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 � 8Ê Ð �o�Ê Ä � � ; �:Q7ATS�Q�:AT�C�&�¤E�57U_E�Y�I_N�S�N^K�B75RILE)I_Q7E�U_E*S�N^67Y�]H<>G_A�<�ILACNH67U*�P�<H�qAT678�57U_E+NHO�ILQ7E}<>��N~]^E à º Ç × ¼ V&:�E}AC6MI_G_NmY757S*E0Ia:«N�Ia`mB¤E*U�NHO�ILE�UZI)O�576RS�ILATN^67U�:Q7ACS�QÍK�ACK�ACSILQRE\K�N~]^E�K�E�6qI_U$N>O�I_Q7E�AC6MILE�GZO�<HS�E*U*·

@ � � D�� ( D���� à º Ç × ¼ � � � � DHG D (m%i��� � ( D (m% D � � � �SD º B¢AC8^5RG_E � � ¼

l x −l l x

δ

U Fx x

−lB¢AT8^57G_E � �

� à º Ç × ¼�½ �CATK� � Á à º Ç Ð �� × Ä �� ¼ Ð Ã º Ç × ¼á ½ Ð Ã�à º Ç × ¼Û Ä Ã � º Ç × ¼Û â��Q7E�6 � U_<>ILATU 0FE�U�ILQ7E)O�N^�T�CN~:AC678+E*UZI_ACK�<>ILE�U*·

� à º ÐX × ¼�½2É × � à º × ¼�½ Ê 8 Ê Ð � �Ê Ä � � ;¾ �ß �  � � �Ã�Ã Ä � �à ÆMÇ ½ Ê

 ºsÊ Ð¥à � � Ä � º � � ¼Z¼ º��MÛ^¼� º × ¼ Ð � º Ðb × ¼ ½ Ê Ð£à � � Ä � º � � ¼ º��S�M¼

� *

@ � � Dq@�� � ��� à º Ç × ¼ �7@ � � ( " � � %�#'� (���� @�� � � ��� � �SD (m% (m% D � � � �SD " º B¢AT8^57G_E � � ¼

−l l x

δδ−1 l x

U Gx x

B¢AT8^57G_E � �� à º Ç × ¼/½ �TACK� � Á à º Ç u�á × ¼ Ð Ã º Ç × ¼á ½ u Ã�à º Ç × ¼� ATK�AT��<HGZ�T`HV � UL<>I_ACU 07E*U*·

� à º ÐX × ¼�½ u Ê Â 8 Ê Ð�� �Ê Ä � � ; × � à º × ¼�½�� Ê Â 8 Ê Ð � �Ê Ä � � ;¾ �ß �  � � �Ã�Ã Ä � �à ÆMÇ ½ Û ºsÊ Ð Û � � Ä � º � � ¼Z¼ º�� à ¼

� º × ¼ Ð � º Ðb × ¼ ½ É º����^¼���7ATIªNHORK�NHI_AT]><>ILATN^6 O�N^GªI_Q7E�O�N^�C�TN~:AC678�<>6F<H�T`mUZACUªATU©AC6�B7��<>S*EH� BFG_N^K º��MÛH¼ <H67Y º�� à ¼ V~ILQ7E�67E�8M<>ILA�]mATIa`NHO�ILQ7E0U_E*S�N^67Y�]><HGZA�<>I_ACN^6RU�S*N^K�E*UO�G_N^K Ð �o�~� � N�N^57G�8^NM<H�©ATU�ILN�S�Q7<HGL<HS�ILE�G_AC��E �o� <HU;<>S*S*5RGL<>I_E*�T`�<>UB¤N^U_UZAC�7�TEH�«¶�I�IL5RG_67U$N^5RIILQF<�I A�O�<�U_Q7N^GJI UZE*8^K�E*6MI ACU�<HY�È_<HS�E*6MI�ILN�<��CNH678�UZE*8^K�E*6MI*V¤I_Q7E*6�ATILU�TE*678HI_QS�<>6767NHI)�¤E}<HGZ�7ATI_GL<HGJ`^�)��Q7ATU;ACU Y757E+I_N�U_N^K�E�AC6MI_G_ACS*<>ILE0AC6MILE�GL<HS�ILATN^67U �¤E�Ia:�E*E�6�I_Q7E}�CNH678�<H67Y�UZQ7N^GJIU_E�8^K�E*6MILU���{|E$:AT�C�X57UZE;ILQ7E$G_E*UZ57�TI_U=NHO ����� �!� "$#&%�#'� (�) +���ILN+B7G_N~]mACY7E$U_N^K�E �TN~:«E�G=��NH5767Y7U«O�NHG�ILQRE]><H�C5RE*U�NHO �o�~�¢¶�I/ACU�5R6RO�N^GZI_576F<>I_E�ILQF<>I�I_Q7EB7G_E�U_E�6qI=<HB7B7GZNM<HS�Q�G_E��m5RACG_E�U�ILQ7ES*NH67U_ATY7E*G_<>ILATN^6�NHOª<)�C<HG_8^E6q57K��¤E*GNHO�S�<HUZE*U��{|E\6RNË:�B7G_NmS*E�E*Y�ILN�<H6F<H��`m�*E;ILQRE\B7<>I_I_E*G_6RU � � �|<>67Y � � ���

