Электродинамика. Нетребко Н.В - 315 стр.

UptoLike

ВУЗ: 

МГУ | Москва

Составители: 

Рубрика: 

Приложение
315
(
)
(
)
[ , ] , ,
div a b b rota a rotb
= −
r r r
r r r
;
(
)
(
)
[ , ] , ,
rot a b b a a b adivb b diva
= ∇ − ∇ + −
r r r r r
r r r r r
;
(
)
(
)
(
)
, [ , ] [ , ] , ,
grad a b b rota a rotb a b b a
= + + ∇ + ∇
r r r r r
r r
r r r r r
;
rot rot a a grad div a
= −∆ +
r r r
.
4.
Если
r
−
r
радиус-вектор, то:
3
div r
=
r
;
0
rot r
=
r
;
r
grad r
r
=
r
;
3
1
r
grad
r
r
= −
r
;
2
r
div
r r
=
r
;
3
0
r
div
r
=
r
;
(
)
[ ,[ , ]] 2 ,
div p r a p a
=
r r r r r
;
(
)
(
)
3 3 5
, ,
3
p r p r r
p
grad
r r r
= −
r r r r r
r
;
[
]
(
)
(
)
3 3 5 3
,
, 3 ,
p r
p r p r r
p
rot grad
r r r r
= − = −
r r
r r r r r
r
(
)
( )
(
)
3 3 5
, ,
1
, 3
a r a r r
grad grad a r
r r r
= −
r r r r r
r r
;
(
)
,
grad p r p
=
r r r
;
( ) ( )
r
grad f r f r
r
′
=
r
;
[
]
( ), 0
grad f r r
=
r
;
[ ] [ ]
( )
, ( ) 2 ( ) ,[ , ]
f r
rot p f r r f r p r p r
r
′
= +
r r r r r r
;
[
]
, 2
rot p r p
=
r r r
;
( )
[
]
,
( ) ( )
r p
rot p f r f r
r
′
=
r r
r
;
(
)
[
]
2
,
n n
rot p r n r r p
−
=
r r r
;
( )
[
]
,
r p
rot p r
r
=
r r
r
; 
Приложение                                                                                    315

         r r       r       r   r      r
                  (           ) (
    div[ a , b ] = b , rot a − a , rotb ;    )
        r r         r r r r r r r r r                           r
    rot[a , b ] = ( b , ∇ ) a − ( a , ∇ ) b + adivb − b diva ;
            r r         r      r       r      r     r r r r r r
    grad ( a , b ) = [b , rot a ] + [a , rotb ] + ( a , ∇ ) b + ( b , ∇ ) a ;
            r     r            r
    rot rot a = −∆a + grad div a .

           r                               r                                       r
4. Если r − радиус-вектор, то: div r = 3 ;                                     rot r = 0 ;
                r                                    r
                r                              1    r
   grad r = ;                           grad = − 3 ;
                r                              r    r
        r                                    r
        r 2                                 r
    div = ;                             div 3 = 0 ;
        r r                                 r
        r r r             r r
   div[ p,[r , a ]] = 2 ( p, a ) ;
            r r                 r r r
          ( p, r ) pr ( p, r ) r
   grad 3 = 3 − 3                     ;
             r       r           r5
           r r                      r r          r r r           r
    rot
        [ p, r ] = − grad ( p, r ) = 3 ( p, r ) r −             p
               3                        3             5
           r                        r             r             r3
              r r                                   r r r
            ( a, r )        1         r r         ( a, r ) r
    grad                =      grad ( a , r ) − 3            ;
                   r3       r3                       r5
                                                                                  r
           r r      r                                                             r
    grad ( p, r ) = p ;                                   grad f ( r ) = f ′ ( r ) ;
                                                                                  r
                       r
    [ grad     f (r ), r ] = 0 ;

          r         r              r f ′(r ) r r r
    rot [ p, f ( r )r ] = 2 f (r ) p +      [ r ,[ p, r ]] ;
                                        r
                                                                                   r r
          r r        r
    rot [ p, r ] = 2 p ;
                                                            r
                                                      rot ( p f ( r ) ) = f ′(r )
                                                                                  [r , p] ;
                                                                                     r
                                                                        r r
          r                     r r
    rot ( p r n ) = n r n − 2 [ r , p ] ;
                                                             r
                                                       rot ( p r ) =
                                                                       [r , p] ;
                                                                          r

Страницы