1、 E= gv 3 d ) )C3(r xdx ydy zdz+ 22rxyz2= +. =a ! h f O ()fx (0, )+ () 0fx na l . 11() ()nnnkafkfx=dxa ! o S222122xyz+= ( 0z ) PS ) M S P (, ,)x yz p3(, ,)SI zxyzdS=. a !B f H = O ()ufr= 0 r 11(1) ( )nnfn=l 711()nfn=? . B1* 7 S/ O T/ .B V 8 7 S 9 - 2 10 - 3 .L l H 8 H 7 S/$ / 5 Y F S a ! uW , = O()fx
2、 ab () () 0fa fb= = (,)x ab H () 0fx() 4()bafxdxf xba0. a ! L “ T O 1()nnfx ax ax a=+null 2n 0(1 1)kakn= # ik H Ms L50ia ()fx n110kkaa+ ()yfx= B. 50(,3)x = . 5 !11xyx+=5(10)0xy= . 64441(1 ) 1dxxx+= + . C7 ! B O()fx (0) 0, (0) 1ff=520020()lim()xxxf tdtf tdt= . 8 ! u 52221xyz+222222(xyzdvabc+)= . 9 Vf
3、 i (, )fxy ,xyt2(, ) (,)f tx ty t f x y= 0(1, 2, 2)P w BO 5 w ) M Z (, )zfxy=(1, 2) 4xf =0P . 10 ! (1 1mna )nmx+nx “ 501nna= . =a ! = O()fx 0,1 (1) (0) (1) (0) 0ff f f = = i (0,1) P() ()ff = . a !40tan , 1nnaxdn= 1 na l 2 21,1nnaa nn+= 2 3 112( 1) 2( 1)nannu 5D221xy+2222()Dxydab+= . 5 ! i O()fx (u f
4、 xyz= )3(),uFt t xyzxyz=5 = ()Ft . 6 ! 0 50xxee dx+= . 7 ! = V O()yfx= (4 ) ( 0)dyyydx= ()yfx= B. 50(,3)x = . 84441(1 ) 1dxxx+= + . C9 ! B O()fx (0) 0, (0) 1ff=520020()lim()xxxf tdtf tdt= . 10 ! (1 1mna )nmx+nx “ 501nna= . =a ! = O()fx 0,1 (1) (0) (1) (0) 0ff f f = = i (0,1) P() ()ff = . a 3 P sY9F
5、x y sY9 F x y g Hg r9 l |9F 34x y+ C Ag 3 s g ? Pg r9 l 9 Kv$ a !40tan , 1nnaxdn= 1 na l 2 21,1nnaa nn+= 2 3 112( 1) 2( 1)nann 1, 1ab . 2 f ) V O 5()fx 1x = (1) 1f =0(1 ) (1 2 sin ) 2 (1 3 tan )limxf xf x f xx+ + = . 3 ! s222(1)( 1)xaxdxxx+TcQ Mf 5 = a . 421limkn nknknenne=+= . 5f 22uxyz=+2 2 ) w (1
6、,1,1)M22zx y= + M ) ELZ _ Z _ nnullMun= . 62111yxdx e dy= . 7 ! O()fx (0) 0f = 222() ( )tF t z f x y dxdydz=+ 222:,0t1x yt z + 520()limtF tt+= . 8 ! 0 H22()Cydx xdyI2x yxy=+null _C2221xy+=n k l5= n . 9 ! w 52224xyz+=22()x ydS+= . 1012!( 1)!( 2)!kkkk k=+= . =a ! uW 0 = O 7 uW =rKl ()fx , a (0, )a ()
7、( 0,)f xMx a (0) ( )f fa Ma+. a 222200sin cos11x xdx dxxx+. a ! f 9 ()fu11() ()xxI f dydz f dzdx zdxdyyy x y=+ 8 . 22 26, 8yx z y x z=+ =2aX2(, )2(0,0)(, ) costtf x y dx x ydy t+= B p . (, )fxy (, )fxyBaXZ H L sZ k p V f iNsZ . 22 3(6 ) (8 ) 0yxydx xxydy+=3()yfx()fxa BO ff wLBH Q L . m Xff f“ d O V