�,+ � +�� (E"�% ��� # � #&% ����� � � �{|E\GZE�O�E�GI_N'B¢AC8^5RG_E � � �«N^6RU_ACYRE*G�ILQ7E�B¤N^AC6MI �0�s�` º_�HÛ^¼ <H67Y º�� �^¼ VR:�E0Q7<~]HE

� Á ½�Ê Ä Â � Á Ä ��º  � ¼ <H6RY / Á ½�Ê Ä ��º  � ¼�â º�� *M¼���

[$UZAC678 º��S�^¼ <H67Y º_� *M¼ O�N^G=ILQRE\BRG_N^BF<>8M<>ILATN^6�K�<HB º�� Á × � Á × / Á ¼ Ð c º�� ÜÁ × � ÜÁ × / Ü Á ¼ VF<>I ��V7:�E0Q7<~]HE� ÜÁ ½ Ê Ä º � Á Ä � Û Ð Â ¼ �¥Ð  � À Ä ��º � � ¼ Ä ��º  � ¼ º����^¼/ Ü Á ½ Ê Ä º � Á Ä � Û Ð Â ¼ � Ä ��º � � ¼ Ä ��º  � ¼�â º�� �M¼

4$EÌ�mI�S*NH67U_ATY7E*G�ILQ7E�ILG_ATB7�CE�U º � ÜÁ × � ÜÁ × / Ü Á ¼ <>67Y º�� À × � À × / À ¼ <HUZU_NmS*A�<�ILE*Y�:ATILQ-ILQ7E�B�NHAC6MI �����©E�I�o� ½ � ß � � �` º��`�M¼ <H67Y º+* �^¼ VRI_Q7E)O�N^�C�TN~:AC6780Q7N^�CY&·

� ÜÁ ½ / Ü Á Ð Â � À Ä ��º  � ¼ º�� �M¼� À ½ / À Ð Û / À � � Ð Â � À Ä ��º  � � ¼ Ä ��º  � ¼ º��^ÉM¼:Q7ATS�Q�I_N^8^E�I_Q7E*G�:A�ILQv� ÜÁ ½ � À <H6RY�/ Ü Á Ä / À ½ Û �CE*<HY�ILN/ Ü Á ½ Ê Ð � � Ä � º � � ¼ Ä ��º  � ¼ º��RÊ˼/ À ½ Ê Ä � � Ä � º � � ¼ Ä ��º  � ¼Ìâ º��MÛ^¼�`�UZ57�7UJILATI_5RILAT678 º��MÛH¼ AC6MILN º��HÉM¼ <H67Y@S*NHK�BF<>G_AC6R8 º��RÊ˼ <>67Y º � �M¼ VR:«E\QF<�]^E

� À ½ Ê Ð � � Ð Â � À Ä � º � � ¼ Ä ��º  � ¼ º��S�M¼� � ½ Ð 8 � Á Ä � Û Ð Â ; � Ä ��º � � ¼ Ä � º � � ¼ Ä ��º  � ¼ º�� à ¼

{|E\6RNË:�S*N^6RU_ACYRE*G�ILQ7E)O�N^�T�CN~:AC6R8�S�<HUZE*U��� � " D ��� � � ���H·�<>U_U_5RK�E � Á = Ð � � ��� �` º� à ¼ VoILQ7ATU<HU_UZ57K�BoILACNH6��TE�<HYRUI_N

