e` 'gn :dhlewt my :ihxt

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Maciej Marczak
5 lat temu
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1 e` 'gn :dhlewt 'qn :'cehq :ihxt :dgtyn :zexexa zeizeà o`k jly zipexhwl`d zaezkd z` meyxl `p,ipexhwl` xèca ef ogana jz`vez z` lawl jpevxa m` , , "n2 ilxbhpiè ilìvpxtic oeayg" a dpiga.zewc 90 :df wlgl onfd.oey`x wlg.(oery/ciip oetlha `l mbe oeaygna `l mbe) xfr xnega ynzydl oi`.dpigad iwlg ipy oia dwqtdd iptl xcgdn z`vl xeq`.wenip `ll dpigad qteh lr 2 dl`y lr epr.lawzi `l wnepn `l oexzt.el` zel`ya oexzt lk wnpl mkilr.zxagna 4-e,1 zel`y lr epr.oextra aezkl xeq`.ddk legk e` xegy hra eazk.zxagnl mitc siqedl e` yelzl oi`.hpcehq xtqne zxagnd ly sc lk lr meyxl yi.zecewp 100 yi lkd jq.dl`yd ci lr meyx dl`y lkl cewipd!dglvda.dl`d zecewpdn 50 raew df wlg :1 wlgl lkd jq.r -a dcewp lka zetivx zeiwlg zexfbp zlra f : R R divwpetd (12%).1. 9 x 4 z 5 f(2, 1, 8) = 7 -e f (2,1,8) = 10î + 20ĵ k :oezp (2, 1, 8) dcewpa g(x, y) = f(x 2 y, x, x y ). i''r zxcben g : R 2 R divwpetd.aeyigd ialy lk z` exiaqd.(1, 2) dcewpa g z` eayg 1

2 C 1 = {(x, y) x 2 + y 2 = 64, x 0 } ici-lr oezpd lelqnd C 1 idi (1%).2.(0, 8) -a miizqne (0, 8) dcewpa ligzn lelqndyk.ely dnbnd z` mbe C 1 lelqnd z` dpekp x`zn xy` xeivd zvayna X epnq [2%](`) oixb htyn zervn`a I = C 1 (e x sin y + 2y)dx + e x cos ydy lxbhpi`d z` aygl dvxp. C P dx + Qdy = D (Q x P y )dxdy [%](a) :ezgqepy. B a `xwp eplawy lebird ivgl.c 2 epnqpy xyi rhw ici-lr C 1 lelqnd z` xebqp jk l.c = C 1 C 2 didi. -l deey B ly ghyd :od ynzyp oday zeivwpetd P (x, y) =.Q(x, y) = -e (b) (.zg` dcewp ly ilily cewip zxxeb dpekp `l daeyz lk df sirqa) [2%](b). `l, ok?b -a zetivx zeiwlg zexfbp zelra dl`d zeivwpetd m`d geqipl m`zda oeekn C m`d,xnelk)?''iaeig'' epeeky xebq mewr `ed C = C 1 C 2 mewrd m`d. `l, ok (?oixb htyn ly 2

3 xy`,icnl miheyt miiehia mdy zeaeyz meyxl mkilr,zepexg`d zevaynd yelya,df sirqa ) [6%](c) (.t miitivitq mixtqn xear sin t e`/e cos t e`/e e t e`/e π e`/e miitivitq mixtqn wx millek (e x sin y + 2y)dx + e x cos ydy = C 1 C 2 B,b sirq itl dxdy =..C 2 lr lxbhpi`d z` l''pd d`vezdn xqgl epilr dzr (e x sin y + 2y)dx + e x cos ydy = C 2 I =. okle megza f(x, y) = x 2 + 4y 2 divwpetd ly ilnipind jxrde ilniqwnd jxrd z` e`vn (1%)..D = {(x, y) : 4x 2 + y 2 25, y 2x + 5}. F (x, y, z) = x (x 2 + y 2 ) î + y 2 (x 2 + y 2 ) ĵ + 2 (ex + y + z 2 ) k ixehwed dcyd oezp (12%).4. Σ = {(x, y, z) : x 2 + y 2 = : x > 0} dveawd dpezp ok-enk :onwlck Ω S e I S e A S xicbp Σ dveawd ly dveaw-zz èdy S ghyn lk xear. S ghynd ghy z` onqn A S (1.miz-d xiv oeekl dpet S-l lnxepd xy`k,i S = F ds (2.Ω S = { ( ) } (y, z) : y2, y, z S dveawd ìd Ω S ( -y jk g(y, z) -e f(y, z) zeyxetn zeivwpet izy e`vn [6%](`) I S = g(y, z)dydz mbe A S = f(y, z)dydz Ω S Ω S I S = ca S.Σ -a lken xy` S l''pk ghyn lk xear S :dgqepd z` egiked [6%](a).S -a ielz `ly reaw èd c xy`k,σ ly dveaw-zz èdy S ghyn lk xear?c ly jxrd dn.g(y, z) -e f(y, z) zeivwpetd izy oia heyt xyw e`vn :fnx - - oey`xd wlgl oel`yd seq - -

