 0x

f xx =

= 0(0, ) 0

y

f yy =

=

0 0f( , )x

x y

0 0f( , )y

x y

fx

fy

47

(3.6.1)

2 f.

1: f=sin(x2y) :

=2xycosx2y =x2cosx y

2 :

2: f=ln(x2+y3) :

2 :

2

2

fx x

fx

= 2 f

y xf

y x

= 2

2

f y y

fy

= 2 f

x yf

x y

=

fx

fy

22 2 2 2

2 4 sin 2 cosf x y x y y x y

x = +

23 2 22 sin 2 cosf x y x y x x y

y x

= +2

3 2 22 sin 2 cosf x y x y x x yx y

= +

24 2

2 sinf x x y

y =

2 3

2f xx x y

= +

2

2 3

3f yy x y

= +

( ) ( )2 2 2 3 2

2 22 2 3 2 3

( )2 2 (2 ) 2( )f x y x x y xx x y x y

+ = =+ +

( ) ( )2 2 2

2 22 3 2 3

2 (3 ) 6f x y xyy x x y x y

= =+ +

( ) ( )2 2 2

2 22 3 2 3

3 (2 ) 6f y x xyx y x y x y

= =+ +

48

K

, x y, . , ( Schwartz), f :

, , ,

, :

(3.6.2)

x y y x, .

3.7.

T(x,y,z) , T0(x0,y0,z0) r0=x0i+y0j+z0k. :

(x0,y0,z0) ' (x,y,z)(9. . ' . .

(9 (x,y,z) (x0,y0,z0) x=x0+x, y=y0+y, z=z0+z x, y, z , .

( ) ( )

2

2

2 2 2 2

2 3 2

2 3

2 3 26 3 3 3 2f

yx y y y y

x yy x yx y

= + + =

+( ) ( ) ( )

2 2f fy x x y

=

fx

fy

2 fx y

2 fy x

2 2f fy x x y

=

49

, .

, /s, s . dT/ds :

(3.7.1)

f=f(x,y,z) df/ds P0(x0,y0,z0) (10 u=i+j+k 2+2+2=1, (. 3.7.1). , P0(x0,y0,z0) u :

x=x0+s y=y0+s z=z0+s (3.7.2)

s P(x,y,z) P0(x0,y0,z0). (3.7.2) :

f (3.7.2), f , s.

f = f(x,y,z) = f(x0+s, y0+s, z0+s) = f(s)

, ( 1 . 3.10), :

(10 u u .

0lim sdT ds s

=

2 2 2 2 2 2 2 2 20 0 0( ) ( ) ( ) ( ) ( ) ( ) x x y y z z s s s s s + + = + + = + + =

0 0 0 0( ) ( ) ( ) ( )df P f P f P f Pdx dy dzds x ds y ds z ds

= + + =

r0 r

P0

P

u

s

. 3.7.1

r

50

(3.7.3)

(3.7.3) : u=i+j+k , :

f (P0) gradf(P0) :

f =gradf = (3.7.4)

f 0.

(x,y,z) :

f =gradf= (3.7.4)

, f 0 :

=f(0)u (3.7.5)

:

f(r: u) (11

r .

(3.7.3) f(x,y,z) :

(11 f(r: u) : f r u.

0 0 0( ) ( ) ( )f P f P f Px y z

= + +

0 0 0(P ) (P ) (P ), , f f fx y z

0(P ) 0(P )0 0 0(P ) (P ) (P )f f f

x y z

+ +i j k

f f fx y z

+ +i j k

0(P )dfds

( )f

ru

51

(3.7.6)

(3.7.5) : df=(fu)ds

1: . :

) u=i = f(r: i)=f(r)i=

) u=j = f(r: j)=f(r)j=

) u=k = f(r: k)=f(r)k=

2: (3.7.5). fu : , - u, , f. , f, f. f f.

3: 3.10.

3.8 f

) (3.7.5) :

cos cosdf f f fds

= = = u u (3.8.1)

df/ds f u, (.3.8.1). |df/ds||f| =0 180 ,

f f fdf dx dy dzx y z

= + +

dfds

( )fx

r

dfds

( )fy

r

dfds

( )fz

r

z

y

x

df/ds r u

O

. 3.8.1

f

52

u f . =0, u=f/|f| df/ds , |f|. : f , df/ds, ( f ), .

1: f=x2y+xz u=2i-2j+k. f r=i+2j-k u. .

: u . ' :

f:

( ) 22f f ff xy z x xx y z

= + + = + + +i j k i j k

r=i+2j-k, . x=1, y=2, z=-1, :

f(r) = 3i+j+k :

=fu0 =(3i+j+k) (2i-2j+k)=2-

f(r), f(r)=3i+j+k : ( )

max

9 1 1 11df fds

= = + + =r

: f (x, y) . . z=f(x,y) (x, y). '

( )u uu

u i j k0 4 4 113

2 2= = + + = +| |

dfds

13

23

13

53

+ =

53

f= . . , z . , . dz ds: dz/ds . , u=f/|f| dz/ds=fu=|f|. (3.8.1) , . z, =180 . 90 , (12 .

.

:

. , , . . , , , . , .

f .

(12 , . z=f(x,y) z .

f fx y

+i j

( ) 2 2 2( , ) ,z f x y z f x y R x y= = = =

2 2 2 2 2 2 2 2

x yfR x y R x y R r

= = ri j

54

, .

E , ( f), f , c.

, - , c Lc . :

Lc={(x,y,z)AR3 / f(x,y,z)=c} (3.8.2) 2: f(r)=f(x,y,z)=

. , f=c =. c>0,

1/c.

f=1/r, - Coulomb . . f , .

f f :

) f ' Lc .

