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18 Q [ + (N 1d]rN = + rd(1 rn 1 cr 9 {@~ 1 r (1 r 2. *" *YŒ $ž Yz^ ŸYzžw $UQW_U -Y_YŒË`Y_Y_^ -Y Œf^ W1ž`b^Ì_UQ UŽ - *Ÿ *UX^ -Yz š $Yz^ -Y YzžlW1 *UQ - -Ỳ [ «Œ Yz - UQUŽ * *Ÿ /Š Ì_UX UŽ * -b^ùx^ *Y_ $Y *ž UX^ÙQU_xZ W_ - -Yz $YZŠÙYz - _ ŒY r < 1 S = 1 r + rd cr 9 }_~ (1 r $Y\Ë9f ŒUX -Ì_ Œl $W1b f b _rxp / Œ 2 r 1 7¹ ; XÀ* XÅ`¾ ¾Ã¹ XÅL XµÙ Ž» Œ^ žw $UQf^ 1«:9Ë9«1 b fb^ùx^ + \W _ * -UX f p z^ -Yz š $Yz^ *UQ u n _ _ŸË`YŒYz^ UQ * * : / _^`W1ž`^Nž`^`W@ -f Œ^ Yp 1ž` f ŸY W@ž` *UX^ x` Œ -Y_ *ž f u n Y Œ _ŸË`Y_Y_^ W1f š / _f $ Y_Y ž` _ * 1 _^ = f(n f(n 1 OPK 48FŒ p 5K #, G³p¾76Œ ¼Z¾X³p¾?9 \³ 1 _ ŒbË`YŒYz^ˆ«@ buxÿ $ : W1f š / _f $ Y S N = Yz b ŸY u n N u n = f(n f(0. n=1 N 1 n(n + 1. n=1 ž` z - 1 _^ xz / _ b f Œf^ f(n f(n m m m S N = f(n k + 1 k=1 7!YzbW_UQYŒ / -l ŸY_ *W_UX $ -Yp ZY -Yz š $Y _ ŒbËlY_Yz^ «:Œ +žl *UXb^ -Yz? $U\ Œ^ ŸY_ *W_UX $ - *Y * -f 1ŸY_Yz bÿ / -š 6.IY z $Ŷ z _^ Z ZYz Nž`^`W@ *b _^Ä *UQ bux^ Ë9UQ *b _ Œ b^ùx^ -Yzf^@ *UQÌ_ - _ f^ùq^5 OPK 48FŒ p 5K #, # G³p¾76Œ 7¾?9 ³ S = n=0 (n + 1/2n. * " " $* $!! f(1 k. cš k=1 ž`žl^œ Y_Y7 *ž9 -ž`w9 $h *Y_ $Y_W9 $rxž / Œ f bu! / *ž` b Œ + *ž`^l^ùž Y_Yz^ G -Yz š $Yz^ -Y_ / _^lw@žl^d -Yz š $Y_Y žlb $ *ž9 * -Yp ZYz^ 7!YzbW@WZY z -UŽ $ z^ *Y_ $Y_^ YZ $YzŠØY_ * Œ _^ Y_^lYz b«1«œ * *b^ùx^ "Z«@ b^ùx^ _ bÿ $bw1f^ ZYzbW_UQYŒỲ xžl *UXb^ *bf $ *bužš :lxw *ž UX^ÙQUXWŒ ZYz wyz $Y_Y_^@ *ž`ž`wœ \ *Y_ $Y un xww@žl^] Œ -Y_ *ž Y_Yz^ ž`wzy_yz^ _ *UŽ * -Yp 1bUX^ *UQ bux^ z "_ ÙQ $ /«1«:_ *UX -_^@ ` Œ Œ 5 -Yz š $Y *ž u n UX^ÙQU5 $Y un *ž UX^ÙQU_xp *Y_ $Y_^ -Yz^l z -YŒYz^ _ bux ZY_^\¹ µ ¾X X¹Ÿ¾Ž Ž¾Ãµš¹w¾? $ $l ŽÅh Z³Z Ø - *UXfŠ *UQ $ *à *ž UX^ÙQ ZYz^ -Yz š $Yz^ $ž Yzžw V _ _ŸË`YŒYz^ *UX wë fuq fb ZYz f -YzŸ *UQ - *Y? z š /UQ $ /«1W9 $UP * ; ZY_fW1ž9 * -Y Yz $ -Y -Yz š $Yz^Ä $ž UQ^Ù Ÿ $UQUX^5 Œ Œ Î -Yz š $Y lyz $Y_Yz^@ -ž`ž5x u n ž9 $ Y un *ž UX^ÙQU_xL _^ W1«9 $UQUQ - -] `½Œ¼Z»» ¹Ÿ¾X Ž¾ µr¹3¾? $ $l ŽÅh Z³7 -Yz š $Ỳ! wë9 _ b fÿ *UQ $ *5 *ž UX^ÙQ py_^ä -Yz š $Yz^ *UQ bux^äž`žlë9uq f buxuq^ _ /UP / b^ùx^ä *YŒYZ $ Y_Y; / -YŒY wyz $Y_Y_^@ *ž` ZY_Y_^ Yz Œ *W1f b b _b pyœyz^ä *Y_ $YŒYz^ Œ Œ Î -Yz š $Yz^ -UX ; Z $f *by\ -UQYŒYz bf bž`w1žz $Ylxl $ž UQ^ÙX ZY un Œ^ fžl _^`^` Œ f bÿ $UP / - «Œ f - *UXfŠ *UQ $ *5 *ž UX^ÙQ ZỲ» µrµ µ zå]?9µšµù³ X¹ÀÃ¹ ¹ µšµ - -Yz š $Yz^Ä $ž UQ^Ù Ÿ $UQ f bu _^ lim u n = 0. n ÎUX _b^7 -Y Y; *UQ $ -Yz -Y% *UPY_Y_ f _ *UX^`Ÿ *UX^ -Yz š $Yz^ *ž UX^lU b $ -Y" Œ^ Z ŽÀÃµÙ³Z¹»?9µÙ Ž¾ µr¹ ; / Œ vn _^ $ž UQ^ÙX ZY $Y u n v n x «:@ un $ž UQ^ÙXUŒ 7!Y_ $ -YŒYp ZY_ $ *rxq vn Ë9b _UX -Ì_ Œ@ $Y u n v n x «:@ u n Ë9f ŒUX -Ì_ Œš. Œf^ÙQ^ˆ lux Y Y Œ^ ¾?Œ `½Œ ŽµÙ Ž¾ µr¹ ;Z ρ = lim n ( un+1 u n cr 9 œz~

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25 `Y _b^ˆw1ž`b^äwpy_w9 $bž` b _f $ *UQf^lUX^ 8Ù - *b 1bYŒYz b9;b^@ *UQÌ_ Y_Yz b Œ^ÄYp ZY_ *ž`žwë9ux^ R *b X V = R dv = b^`^lyz^ R «1 b5 _ b R ;^ R dxdydz b^@ -YZŠÙYz ŸỲ x1b^@ *UXÌŒ -YŒYz b OPK 4 F@ p 5K #, ³Z¾ 6Œ ' Ž¾Ã¹ XÀ76Z¹Ÿ¾ ¾ É. Ð¼Z»»f XÅµÙ ŽµrÀ*³_ ½zÀÃ¹ŸÅ µr¹»f³??1¾ 6Œ¼Z» ¹?9»f¼zµšµÙ X¹Ÿ¾Ž XÅ ¹ ÅlµÙ ØÇzÀ-³_³Z» ¹ Å ³?9»»b³. OPK 4 F@ p 5K #, ³Z¾ 6Œ $`³zÀ*³ ¼Z»b¼Z¹ ½Z¹ Å 'X³µØ³Z¾X¼ZÅ À*³ 'X¼z¹ µšµù³z \³ZÅ ³Z»?l XÅ]µr¹»f³??1¾. z = x 2 + y 2 z = 2y `Y Y_Y Y YŒY :_ *f *UQ f b^@ *UXÌŒ -YŒYz b5 $Y byœ $WŒUQY] *UÄ _ _ŸË`YŒYz^ W p«@ $ bux bf ZYz f -Y_b *UXUQ^ :_ *UQ f bÿ $UQUX^ Ë9fŠ UQ^l $b 1 _^ Y_ $ *r!+*,*.-0/1&*,& /1& %* -"/&"/ 9UXž` Y_Yp ZY_ - -Y -Y_ *WZYŒ / -UX b byœyz^ :_ z /«1^ b^œ -UXÌŒ -YŒYz bf^ I = b f(xdx ŸY_ *W_U b $ -Y ^1ž UXUX -Ÿ $UP / -š. Œ^ˆ«1W9 $ ZY_^` ` by" $Y _ *WŒUX " ^1ž UX -fbwzyz^ä _^`ÌŒUX Ù^@ -UXÌ_ Yz^wË9 Œ _ŸË`Y_Y_^ *žl^œ -UQŸ UQ *h Yp _ _b b by<; *%^Ä _ bu f(x Y_ - *Y f(x ;^ Y_^lYz b«1«@ $ *b^ùq^ˆ ŸYzžl *UXWŒU ;^ Y_ * Œ z _ ŒbË`YŒYz^ˆ^@ž UQUX -b *UQ $ * Ÿ * -Y_ lyz^l -Y _ fb^ b $ *UQUQ - - f(x.gž`^`^ùx -Y_Y_^ˆ bž`w1žz / Œž`W1W_ [, b] {x i, f(x i } " " $ $ $ "?l¼zà*³ 6_³z¹ ½_ 6_³Œ³ Z³p¾ ¾X³ f ; Y_Y Y * ŒW9 $ _ŸË`Y_Y_^ I SK = b :z *U b $ *UQUQ - - ZYzW1b _ b by f( xp $Y\ -Y_YŒË`Y_Y_^ f(dx = (b f(. lž`^`w@ *b z YÄ _ _ŸË`YŒYz^ Y - _W9 * _ŸË`Y :z *U b $ *UQUX^d $ $Y_ $ -Y / Œ - -YzW1f^ ž`žw * -YÄ _ fb^ b $ *UQUQ - -l.ž`^`š ^luž *žlf^ _^ 6Œ Ã¾76z¹ $L¹Ÿ¾ÃµÙ 6Œ³_³ p³zx Ÿ * ;^ Yz - _ :z Z :_^ W_UP $W1 Ÿ / -UXUQ - [, b] f ( + b/2 x b ( ( + b + b I KP cš ` Œ Œ~ = f dx = (b f. 2 2 Œ Œ Ib^@ *UXÌŒ -Y_^lË9ŸY%Y_ * 1b _ŸË`Y_Y_^ *ž` _ Yz b by xz *f *UX^xZUŽ $ Œ f«1^` 9W@žl fwœuxu b $ *UQbËÙX^ WZY_ž9 $ Ỳ xl -Y_YŒË`Y_Y_^ x+b (, f( $Y $?l¼z» ¹Ÿ¾??9ÅlÅl¹6Œ³p¾(IUX b * Y UŽ $Ÿ :z^@ (b, f(b ¹ $L¾X¼ZÅl¹ Å x 6Œ³_³ p³ -Y_Y_ËlY_Yz^ WŒ _ - pyœy $Y Yz b ŸY f Œ f«1^` ; (, f( (( + b/2, f(( + b/2 (b, f(b I Simpson = b 6 I PS = b [f( + f(b]. 2 [ f(4f b b by x 2 + bx + c $Y8 * Z 1 * -Y ( + b?ë9ux b fÿ *UŽ ÎWZY_Yp ZYZ Z ZYZ ly_ z -UX f *žäw@žl pyœ * -Y\ 9 -YŒYz^] z b ; Y_Y_^ Ä $žlž` *UQUX^ Œ -Y_Yz^ [, b] n h = x i x i 1 $f *UQ^5x UŽ * -Y_YŒË`Y_Y_^ I = n i=1 xi 2 x i 1 f(xdx. byl Y_ f ŸY *U _} cš ` :0Œ~ cr 9 ` p~ Ÿ $ *UXŸ *fb^ ] cš _~ + f(b. x 0 = $Y x n = b,zb^ cš ` Œ~

