Задачи по векторному анализу. Михайлов В.К - 92 стр.

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92
Çàäà÷è
4.6. Íàéòè âåêòîðíûå ëèíèè ïîëÿ
ρρρ
a
r
e
r
e
r
=+
2
33
cos sin
θθ
θ
.
4.7. Íàéòè âåêòîðíûå ëèíèè ïîëÿ
ρρ ρ
aebe
z
=+
ρ
ϕ
, ãäå b —
÷èñëî.
4.8. Íàéòè ñåìåéñòâî ëèíèé áûñòðåéøåãî âîçðàñòàíèÿ
ïîëÿ u=r
2
cos
θ
.
4.9. Íàéòè íàïðàâëåíèå áûñòðåéøåãî âîçðàñòàíèÿ ïîëÿ
u
r
=−
cos
θ
2
â òî÷êå Ì (1,
π
/2, 0).
4.10. Âû÷èñëèòü ãðàäèåíò ñêàëÿðíîãî ïîëÿ u=r
2
cos
θ
.
4.11. Âû÷èñëèòü ãðàäèåíò ñêàëÿðíîãî ïîëÿ u
=ρ
+ zcos
ϕ
.
4.3.3. Äèâåðãåíöèÿ
Ïóñòü çàäàíî âåêòîðíîå ïîëå (4.5). Òîãäà åãî äèâåðãåíöèÿ
div
()()()
ρ
a
HHH
aH H
q
aHH
q
aHH
q
=++
1
123
123
1
213
2
312
3
∂
∂
∂
∂
∂
∂
. (4.7)
 ÷àñòíîñòè, â öèëèíäðè÷åñêèõ êîîðäèíàòàõ (q
1
=ρ
, q
2
=ϕ
, q
3
=z;
H
1
= 1, H
2
=ρ
, H
3
= 1):
div
ρ
a
aa aa
z
=+ + +
11 2 3
1
ρ
∂
∂ρ ρ
∂
∂ϕ
∂
∂
;
â ñôåðè÷åñêèõ êîîðäèíàòàõ (q
1
=r, q
2
=θ
, q
3
=ϕ
; H
1
= 1, H
2
=r,
H
3
=rsin
θ
):
div
()
sin
(sin)
sin
ρ
a
r
ar
rr
a
r
a
=+ +
11 1
2
1
2
23
∂
∂θ
∂θ
∂θ θ
∂
∂ϕ
.
                                   Çàäà÷è
       4.6. Íàéòè âåêòîðíûå ëèíèè ïîëÿ
                           ρ         θ eρ + sin θ eρ
                           a = 2 cos                 .
                                  r3     r
                                              r3 θ
                                        ρ    ρ    ρ
       4.7. Íàéòè âåêòîðíûå ëèíèè ïîëÿ a = ρeϕ + bez , ãäå b —
            ÷èñëî.
       4.8. Íàéòè ñåìåéñòâî ëèíèé áûñòðåéøåãî âîçðàñòàíèÿ
            ïîëÿ u = r 2 cosθ.
       4.9. Íàéòè íàïðàâëåíèå áûñòðåéøåãî âîçðàñòàíèÿ ïîëÿ
                cos θ
            u=−       â òî÷êå Ì (1, π/2, 0).
                 r2
      4.10. Âû÷èñëèòü ãðàäèåíò ñêàëÿðíîãî ïîëÿ u = r 2 cosθ.
      4.11. Âû÷èñëèòü ãðàäèåíò ñêàëÿðíîãî ïîëÿ u = ρ + zcosϕ.

                             4.3.3. Äèâåðãåíöèÿ
     Ïóñòü çàäàíî âåêòîðíîå ïîëå (4.5). Òîãäà åãî äèâåðãåíöèÿ

            ρ        1       ∂ (a1 H2 H3 ) ∂ (a2 H 1H3 ) ∂ (a3 H 1H2 ) 
        div a =                            +             +              .   (4.7)
                  H1 H2 H3      ∂q1           ∂q2           ∂q3
 ÷àñòíîñòè, â öèëèíäðè÷åñêèõ êîîðäèíàòàõ (q1 = ρ, q2 = ϕ, q3 = z;
H1 = 1, H2 = ρ, H3 = 1):
                          ρ a ∂a 1 ∂a2 ∂a3
                      div a = 1 + 1 +  +
                              ρ ∂ρ ρ ∂ϕ ∂z ;
â ñôåðè÷åñêèõ êîîðäèíàòàõ (q1 = r, q2 = θ, q3 = ϕ; H1 = 1, H2 = r,
H3 = r sinθ):
              ρ 1 ∂ (a1r 2 )      1 ∂ (a2 sin θ )      1 ∂a3
          div a = 2          +                    +
                 r   ∂r        r sin θ  ∂θ          r sin θ ∂ϕ .




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