Задачи по векторному анализу. Михайлов В.К - 99 стр.

                      ρ ρ                  R

                    ∫ a ⋅ dl = ∫ a ρ d ρ = ∫ 4ρ sin(π / 4)dρ = R 2 .
                    L         L         0

     Ïðèìåð          3.     Âû÷èñëèòü       öèðêóëÿöèþ           ïîëÿ
 ρ         ρ      ρ       ρ
a = ρ sinϕ eρ + ρzeϕ + ρ ez ïî êîíòóðó Ñ : ρ = sinϕ, 0 ≤ ϕ ≤ π, z = 0.
                        3


     Ðåøåíèå. Íà êîíòóðå Ñ èìååì: z = 0, dz = 0, ρ = sin ϕ ,
dρ = cosϕdϕ, 0 ≤ ϕ ≤ π. Òîãäà èñêîìàÿ öèðêóëÿöèÿ

  ρ ρ
                                                                               π
                                                                          sin 3 ϕ
∫ a ⋅ dl   = ∫ (aρ dρ + ρaϕ dϕ + az dz ) = ∫ sin 2 ϕ cos ϕ dϕ + 0 + 0 =
                                                                             3 0
                                                                                  = 0.



                                      Çàäà÷è
                                                   ρ ρ                    ρ
           4.37. Âû÷èñëèòü öèðêóëÿöèþ ïîëÿ a = rer + ( R + r )sin θ eϕ
                 ïî îêðóæíîñòè r = R, θ = π/2 â íàïðàâëåíèè âîçðàñ-
                 òàíèÿ óãëà ϕ.
                                            ρ           ρ           ρ      ρ
           4.38. Âû÷èñëèòü ðàáîòó ïîëÿ a = ρ cos ϕ eρ + ρ sin ϕ eϕ + ρez
                 ïî âèòêó âèíòîâîé ëèíèè L: ρ = R, 0 ≤ ϕ ≤ 2π, z = ϕ.
                                          ρ ρ             ρ       ρ
           4.39. Âû÷èñëèòü ðàáîòó ïîëÿ a = rer + 2θ cos ϕ eθ + ϕ eϕ ïî ïî-
                 ëóîêðóæíîñòè L : r = R, ϕ = 0, 0 ≤ θ ≤ π.
                                           ρ         ρ         ρ
           4.40. Âû÷èñëèòü ðàáîòó ïîëÿ a = sin 2 θ er + sin 2θ eθ ïî ïðÿ-
                 ìîé L: ϕ = π/2, r = 1/(sinθ), π/6 ≤ θ ≤ π/2.
                                                 ρ ρ       ρ      ρ
           4.41. Âû÷èñëèòü öèðêóëÿöèþ ïîëÿ a = zeρ + ρeϕ + ρez ïî îê-
                 ðóæíîñòè ρ = R, z = 0.
                                                    ρ         ρ       ρ
           4.42. Âû÷èñëèòü öèðêóëÿöèþ ïîëÿ a = cos ϕ eρ + ρ eϕ ïî
                 êîíòóðó ρ = sinϕ (0 ≤ ϕ ≤ π).
                                            ρ          ρ         ρ
           4.43. Âû÷èñëèòü ðàáîòó ïîëÿ a = (2 cos θ er + sin θ eθ ) / r 3 ïî
                 ïîëóîêðóæíîñòè L : r = R, ϕ = 0, 0 ≤ θ ≤ π.

                 4.3.7. Âû÷èñëåíèå ñêàëÿðíîãî ïîòåíöèàëà
    Ïðåæäå ÷åì âû÷èñëÿòü ïîòåíöèàë âåêòîðíîãî ïîëÿ aρ â êðè-
âîëèíåéíûõ êîîðäèíàòàõ, íåîáõîäèìî óáåäèòüñÿ, ÷òî ýòî ïîëå


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