� � Ø 8 Û� Ð ÊÛ<; � � Ä � º � � ¼=Ø ���� â º����^¼

4$N~:¿]H<>GZ`�I_Q7E�BF<>I_I_E*GZ6��M`�K�N~]oAT678}ILQ7E�AT6qI_E*GJO�<HS*E�U$<�I ��<>67Y �®<HUUZQ7N~:6�AC6 B�AC8^57GZE � �

l

Variation

B CAEL 1

R

R0

1

L

u(x)

A B Cl

Variation

LE

(R , P , Q ) (R , P , Q )0 0 0 1 1 1

L 1

xu

B�AC8^57GZE � �� �

��Q7ATU�S�N^G_GZE*UZB�N^6RY7U�I_N}ILQ7E)ILE�UZIO�5767S�ILATN^6ª·� à ½ Ó Ô Õ

� à � Ç Ð � Þ �� × � ß �� � Çq� º - × � ¼� à � Ç Ð � Þ �� × � ß �� � Çq� º � × � ¼Ð � À Þ�� �À ß�� � � � à � Ð Ç Ä � Þ �� × ��ß �� � Çq� º � × a ¼ º�� *M¼

º¸� E*E)ILQ7E�GZE*K�<HG_��<�I�ILQ7E�E*6RY�NHO ILQRACU�S�<>U_EH� ¼ 4�N~:�ILQRE��q5F<>6qI_ATIa` � º�» × � ¼ E��q5F<H�TU*·¾ �� º  � � �Ã�Ã«Ä � �à ¼ Æ^Ç Ð Û �;�à º � ¼Ò » Ã�à º � ¼~Ò Ð Û �)�à º � ¼Ò » Ã�à º � ¼~Ò½ ¾ �� º  � � �Ã�Ã Ä � �à ¼ Æ^Ç Ä ¾ �

�º  � � �Ã�Ã Ä � �à ¼ Æ^Ç Ä 8 Ê Ä �o�Ê Ð � � ; � ¾ �� º  � � �Ã�Ã Ä � �à ¼ ÆMÇÐ Û Ê Â

Ê� Á Ð Û Ê ÂÊ� À½ Ê

 ÄÛÂ Ä 8 Ê Ä �o�Ê Ð � � ; � Ê Â ºsÊ Ð§à � � ¼Ð Û Â

ÊÊ Ä Â � Á Ä ��º  � ¼ ÐÛÂ

ÊÊ Ð � � Ð Â � À Ä � º � � ¼ Ä ��º  � ¼½ Ð Û �o� Ä Û Â � Á Ð Û Â � À Ä � º �o� ¼ Ä ��º  � ¼Â º����^¼:Q7E�G_E�AT6�I_Q7E�<H�¤N~]^E0S*N^K�B75RIL<>ILATN^6ªVF:«E0QF<�]^E�5RU_E*Y º � *M¼ <H67Y º��S�M¼ �.�`@ILQ7E\�CN~:�E*G �¤N^576RY º����H¼ O�NHG� � <H67Y@ILQ7E)O�<HS�IILQF<>I Ò � À Ð � Á Ò>½ � Ä � º � ¼ V7:�E�QF<�]^E

� º�» × � ¼ = Ð ����  ⠺�� �M¼4$N~:�� º Ç ¼ � Ê O�N^G Ç�� º � × � ¼ <H67Y � É O�N^G Ç*� º - × � ¼ ��º � × a ¼ � � N �� �;� Æ^Ç ½ � Ä � º � ¼ �9$E�67S*E�:�E0QF<�]HE

