# Дифференциальное исчисление функций нескольких переменных. Скляренко В.А - 65 стр.

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• ## Математический анализ

6. x =
1
2
(e
u
+ e
v
)
2
, y =
1
2
(e
u
e
v
)
2
, z = u 2 v; u
0
= ln (3), v
0
= 0.
7. x = 2
v
cos u + sin u
, y =
1
cos u + sin u
, z = v (cos u + sin u); u
0
=
1
2
π, v
0
= 1.
8. x =
u + v
u
2
+ v
2
, y =
uv
u
2
+ v
2
, z =
1
u
2
+ v
2
; u
0
= 1, v
0
= 2.
9. x = u ch v, y = u
2
sh v
u
2
, z = u
3
+
sh v ch v
u
2
; u
0
= 1, v
0
= 0.
10. x = v + u
2
ln 3 π, y = 3
u
cos v, z = 3
u
sin v; u
0
= 1, v
0
= π.
11. x = 2
v sin u
1 + v
2
, y = 6
v cos u
1 + v
2
, z =
º
2
4
(1 + v
2
); u
0
=
1
4
π, v
0
= 2.
12. x =
»
2 (u
2
3)(1 + v
2
), y =
»
10 (3 v
2
)(u
2
+ 1), z =
º
u
2
v
2
3;
u
0
= 2, v
0
= 1.
13. x = tg
2
v cos u, y = cos
2
v, z = tg
2
v sin u; u
0
=
1
2
π, v
0
=
1
4
π.
14. x =
2
sh u + cos v
, y =
cos v
sh u + cos v
, z = sin v ch u; u
0
= 0, v
0
= 0.
15. x = 3 u + 3 uv
2
u
3
, y = v
3
3 v 3 u
2
v, z = 3 u
2
3 v
2
; u
0
= 1, v
0
= 1.
16. x = 3 u + ln (v u), y = 3 v ln (v u
2
), z = u
2
+ 2 v + u; u
0
= 1, v
0
= 2.
17. x =
º
u
u v
, y = 2
º
v
u v
, z =
1
3
º
uv; u
0
= 4, v
0
= 1.
18. x = 4 ch u cos v, y = 2 ch u sin v, z = 6 sh u sin v; u
0
= ln 2, v
0
=
1
6
π.
19. x = ue
v
, y = e
v
(cos
2
u + 3 sin u), z = e
v
sin
2
u + 4 cos u
; u
0
= π, v
0
= 0.
20. x = 2 u
2
cos v, y = 4 u
2
sin v, z =
º
2u
2
8
v
2
π
+
π
2
; u
0
= 1, v
0
=
1
4
π.
21. x =
ch u
sin v ch u
, y =
sh u
sin v ch u
, z =
1 + u
sin v ch u
; u
0
= 0, v
0
=
1
6
π.
22. x = u
2
+ ln v, y = v
2
ln u, z = uv; u
0
= 1, v
0
= 1.
23. x = u (2 v u 1), y = v (2 u v 1), z = u
2
+ v
2
; u
0
= 3, v
0
= 1.
24. x =
º
1 + u cos 2 v, y =
º
1 + u sin 2 v, z = 2
º
1 + u + cos u;
u
0
= 0, v
0
=
1
3
π.
25. x =
»
3 (u
2
1)(1 v
2
), y = u (1 + sh v), z = v + u
2
ch v; u
0
= 2, v
0
= 0.
26. x =
e
2 u
1 + v
, y =
ve
u
1 + v
, z =
v
1 + e
u
; u
0
= ln 3, v
0
= 2.
27. x =
»
(u + 2)(v 2), y =
»
(u 2)(v + 2), z =
º
3uv; u
0
= 4, v
0
= 4.
65
          (eu + ev )2 , y = (eu − ev )2 , z = u − 2 v; u0 = ln (3) , v0 = 0.
1                    1
6. x =
2                    2
, z = v (cos u + sin u) ; u0 = π, v0 = 1.
v                  1                                       1
7. x = 2               , y=
cos u + sin u      cos u + sin u                                 2
u+v              uv           1
8. x = 2 2 , y = 2 2 , z = 2 2 ; u0 = 1, v0 = 2.
u +v           u +v         u +v
sh v             sh v ch v
9. x = u − ch v, y = u2 − 2 , z = u3 +                  ; u0 = 1, v0 = 0.
u                  u2
10. x = v + u2 ln 3 − π, y = 3u cos v, z = 3u sin v;           u0 = 1, v0 = π.
º
(1 + v2 ) ;
v sin u        v cos u      2                                1
11. x = 2         , y= 6         ,z=             u0 = π, v0 = 2.
»                        »                          º
1+v   2         1+v  2     4                 4
12. x = 2 (u2 − 3) (1 + v2 ), y = 10 (3 − v2 ) (u2 + 1), z = u2v2 − 3;
u0 = 2, v0 = 1.
1             1
13. x = tg2 v cos u, y = cos2 v, z = tg2 v sin u;        u0 = π, v0 = π.
2             4
2               cos v
14. x =                , y=              , z = sin v − ch u;       u0 = 0, v0 = 0.
sh u + cos v      sh u + cos v
15. x = 3 u + 3 uv2 − u3 , y = v3 − 3 v − 3 u2v, z = 3 u2 − 3 v2 ;               u0 = 1, v0 = 1.
16. x = 3 u + ln (v − u) , y = 3 v − ln (v − u2 ) , z = u2 + 2 v + u;                u0 = 1, v0 = 2.
º            º
u           v        1º
17. x =     , y= 2       ,z=       uv; u0 = 4, v0 = 1.
u−v           u−v         3
1
18. x = 4 ch u cos v, y = 2 ch u sin v, z = 6 sh u sin v;                  u0 = ln 2, v0 = π.
6
19. x = uev , y = ev (cos2 u + 3 sin u) , z = ev sin2 u + 4 cos u; u0 = π, v0 = 0.
º          v2 π                    1
20. x = 2 u2 cos v, y = 4 u2 sin v, z = 2u2 − 8 + ; u0 = 1, v0 = π.
π      2                         4
ch u              sh u             1+u                                 1
21. x =                , y=              ,z=              ;    u0 = 0, v0 = π.
sin v − ch u      sin v − ch u     sin v − ch u                            6
22. x = u2 + ln v, y = v2 − ln u, z = uv;         u0 = 1, v0 = 1.
23. x = u (2 v − u − 1) , y = v (2 u − v − 1) , z = u2 + v2 ; u0 = 3, v0 = 1.
º                    º                      º
24. x = 1 + u cos 2 v, y = 1 + u sin 2 v, z = 2 1 + u + cos u;
1
u0 = 0, v0 = π.
»
3
25. x = 3 (u2 − 1) (1 − v2 ), y = u (1 + sh v) , z = v + u2 ch v; u0 = 2, v0 = 0.
e2 u      veu        v
26. x =         , y=     ,z=        ;      u0 = ln 3, v0 = 2.
»                          »                             º
1+v        1+v     1 + eu
27. x =       (u + 2) (v − 2), y =       (u − 2) (v + 2), z =       3uv;         u0 = 4, v0 = 4.

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