Задачи по векторному анализу. Михайлов В.К - 40 стр.

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40
Çàäà÷è
2.61. Ïîòîê ïîëÿ
ρρ
arr= /
3
÷åðåç ñôåðó x
2
+y
2
+z
2
=R
2
, âû-
÷èñëåííûé íåïîñðåäñòâåííî, ðàâåí 4
π
, à ïî òåîðåìå
Îñòðîãðàäñêîãî — íóëþ. Ïî÷åìó?
• Ðåøèòü ñëåäóþùèå çàäà÷è ñ èñïîëüçîâàíèåì òåîðåìû Îñ-
òðîãðàäñêîãî èëè åå ñëåäñòâèé.
2.62. Âû÷èñëèòü ïîòîê ïîëÿ
ρρ
arr= /
3
÷åðåç çàìêíóòóþ ïî-
âåðõíîñòü x
2
+ y
2
+ (z — 2)
2
= 1.
2.63. Âû÷èñëèòü ïîòîê ïîëÿ
ρρ
arr=
2
÷åðåç ñôåðó
x
2
+y
2
+z
2
=R
2
.
2.64. Òî÷å÷íûé çàðÿä, äâèæóùèéñÿ ñ ïîñòîÿííîé ñêîðîñ-
òüþ
ρ
v
, ñîçäàåò â îêðóæàþùåì ïðîñòðàíñòâå ìàãíèò-
íîå ïîëå
ρ
ρρ
Bkvrr=×()/
3
, ãäå k — íåêîòîðûé êîýô-
ôèöèåíò. Âû÷èñëèòü ïîòîê ýòîãî ïîëÿ ÷åðåç çàìêíó-
òóþ ïîâåðõíîñòü x
2
+ (y — 2)
2
+ z
2
= 1.
2.65. Âû÷èñëèòü ïîòîê ïîëÿ
ρ
ρ
a =
ρρ
, ãäå
ρ
ρρ
ρ
=+xe ye
xy
, ֌-
ðåç öèëèíäðè÷åñêóþ ïîâåðõíîñòü x
2
+ y
2
=R
2
, 0 ≤ z ≤ h.
2.66. Âû÷èñëèòü ïîòîê ïîëÿ
ρ
axyyz=−{, ,}2
÷åðåç ñôåðó
x
2
+y
2
+z
2
= 4.
2.67. Âû÷èñëèòü ïîòîê ïîëÿ
ρ
axyz= {, ,}
333
÷åðåç ñôåðó
x
2
+y
2
+z
2
=R
2
.
2.68. Âû÷èñëèòü ïîòîê ïîëÿ
ρ
axyz= {, ,}
333
÷åðåç ïîâåðõíîñòü
êóáà 0 ≤ x ≤ l, 0≤ y≤ l, 0≤ z ≤ l.
2.69. Âû÷èñëèòü ïîòîê ïîëÿ
ρ
axyz= {, ,}
222
÷åðåç ïîâåðõíîñòü
êóáà -l ≤ x≤ l, -l ≤ y ≤ l, -l ≤ z≤ l.
2.70. Âû÷èñëèòü ïîòîê ïîëÿ
axyz= {,,}2
÷åðåç çàìêíóòóþ
ïîâåðõíîñòü S : x
2
+y
2
=z
2
, z=4.
2.71. Âû÷èñëèòü ïîòîê ïîëÿ
ρ
axxzy= {, , }
÷åðåç çàìêíóòóþ
ïîâåðõíîñòü S : z=4 —x
2
—y
2
, z=0.
2.72. Âû÷èñëèòü ïîòîê ïîëÿ
ρ
axz=−{,,}20
÷åðåç çàìêíóòóþ
ïîâåðõíîñòü S : x
2
+y
2
=R
2
, z=0, z=h.
