Задачи по векторному анализу. Михайлов В.К - 80 стр.

                                                                    ρ
      3.30. Íàéòè âåêòîðíûé ïîòåíöèàë ýëåêòðè÷åñêîãî ïîëÿ E
            ðàâíîìåðíî çàðÿæåííîãî ñëîÿ, îïèñàííîãî â çàäà÷å
            3.16.
     • Èñïîëüçóÿ ôîðìóëó (3.14), íàéòè âåêòîðíûå ïîòåíöèàëû
ñëåäóþùèõ ïîëåé, ïðåäâàðèòåëüíî óáåäèâøèñü â èõ ñîëåíîè-
äàëüíîñòè:
             ρ                              ρ
      3.31. a = {111
                   , , };             3.33. a = {6 x ,−15 y,9 z} ;
             ρ                              ρ
      3.32. a = { y , z , x} ;        3.34. a = {0,2 cos( xz ),0} .
      3.35. Äîêàçàòü, ÷òî îäèí èç âåêòîðíûõ ïîòåíöèàëîâ îäíî-
                               ρ                    ρ ρ ρ
            ðîäíîãî ïîëÿ a ïðèâîäèòñÿ ê âèäó A = (a × r ) / 2 . Äî-
            êàçàòåëüñòâî ïðîâåñòè ïóòåì ïîñòðîåíèÿ ñîîòâåòñòâó-
            þùåãî âåêòîðíîãî ïîòåíöèàëà.
                       ρ
      3.36. Ïóñòü p — ïîñòîÿííûé âåêòîð. Äîêàçàòü ñîëåíîè-
                                 ρ ρ ρ
            äàëüíîñòü ïîëÿ a = p × r è ïî ôîðìóëå (3.14) íàéòè
            åãî âåêòîðíûé ïîòåíöèàë.
     • Äîêàçàòü ñîëåíîèäàëüíîñòü ïåðå÷èñëåííûõ íèæå ïîëåé, â
          ρ                       ρ
êîòîðûõ r — ðàäèóñ-âåêòîð, p — ïîñòîÿííûé âåêòîð, u(r) —
çàäàííàÿ ôóíêöèÿ, è ïîêàçàòü, ÷òî èõ âåêòîðíûå ïîòåíöèàëû
                                     ρ      ρ
ìîãóò áûòü ïðåäñòàâëåíû â âèäå A = f ( r ) p , ãäå f (r) — íåêîòî-
ðàÿ ôóíêöèÿ ðàññòîÿíèÿ òî÷êè äî íà÷àëà êîîðäèíàò; íàéòè êîí-
êðåòíûé âèä ôóíêöèè f (r) äëÿ ýòèõ ïîëåé:
             ρ      ρ ρ                     ρ ρ ρ
      3.37. a = r ( r × p ) ;         3.39. a = ( r × p ) / r ;
             ρ ρ ρ                          ρ ρ ρ
      3.38. a = r × p ;               3.40. a = ( r × p ) / r 3 ;
             ρ         ρ ρ                  ρ          ρ
      3.41. a = u( r )r × p ;         3.42. a = ∇u × p .

          3.3. Ëàïëàñîâî ïîëå. Ãàðìîíè÷åñêèå ôóíêöèè
                                ρ
Îïðåäåëåíèå. Âåêòîðíîå ïîëå a ( x , y, z ) íàçûâàåòñÿ ëàïëàñîâûì â
            îáëàñòè V, åñëè â ëþáîé òî÷êå ýòîé îáëàñòè
                           ρ           ρ
                       div a = 0 , rot a = 0 .
     Òàêèì îáðàçîì, â îäíîñâÿçíîé îáëàñòè ëàïëàñîâî ïîëå ÿâ-
ëÿåòñÿ îäíîâðåìåííî ïîòåíöèàëüíûì è ñîëåíîèäàëüíûì.



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