8、f S “ u I n f Hs2311aadxx+Kv . 9 2220limxxtxete dtx+= . 10222() (0 )x Ryr rR+= uW 0 = O 7 uW =rKl ()fx , a (0, )a () ( 0,)f xMx a (0) ( )f fa Ma+. Ba ! 01 121, 1, ( 2)nn nFFFFFn=+ 1 113() 22nnnF 2) 01nnF=21lnnnF= l $ I $ a ! u3( , ) 1, 1 1, ( )D xy x y x f x= l ,( 1)aa a i f k p . 2( 1) () ( 1) ( )D
9、yx fx x f xdxdy+ E= gv 3 SJaYF k5s BaA b5 1. 4 2. sin sinx y+ 3. 4. 0 6 5. 2e1(e 1)6 6. 0yy = 7. 1 8. 0 9. 10. 2sin cosxx x551()5ba . =a h f # ()fx (0, )+111( 1) ( )d ( 1) ( )dnnnnnnaafn fxx fn fx+= + = +xV7 na h . i n1111() ()d () () ()d 0nknkkka fk fx x fn fk fx x+= =+ na / .8 AKV na l . a ! M ,X
10、YZ 5 Z 122xX yYzZ+=V72221(, ,)22xyzxyz = + ZS22122x yz =222(, ,)4xyzx y =#222122zxxx y=222122zyyx y= 2222224d1 dd dd2122xyzzSxyxyxy=+ + =xy 2222 2 2400(1 )d d d (1 ) 2 ( )22 2 28Dx yrIxyrd= = = r. a ()uxf rx r=#222() ()uxrfr frxr =+23x. V 22223y() ()uyrfr fryr =+22223() ()uz rzfr frzr =+. Z 2() () 0
11、fr frr +=. VsZ# 7 () ()fr pr = 20rp p+ = . 12Cpr= 512()Cfrr = s12()Cfr Cr= SHq1() 1frr= . a oUS5 ( , , ) ( sin cos , sin sin , cos )ufxyz fr r r = = . 7 yx zff ftxyz =5 ,xyzf tx f ty f tz= = ( sin )sin cos sin ( sin )sin cos sin ( ) 0xyufr fr txr tyr txyxy = + = + =+ = 22 2cos cos sin cos sin (cos s
12、in cos sin sin cos sin cos )xyzufr fr fr tr = + = + =. 2(sin cos sin cos ) 0tr =# u rf . Ba 0()limxfxax= (0) 0, (0)ffa=. ()fx 0x = # =B #i(0) 0fa = 0 P (0, ) () 0fx . i PN11,() 0,( ) ()1nNf f fnn 11 1() () (0) () ,0ffffnn n= 1n1()lim (0) 01nfnfan= =.#11()nfn=? . a ! V UV 7 S/ H Y t H 5/ ()htddhCt= .
13、! ()x t V 7 S/ H Y tB V ! V 7 51,kkaa+1 110, 0kkaa+02,0CC(1)()kf x 0x = PZhZ97 l . () ( 1)0(0) 0, (0) 0kkffC=()()kf x 5) y7(1) (1)0() (0) 0kkfxf C=(1)()kf x9 L 0x ()()kf x 0x = M # 5 00x W uW (1)0() 0kfxC(1)()kf xN uW iM E gv 3 SJaYF k5s BaA b5 1. 2 2. 12 3. 38 4. 3 = 5. 1039 17!2 6. 441xx+ 7. 2 8.
