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i(M). $C=:F8O i ?@8=8<05B 7=0G5=8O, 70:;NG5==K5 <564C =C;5G:0 M(x,y) =0E>48BAO =0 i-9 1C:25 8 @02=0 =C;N, 5A;8 M =0E>48BAO 2=5 1C:2K.  EMBED Equation.3   8A. 8.6. CABL <K C<55?@545;8BL 7=0G5=8O i 4;O :064>9 B>G:8 2=CB@8 ?@OC3>;L=8:0 LxH. @54?>;>68 =0 @0AAB@5=85 ?@54AB02;5=0 (7040=0 =0 ?;>A:>AB8)=5:>B>@0O 1C:20 Lx, >B=>A8B5;L=> :>B>@>9 B@51C5BAO >?@545;8BL: 0) ?@8=04;568B Lx =0H5<C 65AB2C 87 N 1C:2 8;8 =5B; 1) 5A;8 ?@8=04;568B, B> A :0:>9 87 1C:2 55 A;54C5B >B>645AB28BL. ! MB>9 F5;LN ?@>872545G:0 70 B>G:>9" =58725AB=>9 1C:2K A> 2A5<8 8725AB=K 7=0G8B, GB> 9 B>G:5 1C:2K Lx 1C459 65 B>G:5, => 4;O 1C:2K Li. "0: :0:  5ABL DC=:F8O B>G:8, B> 4;O B>3>, GB>1K A@02=8BL 425 B0:85 DC=:F88, =5>1E>48 A@02=8BL 8E 7=0G5=8O 2 :064>9 B>G:5. >A:>;L:C <K 8=B5@5AC5<AO ;8HL A0<8< @07;8G8O, 0 =5 7=0:0B45;L=KE @07=>AB59, 1C4572>48BL :064CN @07=>ABL 2 :204@0B, 0 70B5< AC<20BL. -BC ?@>F54C@C 6=> >B@078BL 0=0;8B8G5A:8, A>AB0282 2K@065=8O:  EMBED Equation.3 (*) 345: x(x,y) 8 i(x,y)  7=0G5=8O x 8 i 2 >4=>9 8 B>9 65 B>G:5 M(x,y), 0 8=B53@8@>20=85 ?@>872>48BAO ?> 2A59 >1;0AB8 (LxH). >6=> A:070BL, GB> 1C:20 ?@8=04;568B =0H5<C 65AB2C 8 55 A;54C5B >B>645AB28BL A 1C:2>9 Lj 5A;8 Dxj = 0.  A048=0:>2KE 1C:2 2 :064>9 B>G:5 @02=K, 8 25AL 8=B53@0; >1@0I05BAO 2 =C;L. B0:, ;CG8;8 :@8B5@89 >?>7=020=8O 4;O A;CG0O 8450;L=KE 1C:2. K 2848<, B0:81@07> 7040G0 >?>7=020=8O @5H05BAO 2 B@8 >A=>2=KE MB0?0: 0) >?@545;5=8O 7=0G5=8O DC=:F89 (x,y) 2 :064>9 B>G:5 1C:2K. 1) A@02=5=85 =58725AB=>9 1C:2K A> 2A5<8 8725AB=KA@54AB2>AB59 7=0G5=89  4;O 42CE 1C:2 (2KG8A;5=85 45280F89 DC=:F89 A>3;0A=> 2K@065=8O645=85 A@548 2KG8A;5==KE 7=0G5=89 =C;52>3>. $C=:F8N  8<55B A20BL :0: AB5?5=L 70G5@=5==>AB8 (:>MDD8F85=B >B@065=8O) 15A:>=5G=> 9 ?;>I04:8 (B>G:8). =>340 >:07K205BAO 2>76=K;L:> 425 AB5?5=8 70G5@=5==>AB8: 15;>5 8 G5@=>5. ">340  1C45B 8;L:> 420 2>76=KE 7=0G5=8O 0 8 1, A;54>20B5;L=>, >=0 1C45B 42>8G=>9 DC=:F859.  