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1、?12.1?1?1?2?62456yyxxz?)ln(222yxxz?3?yxxyz?4?)(cos)sin(2xyxyz?5?6?)sin(coseyxyzx?()$%&?yxz2tan?7?xyyxzcossin2?8?yxyz)1(?9?10?)lnln(yxz?xyyxz?1arctan?11?12?)(222ezyxxu?zyxu?;?13?2221zyxu?14?zyxu?15?16?1niiiua?xiajiijnjijiijaayxau?,1,?(1)234245yxxxz?II?yxyyz45126?II?(2)223222)ln(2yxxyxxxz?II?2222yxyxy
2、z?II?(3)yyxz1?II?2yxxyz?II?(4)*)2sin()cos(xyxyyxz?II?*)2sin()cos(xyxyxyz?II?(5)sinsin(cosyyxyexzx?II?)sincos(yyxeyzx?II?(6)()$%&?IIyxyxxz22sec2?()$%&?IIyxyxyz2222sec?(7)xyyxyxzcoscos1?IIxyyxxysinsin2?xyyxyxyzcoscos2?IIxyyxxsinsin1?1(8)12)1(?IIyxyyxz?!#+,-?IIxyxyxyxyyzy1)1ln()1(?(9)yxxzln1?II?)ln(1yx
3、yyz?II?(10)?arctanarctanzxy?211xxz?II?211yyz?II?(11)3(222zyxxu?II)(222zyxxe?IIyuxy2)(222zyxxe?IIzuxz2)(222zyxxe?(12)1?IIzyxzyxu?IIyuzyxzxln?IIzuzyxzxy2ln?(13)?23222zyxxxu?II?IIyu?23222zyxy?IIzu?23222zyxz?(14)1?IIzyzxyxu?IIyuxxzyzyzln1?IIzuyxxyzyzlnln?(15)niaxuii,2,1,?II?(16)niyaxunjjiji,2,1,1?II?njx
4、ayuniiijj,2,1,1?II?2.?22),(yxyxyxf?)4,3(xf)4,3(yf?22221,1xyxyffxyx?y?52)4,3(?xf?51)4,3(?yf?3.?2eyxz?02?II?IIyzyxzx?22231e,exxyyzzxyyyII?II2x?02?II?IIyzyxzx?24.?4,422yyxz?)5,4,2(x?x?)5,4,2(2(,)(1,0,1xdx dy dzdx dx dx?)?x8?(1,0,1)1cos(1,0,0)228?2?4?8?5.?1?223),(xyyxyxf?)2,1(?2?)1ln(),(22yxyxf?)4,2(?3?
5、2sin),(yxyxf?)1,0()$%&2,4?(1)?22(,)(6)(32)df x yxyydxxxy dy?dydxdf?8)2,1(?(2)?222222(,)11xydf x ydxdyxyxy?dydxdf218214)4,2(?(3)?23cos2sin(,)xxdf x ydxdyyy?dxdf?)1,0(,dydxdf8282)2,4(?6.?1?2?xyz?xyxyze?3?yxyxz?4?22yxyz?5?222zyxu?6?)ln(222zyxu?(1)?dyxyydxydzxx1ln?(2)?)(1(xdyydxxyedzxy?3(3)dyyxxdxyxydz2
6、2)(2)(2?(4)dxyxxydz2322)(?dyyxx23222)(?(5)222zyxzdzydyxdxdu?(6)222)(2zyxzdzydyxdxdu?7.?yxz2e?)0,1(P)0,1(P)1,2(?Q?12(2,1)(1,0)1(1,1)(,)|(2,1)(1,0)|2PQv vPQ?v?22e,2 eyyzzxxyII?II?1212zzzvvxyIII?IIIv?8.?22yxyxz?)1,1()sin,(cos?v?1?2?3?cossin(2)cos(2)sinzzzxyyxxy?III?IIIv?(1,1)cossinz?I?Ivsin()sin2sincos
7、()244?,(1)?4?)4sin,?4(2)?54?)45sin,5?(3)?37,44?)43sin,3?)47sin,7?9?2?2?),(yxf)2,1()2,1()2,2()2,1()1,1(?1?)2,1(?2?)2,1()2,1()6,4(?1v=(2,2)(1,2)(1,0)?110zzzzxyx2IIII?2?2?IIIIv?2v=(1,1)(1,2)(0,1)?20(1)zzzzxyy2IIII?2?2?IIIIv?)2,1(2zzxyII?II?(1)?)2,2()2,1(grad?f(2)?(4?,6)(1,2)(3,4)?22(3,4)(3,4)534?v?(1,2