� � ¹=º�» × � ¼�½ � º�» × � ¼ Ä ¾ �� � � Æ^Ç ½ Ð ���� Â Ä � Ä � º � ¼=Ù�É��B7GZNË]mAT678}ILQF<>II_Q7E�BF<>I_I_E*GZ6 � � � ��ATU�5767UJI�<H�7�TEH���L��i^`N �J^ ��Y\ktV V&KML�V�LZT�VFl�kt[<W�V5R_Y\[�� z ��� | �Ê ���Q7E/E��mI_GL<$K+57�TI_ACB7�TACS�<�ILAT]HE�O�<HS�ILN^G©O�N^G � à AT6\I_Q7E�UZE*8^K�E*6MI�NHO º � × a ¼ ACU©ILN�K�<>�HE�� à S�N^6MILAC6q57NH57U:A�ILQÍ<�S*N^K�K�N^6�]><H�C5RE�NHO À � <�I � <H67Y ���0¶a6ÍI_Q7ACU$:=<�`^V©I_Q7E}S*N^6MI_AC6q57ATIa`�NHO� Q7NH�CY7U;5RBÍI_NE�G_G_NHGI_E*GZK�US*NH67U_ATUZILAT678�NHO � ß� � :Q7ACS�Q�<HG_E)ILG_<H67UZS*E*6RY7E*6MI�<>�C�T`@UZK�<>�C�&S�N^K�BF<HG_E�Y�:A�ILQ � � <>67YQ7E�67S*E\S�<>6��¤E)AC8^6RN^G_E�Yª�

� �

Û �.�` º��S�^¼ V � º � ¼ Ð � º a ¼�½´Ê Ð ��º �o� ¼ �½´Ê V7UZN�<H6@E��mILG_<�BRACE*S�E)NHO B�E�GZI_57G_�F<�ILACNH6�67E*E�Y7UILN}�¤E<HYRY7E*Y�ILN � à AT6�N^GZY7E*G«I_N}K�<>�HE � º - ¼�½�É <H67Y�� º a ¼�½�É ����Q7ACU�S*<H6��¤E;Y7N^67E �q`�S�Q7NmN^U_AT678<H6�<HBRB7G_N^BRG_A�<�ILE;O�5767S�ILATN^6 � à º Ç ¼ UL<>I_ACUZO `mAT6787·� � à � � = ��º �o� ¼

�× � � Ã�à � � = ��º �o� ¼

� � <H67Y UZBRI º � à ¼�� º � × � ¼Ìâ4�NHILE)ILQ7<>I � � à � � ACU=ILG_<H67UZS*E*6RY7E*6MI�<>�C�T`�U_K�<H�C�ª6RE�<HG=I_Q7E�S*E*6MI_E*G�G_E*8HACN^6@NHO ILQ7E>�UY\[Hr;U_E�8^K�E*6MIº � × � ¼ VRILQRE\E�G_GZN^GAC6MILGZNoY75RS*E*Y��M` � à S*<H6���E��¤N^5767Y7E�Y��M`¤·¾ �

�  � � �Ã�Ã=Ä � �à Æ^Ç =  � � � �� � � � Ä � �� � � � ½ Ê � º � � ¼Ìâ

�«N^K�BF<HGZE*Y�:ATI_Q�ILQ7E}EÌ�RBRG_E*UZU_ATN^6 º���^¼ O�N^G � º�» × � ¼ V&I_Q7E�<H�¤NË]HE�ACU :A�ILQ7AT6"ILQ7E�GL<>678^E}NHO�<HSÌ�S�E*BRIL<H�7�CE�E�G_G_NHG=O�N^G�N^57G�<H6F<>�T`mU_ATU*� 9�E*67S�E\:�E0S*<H6�AC6@E�¦�E*S�I$AT8^67N^GZE � à �{|E�:AT�C��67NHI�G_E�B�E*<>I�ILQ7E\<H�¤N~]^E\G_E�K�<HGZ�qUO�NHG=ILQ7E�G_E�K�<>AC67AT678}<H6F<>�T`mU_ATU*�� � " D ��� � � ��� � <HUZU_57K�E � Á Ø Ð �� ��<H67Yi/ � ½�Ê Ä ��º  � ¼ � º ��Q7E�GZE�<HUZN^6}O�N^G�AC6MI_G_NmY757S*AT678 I_Q7ACU�<>67YILQRE O�N^�C�TN~:AC678\S�<HUZE*UACU/ILQF<�I=:«E)K�AC8HQqI�67NHI=QF<�]^E º��S�^¼ � ILQ7ATU=S�<H6@<>S�IL57<H�C��`�QF<HB7B¤E*6 º UZE*E�� DH� %�#5� (� ¼ �)��Qq57U):�E�S�<>6767NHI�Y7ATG_E�S�IL��`�Y7E*Y75RS*E�ILQRE}UL<HK�E+67E�8M<>I_AT]mATIa`�O�N^G � º�» × � ¼ <HU;AT6 º�� �M¼ �)� N�I�<HS��q�TEILQRACU*Vo:«E�O�57GZI_Q7E*GS�N^67U_ATY7E*G�I_Q7E�BF<>I_I_E*GZ6�I_N�I_Q7E�G_AT8^QMINHO � ���=� ¼