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27 Œœ L[ -U Ê3Yz - b Š UX^lUŽ *UQ fu _ - - Œ^` ž9 *WZY_f^ÙQ^b^Œ -UXÌŒ * Œf^@ *ŸYz bžù ž z *UX -YŒYz^ :_^`^_ b bb *UXUQ^ V Yz bžùxup $UQUX^ xg / _ b fu Ì_UQ^ÙX - _ŸË`Y_Y_^ *Yz *ž`^`^wyzÿ $UP / - Ÿ $ *UXf l Ù^@ -UXÌ_ Y_Y_ fb^yz - _ Œ^ ŸY_ *W_U b *UQ - *Y 1ž` b 1ËlY_Yz^ˆ ZYzb^Ä^lU W b $ *UQUŽ xz / _ *WZY Œ *ž` ZYZ ÎYz bžùxup $UQUX^ V " " " #" $ $'! " [] _^`bž` b z $ -UXŸ / -UX^ b^œ -UXÌŒ -YŒYz bfuq^ I = f(x 1, x 2,..., x n dx n dx 1 dx 1 A n ŠØž` b z * *UXŸ *UX^ Yp ZYz *YŒ * -Y n A n byœ $WŒU f^lux^^1ž UXUQ *Ÿ *UQ $ *L _^ ZYz *f^ ZY_fWŒUQY Œ^`Ì_UQ Ỳ 1 ;Y_^`WZYz bž`ž`w9 *by Œ^ 1b * luxux^ Yz - 1f Œ Ÿ $UP * -Ỳ xp $b^`ì_žl by_ *f *UQUŽ $ -fuq^ W Œ $f $ -UX b«1 - $Y"UŽ -UX^`W1b^ ŸY_ *W_UQ^Œ YZŠÙYZ $Y_^ Ÿ $UP * fuq^ùx^ l []UX^ÙX *UX z^7 ZY_ fb^@ -YÎ -b ž`ž b^@ *UXÌŒ * Œf^@ -byp ZYz -ž`žlëùx^7ë9 UX^l / Ỳ p xq Œ _ŸË`Y_Y_^ W p«@ * :;«1W1 *bž9š b z * *UQb $ *UQ^ f^@ -UXÌ_ - _b^@ *bwzy_yp _ _ /UX^ c ;U * Œ^9Š$Ê? z *UP 7WZY_Yp ZYz Qx Y_žl - Xxb f b ~ «1 fuqb $ /«1W9 $ n 5` 6;fŸ - - Œ^`bž` f _ $ -UXb^ÙX^ b^@ *UQÌ_ Y_Yz b ŸY_ *W_UŽ Y_Y_^ * Z _UX f -Y Yz b by «1W9 $bž` b z $ -UXŸ $ŸY UX^ÙX *UX " UQ _W@W _f^ W1ž` f buxw1b^'ë9 UQ^l *f Œ f bu_x!up $ WZY_W1 *bž` f _ $ -UXŸ $UP * -Y7 Y Y_ž`W9 $UP * -Y b1 b2 1 2 f(x, ydxdy [ n ] b2 w i f(x i, ydy i=1 m j=1 i=1 2 n w i v j f(x i, y j, Ÿ * $Y Z ZYZ (w i, x i (v j, y j Yzb^` _WŒUX - * Œf^9Š $Y f^@ -UXÌ_ - _b^@ * Ÿ / -U Y_ *f ÎW1ž YŒ * -YzW1b^]Ë9 UQ^l * -Ỳ. ZŠ _^Œ Y Y_b $ *UQ^ ^l Q µa?9»f¼ 6Œ³_³ p¼ 'X ŽÅ 1b * `Ü Œ^ x Ÿ - - O(N 2/s Œ^ b^@ *UQÌ_ - _b^œ - b $ *UQbË9UQ^ WŒ _W_ Œ^lYzŸ * fž`w1ž9š z 3 $Y _^`ÌŒUX Y_^Ë9 UQ^l *f w Œ Œ Œ! wyz bž9 -YŒYz^\UŽ -ž`w Z -UXUX^ z NZ b^@ *UQÌ_ - _b^œ - Yz -W1W@žlžl VUQ * ;W9 $rx s 10 Yz - 1 Y_Yz^ ^` Œf^ b^@ *UQÌ_ - _b^œ - b $ *UX $ l. _ $ d *«@«9 $ W_ Œ *WŒUQYŒË9 UQ^l $b _Ÿ *fb^ _^lì_ux fb^ -Y_ * 1f -Y_Y_^ 2 UQ *fš /«1«Ÿ *" UX^ÙX *UQ 10 s ` % _ -W_UPY b - -Y Ë9 UX^l *b _Ÿ * -Ÿ W p«œ -UŽ :z^ žl *UXb^ -YZ -ž`^`^ly_b * fžlw@ž Œ Z $YzŸ / YÄ² ¼ZÅlµÙ!³ZÀÃ»f¼ (Ù \ XÅL XµØ X» cš[ Ê ~Ã % wë9uq - -ˆË9 UQ^l * -Y\[ _^@ -UŽŠ$Ê3Yz - f ŠÙYZ $YZ * *UQ fž Œ^ $UQž` Y_Yp ZY ;9ÌŒUX^ÙQ * ŒbË`YŒYz^ -YZ *žl^`^lyzÿ * fž`w1žly N U _ fb $Y $WZYŒYz ŸYZ -YŒYz^7^lU3 _ fb f bu z "_ b " _[ Ê+ŠÙY i [0, 1] [, b] u i = +U i * ŒW1 * YŒYZ -f z Z /«1 f bu (b b^@ *UQÌ_ Y_Yz bb f bu% Œ^Ä *b f b _b^ N f(u i I M cš ` 0Œ~ = (b N. i=1 žl^ xl *Yz *ž`^l^lyzÿ ž`ž9 * *žz $Y_^ N I M 1Ë` z *žw *Y_ * Œ _^ UQW1 -YzW@ -hy_ * Œ ; f bu_x`wœuq *W1f lyz / _^`^wyz b by I cš ` {@ Œ~ σ(i M = b N 1 2 σ(f, cš ` Œ Œ~

28 ž z ŠØž` b z * *UXŸ *UX^ f^@ -UXÌ_ Y_Y_ fb^ Y Ÿ - - _^ ; $ -Y -b ž` ZY ZYzW1f wx5 / ŒWpY _ _ŸË`YŒYz^Yz - 1f ŒbË`Yl +[ Ê+Š UX^ÙX *UX z^' 1b * lu] _^ σ(f f O(N 1 2 Y_ž`W9 $UP * -Y f^@ *UQÌ_ - _b^@ *ŸYp py_ *žlžl s V ž _ *UX -Y_Y_^ˆŸ * Y_Y_^ s ŠØž` f _ $ -UXŸ $UQUX^ 1«UX *Š W@žlž9 *b 1 _^ $Y z - -UX b " z^ˆž`žl *Nž`^`W@ *b W1ž9 -UX^ F { f(x1,..., x F (x 1, x 2,..., x s = s, jos (x 1,..., x s V cr 9 {l P~ 0, jos (x 1,..., x s V, $Y\ *YŒY_Ë`YŒYz^ N F (x i 1 I M = m(w, xi 2,..., xi s cr 9 {@ _~, N i=1 Ÿ - - _^ ž z *žw *Yp ZYz -ž`žlëùx^ *b byp 1ž`žl * -Yl 5[ _^`bž` b z * *UXŸ $ -Y [ Ê+Š UQ^ÙŽ -UX :Ä _^d lyp ZYzb^`^` Œ f bÿ /Š m(w *UŽ * *ž WZYz wë9uq - *Y Ë9 UX^l * -Y W@žl pyœ * -Y\ 9 {l [ Ê+Š UX^ÙX *UQ z^ 1f - Ùd _^ *b xp / _WZY7 Œ^ 1ž` YZ - _^ Yp py_ *žlžlë9ux^ Ë9 UQ^l / Y] $Y _^ Yzb^lY $žlž` *žlžl $ bž` ŒW@WZYŒY Yz $ -Yp ZYŒ / - Y_ *U LW1ž`b^ -ž` b _WZY_Yp ZYz^ O(N 1 2 O(N x1w1ž`^ˆë9 UX^l *b 2 s s _^ˆ $ž`žl *r [ Ê+Š UX^ÙX *UQ Z Î *UX b 1f«@ /«1 Z 1«@ 1b^ «:@ Œ^`fžl f _ $ *UQb $ *UQ^ Yp py_ *žlž`w9 $bux^ä $b^`ì_žl by_ *f *UQUŽ *UQb $ -YlxP / z *WZY Œ _b pyz 3 Œ f ŸY «Œ 3W p«@ - Î $Y b^@ * _ $Ỳ [ Ê+Š UX^ÙX *UQ W p«@ *UX $ _UQ - -ˆ *Yz *ž`^l^lyzÿ $ bž`w1žz $Ŷ UX!Y_f^lY WZYz^`^lYz -Ŷ Ì_UQ^ÙX - _ŸË`Y\ Y_ -YzŸ $UP / -VW_ _WŒ Yp ZYZŠ -ž`ž9 -UXUX^ x` ZY_Y_^ Yzb^` z * -Y_Y Yz bžùx $ -YlxZ / _WZY _^Äb^Œ -UXÌŒ -Y_^lË9b^ˆWZYz^l^lYz f -Y _ *WŒUl W 5Ÿ -_ fžùx $ Yp pyœy ; w ` ;YŒYZ -Yz $Ỳ x9 w ` ; «ŒY_ ` Î $Y\ w `ªÎYz l _ ŸỲ x?9 \ ŽÀÃ¹Ÿ¾X Xµ+ \ ŽÅh ŽµÙ X» µšxwê wê. buž -UXUX b bf^ùq^ byœ $WŒUX^@ -Y*;«Œx 0:0:l TV l w %Yp 1Ÿ I $Ÿ TV lªîyz lf^` f Œxh² ŽµN `¼P½p¾ ¼$?9 \ ŽÀÃ¹ º ³Z» ÃÅlµÙ ØÇzÀ-³Zµr¹ ¼ZÅlxw ±XY_ËÙ b±"t! -UQ - Qx 0@œ_}9 P 7 PT! *UP *? Žµ³z».bx?9 XÀÃ¹ º ³Z» ; ºX¹ $` Ž¾+¹ Å G¼ZÀÃµrÀ*³ZÅ (h ` "%ÀÃµ ¼$ ºŽ¹ XÅlµr¹?º!¼z $?9µš¹ Å1ÇzxzÊ3Y l *ŸË9Ì_U Í;^`f ŒUX $f /«\T! -UQ - Xx 0:0@ 9 ( - $/ %!# "*,! / 9 b [ f *UQ^ *Y_ $Y_^ *ž UX^Ù f^ùq^ *b ž`ž / Y r 9 8[ f *W *UXž` Y_Yp 1Ÿ / Y -Yz š / _Ÿ / Y Z ZYZ Î *ž 9 [ z - Nžl^`W@ *b _ŸË9UX^ []YŒ±Ž ŸYzž` -b^`f^ä *Y_ $Yz Q 9 {GY_ *W_U" -Yz $YZŠÙYz - _ z n=1 n=r UX^ÙQ 1bY n 1/2 (n + 1, 1/2 ln n=1 (n r! n! n 2 n!, n=1 (ln n n n n/2, n=1 ( 1 + x, b (x , c sin 2 x 1 x n n n! b x 3 + x 3 5x 2 2x 3 7x 2, kun x 0, x j x 2 + 4x + 4 π/2 y cos y sin y y 2 dy, kun x 0. x