14、2224111()15 abc + 9. 10. 42xz=01mm. = aT f 5 0 1 : Hq O 5: i() ( () ()xFx fx f x e=+ () ( () ()xFx f x f x e= ()Fx() ( () ()xFx f x fxe= () ( () ()xFx fx f xe=(0,1) P () 0F = () ()ff = . a 1 04x+ =+#112( 1) 2( 1)nann+. a ! u g xoz221xz+ T 11 () () ()dddxxI ffzxy y yx y z=+xyz =2211 ( ) ( ) 1ddd dddx
15、xf f xyz xyzyyyy+= US V 2221800 6dddrrIryr +=. a cos ( , )yyfxy= (, ) sin ()fxy y Cx= + 7 2(, )2(0,0)sin ( )d cos dttyCx xx yyt+=2200()d cos dttCx x t yy t+ = 20()d sintCx x t t t+=2 22 H p 22() 2 sin 2 cosCt t t t t= 22( , ) sin 2 sin 2 cosf xy y x x x x=+ . Ba ! 425 334(, ) (6 ) (), (, ) (8 ) ()Px
16、y y xy f x Qxy xy xy f x=+ = +QPx y = 324 334 324(8 3 ) ( ) (8 ) ( ) (24 5 ) ( )yxyfx xyxyfx yxyfx+=+h 3y2316 ( ) 8 ( ) 2 ( ) ( ) 0fx xf x y xfx xf x+ =. 21d() 2() 2 (), d, ()()fxxfx fx xfx Cxfx x =0= O sZ 425 2 334 211(6 )d(8 )dxxyCxx xyxyCxy+ . (,)4252 3342(0,0)(, ) (6 ) d (8 ) dxyuxy x xyxx xy xy
17、xy=+ +=3342 34 50010d (8 ) d 25yx5x xy xy x y xy xy+ =+. #sZY34 5510x yxyC+=. a 1 ! LnullOA Z 32yax bx cxd= +5 #(0) 0, (0) 0yy=00() ,()0yx hy x=2200230, ,hhcd a bx x= = = nullOAZ32 322 200 0023 2(3hh hyxx xxx xx= + = 2)x. 2ynF2112 1 2 122 2 2nnn n n nFF F F F F =+ =null1nn 6BZ 122nFF 2112nnF Fy711 1
18、 2 113 3 3 3() () ()22 2 2 2nnn n n nFF F F F F1n + = =null 113() 22nnnF . 21112()3nnF Vn01nnF=l 711ln ( 1) ln 2nFn Vn01lnnnF=? . E gv 3 d O (0) (0) 1 =,5(0)x = . :A .y | Z , V3 0t = (0) 0x = .Z H t p , cos ( ( ) ( ) ( ) 0txtxt t +=, ,NZ H t p ,(0) 2x =2sin ( () () ( () () () 0t xtx t xtxt t +=, V.
19、(0) 3x =7 ! , ,5sin 0 2()0 xxfx= D () ( )Df x f y x dxdy= . :A .y u2(1 cos 2) () ( )fxfy x1:2,0Dx y x x2 + =d , , () ( )Df x f y x dxdy= . 12220()() sinsin()(1cos2xDxfxfy xdxdy dx x y xdy+= = )j8 !_ , O= V sf )34, 43ui jvi= =+nullnullnullnullnullnull(, )fxy P 10, 17PPffuv=5Pdf = . :A 10 .y15dx dy+34
20、655Pfffuxy=,431755Pfffvxy= +=, 10Pfx=, 15Pfy=. Pdf =10 . 15dx dy+9 ) 11(1 ln(1 )nnn=+x l . :A 1 .y H,1) 0x ln(1 ) 112xxx+ . 112( 1)lim lim 1, 112nnnna nran+ +=. 1x = H) l , 1x = H) ? . 10 ! df ,5() ( ) 1( )fxf x x = +) . : 71xt= ,022 20ln ln1()xtI dx dtxa at+=+, 7uata= ,0 222200ln ln ln lnln1() 1()t