, , ( , ), , , . , - Lc , (. 3.8.2). :

r(t) = x(t)i + y(t)j + z(t)k

2 2 2

1 1r x y z

= + +

2 2 222 2 2

1 1c x y zcx y z

= + + =+ +

55

f, , , t, :

f=f(x(t),y(t),z(t))=g(t)=c (3.8.3)

c Lc. (3.8.3) t, :

0= = f (3.8.4)

f =0 t, tP , :

( ) ( )( ) 0P

Pt t

d tf tdt =

=rr

, f -

. , ( r(t)), f

( ) ( ) ( ) ( )dg t f dx t f dy t f dz tdt x dt y dt z dt

= + +

( )d tdtr

( )d tdtr

( )

Pt t

d tdt =r

O

. 3.8.2

x

y

z

f

r(t)

dt

td )(r

P

56

, Lc . f Lc .

(3.8.4),

:

, Lc, , , . , , , c, .

)

c2-c1=c

57

s= (3.8.6)

s, 1 2 cos , ( c |f(r1)| ), =0, . r, ( s), f(r1). f(r1) 1 .

3.9

: f=f(x,y). f z, z=f(x,y) , f.

1: : f=f(x,y)=x2+y2 z=x2+y2, :

1| ( ) | cosc

f

r

Lc1 Lc2

s

r1P1

P2

O

3.8.4

z

r2

y

f(r1)

x

Lc1

Lc2

r

r2

58

. : x2+y2=c c>0 c:

:

59

, . z . Oxy .

2: f=x2-y2. :

(13 . :

.

(13 , ( ).

60

:

: f=f(x,y,z) - , , - . . . f=x2+y2+z2, .

3.10

f(r) (14 R3 r. , ( ' ), u0. f r u , ( ):

(14 R - (,) , , (,) , . :

(pA)(>0)[(,)A]

61

(3.10.1)

1. f(r) , , r u. 2. f(r) , f'(r:u), ( ), r u,

(>0)(r,r1A)(uR3)(>0)[0|r-r1|

62

2. ( ). f(r) , :

f(r: cu)=cf(r: u) cR (3.10.4) : c=0 , (3.10.4) . :

f(r: 0)=

c0 :

, t=hc

h 0, t 0 cf(r: u). 3. ( ). f(r) , r, u, w , f'(r:u) f'(r:w), :

f(r:u+w)=f(r:u)+f(r:w) (3.10.5) : :

f(r+hu) :

, :

f(r+hu+hw)-f(r+hu) = f'(r+hu+hw:hw)

0

63

h 0, : f(r: u+w) = f(r: u)+f(r: w) : f'(r:w) r, :

f(r+hu+hw: w)=f(r:w) 2 3, , , .

4. ( ). f(r) . :

f(r:u+w+v)=f(r:u)+ f(r:w)+ f(r:v) (3.10.6) (3.14) :

f(r:u)=f(r:i+j+k)=f(r:i)+f(r:j)+f(r:k)=

= + + =f(r).u

1: f(r) ' CR3 r(s), s[s1, s2], . g f(r) r(s), .

g(s) = f(100

r(s)) = f(x(s), y(s), z(s)) (3.10.7)

r(s), -

f(r(s))=g(s)

:

lim h 0

( )fx

r ( )fy

r ( )fz

r

( )d sdsr

( )d sdsr

( ) ( )( ( ))dg s d sf sds ds

= =rr

( ( )) ( ( )) ( ( )) ( ) ( ) ( )f s f s f s dx s dy s dz sx y z ds ds ds

= + + + + = r r ri j k i j k

( )f dx f dy f dz df sx ds y ds z ds ds

= + + =

64

(3.7.3).

3.11

1) f=f(x,y) x=x(t), y=y(t). :

: :

(1)

(1)

.

(2)

:

(1):

(3)

(4)

(3), (4) (2):

2) f x, y, z, t, f=f(x,y,z,t). x=x(t), y=y(t), z=z(t) t,

2

2,df d fdt dt

df f dx f dydt x dt y dt

= + 2

2

d fdt

2 2 2

2 2 2

d f d df d f dx f d x d f dy f d ydt dt dt dt x dt x dt dt y dt y dt

= = + + +

,d f d fdt x dt y

2 2

2

d f f dx f dydt x x dt x y dt

= + 2 2

2

d f f dx f dydt y x y dt y dt

= +

2 22 2 2 2 2 2

2 2 2 2 22d f f d x f dx f dx dy f dy f d ydt x dt x dt x y dt dt y dt y dt

= + + + +

65

r(t)=x(t)i+y(t)j+z(t)k

: f:

3) f(x,y,z) 5 (2,1,3), f(2,1,3)=5. :

:

) f(2.1, 0.8, 3.1) ) f(1.9, 0.9, 3).

: :

(1)

) f(2.1, 0.8, 3.1) f(2,1,3) .

x=0.1, y=-0.2, z=0.1 (2)

(2) (1) :

f :

f(2.1, 0.8, 3.1)= f(2,1,3)+f=5+0.9=5.9

) (1.9, 0.9, 3) :

x=-0.1, y=-0.1, z=0

f=2(-0.1)+(-2)(-0.1)+3(0)=0 . f(1.9, 0.9, 3)=5.

df f dfdt t dt

= + r

f f f fdf dx dy dz dtx y z t

= + + + df f dx f dy f dz f d ffdt x dt y dt z dt t dt t

= + + + = + r

( ) ( ) ( )2,1,3 2,1,3 2,1,32, 2, 3f f fx y z

= = =

f f ff x y zx y z

= + +

( ) ( ) ( )2,1,3 2,1,3 2,1,32(0.1) ( 2)( 0.2) 3(0.1) 0.9

f f ff x y z

x y z = + + =

= + + =

66

4) , (2,1,4). : f(x,y,z)=x3+y2+z. ;

: f f. . -f . :

5) : V-, T-, P-, U- . . . :

(1)

V, T . (1) :

V, U (2)

P, T (3)

T, U (4)

U, P (5)

( )( )

(2,1,4)

2

2,1,4

2,1,4

3 2 12 2

f f ffx y z

x y j

= + + = = + + =

i j k

i j k i k

U PT P 0V T

+ =

T P TT P 0V U U

+ = U V VT P 0P T P

+ + = ( )( )P, V VT P 1 0T, U U

= ( )( )V,TT VT P 0

P U U,P + =

67

V, P (6)

: (1) U, P V, T:

U=U(V,T), P=P(V,T) (7)

dU=AdV+BdT dP=CdV+DdT (8)

(9)

(1) :

-TD+P=0 (10)

(2), V, U T, P V, U:

T=T(V,U), P=P(V,U) (11)

(11):

dT=dV+dU, dP=dV+dU (12)

dU=(1/)d-(/)dV, dP=(-/)dV+(/)dT (13)

(8) (13) :

=-/, D=/, (14)

(10) , , , :

(15)

(15) :

+-=0

(16)

(3) .