29 ` } YŒ~ [ z - Nž`^`W@ *b _ŸË9UQ^ ~ xvbÿ~ xgbf ~ xb `~ x `~ x 2 y x 2 + y sin(x/y tn 1 (y/x r = (x 2 + WZY_fW1W1!UX^w $ _b *UŽ $f $ YzŸË9UX -b pyœyz YZ Q L~ byœ $WŒU x $Y y 2 + z 2 1/2 2 f/ x 2 2 f/ y 2 2 f/ x y Nž`^`W@ -f Œf b fu rx9b9 $Ÿ h `±Q~.G z *UPỲ x9ux $ Nž`^lWŒ -f Œ f bu b L f / x y = f / y x ` [ z - Nž`^lWŒ -f Œ^ $ -Yz *b _^ z -b $ *UQ^ Ÿ $ *UXŸË9UQ^ YzbWpYz V $Ÿ byœy_ë9ž9 P f(x, y = x 3 + xy 2 12x y ` œyœ $WŒU f^@ -Y_f^@ -UXÌ_ Y_Y_ fnž`^lwœ / Y 2 x` w~ ^`UX bfw1ž` * -Y f(x, y x x9w1ž`^ YŒ~ 0 x 0 y b f(x, y = x/(x 2 + y 2 f(x, y = (b y + x ` YŒ~ *" * _f -Ylx@UX $ UX b f *f^ 3/2 x 2 b^@ -YZŠÙYz ŸY _^ πb l w~ ;p«@ Z *ž` b Œ $ -Y UX b b ;^ YzW9 $žlb *UX^ *b $UQ fu dz 4πbc/3 ` 0YŒ $WŒU%b^@ *UQÌ_ Y_Yz b ~WZYzbW1f b by ÎU * _^`Š/Ê? _ *UQ ;WZY_Yp _ Œf b ŸY" $YzWZY *žl f ŒW1 *ŸỲ x 2 * _ŸË9b^ 2 + y2 b 2 = y2 b 2 + z2 c 2 = 1 UX^@ *b^ by_žl $UQW1W_UXUP * -Y+ $Y Œ * _f -Y *UX^Yp 1ž` b bylxzuž $ UX b b I = x 2 dx 10 * _ŸË9b^ *b byp 1ž`žl! _^ 0 Yz b ŸY z bg^lux :z^ Œ -Y_Yz^ $Y f ~%Yz^lY_ f«1«@ $ -b *UQ $ *w $Y ŒUX * -Y_Y & K H = - - &H&= - r 1 r 2 K H&= - - &H = - - &H = - H = - I # P D3 n priton 2xn /nd! P n=0 [( 1n /4](x/2 2n P D/ n=1 [( 1n+1 (2x 2n ]/[2(2n!] B $ & 1/2, 1/2, D! 1/2/π B 7 2xy, x 2 2x 2y y 1 cos(x/y, ( x/y 2 cos(x/y y/(x 2 + y 2, x/(x 2 + y 2 x/r, y/r, z/r! 2y, 0, 2x 2, 2, 0 (y 2 + z 2 r 3, (x 2 + z 2 r 3, xyr 3 K& 9 ' ' '., ( 2, 0 (2, 0 (1, ±3 ;? K &= - 3 (b/2 ln[1 + (/b 2 ] + tn 1 (b/! 4( + b + b J L &,= - x = cos t! π p 1 (z/c 2 b p 1 (z/c 2 dz "& M &= D H&- - π/ dv =

30 _

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33 : LYŒ~% 1ž` _ Yz^ _ w 7UXW@ - _ - _^] *ž`ž`^`^wy_ - *Y %ª $Y _^ ŒUXW@ * _ - ; $ -Y ;YŒYz^5 w w~.iy_ * _^ b λb A R «1 Œ z "l!$' " " -" $ " $ $ T+b $ *UQUX^ cn Œ fwœ 1 _^ Ÿ $ *UXUQ^ Y_fW1WZYp _UQW@ * _ - A ~?WZYzž` $ -Y W1ž` fwœux ZYz^xŒ ŒUXW@ * Œ *b^ $ž`žl^œ YzŸ $UQ^ *ž` _ Yz^ / ŒWpY_f^lUX^ b b $ *U Œ _ŸË`Y_Y_^ b _f $ Y_Y Y_fW1WZYp _UQWŒ - _ -f b ŸY c l 0@~ r = + λb. Œ Œ 3 -fupë9už :_^Ä *ž` _ Yz f -Y WZYzW9 * b $ *UX $ c YzbWZY_ - *Y A ~G $Y c ~Ãx` Œ _ŸË`Y_Y_^ˆ *ž` _ z " W1b / Œ * -YŒY c c ` b r = + λ(c. T+b $ *UQUX^ c YzbW1WpYp ŒUXW@ * Œ * ~WZYzž9 * -Y W1ž` bw_uq ZYz^ -YŒ $ Œ^5xÃ / _^lwpy f^@ Y_Y ZY_ $ -YŒYz^ W_ Œ Œ -b *ž` Œ -Y%«1W1 *bw@w Z _UQWŒŠ - _ - _^ z " Œ^ ˆn c l f Œ P~ (r ˆn = 0. žl^ UQË`UX b fb^`ux^ z ";W1f š / _f *UX -YŒYz^ ž` z Œ^ xp^ z wë:_^5xqux $ Î Y_ * _^ z "; Œ _ŸË`Y_Y_^ W1f š / _f $ Y_Y r n = n «:@ ž` * -Y x Ÿ * r ˆn = d Œ^\ -YŒ $ Œ^ W_ -b *ž` _ Y UŽ zÿ $«1«9 Œ / Ỳ ZT!Ÿ / -UXŸË9UX^ x $Y WZYzž` $ -Y d A B W1ž` bw_uq ZYz^ˆ -Y_ * z " _ _ŸË`YŒYz^ «:@?W1b / Œ * -YŒY c ` b P Œ~ r = α + βb + γc, Ÿ * +Ë9UQ f bb * _^Ä lyp ZYzb^`^` _ b bb $ *UX $ *žäw1ž` ZY_ - *Y` b _ α + β + γ = 1 TY_ f b _^Ä«1 Œ z " Œ^ / - W1f š / _f *UX $ -Yp 1Ÿ * -Y _UXW@ - _ - ž` 9Ë9 Œ - *Y c l f Œ~ r c 2 = 2, Ÿ * _^ Yz b b _^ÄW_UQ *W1 b $ *UQUX^ YzbW1WpY% $Y c 7UQW@ * _ -ŸYz bì_ux l -Ÿ Œ^ W Z -UX UX zÿ *«@«1W9 *fuq^ *UX^ -ŒË9U_ ~3 b«@ lf^ UŽ zÿ $«1«9 $žl _ Y_ $ -YÄcNW1ž` ZY` _~3 *YŒY_Ë`YŒYz^Ä $ž` Œ -YŒYz^ˆ *Ÿ / - -ž` b _^ :z -f $ b *UQ - -Ì_UX UŽ * -ŸY_ - *Yl `T!Ÿ / -UXUX^ c YzbW@WZYp ŒUXW@ * Œ * P z - -UX _ $ p d = (p ˆb. OPK 4 F@ p 5K 2, Æ3µš¾Ã¹ $L¹Ÿ¾ÃµØ XÅ $h¹ ŽÅw¹ Å Xµ ¹Ÿ¾ÃÂZÂp¾¾?l¼zÀ*³p¾ÃµÙ³ È5 ¹Ÿ¾ ¾ P = (1, 2, 1 r = +λb = î+ĵ+ ˆk 'X³. b = 2î T+b $ *UQUX^ b«@ lf^ UŽ ĵ + 3ˆk zÿ $«1«9? -YŒ / Y Œ^ p c l f Q{@~ d = ( p ˆn.