21、tuaI dt dt du I aat at a u a+ += =+ +, ln2I aa= . a (10s )XZ logbax x= i L , , p Hq . 1, 0ab ,ab : ! () logbaf xxx=,1ln()lnbbx afxx a = ,101lnbxba=. 00 x x ()fx0xx uW 0 O 1( :()fx ,1 )fx31100141()() 3f x dx dxfx . : O - % T211 100 011() () 1() ()f x dx dx f x dxfx fx= , ( ,#)1)()3) 0fx fx()1)()3)0()
22、fx fxfx , 1033() 4 () 4() ()fx fx dxfx fx+ + . T2()4abab+ , 2111100003()()1() 4() 4fxdx dxfxfxdx dxfx+ , 8 , 1100141()() 3f x dx dxfx. a ! 222:1xyz+3342 82533xy z dv +. : ! ,(, ,) 2 2 5fxyz x y z=+ + 10, 20, 20xy zff f = =, f u(, ,)fxyz = ,AH |K . # 7 , 222(, ,) 2 2 5 ( 1)fxyz x y z x y z=+ + + + 22
23、21 2 0, 2 2 0, 2 2 0, 1 0xy zFxF xF xFxyz =+ = =+ = =+ = = + + =1212 2 1 22(, , ), ( , ,)33 3 3 33PP 7 , f u12() 8,() 2fP fP= (, ,)fxyz Kv 8,Kl 2. (, ,)fxyz3(, ,)f xyzM , f 3(, ,)f xyz u K v 2,Kl 32 . 33342 8225233dv x y z dv dv =+= . E gv 3 d ,5dydx= . :A . 22sinyexZ H p :x22sin0yxxeyx+=, . 22sinyy
24、e= x3X) 111(sin)nnn=l ,5 |S . :A13 .y3111sin6nn n, 13 . 4 , 0()1xxe e= 1)x D ), ,5 =s () ( )Df x f y x dxdy= . :A .y24a () ( )fxfy x u1:2,0Dx y x x2 + =d , , () ( )Df x f y x dxdy=1122() ( ) 4DDf x f y x dxdy a dxdy a= = . 33601limsin 2xxexx8 = . :A1128.y T =sin xx ,3232266 5 70011331lim lim(2 ) 2 6
25、 2 128xxex xex = 1=9 33(, ) 3f xy x y xy=+ l . :A , l1 . (0,0) (1,1) . (0,0) (1,1) 1 . 10 ! 5121aa=,21,1nnnaaa+=+ ,12nnna= . :A 2. y 12211 11 1 111 1 122 2 2 2 2 2nn n nnn n naa a ass += = =+=+ + =+ =+ =122=1112( )224s+ N=a ! ) = 2s = . ()fx (, + () ( )fx fx x=+, O 0, ()xf xe= , p32()f xdx. 2 ,3 x H
26、 , 20,x , :2() ( ) ( 2) ( ) 2xfx fx x fx x x e x =+=+= +, 5 , 32()f xdx=3222()2) 4xexdxe+ =+ . 1aX2200,sin()ttsx edsy tsds=, p22dydx. :2tdxedt= ; 02200sin( ) sin sintttsutytsd2uduudu= =, s222sin , sintdy dytte;dt dx=22dydx=22 22(sin ) 2 (sin cos )tt tte e te t t =+. 2( ) , q (h 2000 12000 , 3?(K ZE
27、). : pf a ! 3JaY , sY xq y q , f 22+ (, ) 6 16 4 2Lxy x x y y=X 3 H ,C q H9 Kv ?Kv 22(, ) 6 16 4 2Lxy x x y y=+ 06xy, p Hq . : !,ab1, 010() logbaf xxx,=1ln()lnbbx afxx a = , 1lnbxba=. 00 x x , ()fx0xx 1x 2(2 2 ) 64bc= 6, 24, 8ab c= =. Ba (10 s ) ! w 222222:1xyzabc+= (, ,)x yz) M 9 w s1dS, US . : M Z222:1xX yY zZabc+=,#2244412x yzabc =+, ! s 1 0z xyD1 xoy g5 1dS=12224442xyzdSabc+=22 22442 2222221(1 )21xyDxy xyabc abcdxyab+ xd7221342222 200222cos311xyDxab r badxdy d draaxy rab=
