# Электродинамика. Нетребко Н.В - 315 стр.

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• ## Электричество и магнетизм

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
r
=
r
;
3
1
r
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
r r r
=
r r r r r
r
;
[
]
(
)
(
)
3 3 5 3
,
, 3 ,
p r
p r p r r
p
r r r r
= − =
r r
r r r r r
r
(
)
( )
(
)
3 3 5
, ,
1
, 3
a r a r r
r r r
=
r r r r r
r r
;
(
)
,
=
r r r
;
( ) ( )
r
r
=
r
;
[
]
( ), 0
=
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
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
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