34 z{ 2-"//1#2/ wt!ÿ / -UXUQ^Ä b«1 `b^äux _b *«1«1 Y@~ $ž` Œ -YŒ / Ỳ x@ w~? -YŒ / Ỳ ²]³zµšÀÃ¹ ¹Ÿ¾Ã¹w _^ _ /UP / -UŽ $ /«bž`w1žz / _ž`w1wœ lxž * -Y W1ž`^`W1b^ Yz bw1f Œ^ *bb * _^ˆ -Y Yz^wY W1b / Œ * -YŒY ž` z - 1 _^ Y_fW1WZY b n A = n. m1 m2 mn _f *UX -YŒYz^\WZYz lëùx b by f^lëùxw9 $b b "l Œ[]Yz $Š Í; *UXb^ W p«@ *UX z^ «:@ UQ *W1b^Œ z $Y A = ( ij (A ij = ij 5 Œ Œ Yz * -fÿ $Ÿ - *Ŷ _^ ZY_YzWZY_ *b 1" $Y m n «9 $ /«1 *b 1"lx1 -Yz^l z -YŒYz^5x1UX $ *U _^ m n YZ - *bb *rxp $Y UX $ Yz * -fÿ *f^ ½Z¹ \ ŽÅl¾Ã¹ ¼ _^ Y_W1 * m n Yz * -fÿ *by; Z pyz " *ž`ž` -UŽ L / Œ w $Y; ZY_f^% / Œ I^`bbËÙX^ Ë9 UQ^l $b z L $Y% pyœ / -f^lux bu UX^@ *f Z ZYZ *Y Yz Q Œ Œ $Y x_^`bb^ Œ^ $LÂp¾ÃµrÂ p 6zµØ¼zÀÃ¹ xq $YÎ / Œ / Œ $Y xœ^`fb^ _^ Z³_³ 6_³ p 6ZµÙ¼ZÀÃ¹ n = 1 m > 1 A n > 1 m = 1 A Œ Œ xz Œ^ UXW1 1b py_ fuq^@ $ *L $WZY_ byœyz -f^ m = n = 1 A 11 WZYz^l - *Yl _Í; *UXb^\ _^\ 1«1Ë`«@ b bb $ Yz $YZ *UQ f ŸY YZ - *bb *b^ «9 / /«@Š? -YzI ZY_Y_WpY_ *b 1fUQ^ UX bu UX^@ $ -fuq^ ž` 9Ë9 Œ $ -Y Yz! / _^l z % ŒUXW@ * Œ *UQf^wỲ. _ f "_b^ m n Yz * -fÿ $ A Œ _ŸË`Y_Y_^ˆYZ $Yz *UQ f ŸY ž` 9Ë9 Œ $ *UX *ž`w9 * ; $ -Y «9 / /«1 ŒUXW@ * Œ *Ÿ / Y n j = 1j 2j W1ž9 -UX^ Œ Œ A = x ( *Y_^` z Y_Yz^ 1 2 n Yz * -fÿ *f^ _ bux ZYz^ Åh Ž» ¹ Z \³ZµrÀÃ¹ ¹Ÿ¾ ¹, ˆ Z ZYZ "UX - /«1Ÿ $UQ^ Y_ - -Y * Z _UQ f bž`w9 $Ÿ - *Yl m = n []Yz * -fÿ *šxp / Œ^`WZY WZY_fW1W15Yz bw1f _ Z ZYz ^l _ b fÿỳ x9 Œ^ˆ^` UX f z^ä^` _ b by YZ * -bb *š Œ Œ YZ - *bb *b^ˆwzyzbw1w1 UX bu UX^@ *f Î Z ZYZ ij = δ ij = Œ^ˆW1«9 $UQUQ - -ˆÂ 6Z¾ ¹6 6- Z \³ZµrÀÃ¹ ¹Ÿ¾ ¹!cN -Y_5bËÙX^@ *f *UQUŽ $ - mj { 0, jos i j 1, jos i = j, Yz * -fÿ $ ~ I c Yz 1 ~Ãx I = c ` b P}_~ UX -W1 * p _ - - Y_ *UŽŠ c ` b Q Œ~

35 %W9 *fw1w YZ - *bb *b^ -ž` fuqu Œ f ŸY\^ÙQ f" YZ - *bÿ $r L 1U _^ UQ *bw_ Œb $ -Y Yzžl 7½z¹ ³XÇ@¼ZÅh³_³Z» ¹ ³zµšÀÃ¹ ¹Ÿ¾Ã¹Ÿ¾ÃµØ³zx` / Œ^`WZY YzfŠ ^` _ b ŸY_ $ -Y ŒfW1W_UPYp ZYZ Yz bw@b z Z ZYZ ÎË9ŸYzÌŒ _^lyœyz bf b ŸỲ % zwœ _ f Yz * -fÿ *b - *Y? $Y Yz ŸYzWŒ _ b Yz * -fÿ $Ÿ - *Y" Z ZYZ "p 1Ÿ / Z _^Yz ŸYZŠl $Y «1 " ž` Œ fÿ $UX 3Y_ fw1b z Qx1 ZY_ $ -YŒYZŠ ZY_ $ *rx9^` _ b bby<; U = 0 0, L = `ÀÃ¹ ½Z¹ ³XÇ@¼ZÅh³_³z» ¹ ³zµšÀÃ¹ ¹Ÿ¾Ã¹Ÿ¾ ¾X³ ^` _ b ŸY_ $ -Y _bw@wœuqyp ZYZ ÎY_ fw1b z Î Z ZYZ ;Ë9ŸYzÌŒ _^lyœyz bf b ŸY; $Y $UQ^Ä 1bUX -UXŸ $b b "\Ë9ŸYzÌŒ zš ^lyœyz bb *f b ŸY -f ŒUXb f ; T = $ " "! " [ YZ * -bb *fuq^ A $Y B ÂQ 9µØ XÅl»f³Z¾76:?9Å -ž` f ŒW9 $UQ^lY Œ^ Yz * -fÿ * S xp / Œ^`WZŸ UX bu S ij = A ij + B ij UX^@ *f Î Z ZYZ WZY_fW1b f bû b^lë9uqw9 $ Yz -UXb b fuœxg X YZ - *bÿ $bux^ pyœ / -f^wyz bw@b z ŸY_ *W_UŽ Y_Y_^ «1 Œ -UXUQ^5 Îž` YŒỲ xvuž * d *ž Yz * -fÿ $bux^ -ž` buxu Œ f ŸYË9 UX^l *b _b Y_Yz^ -Y YZ P w[]yz * -fÿ *fuq^ _ ÙX^l^@«9 * byœ $W1ž :_ *f *UQ f :_^ -Y X Yz * -fÿ *f^ UQ fu UQ^@ *f Î Z pyz S = A B S ij = A ij B ij OPK 4 F@ p 5K 2, # ² ÀÃ¹ µ \³ZµrÀÃ¹ ¹Ÿ¾ ¹ È"6?9Å D = A + 2B A = ( , B = ( , = ( Œ} c l f Zœz~ Yz $ Y_ f ŸY -Yp ZYz b ŸỲ x [ YZ * -bb *UQUX^9Š $Ÿ _ ÙX^`^1«9 * byœ $W1ž Z ZYz 3WŒ ž9 YZ -fb 1b *by% $Ÿ Y_ - * Œ *byz *bf 1Ÿ $ŸỲ ² ³zµšÀÃ¹ ¹Ÿ¾Ã¹ Å(6Œ XÀÃµØ¼z ¹ Åh ŽÅ\¾76Œ³Z»f³Œ³ZÀÃ¹»»f³ -Y *ž`ž W_UQ $ - Yz b byîwpy_fw1w1 Yz * -fÿ *f^ Yz bw@b z λ λ ; b byl Z UX * * b^ùx^ *WZYz ŸY_Y_ *b f ŸY7 Œ^ Ë9b $ * -b `ž9 $Ÿ YŒ * * Œ *byz *bf 1b^ÙX^ˆ byœ $W1ž9 - _ -žl X *% bw_ 1 Œ^ A m n Yz * -fÿ $à $Y B n r Yz * -fÿ $r ;fÿë9ux^ WŒUX * * _ ŸY_ *W1ž`^ -ž` b _W9 $UQ^lY _^ P = AB m r Yz * -fÿ $ xz / _^`WZŸ Yz bw1f _ Z ZYz P? $ []b - - P ij = k=1 UX -W@bW9 $1 ^1«Œ x $Y m = 2 n = 3 r = 2 ( P11 P 12 P 21 P 22 N A ik B kj, i = 1, 2,..., m, j = 1, 2,..., r. x9 -ž` b Œ^ = ( A11 A 12 A 13 A 21 A 22 A 23 B 11 B 12 B 21 B 22 B 31 B 32 c ` b Q Œ~ P 11 = A 11 B 11 + A 12 B 21 + A 13 B 31 P 21 = A 21 B 11 + A 22 B 21 + A 23 B 31 P 12 = A 11 B 12 + A 12 B 21 + A 13 B 32 P 22 = A 21 B 12 + A 22 B 22 + A 23 B 32.