(4) T U . P=P(T,U), V=V(T,U) :

( )( )T, UTT P T 0

P V,P + =

U U P PA , B , C , DV T V T

= = = =

0 + = TV

=

T P TT P 0V U U

+ =

68

dP=dT+dU, dV=dT+dU (17)

(17) dP dU:

dP=(/)dV+(-/)dT dU=(1/)dV-(/)dT (18)

(8) (18) :

=1/, D=-/ (19)

(19) (10) :

1/-(-/)+=0

(-)--1=0 (20)

(21)

P, V T, U.

H (20) :

(5) (6).

1. f(r) = 3x2y-y3z2 f =(1,-2,-1) 2. f ) f(r)=ln|r|=lnr ) f(r)=1/r, r=xi+yj+zk 3. rn =nrn-2r, r=xi+yj+zk 4. , : x2y+2xz=4 =(2,-2,3).

( )( )

P PT U P, VP V P V

T U U T T, UV VT U

= = =

( )( )P,VT, U

( )( )P, V VT P 1 0T, U U

=

69

5. f(r)=x2yz+4xz2 =(1,-2,-1) u=2i-j-2k.

6. f=x2yz3 =(2,1,-1) . .

7. : f1=x2+y2+z2-9=0 f2=x2+y2-z-3=0. =(2,-1,2) , .

8. f(x,y)=3x2+y2 , (x,y) x2+y2 =R2 .

9. ,,, f(x,y,z) = xy2+yz+z2x3 (1,2,-1) 64 .

10. f(x,y) ) 2 (1,2) 1(2,2) ) -2 (1,2) 2(1,1). f (1,2) 3(4,6) f.

11. R P(x,y,z) (,,). R R=.

12. . , ( ), , .

13. f(x,y,z)=xey+yz P(x,y,z) 0(2,0,0) 1(4,1,-2) , s=0.1 .

14. f(x,y,z) v=i+j-k. 23. ) f . ) f i+j.

15. : T=T(x,y)=eysinx. ,

70

"" .

IV

I

4.1

, .. , ' .. , , ( t). .

, - F, :

F: W V (4.1.1) W, V , R2 R3, W=R2 R3 V=R2 R3. :

3: ( ) ( , , ) x y z x y z= + + = F r i j k F r F R (4.1.2) :

, , (- ), Coulomb .

F

1 2 3( ) ( , , ) ( , , ) ( , , ) ( , , ) x y z F x y z F x y z F x y z= = + +F r F i j k (4.1.3)

( )33 2 2 2 2r x y z= = + +r rF

IV

72

F1, F2, F3 x, y, z, .

: "" .

. . F=xi+yj . 4.1.1 .

, .

.

F=-yi+xj. ,

. . ' , (. 4.1.2).

, M, m,

m (x,y,z). :

3MmGr

= F r

. 4.1.1

. 4.1.2

- - 73

, (. 4.1.3), F .

- F 0 (x0,y0,z0) (x,y,z) . , :

x=x0+s y=y0+s z=z0+s (4.1.4)

, , u, , ( u=i+j+k 2+2+2=1). s 0 . (4.1.4) (4.1.3), F , 0 , :

F=F(s)=F1(x+s,y+s,z+s)i+F2(x+s,y+s,z+s)j+F3(x+s,y+s,z+s)k

(4.1.5)

F , s.

4.2

F (x,y,z) u, :

(4.2.1)

F , F1 , F2 , F3 F .

( ) 31 2: dFd dF dFds ds ds ds

= = +FF r u i j k

. 4.1.3

IV

74

=F1u, =F2u, =F3u

(4.2.1) :

( ) ( ) ( )1 2 3d F F Fds = + + F u i u j u k (4.2.2)

(4.2.2) , :

(4.2.3)

F :

(4.2.4)

F, ( F), :

1dFds

2dFds

3dFds

1 1 1 11 1

1 2 2 2 22 2

2

3

3 3 3 3 3 3

F F F FF Fx y z yx z

F F F F FF Fd x y z yF x zds

FF F F F F Fx y z x y z

+ + + + = = = + +

uF u

u

( )

11 1

22 2

3 3 3

FF Fyx z

FF Fyx z

F F Fx y z

=

D F

- - 75

u, :

F(r: u)=D(F)u (4.2.5)

1: f(x,y,z), :

(4.2.5) . u, . , D(F), F, , F. :

) n, ij :

(4.2.6)

. . D(F), :

(4.2.7)

F. (4.2.7) , , "" .

(4.2.7) "" , ( , ), F,

(4.2.8)

1tr

n

iji

=

=

31 2 FF Fx y z

+ +

( )31 2 1 2 3FF F F F Fx y z x y z

+ + = + + + + i j k i j k

IV

76

T "" : F divF , ( div diverge=), F. :

divF=F= (4.2.9)

, :

(4.2.10)

:

(4.2.11)

Maxwell.

) D(F) , ( ), :

(4.2.12)

F : curlF rotF. ( curl curlation= rot rotation=).

F :

F= (4.2.13)

T :

31 2 FF Fx y z

+ +

( ) 0t + =v

0

=E

3 32 1 2 1F FF F F Fy z z x x y

+ + i j k

1 2 3

x y zF F F

i j k

- - 77

curlF=rotF=F=

(4.2.14)

2: , , ( ), :

(4.2.15)

, . , ( i, j, k,), , ( (/x, /y, /z ). :

) f f, .

) F , F. ) F , F.

