JVa20
a
vektor maydon berilgan,
Su
Kroneker belgisi.
o = {1,3,—1}; a,a(A =?
A)
11
B) 12 C) 0 D) 9
JVa21
a
tenzor maydon berilgan,
Su
Kroneker belgisi.
a tJ = i + j; avS y
= ?
A)
4 B) 12 C) 24 D) 0
JV°22
d
3 - rang tenzor maydon berilgan,
StJ
Kroneker belgisi.
djjk = i + j ~ k ; dIJkSjk
=?
A)
{3,6,9} B)
{1,2,3} C) {4,6,8} D) {3,6,-9}
147
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JV»23
d
3 - rang tenzor maydon berilgan,
Sv
Kroneker belgisi.
d tkm
=
i
+ 2
j -
k\ d ilm8ki8ml =
?
A)
{-1,23,8}
B) {15,12,9} C)
{3,8,10} D) {12,12,0}
Jfs24
a 2 -
rang tenzor maydon berilgan,
8V
Kronekerbelgisi.
atJ —
2 + / •
j , QtJ8^ —
?
A)
20 B)
11
C)
32 D)
16
Mustaqil ish variantlaridan namunalar
1- Misol.
u(x,y,z)
maydonning M nuqtadagi S sirtga normal yo'nalishidagi
hosilasini toping (normal oz o'qi bilan o‘tkir burchak tashkil qiladi)
1)
U
=
4 ln(3
+ *2
)
-
8X17,5: x2
-
2y2 + 2r2 = 1,3/(1,1,1).
2)
u
= x * [y + y 4 z , S
: 4 z - 2 x 2
- /
= i,M
(2,4,4).
3)
u
=
-2ln(x2
- 5) -
4xyr,5 :x2
+ .v2
- 2 z J
=
1,3/(1,1,1).
4)
u
= —
x 2y - ^ x 2 + 5 z 2
tS
: z 2 - x
4
2+4yJ- 4 , ^ - 2 , i , l j .
5)
u
=
xz2
- [ x * y , S
:x2 -2_y2
-
3z
+
12
=
0,3/(2,2,4).
6)
u
=
X y f y -
y z 2
, S
:
x 1
+ y
2
= 4z + 9,A /(l,l,l).
7)
u
=
7
InQj- +
x2 j
-
4 w z,5
:
7x2- 4 y 2+4z2 =7,A/(1,1,1).
8)
u
=
a r c t j ^ —
j
+
x : , S : x 1 + y 1 -
2z
=
10,3/(2,2,-1).
9)
u
=
ln(l
+
x2)-xyVr,5:4x2
- y 1
+
z2 =16,A/(l,-2,4).
10)
u
= - j x 1 + y 7 - z , S : x
1
+
y 1
=24z
+
l,3/(3,4,1).
2-Misol.
u ( x ,y ,z )
skalyar maydonning
M
nuqtadagi / yo‘nalishdagi hosilasini
toping.
1)
u = (x 2 + y 2 + z 2)3n, I = i - j + k,
3/(1,1,1).
2)
u = x +
ln(z2 +
y 1 ),[ = -2 i + j - k , M(2,l,l).
3)
u
=
x 2y - J x y
+ r 12, / =
2 j - 2k,M(\,5,-2).
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4)
u = y
ln(l+x2) -
arctgz,
I =
2/ -
3j
-
2k, M
(0,1,1).
5)
u =
jc(ln
y
- arctgz),
/ = 8/ + 4y' + 8
k, M
(-2,1,-1).
6)
« = ln(3 - x2) +
xy1 z ,l = - i + 2 j - 2k,M(\,3,2).
7)
u
=
(sin(x + 2y) + Jxyz,I
= 4/
+3j,M(n! 2,3n
/ 2,1).
8)
u =
x V z - ln(z -1), / = 5»-6y + 2V5*, A/(l,l,2).
9)
i/ = x} + V / + r 1, / =
- j - k,M(\,-3,4).
10)
u = —
----
^ , = J = 2i + k.M(4X-2).
y
x + J y
3- Misol.
Vektor maydonning vektor chiziqlarini toping.
1)
a = 4 y i - 9 x j.
2)
a
= 2
yi + 3xj.
3)
a = 2 xi +
4
yj.
4)
a =
xi + 3
yj.
5)
a =
xi
+
4yj.
6)
a = 3 xi + 6 zk.
7)
a =
4z( -
9xk.
8)
a = 2zi + 3xk.
9)
a = 4yj
+ 8z*.
10)
a = yj + 3zk.
4-
Misol.
a
vektor maydonning
S
sirtning
P x
va
P2
tekisliklar bilan kesishish
qismlaridan o‘tuvchi oqimni toping ( normal yopiq sirtga tashqi
yo‘nalgan).