36 _ OPK 48FŒ p 5K 2, 2 G³p¾76Œ 'X³ È06?9Å P = AB Q = BA Î _ b ŸY WpY_fW1b f bu. YZ - *bb *b f bu+ $Ÿ «1W9 *fw1w: A = YZ * -bb *f b bu z *UXU, B = A0 = 0A = 0 AI = IA = A YZ * -bb *UXb f bu A l[ YZ * -bb *fuq^ˆwœux * * Œ byœ $W1ž` b fu" _^Ä _ Œ A(B = (AB k(ab = (kab = A(kB (A + B = A + B (A + B = A + B []YZ - *bb *bux^ W_UQ $ - _ ŸY_ *W@žÛXG«@ buxuq^l * z -UXž9 Y\ ZYzb lë`y_^œ Yz ŸYzW1ŸỲ x@ - Q x` Yz - _b^ùx^ WŒUX * * _Š AB BA ŸY_ *W1ž`b $ -Y"UQl _ buîupë9uq z - -UX f /«_ z Œ Œ ^lux b Yz * -fÿ *UXb f bu $Y A B Z -UXU AB = BA UX $ $Y 6_¼z?9µÙ¼Z¹& Z³Zµ9 $Y" xl -Yz^l z -YŒYz^5x9UX $ $Y ³ZÅwµš¹6Œ¼Z x@ *Y_^` z Y_Yz^x?9µÙ¼Z¹& Z³Zµr A B AB = BA A B 1žlž` *U _^\^` UX f :_^6_¼z?9µÙ³_³ZµrµÙ¼ZÀÃ¹ 1 ; ž9 -YŒYZ * * _ -UXb b by% _^\UQ *f $ zb^\w_uq *W_UQf^lUX^ AB BA YŒ $U Y W1 ZYz^@ $ - UXWZY_^`fbWZY_ - *Yl []YZ - *bb *b^ z -UX^l - $ x x` Œ^ˆ«1W9 $b^`wœux * -YzŸ *UQ $ * A n n N z^ Yp 1ž` f ŸY _ _ŸË`YŒYz^ YŒ * -Y A n = AAA A. }{{} n kpl z - -UX b "ˆ ³zµšÀÃ¹ ¹Ÿ¾Ã¹ ³zÀ p¼z¹ÿ¾ ¹ ³7?9Å@6zµš¹ ¼z¹ µø³"^`bÿë9ux^ -Yz š $YzW_UQ `f *UX fuq^ WZYzž` $ -YlxÙQ * A c ` b 0Œ~ n exp A = n!. n=0 []YZ - *bb *b^ µrà*³zål¾ $`¼P¼p¾Ã¹ Œ^ YZ * -bb *šx_ / Œ - *Y «9 / /«@ŠG $Y ZY_Y_WZYz -f 1f ; Z ZYZ ; ZYzb lë9ux $ -ž W_UP $WŒUX^ z^5 A A []YZ - *bb *bux^ˆ *ž` b Œ $ -Y z -UŽ -ž` f bu * Yz^w T 1 Œ *f b fu z *UXU A c ` z Œ~ (AB T = B T A T. Í; *UXb^ -Y_ * 1f $ -Yp 1ŸYÄ $b^d -Y YzŸ *by ž`ž`^`^l _W9 $ŸY Z ZYZ Yz * -fÿ *f^ 6Œ¼Z $L»b 6Z¾Ã¹6_¼zÅ '?_Ç@³_³Zµrµr¹ xw / Œ - *Y A / _WZYzb^ÙX^ Yz * -fÿ $b^ Yz bw1f ž`ž`^`^ùx -YŒYz^ W_ fuqw9 $bw_ Œ^Z /ž`ìœyœyz -fw9 *UXUX^x@ *UXW ŽÀÃ ¹ µø XÅ 6Œ¼ZÅ '?_ÇŒ³Œ³Zµrµš¹ UQ fg fbf $Š * Yz * -fÿ $rxp / Œ - -Y c ` 9 P~ A = (A T = (A T 5ž` Œ^`^` _ b bb *UQ $ * -UQYŒYz bb *f b bu YZ - *bÿ $UQf b fu $Y A = A A = A $W1žl 3 Yz -YŒYz^ YZ - *bb *ŸŸ WpY_ -Y_W@ *UX -Ÿ $ Œf 1ŸY «1W9 $f $ zÿ $ ^1ž T UX - _f -Yl %W9 *f^`wœux * -Y_b *f^ä^`bb $ _^ ^ÙQ f"zš Yz * -fÿ *f^ '»6z¹ x. xz / _WZY :z -f *UX b :z^ä $UQ^ÄË`bY_Ì_ _^wy_yz bb f ŸY * $Y_ $UQ 1fUQ^ÄY_ fw1b _ŸË9UX^ $ž Y_^lỲ x A n c ` _ _~ []YZ - *bb *b^ _ /UQ f _^ÄW p«œ * -ŒW_UX Tr A = A 11 + A A nn = _Ÿ $ŸY7 b^ly_b *ž`ž`w9 *bylx9w@ž` *UX^ Tr AB = Tr BA i=1 Tr AB = Tr BA = Tr AB Tr A T = Tr A Tr A = (Tr A. A ii.

37 "!!$ #" " z /UX^d Y YŒYz^ YZ - *bÿ $b^ ½_ XµØ XÀÃ ¹ ÅL³ZÅwµšµr¹ -Y_ _^ bž`w1ž c Yz3 by_žl *UXW_Up~Ãxl / _WZY -f ž`ž det A A WZY_fW1Ÿ / Y Yz bw@b _Ÿ $ 9Y _b^\w1ž`b^ z bw@rx «:@!Ë9UŽ -UX f^ly_^@ $ *w _^ :_ *f *UQ /«Yzb^` ŒYŒ / Y_Y_^ ^ÙQ f" %UŽ -UX f^ly_^@ $ * byœ $WŒUŽ -YŒYz^ Ÿ * _^ A ij -bb *b $ -Y" z *UX :_^ $Ÿ WŒUX - * _ -YŒYz^\ -ž` 3 *UQW@ z b A = n ij A ij = n ji A ji, i=1 i=1 YZ - *bb *b^ Y_ fw1b z -Y A ij ZY_ $ -YŒYp ZY ³z» ¹ ½_ XµØ XÀÃ ¹ ÅL³ZÅwµšµr¹ xž / ŒWZY Œb i ; ZY_Y_WZYz -f 19 $Y j ; ( 1 i+j x` - Yz * -fÿ *f^ Yz * -fÿ $UQf b bu_ c l :Œ~ ž` / -UŽ -YŒYz^ $f *UQ^5xzUŽ * YZ *Š «9 / /«1 -f 1rx` byœ $WŒUŽ Y_Yz^Ä^ _f^] -Y_Yz -Yp ZYz^ YZ * -bb *f^ Ë9UX *UQ b^lyz^@ $ ,j 1 1,j+1 1n A ij = ( 1 i+j i 1,1 i 1,j 1 i 1,j+1 i 1,n i+1,1 i+1,j 1 i+1,j+1 i+1,n. n1 n,j 1 n,j+1 nn OPK 4 F@ p 5K 2, ² ÀÃ¹ µ \³ZµrÀÃ¹ ¹Ÿ¾ ¹ Å A = ³z» ¹ ½_ XµÙ ŽÀÃ ¹ ÅL³zÅlµrµš¹?³z»6z¹ ¼ZÅ 23 ¾?Œ 9µØ XÅ ¾X 6 \³ZµrÀÃ¹ ¹Ÿ¾Ã¹ Åd½Œ ŽµÙ ŽÀÃ ¹ ÅL³zÅlµrµš¹. [ YZ * -bb *bylx` / Œ^`WZÝ Ë9UX *UX f^wyz^@ $ -? Œ^ ^` _ b bylxi -Yz^l z -YŒYz^ ¾Ã¹ Å@Ç>?9»f³_³zÀÃ¹Ÿ¾X 6Z¾Ã¹! UQ *Ÿ *ž`ž` - W@žlf^Ä^` _ b ŸỲ x9 _^ *U Ž¹&(Ø¾Ã¹ Å@Ç>?9»f³Œ³ZÀÃ¹ ÅL XÅUQ f5 :_^`^_ b bf^ùq^5 %UŽ -UX f^ly_^@ *b f bu Z pyz _ _ Y_ - *Y $UQž` Y_Yp ZYZ 3 *ž` b _W9 *UŽ ; A T = A A = A = A YZ - *bb *b^ Ë9UŽ -UX b^ly_^œ * *3 _^ Yz wë9ux^ -YzhWZYz wë9ux^ *Y_ -Y_W1W_UXUQ^\ ZYzb Œ Y b^ùx^ˆw_uq *W_UQ^ :_^ žlž9 $ Y_Y py_f^äë9už -UX f^ly_^@ *b^ UX *Š W1f^x ž9 * -Ÿ UXGË9UX f^ly_^@ *b^ˆf - *UXŸ *Y_ * Œ ŒYl Œ Œ n n Yz * -fÿ $b^äwpy_fw1w15yz bw1f _ W_UX - - z -YŒYz^ ZY_W@b _ b ŸY λ x`^`bf^ λa = λ n %UŽ *UQ A b^lyz^@ -f^'y_ * Œ «9 *«1«ž`ž9 $ -ž YZ * * Y_^lỲ x3w1ž`^ YZ * -bb *f^ *b 1 c -Yz YzWŒUP~ bb Z :_^ * _Ÿ *UXUX^ -f 1bf^dc -Yz YzW1W_UQUQ *UXUQ^w~ ZYzW1b _ b by W_UQ * - z * *ž`^wỳ AB = A B OPK 4 F@ p 5K 2, G³p¾76_ ½Œ ŽµÙ ŽÀÃ ¹ ÅL³zÅlµrµš¹ [ YZ * -bb *f^» ¹ ¹ µšµù¼z \³ZµrÀÃ¹ ¹Ÿ¾Ã¹9UX bwy_ëz /ž`^`ì_ Œ -ž A Yz * -fÿ $b^ˆ * Yz^l Œ^` _f *ž YZ - *bb *A; A = YZ * -bb * dj A Œ^ A ;^Yz bbë9ux *UQ (dja ij = A ji. b^lyz^@ $ -fuq^ ž` / Y Yz^ c l z{@~