B¢ATG_UZI*V�ÈZ57UJI�I_Q7E\<HU_UZ57K�BRILATN^6@N^6 � Á <>67Y�I_Q7E)O�<HS�I�I_QF<>IX = c � ATK�B7��`¤·8 � À Ð Û Ä Â ; ½ 8 � Á Ä � Ä � º � ¼ Ð Û Ä Â ; Ø 8 � � Ð � à Ä � º � ¼ ; Ø �sÊ�� º�� �M¼�«N^67UZACY7E�G�ILQ7E�B¤N^AC6MI.a���¶a6�]mACE�: NHO º��mÊ˼ <H67Y º��MÛH¼ V7A�I�Q7N^�TY7U=ILQF<�I�·

/ Ü À ½�Ê Ä � � Ð 8 � À Ð Û Ä Â ; Ä � º � � ¼ Ä ��º  � ¼Ìâ ºsÊ~É^ÉM¼� AT67S*Ep/ Ü À ½ÑÛ Ð / � :QRACS�Q�E��q5F<>�CU Ê Ä ��º  � ¼ �M`�I_Q7E}<HUZU_57K�BRI_ACN^6"N>O�ILQ7ATU S�<>U_EHVªI_Q7E}<H�¤N~]^E��TE�<HYRUILN}I_Q7E)O�N^�C�TNË:AT678��TNË:�E*G�¤N^5767Y&·

� � ½ 8 � À Ð Û Ä Â ; Ä � º � � ¼ Ä ��º  � ¼=Ø �sÊ�� ºsÊ~ÉRÊ˼4$N~:�57U_AT678}ILQ7E�UL<>K�E)ILE�UZIO�5767S�ILATN^6 º�� *M¼ <>67Y�GZE�<HUZN^67AT678�<HUAC6 � � ���>V7:«E�QF<�]^E>·

� º�» × � ¼ ½ Ð Û �m� Ä Û Â � Á Ð Û Â � À Ä � º �m� ¼ Ä ��º  � ¼Â = Ð � �  º 67NHILE>· �  ¼à É

��Qq57U�ATIQ7N^�TY7U=ILQ7<>I� � ¹=º�» × � ¼ = Ð � � Â Ä � Ä � º � ¼=ÙÚÉU_N�ILQ7<>I�ILQ7ATU�S�<HUZE0ATU<H�CUZN�5767UZIL<H�7�TEH�

� � " D ��� � � �E� � <HUZU_57K�E � Á Ø Ð � c� <H67Y+I_Q7E*GZE=ACU¢< U_QRN^GZI�UZE*8^K�E*6MI º a × � ¼ I_N$I_Q7E=G_AT8^QMI¢N>O º � × a ¼ ���Q7E\S*N^6 0F8^57G_<>ILATN^67U�NHO » <H6RY » à <HGZE\UZQ7N~:6�AT6 B�AC8^57GZE Ê~É �u(x)

L l k

A B C DR

R

0

1

E

Variationl kB C DA L

(R , P , Q )(R , P , Q )1 1 1 2 2 2

uxVariation

B¢AT8^57GZE Ê~É ��©E�I Ò a � Ò>½�� <>67Y�� �=½ � ß�� � � �«NH67U_ATY7E*G�ILQRE�B�N^AT6MIsa <H8M<>AC6ª�¢[$U_AT678;ILQRE º��HÉM¼ <>67Y º�MÛ^¼ <>U_U_NmS*AC<>ILE�Y:ATI_Q�ILQ7E�B7GZN^BF<H8M<�ILACNH6�K�<HB�O�GZN^K �.ILNia�V7:«E�QF<�]^E

� ÜÀ ½ / À Ð Û / À � � Ð 8 � À Ð Û Ä Â ; Ä Â � � Ä � º � � ¼ Ä ��º  � ¼½ Ê Ð � � Ð 8 � À Ð Û Ä Â ; Ä Â � � Ä � º � � ¼ Ä ��º  � ¼ ºsÊ~ÉMÛ^¼¶a6�<HY7Y7A�ILACNH6ªV7<HBRB7�T`mAC6R8�ILQ7E�K�<HB�O�G_N^K a3ILN � <H67Y��q` º+*MÛ^¼ VR:�E0QF<�]HEH·

