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../.. , / 0

../.

0

( 00)

: : 100

(10 2 = 20 )

1. (((( ))))2 2 24 4 xD D y x e + = + = + = + = .

2. (((( ))))2 2

14 2x D xD y x

x+ + = ++ + = ++ + = ++ + = +

.

3. 2 2

( , , )y z xy yz = += += += + (((( ))))2, 1,1

2 3i j k+ ++ ++ ++ + .

4. .v

F dV 2 2 2F x i y j z k= + += + += + += + +

V

0 , , 1x y z .

5. , Re( ), Im( )z z z ? .

6. .

7. , (((( )))) (((( ))))

2 2sin cos

1 2C

z zdz

z z

++++

+ ++ ++ ++ + , C 2z

==== .

8.

1

2

( ) zef z e==== .

9.

20,3

( )2 2

cos ,3 3

t

f t

t t

.

10. ( ) 3 t

f t e ==== .

( 0 )

11. () () (((( ))))2 24 3 6 sin sin 2xD D y e x x+ + =+ + =+ + =+ + = .

() , (((( ))))22 3 25 xD D y e = = = = .

() () 2

2 2

23 4 ln

d y dyx x y x x

dx dx + = + = + = + = .

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() 2 3dx

x y tdt

+ =+ =+ =+ = 2

3 2 tdy

x y edt

+ = + = + = + = .

12. () ()

()

( )r , r xi yj zk= + += + += + += + +

, ( )f r r

. (6)

( )F y z i yzj xzk= + = + = + = +

, S 0, 0, 0, 1, 1, 1x y z x y z= = = = = == = = = = == = = = = == = = = = = C

xoy . (10)

() () F

, (((( )))) (((( ))))1 1 2curl curlF F F = = = =

iiii . (6)

() (((( )))) (((( )))) (((( ))))2 2 2F x yz i y zx j z xy k= + + = + + = + + = + +

0 1,0 1x y

0 1z

. (10)

13. () () ( )f z z ,

2 22 22

2 2( ) ( ) ( )

p pf z p f z f z

x y

+ =+ =+ =+ =

.

()

1w

z==== , ,

.

() ()

1u c==== 2v c==== ,

2zw e==== .

() 1 , ,2i i i+ + + + z

0,1, i w

14.

() ()

(((( ))))

1( )

1f z

z z====

1 1, 1 1 2z z+ < < + + > .

()

(((( ))))

2

0

0cos

d

a ba b

> >> >> >> >

++++ , .

() ()

23 7 1

( )C

z zf a dz

z a

+ ++ ++ ++ +====

, C 2z ==== ,

(3), (1 )f f i ++++ (1 )f i ++++ .

(6)

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() (((( )))) (((( ))))2 2 2 2

cosx dx

x a x b

+ ++ ++ ++ +

, 0a b> >> >> >> > . (10)

15. () ()

, 0 1( )

0, 1 2

t tf t

t

<