38 _ #" $ " _^Ä _ bu Y_ - *Y YZ - *bb * * -UX^5xlUŽ $ ^`fb^ _^ YZ - *bÿ $b^ 6 ÅwµØ X¹Ÿ¾Ã ³zµšÀÃ¹ ¹Ÿ¾Ã¹ xp / z Y A *bb *b^äyp 1ž` b $WZY _^ UX b OPK 48FŒ p 5K 2, ² ÀÃ¹ µ \³ZµrÀÃ¹ ¹Ÿ¾Ã¹ Å 6 - µù Ž¹Ÿ¾Ã \³ZµrÀÃ¹ ¹Ÿ¾ ¹. ;:z^@ -UXŸ YZ - *bÿ $b^ (A 1 1 = A (A T 1 = (A 1 T (A 1 = (A 1 (AB 1 = B 1 A 1 b^lyzÿ *ž`ž`w9 $ŸY ; (AB G 1 = G 1 B 1 A 1. A = A = I UX -W1 :z^\w1ž9 -UX^ A 1 9 1UÎ Œ _ŸË`Y_Y_^ A dj A = A I, A 1 A dj A A = A 1 I, z - * % bb * * A 1 = dj A c ` _}_~. A A = $ "! % # -" $ $'! %$ ÎbU f^ / Y_f^l * *žz /UQ^ Ë9ŸYzÌŒ _^lyœyz bšx«1 ZŠÎ $Y Yz ŸYzW_ Œ W p«@ -UŽ z^ˆ«@ buxÿ *UQ $ * *UXž` Y_Yp 1Ÿ / Y7UQ *f /«1b $ -Y Yzž`W9 *b $ -Y ¾ÃÂZ XµšÀÃ¹ Åh ŽÅwxp / Œ A T = A ³zÅlµr¹Ÿ¾ ÂZ \ ŽµrÀÃ¹ ÅL XÅlxp A T = A ¼zÀÃµØ¼-ÇŒ¼zÅL³Œ³Z» ¹ Åh ŽÅwxz / Œ A T = A 1 À* ³Œ³Z» ¹ Åh ŽÅwxZ / Œ A = A l ŽÀÃ ¹ ¹ µšµr¹ ÅL XÅlxz / Œ A = A ³zÅlµr¹f ` XÀÃ ¹ ¹ µrµš¹ Åh ŽÅwxZ / Œ A = A b YZ - *bÿ $bux^ $Yd * -bë`by_ì_ _^wy_yz b ŸY ^l f /«@W9 *" z^5 9[]Yz * -fÿ * _^ A Yz $Š Yz * -fÿ *fuq^ Œ ÙX b by?9åw¹ µø³œ³zàã¹ ÅL XÅlxp Åh¼ZÀÃ \³_³Z» ¹ xz / Œ A = A 1 AA = A A! *f /«1b 5Ÿ -_W9 $ bf * *UQf^ ^ zb f bu z *UQU w _ * * ŒÌ_ _^wy_yz bÿ $UQ^ YZ - *bb *b^ ËÙŽ *UQ b^lyz^@ * * Œ^]Yzb^lY ±1 f^`w -Y_ lyz^l -Y]^lUX b Yz * -fÿ *f^ Œ _Yzb^lÝ W@b š / _f $ -YŒY\ / ŒW_ *«UX * -b *UX^ $Y Y_^@ *Ÿ $«$Y\ž`^`f -YŒYz -b *UX^ UŽ * -b *UX^ 1 Yz * -fÿ *f^ *ž Y_^lỲ x A = A S + A A x b - $Y x` YzG pyœ / Y_Yp ZY_ $ * A S = 1 2 (A + AT A A = 1 2 (A AT ÙX ff $ -b *UX^ $Y Yz^@ *b ÙQ b * *Ÿ $UQ^ Yz * -fÿ *f^ *ž Yz^lYlx b - A $Y H = 1 2 (A + A A AH = 1 2 (A A

39 *!!$ $* " ($ $*- " " + *f $ zb^ z -W_U YZ - *bb *ŸYz bì_ux l -Y_^ Yz bžù Z ZYZ ;¼Z ¹ ÅL³z¹Ÿ¾Ž³zÀ p¼($¼zå1çœ X» \³Zµr.IYz UQUXW9 *@«1W9 *f^`wœux * -YŒ / YzUX^ UQ * UX *Š W1bW9 $ WŒ _W_ W1 ZYz^@ $ -fwœu by_^ˆ Œ _ŸË`Y_Y_^ˆ Œ fuq ZYz^ $UQ f ŸYzb^ÙX^ T+«1 $ /«1 _UQWŒ - _ -by xz / _WZY * z -UXž9 * -YŒY _ Œ *«1 _^ x 0 c l _ Œ~ Ax = λx -Yz^l z -YŒYz^ Yz * -fÿ $b^ ¼z ¹ Åh³Z¹Ÿ¾ p 6zµØ¼zÀÃ¹6p¾Ã¹h $Y $WZY_ byœyz -by ;^ ¼z ¹ Åh³Z¹Ÿ¾X³ZÀ Z¼ 6Z¾Ã¹ z "dc ` Œ _ŸË`Y_Y_^ W1b / Œ * -YŒY ž` z - 1 _^ A λ A c ` Œœ_~ (A λix = 0 Yz5UQW9 bb * *Ÿ $UP / - z "_ -«@ _W9 $. _ f «1 Œ z "_ -«1 % Y_W@ * b^ ( 11 λx x n x n = ( 22 λx n x n = 0 n1 x x ( nn λx n = 0 z b " _^ÄUXfŠš - *b 1bYŒYz bl YZ -WZYzŸ $ž / Œ I $Ÿ ZYzb^ / Œ WŒUX - * Œf^lËÙŽ *UQ 11 λ 12 1n λ 2n n1 n2 nn λ ˆ¾X 6?9»f³_³zÀÃ¹ ÂQ 1µ» zå *ž`^`^luž $ -ž * Œ _ŸË`Y_Y_^ W1f š / _f $ Y_Y det(a λi = 0. b^lyz^@ * *L _^ ^` _ b ŸỲ x1 - Q. :0 = 0. c l _ Œ~ UQ * - _b^lë9ux *UQ b^lyz^@ $ -G Œ^ *ž`ž` -UXUQ^ ;^`^lux^dy_ $ *UXUQ^ _ b«1^` rx1 / _ b bŷ _^d * -UX^ /ž`žl $ Ỳ 1UQW@žl byœyz -f«1 9Š z "_^dc l _ Œ~+ -Yz *WZYzŸ $žl^ly -Y_YŒË`Y_Y_^ˆ *fÿ λ n WZY Yz buž * -Y n YZ - *bb *b^ n b^lyzÿ -Yz - _ z $Yl Œ ž`^ b^ly_b -Yz - _ _ + _^ A "Z«9Ë9UŽ * /«_x1 ZY_ $ -YŒYp pyz? b^ly_b * _UQWŒ - _ - *YŒY_Ë`YŒYz^ YZ -WZYzŸ $U Y_ f ŸY bf^luqy_y_ *b^ùq^ z "_ -«1 ;Q / Œ λ i _^7 / ŒW1f^ f^ly_b -Yz - _ wxœ^lfb^ $f ZY_ $ -YŒYp py b^lyzÿ * _UXW@ - _ - x i _ Œ *«1 _^ / ŒW1f^'UQ Šr * f% -Yz *WZYzŸ *ž5 * b^lyzÿ * _UXW@ - _ - UQf Z f^ly_b * _UQW@ * _ -Œ $Y _^ / _W1b^ *WZYz ŸY_Y_ *rx1^`fb^ α *%^lw@b^ˆ -Y Yz^lYÄÅL¼zÀÃ ¹ µrµù³_³" b^lyzÿ * _UXW@ - _ - «1W1 *bw:_^ OPK 4 F@ p 5K 2, ( Æ3µš¾Ã¹ ³zµšÀÃ¹ ¹Ÿ¾Ã¹ Å Ax i = λ i x i :_ p«9ë9««@w9 *fw Œ $f $ -UXŸ $UP / -šxg / Œ ˆ^` A(αx i = α(ax i = α(λ i x i = λ i (αx i. A = ¼z ¹ Åh³Z¹Ÿ¾X³ZÀ Z¼Zµ 'X³ÅL¼zÀÃ ¹ µù ŽµA?9µ3¼z ¹ Åh³Z¹Ÿ¾ Z 6zµØ¼zÀÃ¹ µ. b^lyzÿ -Yz - _ z? _ _b ZYZ _ b by ½_ ØÇ@ ŽÅh ŽÀ-¼Z¹ µa?9ål X¹ µø³zx9ux b ŒUXW@ * Œ *r * -YzŸ *fw9 *h Z f $ zb^ x i b^lyzÿ $žlž9 $ Y 1«1Ë`«@^@ zux^ b^ly_b -Yz - ZY_ $ -Y_Y WZYzW9 $L Yz žl *UQY %$* " " " -" $ $ " %$ $!!" *% bw_ 1 Œ^ ^lux b Yz * -fÿ *X $Y A S *Y YŒYÎWŒUX * -Yz bž`w1žly Œ fuq ZỲ xz - z^`^ _ b fb^ùx^ Yz * -fÿ *š p[]yz - *bÿ $b^ A W1ž` ZY YZ - *bb * _^ ;^ :_ *f $ _ - -(6:? Z³:?=6Z¾X Ã¾ ¾X³ A S c l :0Œ~ A = S 1 AS. 7 z b * - Œ / YZ Y_Yz^xXUŽ * W1ž` ZYŒY ;^7 -Y_WZYzŸ $b^ ;W9 $r Z $Y Z ZYZ -Y Y_YWŒUX * -Y_ fž`w1žlyî _ bužš S 1 A A A B 1ŸY Yz * -fÿ $U$ $Ỳ x- $Y xz -Yz^` _ -Y_Y_^ ;^% $Y ;^ Œ fuq py_^ä 6 P¹& Z³Z»f XÅlµrµØ 'X³ WŒUQ *W_UX^ :z^!w1 1b py_ fuq^@ *UXb f ŸY B = S 1 AS A B Yz * -fÿ $UQf b ŸY Œ^Ä -Y Y; z bw19 $Ÿ Ë9UŽ -UX b^ly_^œ * *r L _^ f^ly_b $Š