� � ½ / � Ð Û / � � � Ä Â � � Ä � º � � ¼ ºsÊ~ÉS�M¼�«N^K�BF<HGZAC678 ºsÊ~ÉMÛ^¼ <H6RY ºsÊ~ÉS�M¼ <>67Y�5RU_AC6R8 ºsÊ~É^ÉM¼ <>67Yv/ Ü À Ä / � ½¿Û 8HAT]^E>·

/ � ½ Ê Ð � � Ä 8 � À Ð Û Ä Â ; Ä � º � � ¼ Ä ��º  � ¼Ìâ ºsÊ~É à ¼� � ½ 8 � À Ð Û Ä Â ; Ä � º � � ¼ Ä � º � � ¼ Ä ��º  � ¼ ºsÊ~É��^¼

:Q7ATS�Q��TE�<HY�ILN� � ½ � ÜÀ ½ Ê Ð � � Ð�� � Ä Â � � Ä � º � � ¼ Ä � º � � ¼ Ä ��º  � ¼Ìâ ºsÊ~É *M¼

à Ê

4$N~: ]><HGJ`�I_Q7E+BF<�I_ILE�G_6"�M`�K�N~]mAC6R8�ILQRE+AC6MI_E*GZO�<>S*E*U <>I �²<H67Yqa <HU UZQ7N~:6�AC6 B¢AC8^5RG_E Ê~É ����Q7EILE�UZIO�5767S�ILATN^6 ��57U_E�Y�ATU�ILQq57U*·� à ½ Ó Ô Õ

Ð � à � Ç Ð � Þ �� × � ß �� � Ç�� º � × � ¼� À Þ�� �À ß�� � � � à � Ç Ð � Þ �� × ��ß �� � Ç�� º � × a ¼� À Þ�� �À ß�� � � � à � Ð Ç Ä �FÞ��� × ��ß��� � Ç�� º a × � ¼ºsÊ~É��^¼

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<H67YÛ��Rº : Þ À ¼ �>ß � � � á :Cß À Ä Û��Rº : ¼ �>ß � � � � � á :ÏÞ À Ä ���7º :ÏÞ À ¼*ºsÊ Ä �>ß � � � � ¼ Ä �7º : ¼*ºsÊ Ä �>ß � � � � � � ¼�� á :Ð Û �7º : ¼��7º :ÏÞ À ¼� � � º : ¼ á :Cß À Ä � º : Þ À ¼ á :ÏÞ À Ä 8 - º : ¼ Ä - º :ÏÞ À ¼Û ; á : �½ �Rº :ÏÞ À ¼�ºJÊ Ð � ß � � � ¼ � Ð �7º : ¼*ºsÊ Ð � ß � � ��� � ¼ �Ð �7º : ¼ �Rº : Þ À ¼� 2 - º : ¼ Ð Û � º : ¼ Ð º - º :ÏÞ À ¼ Ð Û � º :ÏÞ À ¼Z¼ 6 ºsÊËÛ^Û^¼

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� º � ¼ � Ð Û � � º � ¼�º Ä � º � ¼s¼ � ½ � º � ¼ Ð � � º � ¼ D º � ¼<>67Y � º � ¼ � Ð Û Ê� � º � ¼�º Ä � º � ¼s¼ � ½ � º � ¼ Ð Ê� � º � ¼ D º � ¼��Q7E0<H�¤NË]HE��CE�<>Y7U=ILN Û7º Ä � º � ¼J¼ � ½ D º � ¼ <H67Y�� º � ¼ � ½ � º � ¼ ��9�E*67S�E� º � ¼�½ � º � ¼Û º Ä � º � ¼J¼ ß À D º � ¼ N^G�E��m5RAT]><H�CE�6MIL�T` � º � ¼ ß À � º � ¼�½ � º � ¼ ß À � º � ¼Û º Ä � º � ¼Z¼ ß À D º � ¼Ìâ

à �

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A B C D EA B C D E

VariationVariationu(x) u (x)x

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u (x)

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