Задачи по векторному анализу. Михайлов В.К - 56 стр.

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56
• Äîêàçàòü ðàâåíñòâà, â êîòîðûõ
ρ
a
è
ρ
c
— ïîñòîÿííûå âåê-
òîðû,
ρ
r
— ðàäèóñ-âåêòîð, u(r) — ñôåðè÷åñêîå ñêàëÿðíîå ïîëå:
2.117.
rot( ) ( ) /rc r c r
ρρρ
=×
; 2.121.
rot( )
ρρ ρ
cr c×=2
;
2.118.
rot( ) ( )rc rr c
3
3
ρρρ
=×
; 2.122.
rot[( ) ]
ρρ ρ ρρ
cr a ac××=×
;
2.119.
rot( )
ρρρ ρ ρ
rac a c⋅=×
; 2.123.
rot[( ) ] ( )
ρρ ρ ρρ
cr r cr××= ×3
;
2.120.
rot( )
rcr c r⋅=×
; 2.124.
rot[()] ( )/urc u r c r
ρρρ
=′ ×
.
• Äîêàçàòü ðàâåíñòâà, â êîòîðûõ
ρ
a
è
ρ
b
— ïðîèçâîëüíûå
âåêòîðíûå ïîëÿ:
2.125.
rot( ) ( div div ) [( ) ( ) ]
ρ
ρ
ρ
ρρ
ρ
ρ
ρρ
ρ
ab abba baab×= − + ⋅∇−⋅∇
;
2.126.
rotrot grad(div )
ρρρ
aaa=−∆
, ãäå
∆∆∆∆
ρ
aaaa
xyz
= {, ,}
.
2.127. Ïîëüçóÿñü ðàâåíñòâîì çàäà÷è 2.126, âû÷èñëèòü
rotrot
ρ
a
ïðè
ρ
azxy= {, , }
22 2
.
2.128. Äîêàçàòü, ÷òî åñëè âåêòîðíîå ïîëå
ρ
auv=∇
, ãäå u è
v — ñêàëÿðíûå ïîëÿ, òî
ρρ
aa⊥ rot
.
2.129. Äîêàçàòü, ÷òî ïîëå
ρρ
ac u=×∇
, ãäå
ρ
c
— ïîñòîÿííûé
âåêòîð, ÿâëÿåòñÿ ñîëåíîèäàëüíûì.
2.130. Äîêàçàòü, ÷òî åñëè ïîëÿ
ρ
a
è
ρ
b
áåçâèõðåâûå, òî
ïîëå
()
ρ
ρ
ab×
— ñîëåíîèäàëüíî.
2.131. Äîêàçàòü, ÷òî ïîòîê ðîòîðà ïîëÿ
ρ
a
÷åðåç ëþáóþ
çàìêíóòóþ ïîâåðõíîñòü S ðàâåí íóëþ.
2.132. Óðàâíåíèÿ Ìàêñâåëëà, ñâÿçûâàþùèå ýëåêòðè÷åñêîå
ïîëå
ρ
E
è ìàãíèòíîå ïîëå
ρ
B
, èìåþò âèä:
rot ,
rot .
ρ
ρ
ρ
ρ
E
B
t
B
c
E
t
=−
=
∂
∂
∂
∂
1
2
Ïîêàçàòü, ÷òî åñëè ïîëå
ρ
E
ñîëåíîèäàëüíî, òî èç
ýòîé ñèñòåìû ñëåäóåò âîëíîâîå óðàâíåíèå:
                                                 ρ ρ
     • Äîêàçàòü ðàâåíñòâà, â êîòîðûõ a è c — ïîñòîÿííûå âåê-
       ρ
òîðû, r — ðàäèóñ-âåêòîð, u(r) — ñôåðè÷åñêîå ñêàëÿðíîå ïîëå:
                   ρ       ρ ρ                               ρ ρ         ρ
     2.117. rot( rc ) = ( r × c ) / r ;      2.121. rot( c × r ) = 2c ;
                     ρ          ρ ρ                           ρ ρ ρ ρ ρ
     2.118. rot( r 3 c ) = 3r ( r × c ) ;    2.122. rot[( c × r ) × a ] = a × c ;
                 ρ ρρ ρ ρ                                    ρ ρ ρ              ρ ρ
     2.119. rot( r ⋅ a )c = a × c ;          2.123. rot[( c × r ) × r ] = 3( c × r ) ;
                 ρ ρρ ρ ρ                                        ρ         ρ ρ
     2.120. rot( r ⋅ c )r = c × r ;          2.124. rot[u( r )c ] = u ′( r × c ) / r .
                                                   ρ        ρ
     • Äîêàçàòü ðàâåíñòâà, â êîòîðûõ a è b — ïðîèçâîëüíûå
âåêòîðíûå ïîëÿ:
                 ρ ρ           ρ      ρ ρ    ρ        ρ       ρ ρ         ρ
     2.125. rot( a × b ) = ( a div b − b div a ) + [( b ⋅ ∇ )a − ( a ⋅ ∇ )b ] ;
                     ρ                ρ    ρ            ρ
     2.126. rotrot a = grad(div a ) − ∆a , ãäå ∆a = {∆a x , ∆a y , ∆a z } .
                                                                                     ρ
     2.127. Ïîëüçóÿñü ðàâåíñòâîì çàäà÷è 2.126, âû÷èñëèòü rotrot a
                    ρ
            ïðè a = {z 2, x 2, y 2} .
                                                                 ρ
     2.128. Äîêàçàòü, ÷òî åñëè âåêòîðíîå ïîëå a = u∇v , ãäå u è
                                                ρ         ρ
            v — ñêàëÿðíûå ïîëÿ, òî a⊥ rot a .
                                          ρ ρ                  ρ
     2.129. Äîêàçàòü, ÷òî ïîëå a = c × ∇u , ãäå c — ïîñòîÿííûé
            âåêòîð, ÿâëÿåòñÿ ñîëåíîèäàëüíûì.
                                                   ρ        ρ
     2.130. Äîêàçàòü, ÷òî åñëè ïîëÿ a è b áåçâèõðåâûå, òî
                       ρ ρ
            ïîëå ( a × b ) — ñîëåíîèäàëüíî.
                                                                 ρ
     2.131. Äîêàçàòü, ÷òî ïîòîê ðîòîðà ïîëÿ a ÷åðåç ëþáóþ
            çàìêíóòóþ ïîâåðõíîñòü S ðàâåí íóëþ.
     2.132. Óðàâíåíèÿ Ìàêñâåëëà, ñâÿçûâàþùèå ýëåêòðè÷åñêîå
                    ρ                            ρ
           ïîëå E è ìàãíèòíîå ïîëå B , èìåþò âèä:
                                               ρ
                                   ρ        ∂B
                                  rot E = − ∂t ρ,
                                   ρ
                                  rot B = 12 ∂E .
                                          c ∂t
                                         ρ
               Ïîêàçàòü, ÷òî åñëè ïîëå E ñîëåíîèäàëüíî, òî èç
               ýòîé ñèñòåìû ñëåäóåò âîëíîâîå óðàâíåíèå:




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