40 Y {Œ [ YZ * -bb *fuq^ _ fÿ *UŽ +Yz bì_uq ` Yz b fÿ $UX _ _ «1 $ /«1 Z YZ * -bb *fw1ž` ZY_ž`W9 $Ÿ * -Y ž`ž9 * *ž YZ * * b^ly UQ f b^1 py_ *ŸYz^@ * *UXb^lYl [ fw _ f YZ - *bÿ $b^ W1ž` ZY YZ - *bÿ $bux^ / * -Y7 Œ^ˆË9ŸYzÌŒ _^lyœyz bf^lux^ YZ - *bÿ $ A Λ x1 X 9 Œ^ Œ fu YŒ * -Y :_^`^Œ f b Š ^`UX^ YZ - *bb * * -UX^5x9UX $ S *Y_^` z Y_Yz^x@UX $ *% Œ^ *YŒY_Ë`YŒYz^ Λ ž`^ UQ *W1f :_^ y ; b by λ λ Λ = 2 0 = S 1 AS, 0 λ n A Œ^ ½Z¹ ³XÇ@¼ZÅh³Z» ¹Ÿ¾X¼z¹ µ? Z³Z % buxÿ *UQ $ * _^ Œ _ YZ - *bb *b^ A Ë9ŸYzÌŒ _^lyœyz bf^lux^w1ž` ZY A n = SΛ n S 1. YŒ * -Y YZ - *bÿ $r? UQ $ - S 1 Ax = S 1 ASS 1 x = ΛS 1 x = λs 1 x. «9 $ /«@ ŒUXW@ * Œ *ŸY S 1 x x` _ ŒbËlY_Yz^ˆW1f š / _f $ Y_Y Λy = dig (λ i y = λy. Yz b _ c ` c ` _ Œ~ c ` ` P~ YZ - *bÿ $b f ŸY S 1 ^1«Œ Œ^ UX -? *ž`ž` -+W1ž`b^ fw :_^ UX bu UX^@ -UXŸ / λ λ i x^lfb^ z "_ -«1 z b "Ä Œ^ py_f^d - *b 1bYŒYz b! YZ *WZY_ Š $ž! Šr * f YZ -WpY_b *ž8 Œ^ Œ fu y 1 = y 2 = = y n Y_ - *Y ZYzb^5xh / Œ b^lyzÿ -Yz - _ = 0 Œ^ / ŒW1f^ λ Ë9bY_Ì_ Œ^lY_Y_ fbux bu UX^@ -UXŸ / λ i.g _Ÿ *f^ *Y_^` 1UX^ [ YZ * -bb *f^ Ë9ŸYzÌ_ Œ^lY_Y_ fÿ *UX^W1ž` ZY Yz * -fÿ $b^ Yz bw@b z A λ i Z ZYZ Î *UX^Ä f^wyzÿ *Y_ * Œ z Q ;UX *UX f^wyz^@ *b^ $Y; _ /UQ^ f^wyzÿ $ž`žlw1 *bux^ UQ *žl $ *UQUX b by Œ^ *% _ 3 ŒUXW@ * Œ *f x i. z b Œf^ det Λ = Tr Λ = b^lyzÿ -Yz - _ 1 _^ λ i bff $ /«1 Z n λ i = det A i=1 n λ i = Tr A. i=1 U = (x 1 x 2... x n = YZ - *bÿ $b^ f^ly_b -Yz - _ _ Q.GY_ *WZYŒ / -UX b byœyz^ A x 11 x 1n x n1 x nn. Yz * -fÿ $ŸY c _~ c l Œ~ UX * * Y_ f ŸY ZYŒ $U Y_ Y * _Ÿ *f^ *Y_^` 1UX^ b^lyzÿ * _UXW@ - _ -fuq^ * AU = (Ax 1 Ax 2... Ax n = (λ 1 x 1 λ 2 x 2... λ n x n = UΛ. YZ - *bÿ $b f ŸY U 1 *YŒY_Ë`YŒYz^ ž` / Y Λ = U 1 AU, Yz * -fÿ *5Ë9bY_Ì_ Œ^lYz bb * _ A ;^5! +*,*.-"/1#%/1 +*,&*4%*,5+/ '+ 2-0/1/1#%*,/15151* 1UXž` Y_Yp ZYŒ * -Y W _ * -UX b " z^ lyz / _ *UX byl YZ - *bÿ $U$ $YÎ^1ž UXUQ *Ÿ $UQ^ ŸY_ *Y_ Ỳ xp $UQW Î Yz -WpYŒ / -UX b byœyz^ YZ - *bÿ $b^ UQ *b by_b *ŸY

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47 {1œ (φ + ψ φ + ψ (f + g f + g (f + g f + g (φψ φ ψ + ψ φ (f g f ( g + g ( f + (f g + (g f (φf φ f + f φ (f g g ( f f ( g (φf φ f + φ f (f g f( g g( f + (g f (f g φ 0 ( f ( f ( f 2 f 7UXW@ * Œ *bux^ Ë9 hux -UX^@ -byœyz bf ŸY_ *W_UQ^`^lYz^ byœ $W1žl :_^Œ z l $Y _UQWŒ - _ -fwœux^@ $ -"l $Y Z ZYZ 7 *WpYŒYz ŸY_Y_ *bw_uq^œ * * $Y φ ψ f g wyœ~ 1«1 bf^@ -UX Œ -Ë`f^lYŒYZ - ρ x φ x z w~ Yz b f ŒW_ 1 _ Ë9b^lY_Yz *f r x θ x φ OPK 4 F@ p 5K 2, 2 ² - ÀÃ¹ µ p 6ZµÙ¼ZÀÃ¹6Œ XÅlµ Å À*¼P¼ZµrµØ¼zÀÃ¹. f = x 2 y 2 z 2 î + y 2 z 2 ĵ + x 2 z2ˆk _ $ - _ -f^ f^@ -ž` -fb 1f^lUX^ *ž` bw1f^@ Y bb * /«@«W_UQ^Œ z^ «_ * *UQb *«1«Œ -UXUQ^5 + $ UX -W1fW9 *^`UQ $ *UQUX^ ^` UXžl *W_UQ^@ z^ - 1 z * * _ - f $ Y_YÄ $UQ^ W1ž` Y_^` UXž` $ -Y W1žl * -YzW1b^ Ÿ / -UXUP * ` h UX^@ $ :lx_ / Œ^`WZY - 1 z $ - _ - Œ^ ^` Œ f ŸỲ xh *Y_^` z Y_Y_^ $hâ ZÀÃµÙ Žµrµ z 6p¾Ã¹ $ " $ $ " " "?Ë9UX b fÿ *UQ - - WZY Yz buxuq - -Y UQ -YŒYZ $ - _ - :_ *f *UQ -fb^ˆwzy_ $ -UXŸ $Ÿ * -Y W_ 1 _ Ë9b^lY_Yz *UXŸ - *Ylx ž9 $ Y žw $UQf^ _^Ä _^lì_ux fš Y_^Ä bž` Œ^Œ -UXUQ^Ä 1ž` ŒW9 $ 1ž` Yz $ Yp py_^ˆ 1«1Ë`«@ b bb $ W p«@ $ * _Ÿ *UX^` ŸYzŸ / Ỳ x9 Œ^`Ì_UQ Yz^ÄÌ_UQ UŽ - *ŸY_Y_^ˆ * b ZY_Y WŒ 1 _ Ë9f^wY_YZ * * z š /UQ $ *UQ :l? $ UX -W@bW9 $ÎYz * UQf b bu $l³z»»b¼ 6Œ¼Q¼zÀ*½Z¹ Åh³_³Zµr¹Ÿ¾ÃµØ¼ Œ^ bž` _^l^` _ b fb^`uq^ ZY_ fb^@ -Y Yz b b zš *«UX * -by_^ˆ 1ž` ŒW9 $rx1 -Y_5Ë9bYz * Ÿ *f b fu _ buxw1«1«1 fuqf b bu ¾ÃÂZ» ¹ ÅwµØ XÀÃ¹6Œ¼Q¼zÀ*½Z¹ Åh³_³Zµr¹Ÿ¾ÃµØ¼z žl pyœ * -Y!l ŒY7 Œ^ UQ *f *UŽ * /«WZYz * *UQUQ *b *UX^WŒ 1 _ Ë9f^wY_YZ -b $ * Œ^ Ÿ / -UXUX^ UP $f /«9 $«1 bf^@ *UQ *bw_ 1 Œ -Ë9fŠ P = (x, y, z ^wy_yz * *bux^ c UŽ zÿ $«1«9 Î *«@ bb^œ -UX -f^ Y_W1 *UX bÿ / YŒ~Ãx cnw1bux * * _W1ž` YŒ~G $Y cnuž zÿ $«1«9 W1bf^`^l -UŽ /«9 / ^` _ b ŸYZ -YŒ / YŒ~ ρ φ z Yp 1ž` b ŸỲ,_ lë:_^5x9už * c l }Œ _~ x = ρ cos φ, y = ρ sin φ, z = z, Ÿ * -Y_Y\Y_ * Œ z x Y_ * Œ z $Y w % Œ *WZY * / _f $ -Y Yz b bÿ _ Œ^dc ` }_ Œ~3 by_žl $W@Š ρ [0, [ φ [0, 2π] z ], [ WŒUXUX Y_fW1WZYp _UQW@ * _ -b^ ŸYzžw $UQW@WŒUXUP $UQUX^ $Y Ë9UX -f Œ _ Yz b by ;^5x ;^ $Y ;^Ä -UXUX^Ä *YŒY_Ë`YŒYz^\ ZY_ $ -YŒYp ZY_ $ * ρ φ z