4 e` 'gn :dhlewt 'qn :'cehq :ihxt :dgtyn :zexexa zeizeà o`k jly zipexhwl`d zaezkd z` meyxl `p,ipexhwl` xèca ef ogana jz`vez z` lawl jpevxa m` , , "n2 ilxbhpiè ilìvpxtic oeayg" a dpiga.ipy wlg.(oery/ciip oetlha `l mbe oeaygna `l mbe) xfr xnega ynzydl oi`.zewc 90 :df wlgl onfd.dfd wlgd meiq iptl xcgdn z`vl xeq`.wenip `ll,df qteha 7 dl`y ly ` sirqe 6-e 5 zel`y ly epr.lawzi `l wnepn `l oexzt. wenip mr,zxagna 7 dl`y itirq x`y lr epr.oextra aezkl xeq`.ddk legk e` xegy hra eazk.zxagnl mitc siqedl e` yelzl oi`.hpcehq xtqne zxagnd ly sc lk lr meyxl yi.zecewp 100 yi lkd jq.dl`yd ci lr meyx dl`y lkl cewipd!dglvda.dl`d zecewpdn 50 raew df wlg :2 wlgl lkd jq miiwzn f : R 2 R dtivx divwpet lk xeary dpekzd zlra ìd R 2 -a D dveawd (10%).5 ( 1 ) x 2 ( ) ( x)/2 f(x, y)dy dx + f(x, y)dy dx = f(x, y)dxdy. 0 y=0 1 y=0 D o`k D dveawd z` exiiv [2%](`) f(x, y)dxdy = D y= x= lxbhpià zeleab z` enilyd f(x, y)dx dy. [8%](a) 4

5 . R -a dreaw dcewp (x 0, y 0, z 0 ) idz (18%).6. V δ = {(x, y, z) R : (x x 0 ) 2 + (y y 0 ) 2 + (z z 0 ) 2 < δ 2 } dveawd z` xicbp δ > 0 lk xear -l deey.zepekpd zeaeyzd z` X-a epnq,miàd mitirqd lka V δ 1dxdydz yleynd lxbhpi`d δ > 0 lk xear [4%](`). δ/, 2π, 4π, 4πδ 2, πδ 2, 2πδ, 4πδ /. δ > 0 lk xear V δ a ziliaxbhpi` ìdy f : R R divwpet dpezp [7%](a).iaeig δ lk xear φ(δ) = 15 f(x, y, z)dxdydz 4πδ xicbp V δ :dàd dpekzd zniiwzn,miieqn iaeig ɛ xeare,miieqn iaeig δ xeary gipp (x x0 ) 2 + (y y 0 ) 2 + (z z 0 ) 2 < δ :m`. f(x 0, y 0, z 0 ) ɛ < f(x, y, z) < f(x 0, y 0, z 0 ) + ɛ :if` Af(x 0, y 0, z 0 ) Aɛ N φ(δ) Af(x 0, y 0, z 0 ) + Aɛ N ik gxkda raep ef dpekzn. 5, 5/, 20, 5π, 2/, 8, 2π -l deey A xy`k., 2, 1, 0 -l deey N e.(x 0, y 0, z 0 ) dcewpa dtivx a sirqa zxèznd f divwpetdy gipp [7%](b).dàd dniyxdn lim δ 0 φ(δ) leabd ly jxrd z` exga, 2f(x 0, y 0, z 0 ), 5πf(x 0, y 0, z 0 ), 5f(x 0, y 0, z 0 ), 20f(x 0, y 0, z 0 ), 0. 5f(x 0, y 0, z 0 ), 8f(x 0, y 0, z 0 ), 2πf(x 0, y 0, z 0 ), 4π, 2π..f(x 0, y 0, z 0 ) < 0 m` -l deeye f(x 0, y 0, z 0 ) > 0 m` + -l deey l"pd leabd..miiw `l l"pd leabd dxeary (x 0, y 0, z 0 ) -a dtivxe V 1 -a ziliaxbhpi` f divwpet zniiw 5

6 [2%](`) (22%).7 (.ef dl`ya jk-xg` yexcy n1 `''eecgn xneg lr zxekfz llek df sirq) ly dcewp lka ipye oey`x xcqn zetivx zexfbp zlra φ : R R cg` dpzyn ly divwpet dpezp.r (1). φ(t) = φ(0) + φ (0)t φ (s)t 2 -y jk s R miiw t R lkl if` (.dpekpd daeyzd z` X-a o`k epnq) miniiwn cinz (1) `gqepa t -e s mixtqnd. s -e 0 oia `vnp t, t -e 0 oia `vnp s, s < t, s > t, s = 0, s = t.zxagna ef dl`y itirq xzi lr epr ipyde oey`xd xcqdn zeiwlgd dizexfbp lky jk,f : R R divwpet dpezp [8%](a).R -a dcewp lka zetivxe zeniiw.φ(t) = f(ht, kt, lt) i''r cg` dpzyn ly φ divwpet xicbpe (h, k, l) R dreaw dcewp xgap.mkzaeyz z` ewnp.f ly dizexfbpa zeielzd φ (t) -e φ (t) zexfbpd xear ze`gqep e`vn.zecewp 2 lawi "b sirq lr dper `l ip`" yexta azeke df sirq lr zeprl rcei `ly hpcehq [12%](b) dizexfbpae f -a mielzd mireaw,a,a 2,A 1,A 0 e`vn a -e ` mitirq zxfra [[10%]] (I) P znieqn dcewpa f zexfbpa miielzd B -e B 2,B 22,B 1,B 12,B 11 mireawe (0, 0, 0) dcewpa (2).f(h, k, l) = A 0 + A 1 h + A 2 k + A l + B 11 h 2 + B 12 hk + B 1 hl + B 22 k 2 + B 2 kl + B l 2 (b) -y jk.(mincwnd zxyr lk xear zeyxetn ze`gqep lawl mkilr)?(h, k, l) -e (0, 0, 0) zecewpl qgia P dcewpd ly mewind iabl xen`l elkez dn [[2%]] (II) - - ipyd wlgl oel`yd seq - - 6

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