1)
a
=
xi + yj + zk,S : x 2 + y 2 =
1,P,
:z = 0,P2 :z = 2.
2)
a = xi + y j - z k , S :x* + y 2 = \,Pt :z = 0,P7 :z =
4.
3)
a = xi + yj + 2 z k ,S : x 2 + y 2 = \,PX :z = 0,P2 :z =
3.
4)
a = xi + yj
+
z 2k,S : x 2 + y 2 =
1,
Px : z =
0,
P2 : z =
1.
5)
a = xi + yj + xyzk,S:x 2 + y2
=1,/*,
:z = 0,P2 :z = 5.
6)
a = (x-y)i + (x + y)j + z2k,S
:x2
+y2 = \,PX :z
=
0,P2 :z = 2.
7)
a = ( x + y ) i- ( x - y ) j + xyzk,S:x2
+><2 =1,P,
:z = 0,P2
:z = 4.
8 )
u = ^x, + J'}’! )< + (y, + x ^y)J + ztk .S :x ^ + y , = l.P, :z = 0,P, :z = 3
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9)
a = xi + yj + smzk,S : x 2
+j-2 =1
,PX
:z = 0,P2 :z =
5.
10)
a = xi + yj + k,S :x 2 + y 2
=1,P,
:z = 0,P2 :z =
1.
5- Misol.
a
vektor maydonning S sirtning P tekislik bilan ajratilgan qismidan
o ‘tuvchi oqimini toping (normal sirtlar bilan chegaralangan yopiq
sohaga tashqi)
1)
+
+
+
+ y = r5(r> 0 ),P :r = l.
2)
a = y i - x j + k ,S :x2 + y 2 = z 2( z > 0 ) ,P : z = 4.
3)
a = xyi - x 1 j
+
3k,S : x 2 + y 2 = z 2(z. >
0),
P : z =
1.
4)
a = x:i + yzj + (:2- \ ) k , S : x 2+y2 = zi (z>0),P:: = 4.
5)
a = y 2x i - y x 2j + k,S
:x 2
+ y 2 = z 2(z >0),P: z =
5.
6)
a = (xz + y)i + ( y z - x ) j + (z2 - 2 )k,S :x2 + y 2 = z 2(z > 0),P : z =
3.
7)
a = xyzi + x 2zj + 3k,S :x 2 + y 2 = z 2( z tO ) ,P : z = 2.
8)
a = (x + xy)i + ( y - x 2) j + ( z - \ ) k , S :x2 + y 2 = z 2(z
£ 0),
P :z =
3.
9)
a = (.r + j ’)'+ 0 '-Jf)y + (^ -2 )Jt,5 :x 2 +„vJ =
z 2(z > 0),P:: = 2.
10)
a = xi
+
yj + (z - 2 ) k , S : x 2 + y 2 = z 2( z > 0 ) , P : z = 2.
6-Misol.
a
vektor maydonning P tekislikning 1-oktantadagi qismidan o‘tuvchi
oqimni hisoblang (normal z o‘qi bilan o ‘tkir burchak tashkil qiladi)
1)
a = xi + yj + z k , P : x + y + z =
1.
2)
a = y j
+
zk. P
:
x + y + z =
1.
3)
a = 2xi + yj + zk,P : x + y + z = \.
4)
a = xi+ 3yj+ 2 z k ,P : x + y + z = \.
5)
a = x i + 3 y j ,P : x + y + z = \.
x
6)
a = xi + yj + zk,P : — + y + z =
1.
7)
8
)
x
a = xi+ 2yj + z k ,P : — + y + z =
1.
X
a = y j + 3 z k ,P : — + y + z = \.
g\
a = xi + yj + z k ,P : x + — + — = \.
>
2
3
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10)
7- Misol.
Yopiq sirtdan o'tuvchi oqimni toping (normal tashqi)
1)
a = (ex
+ 2
x)i + exj + eyk,S : x + y + z - l , x = 0,y = 0,z = 0.
2)
a =
(3
z' +x)i + (e* - 2 y ) j + (2 z-xy)k,S :x2 + y 2 = z 2,z = \,z
= 4.
3)
a =
(\n
y + lx)i+)s\x\z- 2y)j +
(ey
- 2z)k,S :x2 +y2 +z2 = 2x + 2y + 2z-2.
4)
a = (cosz + 3x)i + (s\n z - 2 y ) j + (ey - 2z)k,S :z 2 = 36(x2 + y 2),z =
6.
5)
a =
(
e'’ - x)i + (x: + 3y)j
+ (r +
x2)k,S :2x + y+ z = 2,x = 0,y = 0,: = 0.
6)
a = (6x
- cos
y )i
-
(ex + z ) j~ ( 2 y + 3z)k,S :x 2 + y 2 = z 2,z = \,z = 2.
