Задачи по векторному анализу. Михайлов В.К - 52 стр.

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íàïðèìåð, åñëè u è v — äâà ñêàëÿðíûõ ïîëÿ, òî
∇(uv)=v∇u+u∇v.
5. Åñëè îïåðàòîð ∇ äåéñòâóåò íà êàêîå-ëèáî ïðîèçâåäå-
íèå ñ ó÷àñòèåì âåêòîðíîé âåëè÷èíû, òî ñíà÷àëà ó÷è-
òûâàåòñÿ åãî äèôôåðåíöèàëüíûé õàðàêòåð, à çàòåì
óæå âåêòîðíûé.
6. Åñëè â ñëîæíîì âûðàæåíèè îïåðàòîð ∇ äåéñòâóåò òîëü-
êî íà îäíó èç âåëè÷èí, òî îíà îòìå÷àåòñÿ èíäåêñîì
ó îïåðàòîðà ∇, êîòîðûé â îêîí÷àòåëüíîì âûðàæåíèè
ñíèìàåòñÿ.
7. Âñå âåëè÷èíû, íà êîòîðûå îïåðàòîð ∇ íå äåéñòâóåò,
â îêîí÷àòåëüíîì ðåçóëüòàòå ñòàâÿòñÿ âïåðåäè íåãî.
Ïðèìåð 1. Ïóñòü u(x,y,z) — ñêàëÿðíîå ïîëå,
ρ
axyz(, ,)
—
âåêòîðíîå ïîëå. Òîãäà
div( ) ( ) ( ) ( ) ( ) ( )ua ua ua ua u a u a
au au
ρρρρ ρ ρ
=∇⋅ =∇ +∇ = ∇ ⋅ + ∇ ⋅ =
uaauuaa u() ()div grad∇⋅ +⋅∇ = +⋅
ρρ ρρ
.
Ïðèìåð 2.
rot( ) ( ) ( ) ( ) ...ua ua ua ua
au
ρρ ρ ρ
=∇× =∇ × +∇ × =
Òåïåðü ðàáîòàåì ïî ïðàâèëàì âåêòîðíîé àëãåáðû, ïåðåìåùàÿ
ñêàëÿð u ëèáî ïîä «æäóùèé» åãî îïåðàòîð ∇, ëèáî âûíîñÿ åãî
çà ∇ âëåâî:
... ( ) ( ) rot grad=∇× +∇ ×= −×ua uauaa u
au
ρρρρ
.
(Çäåñü ó÷òåíà àíòèêîììóòàòèâíîñòü âåêòîðíîãî ïðîèçâåäåíèÿ).
Ïðèìåð 3.
div( ) ( ) ( ) ( ) ...
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ab ab ab ab
ab
×=∇⋅×=∇⋅×+∇⋅×=
Òåïåðü èñïîëüçóåì âåêòîðíûé õàðàêòåð îïåðàòîðà ∇, ò. å. ïðàâèëà
ðàáîòû ñî ñìåøàííûì ïðîèçâåäåíèåì òðåõ âåêòîðîâ, ïîäâîäÿ
âåêòîðû
ρ
a
èëè
ρ
b
ïîä «æäóùèé» èõ îïåðàòîð ∇ è ìåíÿÿ ìåñòàìè
îïåðàöèè «⋅» è «×»:
... ( ) ( ) ( ) ( ) rot rot=∇×⋅−∇×⋅=⋅∇×−⋅∇×=⋅ −⋅
ab
ab ba b a a b b aa b
ρ
ρρ
ρ
ρ
ρρ
ρρ
ρρ
ρ
.
              íàïðèìåð, åñëè u è v — äâà ñêàëÿðíûõ ïîëÿ, òî
              ∇(uv) = v∇u + u∇v.
         5. Åñëè îïåðàòîð ∇ äåéñòâóåò íà êàêîå-ëèáî ïðîèçâåäå-
              íèå ñ ó÷àñòèåì âåêòîðíîé âåëè÷èíû, òî ñíà÷àëà ó÷è-
              òûâàåòñÿ åãî äèôôåðåíöèàëüíûé õàðàêòåð, à çàòåì
              óæå âåêòîðíûé.
         6. Åñëè â ñëîæíîì âûðàæåíèè îïåðàòîð ∇ äåéñòâóåò òîëü-
              êî íà îäíó èç âåëè÷èí, òî îíà îòìå÷àåòñÿ èíäåêñîì
              ó îïåðàòîðà ∇, êîòîðûé â îêîí÷àòåëüíîì âûðàæåíèè
              ñíèìàåòñÿ.
         7. Âñå âåëè÷èíû, íà êîòîðûå îïåðàòîð ∇ íå äåéñòâóåò,
              â îêîí÷àòåëüíîì ðåçóëüòàòå ñòàâÿòñÿ âïåðåäè íåãî.
                                                                             ρ
     Ïðèìåð 1. Ïóñòü u(x,y,z) — ñêàëÿðíîå ïîëå, a ( x , y, z ) —
âåêòîðíîå ïîëå. Òîãäà
           ρ            ρ            ρ            ρ              ρ                ρ
     div( ua ) = ∇ ⋅ ( ua ) = ∇ a ( ua ) + ∇ u ( ua ) = u( ∇ a ⋅ a ) + (∇ u u ) ⋅ a =
                             ρ ρ                      ρ ρ
                      u( ∇ ⋅ a ) + a ⋅ ( ∇u ) = u div a + a ⋅ grad u .
       Ïðèìåð 2.
                   ρ              ρ               ρ             ρ
             rot( ua ) = ∇ × ( ua ) = ∇ a × ( ua ) + ∇ u × ( ua ) =...
Òåïåðü ðàáîòàåì ïî ïðàâèëàì âåêòîðíîé àëãåáðû, ïåðåìåùàÿ
ñêàëÿð u ëèáî ïîä «æäóùèé» åãî îïåðàòîð ∇, ëèáî âûíîñÿ åãî
çà ∇ âëåâî:
                           ρ                 ρ         ρ ρ
            ... = u( ∇ a × a ) + ( ∇ u u ) × a = u rot a − a × grad u .
(Çäåñü ó÷òåíà àíòèêîììóòàòèâíîñòü âåêòîðíîãî ïðîèçâåäåíèÿ).
     Ïðèìåð 3.
              ρ ρ             ρ ρ               ρ ρ               ρ ρ
         div( a × b ) = ∇ ⋅ ( a × b ) = ∇ a ⋅ ( a × b ) + ∇ b ⋅ ( a × b ) =...
Òåïåðü èñïîëüçóåì âåêòîðíûé õàðàêòåð îïåðàòîðà ∇, ò. å. ïðàâèëà
ðàáîòû ñî ñìåøàííûì ïðîèçâåäåíèåì òðåõ âåêòîðîâ, ïîäâîäÿ
             ρ          ρ
âåêòîðû a èëè b ïîä «æäóùèé» èõ îïåðàòîð ∇ è ìåíÿÿ ìåñòàìè
îïåðàöèè «⋅» è «×»:
             ρ ρ              ρ ρ ρ              ρ ρ            ρ ρ           ρ ρ         ρ
... = (∇ a × a ) ⋅ b − (∇ b × b ) ⋅ a = b ⋅ (∇ × a ) − a ⋅ (∇ × b ) = b ⋅ rot a − a ⋅ rot b.




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