Задачи по векторному анализу. Михайлов В.К - 20 стр.

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1.54. Ïîêàçàòü, ÷òî â ëþáîé òî÷êå ýëëèïñà 2x
2
+y
2
= 3 ïðî-
èçâîäíàÿ ñêàëÿðíîãî ïîëÿ u=y
2
/x ïî íàïðàâëåíèþ
íîðìàëè ê ýëëèïñó ðàâíà íóëþ.
1.55.  êàêîé òî÷êå Ì ãðàäèåíò ñêàëÿðíîãî ïîëÿ
u=x
2
+ 2y
2
+ 3z
2
+xy+3x —2y —6z ðàâåí íóëþ?
1.56.  êàêèõ òî÷êàõ ïðîñòðàíñòâà ãðàäèåíò ñêàëÿðíîãî ïîëÿ
u=x
3
+y
3
+z
3
—3xyz :
a) ïåðïåíäèêóëÿðåí îñè z;
á) ðàâåí íóëþ?
1.57.  êàêèõ òî÷êàõ ïðîñòðàíñòâà ìîäóëü ãðàäèåíòà ñêà-
ëÿðíîãî ïîëÿ u=ln(1/r) ðàâåí åäèíèöå?
1.58. Âû÷èñëèòü ïðîèçâîäíóþ ñêàëÿðíîãî ïîëÿ u=1/r
â òî÷êå Ì (2, 2, 1) ïî íàïðàâëåíèþ åãî ãðàäèåíòà
è âäîëü îñè z.
1.59. Âû÷èñëèòü ïðîèçâîäíóþ ñêàëÿðíîãî ïîëÿ u = xyz
â íàïðàâëåíèè ãðàäèåíòà ñêàëÿðíîãî ïîëÿ v =lnr
â òî÷êå Ì (2, 2, 1).
1.60. Âû÷èñëèòü ïðîèçâîäíóþ ñêàëÿðíîãî ïîëÿ u=x
3
y
2
z
â òî÷êå Ì (1, 1, 1) â íàïðàâëåíèè ãðàäèåíòà ïîëÿ
v=x
3
+y
2
+z â ýòîé òî÷êå.
1.61. Âû÷èñëèòü ïðîèçâîäíóþ ñêàëÿðíîãî ïîëÿ u (x, y, z)
â íàïðàâëåíèè ãðàäèåíòà ñêàëÿðíîãî ïîëÿ v (x, y, z).
Ïðè êàêîì óñëîâèè îíà ðàâíà íóëþ?
• Âû÷èñëèòü ãðàäèåíòû ñëåäóþùèõ ñêàëÿðíûõ ïîëåé, â êî-
òîðûõ
ρ
a
è
ρ
b
— ïîñòîÿííûå âåêòîðû, à
ρ
r
— ðàäèóñ-âåêòîð:
1.62. u=r; 1.66.
uabr= ()
ρ
ρ
ρ
.
1.63.
u
r
=
1
2
; 1.67.
uarbr=⋅ ⋅()()
ρρ
ρ
ρ
;
1.64.
ur
n
=
; 1.68.
u
ar
r
=
⋅
ρρ
3
;
1.65.
ure
z
=⋅
ρρ
; 1.69.
uarb=×⋅()
ρρ
ρ
;
1.70.
uar=×()
ρρ
2
;
1.71. Äîêàçàòü, ÷òî â ëþáîé òî÷êå óãîë ìåæäó ãðàäèåíòà-
ìè äâóõ ñêàëÿðíûõ ïîëåé u = 1/(x + y + z)
è v = exp(x + y — 2z) ðàâåí
π
/2.
     1.54. Ïîêàçàòü, ÷òî â ëþáîé òî÷êå ýëëèïñà 2x2 + y2 = 3 ïðî-
           èçâîäíàÿ ñêàëÿðíîãî ïîëÿ u = y2/x ïî íàïðàâëåíèþ
           íîðìàëè ê ýëëèïñó ðàâíà íóëþ.
     1.55.  êàêîé òî÷êå Ì ãðàäèåíò ñêàëÿðíîãî ïîëÿ
           u = x2 + 2y2 + 3z2 + xy + 3x — 2y — 6z ðàâåí íóëþ?
     1.56.  êàêèõ òî÷êàõ ïðîñòðàíñòâà ãðàäèåíò ñêàëÿðíîãî ïîëÿ
           u = x3 + y3 + z3 — 3xyz :
           a) ïåðïåíäèêóëÿðåí îñè z;
           á) ðàâåí íóëþ?
     1.57.  êàêèõ òî÷êàõ ïðîñòðàíñòâà ìîäóëü ãðàäèåíòà ñêà-
           ëÿðíîãî ïîëÿ u = ln(1/r) ðàâåí åäèíèöå?
     1.58. Âû÷èñëèòü ïðîèçâîäíóþ ñêàëÿðíîãî ïîëÿ u = 1/r
           â òî÷êå Ì (2, 2, 1) ïî íàïðàâëåíèþ åãî ãðàäèåíòà
           è âäîëü îñè z.
     1.59. Âû÷èñëèòü ïðîèçâîäíóþ ñêàëÿðíîãî ïîëÿ u = xyz
           â íàïðàâëåíèè ãðàäèåíòà ñêàëÿðíîãî ïîëÿ v = lnr
           â òî÷êå Ì (2, 2, 1).
     1.60. Âû÷èñëèòü ïðîèçâîäíóþ ñêàëÿðíîãî ïîëÿ u = x3y2z
           â òî÷êå Ì (1, 1, 1) â íàïðàâëåíèè ãðàäèåíòà ïîëÿ
           v = x3+y2+z â ýòîé òî÷êå.
     1.61. Âû÷èñëèòü ïðîèçâîäíóþ ñêàëÿðíîãî ïîëÿ u (x, y, z)
           â íàïðàâëåíèè ãðàäèåíòà ñêàëÿðíîãî ïîëÿ v (x, y, z).
           Ïðè êàêîì óñëîâèè îíà ðàâíà íóëþ?
    • Âû÷èñëèòü ãðàäèåíòû ñëåäóþùèõ ñêàëÿðíûõ ïîëåé, â êî-
      ρ    ρ                               ρ
òîðûõ a è b — ïîñòîÿííûå âåêòîðû, à r — ðàäèóñ-âåêòîð:
                                                  ρρρ
     1.62. u = r;                     1.66. u = ( abr ) .
                  1                            ρ ρ ρ ρ
      1.63. u =      ;             1.67. u = ( a ⋅ r )( b ⋅ r ) ;
                  r2
                                              ρ ρ
                                             a ⋅r
      1.64. u = r n ;              1.68. u =       ;
                                               r3
                ρ ρ                            ρ ρ ρ
      1.65. u = r ⋅ ez ;           1.69. u = ( a × r ) ⋅ b ;
                                               ρ ρ
                                   1.70. u = ( a × r )2 ;

      1.71. Äîêàçàòü, ÷òî â ëþáîé òî÷êå óãîë ìåæäó ãðàäèåíòà-
            ìè äâóõ ñêàëÿðíûõ ïîëåé u = 1/(x + y + z)
            è v = exp(x + y — 2z) ðàâåí π/2.

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