dwihnznl dhlewtd l"hn - oeipkhd g"qyz sxeg 104014 'z `"ecg 10..008 '` cren ziteq dpiga oexzt [10%] :1 dl`y.(0, 0) dcewpd zaiaqa zeneqg ody zeiwlg zexfbp zlra f(x, y) idz.(0, 0) dcewpa dtivx f ik gked.lim f(x, y) = f(0, 0) ik ze`xdl jixv zetivx gikedl ick x 0 y 0.lim f(x, y) f(0, 0) = 0 ik gikep :odly minqgd z` M 1, M -a onqpe,zeneqg ody zeiwlg zexfbp zlra f(x, y) oezpd itl x 0 y 0 f x (x, y) M 1, f y (x, y) M.(zeneqge zeniiw zeiwlgd zexfbpd day daiaq dze`) (0, 0) ly daiaqa dcewp (x, y) idz,okae f(x, y) f(0, 0) = f(x, y) f(0, y) + f(0, y) f(0, 0) f(x, y) f(0, y) + f(0, y) f(0, 0) = - yleynd oeieey-i`a ynzyp :('fpxbl zgqep) miipiad jxr htyn ynzyp dzr,dxifb divwpet `id f(t, y), f(0, t) zeivwpetd on zg` lk okl,zeiwlg zexfbp zlra f oezpd itl :mitwz htynd i`pz okle,(t cigid dpzynd ly) dtivx okle = f x ( x, y)x + f y (0, ȳ)y zeiwlgd zexfbpd zeniqga ynzyp.daiaqd jeza miipia zecewp od x, ȳ xy`k M 1 x + M y. 0 f(x, y) f(0, 0) M 1 x + M y.'uieecpqd llk itl lim x 0 y 0 mekiql f(x, y) f(0, 0) = 0 okl.l"yn 1
.(1, 0) dcewpd on xzeia zewegxe xzeia zeaexwy x + y 9 4 [15%] : dl`y = 1 dqtil`d lr zecewpd z` `vn.g(x, y) = x 9 + y 4 1 onqp. (x 1) + y i"r oezp (1, 0) dcewpd on (x, y) dcewp ly wgxnd reaixd zivwpet ik) wgxnd reaixl menxhqw` z`ivnl dlewy ilnxhqw` wgxn z`ivn divwpetd ly oeviwd zecewp z` `evnl witqn okle,(ynn dler zipehepen u(t) = t f(x, y) = (x 1) + y g(x, y) = 0. ueli`l setka g m`d wecap.(minepilet od ik) R lka 1 od f, g zeivwpetd.'fpxbl iltek zhiya ynzyp ( ) :qt`zdl leki x g = 9, y. 4.ueli`d z` miiwn eppi` x = y = 0 la`,x = y = 0 xear wx qt`zn g.miniiwzn 'fpxbl iltek htyn i`pze qt`zn `l hp`icxbd xnelk ( ) x F = f(x, y) + λg(x, y) = (x 1) + y + λ 9 + y 4 1 F x = (x 1) + λx 9 = 0 F y = y + λy 4 = y ( 1 + λ 4 'fpxbl zivwpet xicbp,laewnk.dly zeihixwd zecewpd z` `vnpe ) = 0 - zeiexyt` izy opyiy raep F y -l d`eeynd on,x = ±3 raep ueli`d on f`e y = 0-y e`.y = ± 8 5 lawp ueli`d one x = 9 5 raep F x xear d`eeynd on.λ = 4 gxkda f`e y 0-y e` f(3, 0) = 4, f( 3, 0) = 16, f md el` zecewpa f ikxr.zeihixw zecewp rax` ep`vn ( ) 9 5, ±8 = 16 5 5 zlawn (dtivx `idy) f divwpetd,q`xhyxiiee htyn itl.dneqge dxebq dveaw `id dqtil`d zeihixwd zecewpd z` xewql witqn okl,dly meniqwnde menipind ikxr z` dqtil`d lr izya lawzn ilnipind wgxnde,( 3, 0) dcewpa lawzn ilniqwnd wgxnd.( 9, ± ) 8 5 5.lirl epnyxy :ziteq daeyz zecewpd
:3 dl`y zniiwny,r lka dtivx divwpet g(t) idz 0 g(t) dt = 1, / 0 g(t) dt = g(t) dt = 3, / / g(t) dt = 4 x y= 1 Y. = { (x, y) x + y 1 } megzd idi.