4.3

) : ' F , () ' . F=xi+yj 4.1.1 F=2>0 . F=-xi-yj F=-2

IV

78

v(x,y,z) (x,y,z). F(x,y,z)=(x,y,z)v(x,y,z) :

[ ] [ ] [ ]

= F

(4.3.1)

v (x,y,z). F, F, (x,y,z).

: x, y, z, (. 4.3.1). x, y, z , ' F (15. (x,y,z). : F(x,y,z)j(xz) , j .

, F(x,y+y,z)j(xz) j .

[F(x,y+y,z)-F(x,y,z)]j(xz) (16 (15 F . F . (16 [F(x,y+y,z)-F(x,y,z)]j(xz) , j . [F(x,y+y,z)-

O

x

y

z

z

y A

. 4.3.1

B

x

- - 79

, j.

:

[F(x+x,y,z)-F(x,y,z)]i(yz) [F(x,y,z+z)-F(x,y,z)]k(xy) . xyz , :

=

+ =

F(x,y,z)]j(xz) , j .

000

[ ( , , ) ( , , ] ( )lim xyz

x x y z x y z y zx y z

+ + F F i

[ ( , , ) ( , , ] ( )x y y z x y z x zx y z

+ + + F F j

[ ( , , ) ( , , ] ( )x y z z x y z x yx y z

+ + = F F k

0[ ( , , ) ( , , )]lim x

x x y z x y zx

+ = +F F i

0[ ( , , ) ( , , )]lim y

x y y z x y zy

+ + +F F j

0[ ( , , ) ( , , )]lim z

x y z z x y zz

+ + F F k

1 10

[ ( , , ) ( , , )]lim xF x x y z F x y z

x + = +

2 20

[ ( , , ) ( , , )]lim yF x y y z F x y z

y + + +

3 30

[ ( , , ) ( , , )]lim zF x y z z F x y z

z +

IV

80

=F(x,y,z) (4.3.2)

, F (x,y,z) B(x,y,z).

:

F(x,y,z)>0 : ) (x,y,z) ""

) (x,y,z) .

F(x,y,z)

- - 81

) : F , ,

, , =k. R , (x,y,z) , (. 4.3.2). :

r = Rcosti+Rsintj+zk

v= =(-Rsinti+Rcostj)=-yi+xj

v=x y zy x 0

i j k

=2k=2

, , v 2=, .

v .

F , ( ), :

(F).n= (4.3.5)

S C, n S ,

.

ddtr

S 0C

1lim dS

F rv

y

z

t

R

x

r

. 4.3.2

IV

82

F , F .

F :

(4.3.6)

- : . , , v ' , . , , . . , ' . .

4.4

. ,

,

. ..

, Q,

.

, , . :

,

. ,

01limV

S

dSV

= F n Fw

1fr

=

0

14

Qfr=

Mf Gr

=

3

14 4

Q Qfr r

= = = rE

3

Mf G GMr r

= = = rE

- - 83

q m, q m :

,

, :

: F ; ; :

F f f=F; : , F=0. f, F f f .

F=F1i+F2j+F3k f=F :

=F1i+F2j+F3k

(4.4.1)

(4.4.1) :

(4.4.2)

:

34Qq

r=rF 3

MmGr

=F r

f f fx y z

+ +i j k

1f Fx

= 2

f Fy

= 3

f Fz

=

21Ff

y x y

=2

2Ffx y x

=

21Ff

z x z

=2

3Ffx z x

=

22Ff

z y y

=2

3Ffy z y

=

IV

84

F=0 (4.4.3) (4.4.3) f f=F. : f=F F = (f)=0. , , (4.4.3) f, f=F . f1 , :

(4.4.4)

f1 , :

,

(4.4.4) :

f1(x,y,z)= (4.4.5)

(x0,y0,z0) (x,y,z) x. f1 (4.4.5) f1 /x=F1 f1 /y=F2 f1 /z=F3 . f1+f2 , f2 y z. f1+f2 :

:

3 2 0F Fy z

=

31 0FFz x

=

2 1 0F Fx y

=

11

f Fx

=

12

f Fy

=

13

f Fz

=

0

1( , , )x

x

F t y z dt

1 21

( )f f Fx

+ =

1 22

( )f f Fy

+ = 2 12f fFy y =

- - 85

F2- =g(y,z)

( ) ( )0

2 , ,y

y

f y z g t z dt= (4.4.6) f1+f2 (4.4.1), :

f3(z) :

f3(z)=F3-

z. :

h(z)=F3-

: ( ) ( )0

3

z

z

f z h t dt= (4.4.7) : F F=0, f=f1+f2+f3 f=F :

21 2 1 2 1 2 1

2 0f F f F f F FF

x y x x y x y x x y

= = = =

1fy

1 21

( )f f Fx

+ = 1 2 2( )f f Fy

+ =

1 2 33

( )f f f Fz

+ + = 1 2( )f fz

+

1 23

( )F -

f fx z

+ = [ ]3

1 2F f fx z x

+ = 3 1 0F F

x z =

1 23

( )F -

f fy z

+ = [ ]3 1 2F f fy z y

+ = 3 2 0F F

y z =

1 23

( )-

f fF

z

+

1 2( )f fz

+

IV

86

,

,

(4.4.8)

:

F=xyz(2z+3x)i+z(x2z-3y2+x3)j+y(2x2z-y2+x3)k. F=0. f f=F. :

fx

=F1 =xyz(2z+3x) (4.4.9)

fy

=F2 = z(x

2z-3y2+x3) (4.4.10)

=F3 = y(2x2z-y2+x3) (4.4.11)

(4.4.9) x y z :

f= +c1(y,z)= +c1(y,z)=x2yz2+x3yz+c1(y,z)

(4.4.12)

c1 y z.

(4.4.12) (4.4.10) :

2 2 3 1( , )f x yz x yz c y zy y

= + + =F2 = z(x

2z-3y2+x3)

x2z2+x3z+ 1( , )c y zy

=x2z2-3y2z+x3z 1( , )c y zy

=-3y

2z

c1(y,z)= +c2(z)=-y3z+c2(z) (4.4.13)

( )0

1 1, , ( , , )x

x

f x y z F x y z dx=

( )0

12 2,

y

y

ff y z F dyy

=

0

1 23 3( )

z

z

f ff z F dzz z

=

fz

1Fdx (2 3 ) xyz z x dx+

3 2y zdy

- - 87

c2 , z.