MB>G:0 i-9 1C:25 8;8 =5B 8 65B 1KBL =0720=0 ?>MB>9" DC=:F859. >AB@8 1C45B >7=0G0BL B>340 2K@065=85 (*); "0: :0: 8=B53@0; =5 @025= =C;N B>;L:> B0<, 345 8<5NBAO M;59 8 >BACBAB2CNB M;5@>9 1C:2K 8 =0>1>@>B, B> Dxi 1C45B @02=> AC<I048 1C:2 Lx, :>B>@0O O2;O5BAO >1I59 A Li 8 ?;>I048 1C:2K Li, =5>1I59 A Lx. /A=>, GB> 5A;8 425 1C:2K >48=0:>2K, B> >=8 848=0:>2K5 >G5@B0=8O 8, A;54>20B5;L=>, Dxi = 0, (x = i). 5;8G8=0 ?;>I048, =5>1I59 4;O 42CE 40==KE 1C:2, O2;O5BAO, :0: <K 2848<, 206=59H8@>AB0284>1=K5 (*), 4;O 2A52>76=KE ?0@=KE A>G5B0=89 87 =0H8E N 1C:2:  EMBED Equation.3  '8A;> B0:8E A>AB>O=89 1C45B, >G5284=>  EMBED Equation.3  Dij E0@0:B5@87C5B 25;8G8=C @07;8G8O ACI5AB2CNI53> <564C 1C:20<8 Li 8 Lj. '5;LH5 @07;8G0NBAO 1C:2K ?> A2>59 :>=D83C@0F88, B5;LH5 C =8E =5>1I0O ?;>I04L Dij. -B> >7=0G05B, GB> 2KG8A;82 7=0G5=8O Dij 4;O =0H53> 0;D028B0, 6=>, 27O2 =08G5B0=8O =081>;55 ?>E>68E 1C:2, A0?0A=K5" 9AB25 A;54C5B >1@0B8BL >A>1>5 2=8G5B0=8O. A;8 27OBL >4=C 1C:2C 2 >B45;L=>AB8, 0  @0AA8G=CN DC=:F8N:  EMBED Equation.3 . B;8G85 i-9 1C:2K >B 1C:2K (0), 7=0G5=8O  :>B>@>9 2 :064>9 B>G:5 5ABL =C;L, >?@545;O5BAO 25;8G8=>9 ?;>I048 i-9 1C:2K. =0;>38G=>, G5@=0O 1C:20 =0 15;>=5:  EMBED Equation.3 . > A8E ?>@  @0AA<0B@8205<K5 1C:2K 8@=5G=>, @07;8G=K5 >BB8A:8 >4=>9 8 B>9 65 1C:2K >B;8G0NBAO 4@C3 >B 4@C30, 8=>340 4>2>;L=> 7=0G8B5;L=>. # @50;L=>9 1C:2K 3CB >BACBAB2>20BL =5:>B>@K5 M;5 (=5?@>?5G0B:8) 8 3CB ?>O2;OBLAO =5:>B>@K5 ";8H=85" CG0AB:8  3@O7L. !@02=89:  EMBED Equation.3 , 345  EMBED Equation.3  DC=:F8O i 4;O 8450;L=>9 1C:2K. 5;8G8=0, >?@545;O5A;54=8?@545;5==K1@07>20BL "?><5E8" 8;8 "HC7=8:0NI85 ?@8 GB5=88. A;8 25;8G8=0 Dij (4;O 8450;L=KE 1C:2) E0@0:B5@87>20;0 @07;8G85 @0AAB>O=89 <564C @07;8G=K<8 1C:20 Dii E0@0:B5@87C5B >B:;>=5=85 @50;L=>9 1C:2K >B 8450;L=>9 70 AG5B A;CG09=KE HC2. 0AAB@8< 45280F8N :0: DC=:F8N 42CE ?5@525B ?0@05:F859 1C:2K =0 >AL Y :  EMBED Equation.3 . !>>B25BAB25==>, ?