(df sirqa wnpl jxev oi`) megzd ly dviwq xiiv [1%] (`). g(x + y) dxdy ayg [9%] (a) x+y=1 x+y= 1 x y=1 X (`) (a) i"r lewy ote`a megzd z` aezkl ozip = { (x, y) 1 x + y 1, x y 1 } :zecgein zehpicxe`ewl xearl rivp u = x + y, v = x y mpn`e,qt`zn `l dly o`iaewridy i`pza r"gg recike 1 `idy zix`pil dwzrd idef (u, v) (x, y) = 1 1 1 =..letkd lxbhpi`a mipzyn iepiyl ef dwzrda ynzydl lkep jkitl g(x + y) dxdy = g(u) (x, y) (u, v) dudv i"r oezp megzd xy`a = { (u, v) 1 u 1, v 1 }. (x, y) (u, v) = (u, v) (x, y) = + 1 `ed lxbhpi`a yexcd o`iaewrid R lka 1 `id zix`pild dwzrdd ik xekfp) zeketdd zeivwpetd htyna epynzyd xy`k g(x + y) dxdy = 1.(mitwz htynd i`pz okl,qt`zn `l dly o`iaewride g(u) dudv = 1 dv g(u) du = 1 3 = 3. - mekiql 3
( ) ( ) y x F (x, y, z) = x + y + cos y î + x + y x sin y ĵ + z ˆk,y + z = 0 xeyind mr x + z = 1 lilbd ly jezigd mewr `ed xy`k [15%] :4 dl`y ixehweed dcyd oezp.y xiv ly iaeigd oeeikd on lelqnd lr milkzqn xy`k oeryd ibegn oeeika F d r ayg xy`a, F = G + H dxeva oezpd dcyd z` meyxp G(x, y, z) = y x + y î + x x + y ĵ H(x, y, z) = cos y î x sin y ĵ + z ˆk ik) ely l`ivphet zivwpet `id U(x, y, z) = x cos y + 1 3 z3 ik,r 3 lka xnyn dcy `ed H dcyd.( U = H zniiwn xy` R 3 lka 1 divwpet idef H d r = 0 okl. xebq lelqn lkl z` siwn xy` xebq lelqn lkl G d r = π miiwn xy` "mqxetnd dcyd" `ed G dcyd ly iaeigd oeeikd on lelqnd lr milkzqn xy`k,oeryd ibegnl cbepnd oeeika zg` mrt Z xiv.z xiv Z.oeeik dfi`ae Z xiv z` siwn oezpd lelqnd m`d wecap x +z =1 X y+z=0 Y xexaa mi`ex ea,sxevnd xeiva opeazp :1 dhiy oeryd ibegn oeeika Y xiv z` siwn xy` lelqndy,y xiv ly iaeigd oeeikd on eilr milkzqn xy`k milkzqn xy`k oeryd ibegn oeeika Z xiv z` siwn.z xiv ly iaeigd oeeikd on eilr opeazpe lelqnd ly divfixhnxt meyxp : dhiy.xy xeyin lr ely lhida oeryd ibegn oeeika oeeknd) lelqnd zivfixhnxt,lynl,`id (Y xiv ly iaeigd oeeikd on eilr milkzqn xy`k z = cos t, x = sin t, y = z = cos t, 0 t π xy` dcigid lbrn `edy x = sin t, y = cos t, 0 t π mewrd `ed XY xeyin lr lhidd.oeryd ibegn mr `ed epeeik,"ilily"d oeeika zg` mrt Z xiv z` siwn xy` xebq lelqn `ed oezpd lelqnd :mekiql F d r = G d r + `id ziteqd daeyzd okle, H d r = π + 0 = π. G d r = π okl 4
:5 dl`y.t lkl zetivxa dxifb divwpet b(t) idz z = y + xb(z) d`eeyndy d`xd [5%] (`).(x 0, y 0, z 0 ) = (0, 0, 0) dcewpd zaiaqa z(x, y) dcigi divwpet dxicbn.u(x, y) = f(z(x, y)) xicbpe,t lkl dxifb f(t) idz [4%] (a).dzaiaqae (x 0, y 0 ) = (0, 0) dcewpa dxifb u ik d`xd.ef daiaqa u (x, y), u(x, y) z` ayg [6%] x y (b) F (x, y, z) = y + xb(z) z onqp (`).F (x, y, z) = 0 d`eeynd jezn x, y ly divwpetk z z` ulgl mipiipern ep` xnelk :zenezqd zeivwpetd htyn i`pz z` wecap,f (0, 0, 0) = 0 (1) 1 zeivwpet ly yxtde mekq,dltkne,(oezpd itl) zetivxa dxifb b(z) ik,r 3 lka F 1 (), 1 divwpet `ed F z (0, 0, 0) = xb (z) 1 = 0 (3) (0,0,0).(miiw b (0) jxrd hxta okle zetivxa dxifb b(z) ik al miyp).yxcpk z(x, y) dcigi divwpet zniiw ok` okle miniiwzn zenezqd zeivwpetd htyn i`pz,t lkl dxifb f(t) oezpd itl (a).(zenezqd zeivwpetd htyn ly d`vezk) dzaiaqe (0, 0)-a dxifb z(x, y) oke.dzaiaqe (0, 0)-a dxifb u(x, y) zxyxyd llk itl okl itk mitwz mdi`pz xy`) zenezqd zeivwpetd htyn ze`veze zxyxyd llk it lr (b) - ('a,'` mitirqa epi`xy u x (x, y) = f (z(x, y))z x z x = F x = b(z) F z xb (z) 1 u x (x, y) = b(z(x, y))f (z(x, y)) xb (z(x, y)) 1 u y (x, y) = f (z(x, y))z y z y = F y 1 = F z xb (z) 1 u y (x, y) = f (z(x, y)) xb (z(x, y)) 1 5