(4.4.12) (4.4.13) :

f=x2yz2+x3yz-y3z+c2(z) (4.4.14)

(4.4.14) (4.4.11) :

=2x2yz+x3y-y3+ c2(z)= F3 =y(2x2z-y2+x3)

c2(z)=0 c2(z)=c=.

: f=x2yz2+x3yz-y3z+c

1: F , F=0, , , F=0, . : F=x3yi+yx2j+xzk. F, F .

: F , f F=f. -:

(1) (2) (3)

f , , , . F , F1, F2, F3 x, y, z . :

(1) y :

(4)

(2) x :

fz

ddz

ddz

31

f F x yx

= =2

2f F yxy

= = 3f F xzz

= =

2 2f fx y y x

=

23f x

y x =

IV

88

(5)

f

F=f. F . . . . F=yk, , curlF=i0.

4.5.

: F=F1i+F2j+F3k , G=G1i+G2j+G3k :

F=G (4.5.1) : F=0 (4.5.2) F=G F=(G)=0. , , (4.5.1) G.

(4.5.1) :

, , (4.5.3)

G1, G2, G3. (4.5.1) :

G = G1i+G2j+G3k

G1 =0 G2= G3=-

2

2f xyx y

= 2 2f fy x x y

3 21

G G Fy z

=

312

GG Fz x

=

2 13

G G Fx y

=

0 0

3 1 0( , , ) ( , , )x z

x z

F t y z dt F x y u du 0

2 ( , , )x

x

F t y z dt

- - 89

1: G, (4.5.1), G+, , (4.5.1). :

(G+)=G+()=F+0=F

4.6

f F G .

1.(f+)=f+ grad(f+)=gradf+grad 2. (F+G)=F+G div(F+G) = divF + divG 3. (F+G)= F+G curl(F+G)=curlF+curlG 4. (fF)=(f)F+f(F) div(fF)=gradf.F+fdivF 5. (fF)=(f) F+f(F) curl(fF)=gradfF+fcurlF 6. (FG)=G(F)-F(G) div(FxG)=GcurlF-FcurlG 7. (FG)=(G)F-G(F)-(F)G+F(G) 8. (FG)=(G)F+(F)G+G(F)+F(G)

9. (f)=2f=2 2 2

2 2 2

f f fx y z

+ + (17

2= o Laplace.

10 (f)=0 curlgradf=0 11. (F)=0 divcurlF = 0 12. (F) = (F) - 2F

(17 f(x,y,z) 2f(x,y,z)=0 .

2 2 2

2 2 2x y z + +

IV

90

4.7

1) , :

+(v)=0 (1)

=(x,y,z,t) v=v(x,y,z,t) . :

+()v+v=0 (2)

+v=0 (3)

(4)

Stokes.

: :

(fF)=(f)F+f(F) f=, F=v (v)=()v+(v). (1) :

+(v)= +()v+v=0

(2).

(3), v :

v=dx/dti+dy/dtj+dz/dtk, (2) :

+()v+v= + + v=

= + + v= +v=0

.

t

t

DDt

D dx dy dzDt x dt y dt z dt t

= + + +

t

t

t

t

dx dy dz

x y z dt dt dt + + + + i j k i j k

t

dx dy dz

x dt y dt z dt + +

DDt

DDt

- - 91

2) : v=(y,0,0). . . t=1/3, t=0 x=0, x=1, y=0, y=1, z=0, z=1.

: , ,

+(v)=0

v=0. v v=0 . . - . 4.7.1.

- . t=0 , (). () . t=1/3 () (), , . 4.7.2.

3) : v=(x,0,0). . . t=1, t=0

t

x

z

y

. 4.7.1

x

y

z

1

1

1 t=0

t=1/3

. 4.7.2

IV

92

x=0, x=1, y=0, y=1, z=0, z=1.

: . 4.7.3. x=0, , , OX |x|. v=10 .

v(x,0,0) . :

c -, t=0.

, x=cet c=0

. x=0. , .. CKLH, x=1 t=0.

t=0 1=ce0 c=1 x(t)=et.

t=1 x=et=e MCKL

ABOMGDEF, GDEF x=e, . 4.7.4. V=e11=e. 4) o ,

T=T(x,y,z). (1)

:

( ), 0, 0 , 0, 0 cet

d dx xdt dt

dx x xdt

= = = = =

rv

. 4.7.4

x

y

z .4.7.3

- - 93

(2)

s0. : (3)

: . - . . f, F, , , : f=F. f . , . , , , , . :

T(x,y,z) (x,y,z), T(x+x y+y, z+z) (x+x y+y, z+z). s

.

( ) ( ), , , ,T x x y y z z T x y zTs s

+ + + =

dT dTds ds

= r

s

IV

94

(x,y,z) (x+x y+y, z+z). (x,y,z):

:

1. :

) ) =0 ) 2

2. r=xi+yj+zk r=|r| n div[rnr]=0

3. =(x,y,z), d/dz=0 (0,0,z)=0. :

F(x,y,z)=(x3+3y2z)i+6xyzj+k

f. f.

4. F=4u. -i , u=yj+zk, .

5. f(r), f(r)r .

6. ) 2f(r)=

T T Td dx dy dzx y z

= + +

0lim sdT T dx T dy T dz

s ds x ds y ds z ds = = + + =

T T T dx dy dz dTx y z ds ds ds ds

= + + + + = ri j k i j k

3 0r =

r3r

r 1 0

r =

2ue

2

2df dfdr r dr

+

- - 95

) f(r) 2 f(r)=0 7. :

) 2[lnr]=1/r2 ) 2rn=n(n+1)rn-2 ) [r3r]=6r3 8. F=F1i+F2j+F3k . (x,y,z), F . , F . , F1(x,y)dx+F2(x,y)dy=0 .