8、)3412255f I45?2?2?I?10.?1?2?)sin(22xyyxz?()$%&?22221byaxz?3?zyxyzxyzyxu5264332222?)1,1,1(?(1)?)cos()sin(2),cos(2(grad23xyxyxyyxyyxz?(2)2,2(grad22byaxz?(3)?grad(236,4342,645uxyyxzzy?)5,9,11()1,1,1(grad?u11.?xyyxf?),(5?,?)0,0(),(?yx),(gradxyf?;?,?)0,0()sin,(cos?v?cos,sinxtyt?:?:?221(,)(0,0)cossinsin22
9、fxyfx ytt?:?:?3,44?)1,1()1,1(?12?3),(xyyxf?)0,0(ieie?2,1?i?3(,)(0,0)(,)(0,0)lim(,)lim0(0,0)x yx yf x yxyf?3000(0,0)lim0 xxxfx:?:2?:?3000(0,0)lim0yyyfy:?2:?:?)0,0()sin,(cos?v?0(0cos,0sin)(0,0)limtffttft?I?Iv30cossinlimtttt?2?330sin2lim2tt?sin20?2k?sin20?2k?ieie?2,1?i?13.?0,0,0,),(222222yxyxyxxyyxf 6?
10、)0,0(?22220(,)(0,0)xyxyx yxy?(,)(0,0)lim(,)x yf x y?22(,)(0,0)lim0(0,0)x yxyfxy?20000(0,0)lim0 xxxxfx:?:2?:?:?20000(0,0)lim0yyyyfy:?2:?:?:?)0,0(0,0)(0,0)(0,0)(0,0)xyfxyffxf?:?:?:?:y=(,)fxy:?22x yxy:?:22()oxy?:?:?)0,0(14.?0,0,0,1sin)(),(22222222yxyxyxyxyxf?),(),(yxfyxfyx)0,0(?222201(0)sin00(0,0)lim0
11、xxxxfx:?:?:?:?(,)(0,0)x y?22222222121(,)2 sincos,0 xxfx yxxyxyxyxy?700lim(,)limxxxfx y?x y?22111(2 sincos)222xxxx?(,)xfx y?)0,0(,)yfx y?)0,0(0,0)(0,0)(0,0)(0,0)xyfxyffxf?:?:?:?:y=22221()sinxyxy?22()oxy?:?:?)0,0(15.?0,0,0,2),(2222422yxyxyxxyyxf?)0,0(?)sin,(cos?v?0(0cos,0sin)(0,0)limtffttft?I?Iv232422
12、02cossinlim0,(cossin)tttt?2?2/?)0,0(,)x y?2xky?)0,0(2424420022lim(,)lim1yyx kykykf x yk yyk?k?16?1?xyzarctan?22222,yzyxzxzIIIIIII?2?)cos()sin(yxyyxxz?22222,yzyxzxzIIIIIII?3?xyxze?2323,yxzyxzIIIIII?8?4?)ln(czbyaxu?22444,yxzxuIIIII?5?qpbyaxz)()(?qpqpyxzIII?6?zyxxyzu?erqprqpzyxuIIII?(1)?2222y11zyxxxyyx
13、I&)?$I?%(&)?$%(222111zxyxxyyxI&)?$I?%(&)?$%(?22222)(2yxxyxz?II,?IIIyxz222222)(yxxy?,?II22yz222)(2yxxy?(2)?(1)sin()cos()zyxyxxyxI?I?(1)cos()sin()zxxyyxyyI?I?)sin()cos()2(22yxxyxyxz?II,?IIIyxz2)sin()1()cos()1(yxxyxy?,?II22yz)sin()2()cos(yxxyxy?(3)?2exyzxyI?I?232exyzxyI?I?IIIyxz22(2)xyxx y e?xyeyxxyyxz
14、)42(2223?III,xyeyxxyxz)3(3223?III?9(4)?1()uaxbyczaxaxbyczxaxbyczII?I?I?2222()()()uaaxbycza2xaxbyczxaxbyczII?I?I?323332()2()()uaaxbycza3xaxbyczxaxbyczII?I?I?43443 2()6()()uaaxbycza44xaxbyczxaxbyczI2I?I?I?3322uuxyy xII?III I=22332()2()()aaxbycza baxbyczyaxbyczI?I?442222222443 2()6()()uua baxbycza bxyy