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51 }9!" $ #" #" LY@~3 9ž` /UX $ -ž f^@ Ỳ x9 w~ Yp Œ _b^ ž` *UX^Ä 1fb ZYzb^@ *UXÌŒ -YŒYz bfuq^\ -Y YzžlW1 *UQ - -Ỳ x «:@?b^@ *UQÌ_ Y_Yz bf + -YŒ bžùxÿë9uq^ «1 f Œ _ŸË`Y_Y_^ ŸY_ *W_UQY ^`bf^ $WZY_ byœyz - Š W1ž`b^ ŒUXW@ * Œ *bw_uq^@ *b f buxw1b^5x` -ž` f ŒW9 $UQ^lY Œ f buq - -Y / _WŒ ˆ *WpY_ byœyz -5 YzV _UQWŒ - _ -š h Î^wYz b _Ì_Ÿ *UQ $ * c ` :@~WZYz^w * -Ỳ x b^9š Yzb^@ *UXÌŒ -YŒYz b _UQW@ * _ - ž`ž9 * *žz /bux^ä«1 f Z pyz V / z -Y_W1f^ $UQž` Y_Yp 1b $ -Y ž` 9Ë9 Œb $ -Y<; c φds, f ds, f ds. S S S T+f^@ -Y xœ / _^`WZY «@ bvf^@ -UXÌ_ Y_Y_ fg byœ $WŒUŽ Y_Yz^x` _ ŒG Œ f ŸY Yp Œ _b^ -YzV *ž` /UX $ -ž c UQ *Wh Lf^@ -UXÌ_ - _b^@ *ŸY *ž` /UX *ž`^ b^9š S ^wyz^]«1 bl $Yz * -Y ~ÃxŒ $Y«1 f z Œ fuq ZYZ ;b^@ *UXÌŒ -YŒYz b % Z pyz ;«1 fuqb *UQ $ *IWZYzW9 *f^`wœux * -Y_b *by b^@ *UXÌŒ -YŒYz bu/ $Yl 7UXW@ * Œ *ŸË9 Š NUQ *UQ^@ *ŸY_Yz b Œ^ S ds ; _ b^œ YzUQ fu UQ^@ *b^ b^œ YZŠÙYz ŸYz^5x ;^5x * -Y_f^ÙQ^] _UQW@ * _ -WŒ *Ÿ *ž` _ Y_ - *Y b^@ -Y_Y pyœ / Y_Y_^5x Y_Yz bb _UXW@ - _ -f^ *ž`ž`^@ -Ý *b ž`žds _^`WŒ b^@ -Y $žl /UX $ *ž cnw1ž9 -UX^ W1ž` ZY_ - *Y l ŒYŒ~ ZYzYp _ Œf^ ds = ˆndS c ` z w~ž 1ž` /UŽ -ž`^ f^l^lyz^d^` Œ Y_Y_ fb _UQW@ * _ $ Œ * -YŒY Yzb^lY 8$ž` zb^@9 b^`^lyz^ *b :_^l *ž` fwœu Y_ $ -Y *b byzš 1ž`žwË9UQ $ -Yl 5 Œ _ UQ^ f^`^wyz^ -Y Yzž`W9 *UQ - *YÄ * -YŒYz^ _UXW@ - _ -f^ Œ * _f $ -Yp ZY_^8$ _bw_uqy_^ W _ËÙX^d - z^`^œ^@9 ž9š WZYŒYz^ b^@ -YŒY YZ $YŒYp py_^ *UQž`^lYzW p«1 z^ W1bUX * * Œ *ž`ž`^l^lyz^ *UQUX^ - Q 1W1ž`^ ŒfWŒUQYz^ W ŒË9UX^ˆ $ Œ UŽ 3 Œ * _f $ YZŠ ZYz W1bUX * $ž`žl^`^lyz^ä -UXž`^lY_W p«@ : -W1f^5x UQž`WZYz $ Œ * -YŒY ŒUXW@ * Œ *b^ˆ *ž`ž`^`^wyz^5 w 9UXW Yp _ _b^ UX $ *ž` /UŽ * *ž f^@ -Y -Y_Yp ZYz " Œ f ŸYÄWpY_ lë9up / - -Y_!žl *UQY Y_ $ *WŒ _^`^luxw@ *bf 1Ÿ $ŸY cn 1 $ P 5W1ž` ZYz^ ` _Y * Œ *žw -~Ž G 1UX^ * $YŒYz^dYp _ Œ Š UQ f ŸY b^`^ly_ f ŸY *žl fuqu; _ b ŸY WpY_W9 $ ž` _ f -Y cnuq _ lfžl $ŠØ^lYzž` wy _^\UQ * UQ *W1W1 f^`^wy_ $ -Ỳ xž / _ b by Œ^\ ZYzb^«1W1 * ž` _ bÿ~ž T+ * _ /b * _f -Y_UQ - *Y b^@ -Y WZYz * *UXŸ *b *UQ *UX b fu Šš Y_ * _ b fuäcnw1ž` ZY l YZ Y_Yz^xlUŽ * S xy ds ;^ * _ /UXW@ *b da Œ^ x Ÿ - - da = cos α ds α _^ W1ž` Y z ŠÙYzW9 *UX bf^d $žlž`^@ -YzŸ *UX^ «1W9 $bw1w:z _UQW@ * _ -b^\ $Y b^`^ly_^ ^` _ Y_Yz bb^ _ fb f ` l 9 -UX^ / _WZYzŸ *UQ - *Y ;^ b $ *UQUQ - - _^ S Œ Œ ^1«@ ds = da cos α = da ˆn ˆk. b^œ Y S Œ _ŸË`Y_Y_^ W1b / Œ * -Y_Y z "_^ f(x, y, z = 0 x Ÿ * $Y ŸY_ *W_UŽ Y_Y_^ b^`^ly_ f ŸY *b^`w_uq $ YzŸ $UQ^lY f f/ z b^@ *UQÌ_ Y_Yz bb^ly Šr -YŒ $ Œ^ÄY_ fžluxuq - *Y,_f^ S xy R ds = f da f da = f ˆk ( f/ z, UX^ÙX *UQ f bux^ f^l^lyz^ ^` Œ b^@ -Ỳ Y_Y_ fh Œ^ ˆn = f/ f $Y c l Œ Œ~ b^@ -Yzb^@ *UQÌ_ Y_Yz bi _ _ŸË`YŒYz^ W1b / Œ * -YŒŶ WZY_WŒŠ OPK 4 F@ p 5K 2, ³Z¾ 6Œ ¹ ÅwµØ ØÇ_À*³_³z» ¹ I = dsèw ¹Ÿ¾ ¾ 'X³ ¼ZÅ $?l¼z» ¹ $l³z»»f¼zå È $L¹ ÅwµØ³-. S f f = xî S x 2 +y 2 +z 2 = 2 z 0 []«:Œ b^œ Yzb^@ *UXÌŒ -YŒYz b ˆ Z pyz N«9 *fbwzy_ - *Y8W_UQ *W_UQb *UQ - d Œ -Y_ - *Yl %c $ zÿ / -Y Yzžw / Y b UX^lUXU UP $ UX -W1fW9 *z byœ $WŒUŽ * -YzUP * -Y b^`^lyz^ WŒ _WŒ _^ly_b * ZYz Yzžl $ -Ylx ρ(rdsx Ÿ - - S ρ _^ ZYz Yzžw / -f ÙQ«9 X.G _Ÿ $ -Y žl z Œ fuq 1bY - *U_ b^œ Yzb^@ *UXÌŒ -YŒYz bu/ $Y%UQ *fb^@ /«1«?l¼z¹ ÅlµÙ rç_à*³œ³z»f X¹Ÿ¾ ¾Ž³zx@ *b f "f^@ *UQÌ_ Y_Y_ f@w1ž` ZY_Y" _UQWŒ - _ -fwœux^@ _^7 1ž` z Y f^l^lyz^ ds

52 ž }_ lt!b^`^ly_^ S - z /UQW@ *b xy Šr -YŒ $ Œ^ÄY_ fžùquxw9 * R ds _^ b^@ -YZŠÙYz bw1f w %$ #" #". f ŸYp 1ž`žl *f^@ *UQÌ_ Y_Y_ f Œ _ŸË`Y_Y_^ z - -UX b " / ]UQ^`^lY_ Y]Yz - ZYZ $ Yp ZYz b bŷ -Yp ZY_ f ŸỲ ÎU Z ZYz pyzš+ $Y f^@ Yzb^Œ -UŽŠ Ì_ Y_Yz bu/ $Y «1W1 *b^`w_uq $ YzŸ $U bylxl *f b -f ŸYp 1ž`žl -Yz bw@b _^d $WZYz ŸY_Y_ *r.gy_ * 1f -Y_Y_^ *fÿ WZYz lë9uq^@ /«@«b *" *b byzš $b^@ *UQÌ_ Y_Yz bu/ $Ylx c l 1œz~ φdv, dv, V V / _Ÿ / Y UPË9UX b bf^ùq^ _^ $WZYz ŸY_Y_ *fšh $Y% z bw@ zb^ùx^ _UQW@ * _ -Ÿ $ž`žl *UŒ ž9 *UQ^ WZY Yz buxup * -Y 9 ` Î - 9Ë9UŽ * *bf^5xœw_ Œ bž` f _ $ -UXŸ $UQ^ Y_ fžùqux^ *b V _^ * fÿ $ * _Í% $UQf^ V dv Œ^` ž9 -WpY_b *UX^ WZY Y_ fuqux^ W1«1 *UXUP * Î Œ f buq - -Y _ f ŸYzŸ *UX^ b^@ *UQÌ_ Y_Yz bb^7 byœ $WŒU b^ùx^ UŽ *UQ^`W1f^ Yz^wYz b«@«@ * *Ÿ $UP / -@ Œ _ V _ b by\ 1«1 1f^ lyz^`wzy_ byœỳ L 9fW9 *G Œ^\ / Œ *W1žl ÎW Z -UX : UQ * * *b byp 1ž`žw $b^@ *UXÌŒ -YŒYz b b^@ -Yzb^@ *UQÌ_ Y_Yz bb^ Yp 1ž` b ŸY ; $Ÿ -_ z^ bux^ùq^ WZYz * *b _^5x1 / Œ^`WZY Œ Z $Y Œ^ f^`^wyz b by ;^ V ds x *b byp 1ž`žw _^ dsx9 $Y]W_ _WŒ ;^ S dv = 1 3 r S $Ÿ - z^l $ž` bw_u Yz^ˆYp ZY_ *ž`žwë9ux^ˆ *b byp 1ž`žw _^ $bb V = 1 3 r ds. S. z^\ -ž` b _W9 $UQ^ z -W_UX«9 + _^ W1ž`f *UQ^`W1f^ ž`žwyz b by *U%^` * _f^ Yz lë9 Œ f bÿ / Y_Y ÌŒ -YŒË9fUQ^@ *b^ c l {@ Œ~Ãx1Ë9b _UQ $Š Ì_UX^w * *f^ c ` { 0@~I $Y - 1 z $ - _ -f^ c ` }9 P~UQ * * Ÿ $UQ^]W_ 1 Œ -Ë9b^lYŒYZ $ - _ /UP / -UX _ $ *b YZ * * b^ly\ìœux UX $Š *Ÿ $b^ly by_žl $UQW1W_UXb^lY GYzžl *UXW1W_UQbËÙX^] ZYz *f^ly_f^lux^ / * Œ `f *UŽ Y_Y_^5 5 1WZYz ŸY_Yz -bw_ux^@ z^ $YÄ _UQW@ * _ -bw_ux^@ z^ φ f Ì_ Y_Ë9bUX^@ $ -šx`ë`f ŒUX -Ì_UX^w * *_ $Ÿ - 1 z * * _ -h _ ŒbË`YŒYz^ˆW@b š / _f $ -YŒY f^@ *UQÌ_ Y_Y_ fuqfw9 * c l Œ Œ~ φ = lim ( 1 φds V 0 V f = lim ( 1 V 0 V f = lim ( 1 ds f. V 0 V f ds c l 0Œ~ Ø - *U?Y_ *ŸY_ - *Y YZ -U Y_Yz $ -b *UQ - *Y W1b $Y_ f bÿ $ž`žwë9uq - *YÎUQË9UQ f bb *UŽ I Z ZYZ ÌŒ -YŒË9fUQ^@ *b^5xqë9b _UQ *ÌŒUX^l - *f^3 $YÎ - 1 z * * _ -b^ *f *UQ Ë` hux -UX^@ *ŸY_Y_ fb ŸYzžl *UXW1W_UQbËÙX^ˆ * $Y_ $ -Ỳ - #" $ $'! %$'- " $ c l œz Œ~ ZŠ 7¹& Z ŽÀrÇŒ XÅ`¾ ¾Ã¹»f³:?1¾X? bff $ : - _Ÿ $bf^w *Y f^`^wyz^ UX^ÙQ _^ _UQW@ * _ -bw_ux^@ _^ $Y7 ŒUXW@ * Œ *bw_uq^@ z^ Ë9b _UQ $Š S Ì_UX^w * *f^ä ŸY_ *W_UŽ * *ž`^wy ;^ $Ÿ :z^w - $žl fwœu Yz^ *b byp 1ž`žlËÙX^ «1 ff - *U>; S c l œ9 P~ V fdv = S f ds.

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