9. F G , FG .

10. Mxwell :

E=- E=0

B= B=0

. :

2u= : (F)=-2F+(F)

11. ;

1) culr divF 2) curl curlF 3) grad gradf 4) div divF 5) div gradf

6) grad divf 7) grad divF 8) grad curlF 9) div curlF 10)curl gradf

( 1) OXI 2) NAI 3) OXI 4) OXI

5) NAI 6) OXI 7) NAI 8) OXI

9) NAI 10) NAI

t

B

Et

2

2

ut

V

5.1

, , . R. , , ( ). :

) , ,

) , .

, : 1) ,

2) ,

3) , , .

, .

98 V

5.2 C, . 5.2.1, :

r(t)=x(t)i+y(t)j+z(t)k t[t1 , t2] (5.2.1) f(x,y,z),

C. n P0=A, P1, . . . , Pn=B. k(xk,yk,zk)

k=0, 1, 2, , n-1. :

(5.2.2)

sk

. n sk . , , f C. ( C,

). : (5.2.3)

:

( ) 1max( ) 0 0

, , lim ( )k

n

n k ks kC

I f x y z ds f s = = = (5.2.4)

To C :

) " ":

r=r(s)=x(s)i+y(s)j+z(s)k

s[s1 , s2]

pAB

pAB pAB

pk k 1P P +

1

1( )

n

k kk

I f s=

= p

k k 1P P +

pAB

pAB( , , )

C

I f x y z ds= x

y

z

OP0=A

Pn=B

P1

P2

Pk

Pk+1

k

.5.2.1

99

( s r(s) ), (5.2.3) :

(5.2.5)

) " ":

r=r(t)=x(t)i+y(t)j+z(t)k t[t1 ,t2] :

(5.2.6)

(5.2.3) :

(5.2.7)

(5.2.5) (5.2.7) .

1: ) f(x,y,z)=1 (5.2.3) C. , XY, z(s)=0 z(t)=0.

) C , (. -, : .

5.3. .

) f1 , f2 k1 , k2 , :

(5.3.1)

2

1

[ ( ), ( ), ( )] s

s

I f x s y s z s ds=

( ) ( ) ( ) ( )2 2 22 2 2 dx dy dzds dx dy dz dt t dtdt dt dt

= + + = + + = r

2

1

2 2 2

[ ( ), ( ), ( )]

t

t

dx dy dzI f x t y t z t dtdt dt dt

= + +

C

fdsv

[ ]1 1 2 2 1 1 2 2C C C

k f k f ds k f ds k f ds+ = +

100 V

) f , :

(5.3.2)

) (5.3.3)

) , ( ), , :

(5.3.4)

ds . :

,

dt>0 t1 t2 (5.2.7) t1t2 , , ( - 4).

:

1) f(x,y,z)=x-z : r(t)=costi+sintj+tk t[0,/2] : (5.2.7) :

x(t)=cost dx(t)/dt=-sint, y(t)=sint dy(t)/dt=cost, z(t)=t dz(t)/dt=1 :

2) f(x,y)=x+y2 , =(0,0) =(1,1) :

pAB

p ppAB A fds fds fds= +

C C

| |fds f ds

ppAB BAfds fds=

2 2 2dx dy dzds dtdt dt dt

= + +

[ ] [ ]/2/2 2 20 0

2 22

0

cos sin cos 1 2 cos

2 sin 2 12 8

I t t t t dt t t dt

tt

= + + = =

= =

101

) x=t, y=t, t[0,1] ) x=sint, y=sint, t[0,/2] ) x=t2, y=t2, t[0,1] : ) :

) :

) :

3) x=f(t), y=f(t) f(0)=0, f(1)=1. f(t) [0,1] 52/6 f(t).

:

2

1

2 2

( , ) [ ( ), ( )]

t

Ct

dx dyI f x y ds f x t y t dtdt dt

= = + =

( ) 11 2 320 0

5 21 1 22 3 6t tt t dt

= + + = + =

2 22 2 2

0 0

sin sin 2cos 2 sin sin (sin )I t t tdt t t d t

= + = + = 2 3 2

0

sin sin 5 222 3 6

t t

= + =

( ) ( )1 12 22 4 2 4 20 0

12 4

0

2 2 8

52 2 26

I t t t t dt t t t dt

t t tdt

= + + = + =

= + =

( ) ( )1 122 20 0

( ) ( ) 2 2 ( ) ( )f t f t f t dt f t f t f t dt + = + =

102 V

1: 2, ( 3), , .

4) :

: r(t)=Rcosti+Rsintj+tk (-R,0,) (R,0,0).

: :

(-R,0,) t1= (R,0,0) t2=0. ds0, ( ), dt . t . . , . : t1=>t2=0 :

5.4

1)

C p=p(x,y,z). :

( ) ( ) ( )12 31

2

0 0

1 1 52 ( ) ( ) 2 2 22 3 2 3 6

f t f tf t f t df t

= + = + = + =

( )2 2 2AB

I x y z ds= + +

( ) ( )

2 2 2

2 2 2 2 2sin cos

dx dy dzds dtdt dt dt

R t R t dt R dt

= + + = = + + = +

( )( )

11

22

2 2 2 2 2 2 2 2 2 3

00

2 2 2 2 2

13

33

tt

tt

I R t R dt R R t t

R R

=== =

= + + = + + = = + +

103

(5.4.1)

r(t)=x(t)i+y(t)j+z(t)k t[t1, t2] , :

(5.4.2)

2)

Pk(xk ,yk zk), k=1,2,...,n n mk, k=1,2,...,n, ( ), ,x y z :

1

1

n

k kk

n

kk

x mx

m

=

=

=

1

1

n

k kk

n

kk

y my

m

=

=

=

1

1

n

k kk

n

kk

z mz

m

=

=

=

(5.4.3)

:

(5.4.4)

. (5.4.5)

3) , .

m , md2, d .