15、xaxbyczyaxbyczII2I?I III?I?(5)()()p qpqpqqppqpqpqzzyxaxyxyxy?&)&IIIII?$IIIIII%(%b)()()ppqqpqdxadybdxdy?!p q=?(6)?x?y?z?Leibniz?()()()()()(p q rpxqyrzpxqyrzpqrpqrpqruxeyezedxedyedz)exyzxyzdxdydz?IIII?IIIIII?=()()()xyzxp eyq ezr e?2?2?=()()()x y zxpyq zr e?17?1?)ln(xyxz?z2d?2?)(sin2byaxz?z3d?3?)(e222z
16、yxuzyx?u3d?4?yzxsine?zkd?(1)(ln()1)xdzxydxdyy?10222221dyyxdxdyydxxzd?(2)?2sin()cos()()sin2()()dzaxbyaxby d axbyaxby adxbdy?222cos2()()d zaxby adxbdy?33)(2sin4bdyadxbyaxzd?(3)?222()()(222)x y zduexyzdxdydzxdxydyzdz?22222()()2(222)()x y zd uexyzdxdydzxdxydyzdz dxdydz?22222dxdydz?2()()x y zexyzdxdydz?3
17、d u=222326()()xdxydyzdz dxdydz?2226()()dxdydzdxdydz?2223dx3222)66(dyyzyx?(66)x y zexyzx?dz)66(3222dzzzyx?+dydxyxzyxezyx2222)224(3?dzdyzyzyx2222)224(?dxdzxzzyx2222)224(?2222)242(dxdyyxzyx?2222)242(dydzzyzyx?)242(2222dzdxxzzyx?2226(222)x y zexyzxyz dxdy?(4)()kkd zdxdyxyII?II0sinixk ikikik iikeydxdyixy
18、?i?&)II?2$II%(?0sin2kxiikkieydx dyi?k i?&)?&)?$%(%(?18?),(yxfz?xyyxz?II11sin?3sin2),0(yyyf?),(yxf?x?111(,)sinln(1)()f x yxyxyg yy?,?3sin2),0(yyyf?3()2sing yyy?3)1ln(1sin)2(),(yxyyyxyxf?19?1?)sin(e2nyzxkn?22yzkxzII?II?2?yxzuarctan?Laplace?0222222?II?II?IIzuyuxu?1?2cos()kn xznenyy?I?I?2222sin()kn xzn
19、enyy?I?I?zxI?I22sin()kn xkn eny?=22zkyII?2?uxI?I22111yzzyxyxy22?2?&)?$%(?22uxI?I222()2xyzxy?uyI?I222211xxzzyxyxy&)22?$?&)%(?$%(22uyI?I222()2xyzxy?uxI?Iarctanxy?220uzI?I?0222222?II?II?IIzuyuxu?20?trttrf42e),(?f()$%&IIII?IIrfrrrtf221?12ftI?I212241()4rtttre?frI?I21412rttre?2frrrII&)$II%(=21224431()24rt
20、trtre?23?21?1?T),sin,cos()(cxxbxax?f4?x?2?T23)tan,cot3(),(zyxzexzyxy?f()$%&4,2,1?3?T),sin,cos(),(vvuvuvu?g),1(?(1)()(sin,cos,)Txax bx c?f,()4?T),22,22(cba?(2)2223cotcsc(,)32 tansecyyezezx y z2xyzyz&)?$%(f?(1,2,)4?2232348ee&)?$%(?(3)?cossin(,)sincos01vuvu vvuv?&)$?$%(g),1(?()$%&?101001?22?33:RR?f?1?z
21、zyxfyzyxfxzyxf?),(,),(,),(321?f?2?f?3?f)(),(),(diag(zryqxp?1?1(,)(1,0,0)fx y z?2(,)(0,1,0)fx y z?3(,)(0,0,1)fx y z?13?f(2)?1(,)(1,0,0)fx y z?1(,)f x y z?y?z?11(,)f x y zxC?22(,)fx y zyC?33(,)fx y zzC?(3)?1(,)(),0,0)fx y zp x?1(,)f x y z?y?z?1(,)()f x y zp x dx?F?2(,)()fx y zq y dy?F?3(,)()fx y zr z dz?F?14