διανυσματικη αναλυση

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    2013

  • Email: [email protected]

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  • 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

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    VIII155

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    IX189

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

    215

    21510.121510.2,21510.3LAGRANGE22710.4232235

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    249

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    I 431432III434IV436V436VI438V439V440GREEN,STOKES,GAUSS441X442

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

    MAPLE500

    531

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  • 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