y dx + z dy + x dz Z (0,0,1) [15%] :6 dl`y z` (zxg` jxca `le) qwehq htyn zxfra ayg.xeivay lelqnd `ed xy`k Y (0,1,0) X (1,0,0).(mepilet `ed ely aikx lk ik) R 3 lka 1 `ed, F = y î + z ĵ + x ˆk epnqpy,dcyd.oirhewnl wlg okle,yleyn `ed lelqnd z` eilr rawpe,ezty -y edylk oirhewnl wlg ghyn xgap qwehq htyna ynzydl zpn-lr.zipnid cid llk i"tr lnxepd oeeik lnxepd oeeik mr (wlg ghyn `ed xeyind) x + y + z = 1 xeyind ly wlgd z` xgap lynl.(,, ). F d r = F ˆn ds okle,miniiwzn qwehq htyn i`pz S F î ĵ ˆk = x y z = ( z, x, y). y z x i"r oezp (xeyind ly yleynd wlgd xnelk) S ghynd S = {(x, y, z) x + y + z = 1, (x, y) }, = {(x, y) x 0, y 0, x + y 1} :ghynd ly zixhnxt dbvd meyxp R(x, y) = xî + yĵ + (1 x y)ˆk = R y R x = (z x, z y, ) = (,, ) S F ˆn ds = ( z, x, y) (,, ) dxdy = = (x + y + z(x, y)) dxdy = dxdy = 1 = 1 okle.(z(x, y) = 1 x y -y dcaera epynzyd xy`k) 6
[0%] :7 dl`y.(, 4, 0) dcewpd jxc xaery ghyn S idi i"r oezp `ed ike,ghynd lr (x 0, y 0, z 0 ) dcewp lka wiyn xeyin miiw df ghynl ik reci (x 0 + z 0 )(x x 0 ) (y 0 + z 0 )(y y 0 ) + (x 0 y 0 )(z z 0 ) = 0.S ghynd z` zx`znd d`eeyn `vn xy`k F (x, y, z) = 0 dxevd on d`eeyn i"r x`ezn xy` S ghyn ytgp ep`,f 1 (1).ghynd lr (x 0, y 0, z 0 ) dcewp lka F (x 0, y 0, z 0 ) 0 ().(x 0, y 0, z 0 ) dcewp lka wiyn xeyin ok` yi ghynl ik migihan el` mi`pz ipy i"r oezp (x 0, y 0, z 0 ) dcewpa ghynl wiynd xeyind,l"pd zegpdd zgz F x (x 0, y 0, z 0 )(x x 0 ) + F y (x 0, y 0, z 0 )(y y 0 ) + F z (x 0, y 0, z 0 )(z z 0 ) = 0 miiwzn oezpd itle F x (x 0, y 0, z 0 ) = x 0 + z 0, F y (x 0, y 0, z 0 ) = y 0 z 0, F z (x 0, y 0, z 0 ) = x 0 y 0 dxevd on divwpet lk ik ze`xl lwe,f divwpetd z` miytgn ep` F (x, y, z) = 1 x + xz 1 y yz + ( ). F (x, y, z) = (x + z, y z, x y),mpn`e, reaw lkl,dni`zn divwpet `id.f (x, y, z) = 1 x + xz 1 y yz + = 0 `id ghynd z`eeyny o`kn,(, 4, 0) dcewpd jxc xaer ghynd ik oezpd zxfra rawp reawd z` :F (, 4, 0) = 0 i`pzd on xnelk F (, 4, 0) = 1 4 + 0 1 16 0 + = 6 + = 0. 1 x + xz 1 y yz + 6 = 0 `id zyweand d`eeynd okle, = 6 okl (x 0, y 0, z 0 ) dcewp lka wiyn xeyin ok` yi lirl `gqepd i"r x`eznd ghynly wecal x`yp,mepilet F ik F 1 (1) dl`k zecewpa la`,x = y = z oday zecewpa wx qt`zn F :hp`icxbd z` wecap () F (a, a, a) = 1 a + a 1 a a + 6 = 6 0 :eilr i"r x`eznd ghynl okle,qt`zn hp`icxbd day F (x, y, z) = 0 zniiwny dcewp s` oi` okl.yxcpk,eilr (x 0, y 0, z 0 ) dcewp lka wiyn xeyin ok` yi 1 x + xz 1 y yz + 6 = 0 d`eeynd,xekfke, F xnynd dcyl l`ivphet zivwpet ep`vn mvra ep` ( ) alya ik al miyp :dxrd.reaw ick cr zrawp l`ivphetd zivwpet 7