2.73. Âû÷èñëèòü ïîòîê ïîëÿ
ρ
ayzxzxy= {, ,}
:
                              Çàäà÷è
                            ρ ρ
      2.61. Ïîòîê ïîëÿ a = r / r 3 ÷åðåç ñôåðó x2 + y2 + z2 = R2, âû-
            ÷èñëåííûé íåïîñðåäñòâåííî, ðàâåí 4π, à ïî òåîðåìå
            Îñòðîãðàäñêîãî — íóëþ. Ïî÷åìó?
     • Ðåøèòü ñëåäóþùèå çàäà÷è ñ èñïîëüçîâàíèåì òåîðåìû Îñ-
òðîãðàäñêîãî èëè åå ñëåäñòâèé.
                                          ρ ρ
      2.62. Âû÷èñëèòü ïîòîê ïîëÿ a = r / r 3 ÷åðåç çàìêíóòóþ ïî-
            âåðõíîñòü x2 + y2 + (z — 2)2 = 1.
                                                  ρ 2ρ
      2.63. Âû÷èñëèòü ïîòîê ïîëÿ a = r r ÷åðåç ñôåðó
            x2 + y2 + z2 = R2.
      2.64. Òî÷å÷íûé çàðÿä, äâèæóùèéñÿ ñ ïîñòîÿííîé ñêîðîñ-
                  ρ
            òüþ v , ñîçäàåò â îêðóæàþùåì ïðîñòðàíñòâå ìàãíèò-
                          ρ    ρ ρ
            íîå ïîëå B = k ( v × r ) / r 3 , ãäå k — íåêîòîðûé êîýô-
            ôèöèåíò. Âû÷èñëèòü ïîòîê ýòîãî ïîëÿ ÷åðåç çàìêíó-
            òóþ ïîâåðõíîñòü x2 + (y — 2)2 + z2 = 1.
                                            ρ    ρ    ρ    ρ    ρ
      2.65. Âû÷èñëèòü ïîòîê ïîëÿ a = ρρ , ãäå ρ = xe x + ye y , ÷å-
            ðåç öèëèíäðè÷åñêóþ ïîâåðõíîñòü x2 + y2 = R2, 0 ≤ z ≤ h.
                                              ρ
      2.66. Âû÷èñëèòü ïîòîê ïîëÿ a = {xy ,2 y,− z} ÷åðåç ñôåðó
            x2 + y2 + z2 = 4.
                                               ρ
      2.67. Âû÷èñëèòü ïîòîê ïîëÿ a = {x 3, y3, z 3} ÷åðåç ñôåðó
            x2 + y2 + z2 = R2.
                                        ρ
      2.68. Âû÷èñëèòü ïîòîê ïîëÿ a = {x 3, y3, z 3} ÷åðåç ïîâåðõíîñòü
            êóáà 0 ≤ x ≤ l, 0 ≤ y ≤ l, 0 ≤ z ≤ l.
                                       ρ
      2.69. Âû÷èñëèòü ïîòîê ïîëÿ a = {x 2, y 2, z 2} ÷åðåç ïîâåðõíîñòü
            êóáà -l ≤ x ≤ l, -l ≤ y ≤ l, -l ≤ z ≤ l.
                                           ρ
      2.70. Âû÷èñëèòü ïîòîê ïîëÿ a = {2 x , y , z} ÷åðåç çàìêíóòóþ
            ïîâåðõíîñòü S : x2 + y2 = z2, z = 4.
                                           ρ
      2.71. Âû÷èñëèòü ïîòîê ïîëÿ a = {x, xz , y} ÷åðåç çàìêíóòóþ
            ïîâåðõíîñòü S : z = 4 — x2 — y2, z = 0.
                                          ρ
      2.72. Âû÷èñëèòü ïîòîê ïîëÿ a = {2 x ,0,− z} ÷åðåç çàìêíóòóþ
            ïîâåðõíîñòü S : x2 + y2 = R2, z = 0, z = h.
                                           ρ
      2.73. Âû÷èñëèòü ïîòîê ïîëÿ a = { yz , xz , xy} :


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