0 OXYZ, OX, IOY, IOZ X, Y, Z OXY, IOXZ, IOYZ

( , , )C

M p x y z ds=

2

1

2 2 2

[ ( ), ( ), ( )]

t

t

dx dy dzM p x t y t z t dtdt dt dt

= + +

1 ( , , ) ,kC

x xp x y z dsM

= 1 ( , , ) ,kC

y yp x y z dsM

= 1 ( , , )kC

z zp x y z dsM

=

C

( , , )M p x y z ds=

AA

104 V

OXY, OXZ, OYZ . : (5.4.6)

(5.4.7)

(5.4.8)

(5.4.9)

(5.4.10)

(5.4.11)

(5.4.12)

:

(5.4.13)

, , ' d (x,y,z) , .

1: , :

r(t)=costi+sintj+btk 0t

105

s(t)=| r(t)|=|-sinti+costj+bk|=

23 2

2 2 2 2 2 2 2 2

0

82 3

bM a b t a b dt a b a

= + + = + +

)

:

:

2 2a b+

( )2 2 21 1( , , )kC C

x xp x y z ds x x y z dsM M

= = + + = 2 22 2

2 2 2 2 2 2 2 2

0 0

1 cos cos cosa a ba t a b t a b dt a t b t t dtM M

+ = + + = + 2

21 0

0

cos sin | 0I tdt t

= = =2 2

2 22 2 2 22 0 0 0

0 0

222

0 00

cos sin sin | 2 sin 2 cos

2 cos | 2 cos 4 2sin 4

I t tdt t d t t t t tdt td t

t t tdt t

= = = = =

= = =

2 2 2 22

2 2 22 3 2

4 64 8 3 42 3

ka a b ab abx b

M a ba b + = = = ++

( )2 2 21 1( , , )kC C

y yp x y z ds y x y z dsM M

= = + + = 2

2 2 2 2 2

0

22 22 2 2

0

1 sin

sin sin

a t a b t a b dtM

a a b a t b t t dtM

= + + = + = +

22

1 00

sin cos | 1 ( 1) 0I tdt t= = = =

106 V

= -42+(2-2)=-42

) d2=x2+y2=2.

Iz=

5.5

W, F r, (. 5.5.1), :

W=Fr=|F||r|cos , . , , ( .

[ ]

2 222 2 2 2

2 0 00 0

2 222 200 0

sin cos cos | 2 cos

4 2 sin 4 2 sin 2 sin

I t tdt t d t t t t tdt

td t t t tdt

= = = + =

= + == +

2 2

04 2 co s |t + =2 2 2 2 2

2 22 2 2

2 3 2

4 64 8 3 42 3

ka a b ab aby b

M a ba b

+ = = = ++

( )2 2 21 1( , , )kC C

z zp x y z ds z x y z dsM M

= = + + = 2 22 2

2 2 2 2 2 2 3 3

0 0

1 a bbt a b t a b dt a bt b t dtM M

+ = + + = + =

( )

22 2 2 32 4 2 2 3 4

2 3 20

2 2 2

2 2 2

1 2 4 82 4 2 3

3 23 4

a b a b bt t a b bM a b

b b aa b

+ = + = + = ++= +

2 2 2 2( , , ) ( , , ) ( , , )C C C

d p x y z ds a p x y z ds a p x y z ds a M= = =

F

r

. 5.5.1

107

O

x

y

z

P0=A

Pk

Pk+1 Pn=B

rA

rk

rk+1rk

. 5.5.2

), C.

C, (. 5.5.2), : r(t)=x(t)i+y(t)j+z(t)k t[t1 ,t2] r(t1)=rA r(t2)=rB

F(x,y,z)=F1(x,y,z)i+F2(x,y,z)j+F3(x,y,z)k

F. C

k k+1 rk rk+1 rk=rk+1-rk. F k k+1 F , ( k k+1 F ), Fk, Fk F ' kk+1. W :

Wk=Fk rk (5.5.1) :

W= (5.5.2)

(5.5.2) n max|rk|0

(5.5.3)

:

pAB

n

k kk 1=

F r

max| | 0 1

limk

n

n k kk

W = = r F r

108 V

(5.5.4)

F C. :

[ ] [ ] [ ]2

1

1 2 3( ) ( ) ( )( ), ( ), ( ) ( ), ( ), ( ) ( ), ( ), ( )

t

t

dx t dy t dz tF x t y t z t F x t y t z t F x t y t z t dtdt dt dt

= + +

(5.5.5)

1: F=(3x2+6y)i-14yzj+20xz2k. :

0=(0,0,0) 1(1,1,1) , :

) x=t, y=t2 , z=t3

) , (0,0,0) (1,1,1).

) (0,0,0) (1,0,0) (1,1,0) (1,1,1).

:

) x=t, y=t2 , z=t3 r(t)=ti+t2j+t3k r(t)=i+2tj+3t2k

=

:

max| | 0 1lim

k

n

n k kkC

d = = rF r F r

C C

dd dtdt

= = rF r F

Cd F r

( )210

1

1 2 2 5 7 2

03 6 14 20 2 3

tP

Pt

dd dt t t t t tj t dtdt

= = + + + + = rF r F i j k i k1 2 6 9 3 7 10 1

009 28 60 3 4 6 | 5t t t dt t t t + = + =

[ ] [ ]1 10

1 1

0 0

1 2 3

2 21 2 3 (3 6 ) ( 14 ) 20

P P

P P

P P

P P

d F F F dx dy dz

F dx F dy F dz x y dx yz dy xz dz

= + + + + == + + = + + +

F r i j k i j k

109

x=t dx=dt, y=t2 dy=2tdt, z=t3 dz=3t2dt 20t7(3t2)dt=3t3-4t7+6t10

) (0,0,0) (1,1,1) : x=t, y=t, z=t

=

) (0,0,0) (1,0,0) :

y=0, z=0 dy=0 dz=0 0x1 :

1 2 3 1

1 00I (3 6 0) | 1

x

xx dx x

=== + = =

(1,0,0) (1,1,0) :

x=1 dx=0, 0y1, z=0 dz=0 1

2 0I 14 0

y

yyzdy

=== =

(1,1,0) (1,1,1) : x=1 dx=0, y=1 dy=0, 0z1

I3=

:

2: (-1,0) (1,0) F=yi+xj :

1) , 2) ,

3)

: 1) y=0 dy=0

2) :

1 2 2 5

0(3t 6t )dt 14t (2tdt)+ + |01 5=

( )1 2 2 3C 0

3 6 14 20d t t dt t dt t dt = + + = F r( ) ( )1 12 2 3 2 3

0 0

133 6 14 20 6 11 208

t t t t dt t t t dt+ + = + =

1 12 2 3 100 0

20 2020 20 |3 3

z z

z zxz dz z dz z

= == == = =

F r = + + = + = d I I IPP01 1 2 3 1 203 233

21y x= 21y x=

( ) ( ) 0C C C

d y x dx dy ydx xdy = + + = + = F r i j i j

110 V

x=rcost, y=rsint dx=-rsintdt, dy=rcostdt. r=1 t0. :

3) 2 t 2.