7)
a = (4x - 2 y 1 )i +
(ln r - 4 v)y +
:x2 +y2 +:2 =2x + 3.
8)
a = (l + J z ) i + ( 4 y - - J x ) j + x y k , S : z 2 = 4 (x2 + y 2),z = 3.
9)
a = ( J : - x ) i + ( x - y ) j + (y2- : ) k , S : 3 x - 2 y + : = 6,x = 0,y = 0,: = 0.
10)
a = (yz + x)i + (x2 + y ) j +
(
jo
'2 +
z ) k , S : x 2
+
y 2 + z 2 = 2z.
8-Misol.
Yopiq sirtdan o'tuvchi oqimni toping (normal tashqi)
1)
a = (x + z)i + (z + '
2)
a = 2 xi + z k , S :
z = 3x2 + 2 y 2
+1,
x 2
+
y 2
= 4 ,z = 0.
3)
a = 2 x i
+ 2
y j + z k , S :
\y = x 2, y = 4x2, y = \(x> 0),
jz = y,z = 0.
4)
a = 3 x i - z j , S :
z = 6 - x 2 - y 2,
z 2 = x 2
+
y 2 (z >
0).
fjc2
+ y 2
=2
y,
6)
a = x i - ( x + 2 y ) j + y k , S :
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z = x + y
+ l,z = 0,
7)
a = 2 ( z - y ) i + ( x - z ) k , S : ^ x 2 + y 2
= J
. „ . f ? = 4 - 2 ( x J + / ) ,
g)
« = * ' + * ; - > 'M - { z=2(Jci + / ) .
9)
+
2
2
z = x 2
+ y ,
10)
a = 4 x i - 2 y j - z k , S :
3x + 2y = \ 2Jx + y = 6, y = 0,
x + y + z = 6,z = 0.
9-Misol.
Yopiq sirtdan o‘tuvchi oqimni toping (normal tashqi)
r = x2 + / , r = l,
1)
a = x 2i + x j + xzk, S: x = 0,y
= 0,
(1-oktaiit)
2)
a = (x2 + y3)i + (y2 +z2)j + (y
2
+z*)k,S
. ) x 2+ y 2 =l,
z = 0,z = l.
3)
a = x 2i + y 2j + z 2k , S :
4)
a = x 2i + y j + x / z k , S :
5)
a = x z i + z j + y k , S :
j
x2
+ y2 + z2 =4,
x 1 + y2 = z 2(z>
0).
x2
+ y 2 + z 2
= 1,
z = 0(z £ 0).
x 2 + y 7 = l - z ,
z = 0.
„
(x+ y + z
=
2,x = 1,
'
}
’
|x = 0,>' = 0,z = 0.
.2.- , ..2 .• ,
2
,
c . [
z = x 2 + y 2 + z 2,
7)
a = x i + y j + z k, S \
I
z =
0(z 2: 0).
8)
a = x 3i + y * j + z 3k, S : x 2 + y 2 + z 2 =1.
9)
a = (zx + y)i + ( z y - x) j + (x * + y 2)k
s
\x2 + y 2 + z2 =\,
’ > = <
0(z £ 0).
10)
a
=
y*xi + z 2y j + x *zky S : x 2 + y 2 + z 2 =\
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10- Misol.
F kuchning
L
chiziq bo‘ylab A/nuqtadan
N
nuqtaga ko‘chishdagi ishini
toping
1)
F = (x2 - 2y)i + ( y 2
-
2x)j, L : MN,M(-4,0),N(0,2).
2)
F
= (xJ + 2
y)i + ( y 2
+
2x) j , L : MN,
A/(-4,0), JV(0,2).
3)
F = (*J +
2y)i + ( y2 +2x)j, L :
2 - ^ - = y,M(-4,0), AT(0,2).
O
4)
F = (x + y)i + 2xj, L : x 2 + y2 =4(y>0),M(2,0),N(-2,0).
5)
F
= x3/ -
y 3j , L : x 2 + y 2 =
4(x > 0,> > 0),
M(
2,0),
N(0,2).
6)
F = (x + y)i + ( x - y ) j , L : y = x 2,M(-l,l),N(\,\).
7)
F = x 2y i - y j , L : MN,M(-l,0),N(0,l).
8)
F = (2x - y)i + (x2 + x)j, L : x 2 + y 2 = 9 (yZ 0),M(3,0),N(-3,0).
2
9)
F = (x + y)i + ( x - y)j, L : x 2 + ^ - = l(x>0,y> 0), M(\,0), N(
0,3).
10)
F = y i - x j , L : x 2 + y 2 =\(y>0),M(\,0),N(-l,0).
11- Misol.
Maydoning yopiq kontur bo‘yicha sirkulyatsiyasini toping
2)
a = - x 2y }i
+
j + zk, F :
1)
a = y i - x j + z k, r :
'
V2
V2
x
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