@>5:F859 1C:2K =0 >AL X 1C45B 8=B53@0;, 7028AOI89 >B ?0@0, GB>  EMBED Equation.3 . "0: :0:  EMBED Equation.3 , B> ?@8  EMBED Equation.3   EMBED Equation.3 , B.5. Fi(x) = C. "0:81@07><, B0< 345 =0 A09 1C:25 =5 ?@>8AE>48B 87;L >A8 X, 8E =5B 8 2 ?@>5:F88. 0>1>@>B 87=D83C@0F88 ?@>5:F89 A>>B25BAB2CNB 87<5=5=8O20B5;L=>, ?@>5:F88 1C:2 3CB 1KBL 8A?>;L7>20=K 4;O >?>7=0=8O. 8.4.2. @0:B8G5A:0O AE5340 (x,y) O2;O5BAO 42>8G=>9 DC=:F859, F(x) 2K@0605B 70:>= 87I048 1C:2K 24>;L >A8 X. =0;>38G=> F(y) 5ABL 70:>= 87I048 24>;L >A8 Y. @C3820;CG05BAO 2 @57C;LB0B5 AC<20=8O 2A5E 7=0G5=89 (x,y) ?@8 D8:A8@>20==>< x0, 70B5B>@>2>;CG5==0O B0:81@07>4=>7=0G=>9, B.5. 65B 1KBL 2K@065=0 2 2845 S = F(x). ( 8A. 8.7).  8A. 8.7. @8 8A?>;L7>20=88 ?@>5:F89, 2708 >4=>7=0G=K1@07>< A2O70==KE A 1C:20;CG05BAO 2K83@KH 2 :>;8G5AB25 42>8G=KE M;52, =5>1E>48AB@ 2A53> ?>;O, =0 :>B>@>5 =0=5A5= A8;, ?@>872>48BAO 48A:@5B=>, B.5. ?> AB@>:0;1F05:F8O 70405BAO A>2>:C?=>ABLN A2>8E AB>;1F>2 (AB@>:):  EMBED Equation.3  345 n  G8A;> AB>;1F>2 (AB@>:). 0B5872>48BAO :20=B>20=85 ?> ?@028;C  EMBED Equation.3  345  EMBED Equation.3   ?>@>3>2>5 7=0G5=85 4;O 40==>3> AB>;1F0 ?@>5:F88. 20=B>20=85 ?@>872>48BAO :20=B>20B5;5B>@K9 ?>40NBAO A>>B25BAB2CNI85 C@>2=8 >B 1;>:0 MB0;>==KE =0?@O65=89 - ( 8A. 8.8.). "450;L=K5" 7=0:8, A :>B>@K872>48BAO A@02=5=85, 70?8AK20NBAO 2 1;>: ?04>2 ?@>5:F89 >?>7=02053> 7=0:0  EMBED Equation.3  A "8450;L=K872>48BAO ?>A;54>20B5;L=>, @07@O4 70 @07@O4> =5A>2?045=89 a A :064K2.  8A. 8.8. 284C B>3>, GB>  65B ?@8=8;L:> 7=0G5=8O 1 8 0, >?5@0F8O 2KG8B0=8O 8 2>72545=8O 2 :204@0B 42CE  >ACI5AB2;O5BAO ?CB565=8O ?> 4C;N 2  EMBED Equation.3 . >4AG5B G8A;0 =5A>2?045=89 2545BAO AG5BG8:>< !G0. 0;55 A?5F80;L=0O AE5?@545;O5B <8=8 0 8 2K405B 53> =0 AE540 AE5;CG05BAO A83=0;, :>B>@K9 65B ?@8=8MDD8F85=B ?@>?>@F8>=0;L=>AB8. A;8 Z = 1, B> @5H5=85 2K=>A8BAO 2 ?>;L7C B>3> 7=0:0, A :>B>@K;CG5=> amin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