, (-1, 0), (1, 0). . 5.7 .

5.6

C , r=r(s), :

(5.6.1)

F , F.T(18 :

(5.6.2)

F C.

C ,

(18 =dr/ds |dr|=|ds|

( )0 02 2 2 2C C

00

sin cos cos sin

sin 2cos 2 02

d ydx xdy tdt tdt t t dt

ttdt

= + = + = =

= = =

F r

( )2 22 2 2 2C C

22

sin cos cos sin

sin 2cos 2 02

d ydx xdy tdt tdt t t dt

ttdt

= + = + = =

= = =

F r

=F 0G

C C C C( )dd ds ds s t dt

ds = = = rF r F F T F T

Cd F r

111

(5.6.3)

F C.

5.7. ) F , f: F=f. C, :

(5.7.1)

1 , 2 C, :

(5.7.2)

C, 1 2 . C , :

(5.7.3)

, ( ).

) F C r=r(t). A Newton:

(5.7.4)

F C :

=

C

d F rv

C C

C

C

f f fd f d dx dy dz dfx y z

= = + + =

F r r

( ) ( )2 21

1

P PP 2 1C P| P Pd df f f f = = = F r

C

d F rv

( ) 22d dm mdt dt= =v rF r

[ ]2

1

C C C

1( ( )) ( ) ( ) ( ) ( ) ( )2

t

t

dd t t dt m t t dt m t t dtdt

= = = F r F r r v v v v

112 V

= (5.7.5)

F , f: F=f, :

(5.7.6)

(5.7.5) (5.7.6) :

=f(r(t2))-f(r(t1))

- f(r(t1))= - f(r(t2)) (5.7.8)

-f , , ( ), . , .

:

1: F , :

) F f.

) C1 C2

r1(t) r2(t) .

) =0 C .

) F=0

2

1

2 2 22 1

1 1( ) ( ) ( )2 2

t

t

dm v t dt m v t v tdt

=

2 22

11 1

2 1C( ( )) ( ( )) ( ( )) | ( ( )) ( ( ))

t t ttt t

d f t d df t f t f t f t = = = = F r r r r r r r

2 22 1

1 1( ) ( )2 2

mv t mv t

21

1 ( )2

mv t 12

22mv t( )

1 21 2C C

d d = F r F r

C

d F rv

113

: Fdr .

, , :

2: F , :

) F=0 F. )

.

)

.

) F G, F=G. To G F , 4.5.

VIII.

5.8

, :

1) , 2) , 3) (5.8.1)

.

1: m0 . (. 5.8.1). F, C, :

C

d F rv

S

dF S

S

0d = F Sw

C( ( ))s ds F r C ( ( ))f t d r r C ( ) d F r r

m0

x

y

z

dF

r(t) dm=pds

. 5.8.1

C

114 V

x=x(s), y=y(s), z=z(s) p=p(x,y,z).

: :

003 3

CC

( ) ( , , )m dmG Gm p x y z dsr r

= = = rF r

5.9

1) m

, : .

r .

:

. :

. :

03G

m dmdr

= F r

[ ]0 32 2 2 2

C

( ( ), ( ), ( )) ( ) ( ) ( )( ) ( ) ( )

p x s y s z sGm x s y s z s dsx s y s z s

= + + = + +

i j k

[ ]

[ ]

0 32 2 2 2

C

0 32 2 2 2

C

( ( ), ( ), ( )) ( )( ) ( ) ( )

( ( ), ( ), ( )) ( )( ) ( ) ( )

p x s y s z sGm x s dsx s y s z s

p x s y s z sGm y s dsx s y s z s

= + + +

+ + + +

i

j

[ ]0 32 2 2 2

C

( ( ), ( ), ( )) ( )( ) ( ) ( )

p x s y s z sGm z s dsx s y s z s

+ + +

k

MV Gr

=

2

MmGr r

= rF

MV Gz

=

2

MmGr

= F k

115

2) .

.

: . F=0. . . 1 . 5.7 . , .

r. : x=rcost, y=rsint dx=-sintdt, dy=costdt :

, , , .

, , =0

. ..

(x0,y0) r , ( ), :

3 2C

rr r

zz z

GMm GMm Mmd F dz dz G Vz z r== =

= = = = = F r

2 2

y xx y += +

i jF

C

d F rv

2 2 2 2

22

2 2 0

0

sin cossin cos 2

CC

t

y xd dx dyx y x y

r t r tr t r t dt dtr r

=

= + =+ +

= + = =

F rv v

C

d F rv2 20 0r x y< +

(x0,y0) r

x0

y0

116 V

=0.

1. . z=ysin(zy) (0,0,0)

0, ,2 2 .

2. C , x2+y2+z2=2 x2+y2+z2=2 . F=5r3r =5-5 .

3. OXY :

F(x,y)=(2xey+y)i+(x2ey+x-2y)j

r(t) = cos3ti+sin2tj 0t/2

4. F=(x+z)i-(y+z)j+(x-y)k. F 1(0,0,1) 2(1/2,-1,1) r(t) = sin2ti-tantj+k

5. 2 2

2 2 1x y + =

: ] .

C

d F rv

2re=F r

Cd F r

1 [2

y x= F i j