40

HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

  • Upload
    others

  • View
    9

  • Download
    0

Embed Size (px)

Citation preview

Page 1: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

����������� ���������� ������� ���������� �"!$#%���� ��&'�����()�*� ��+�,&-�.� /��0�1�)23�54�������6�����7 �84 ���������*9

:5;.<=;.>@?BADCFE=G

HJIFK�LMIFNPORQ S8TFUJUFV

WYXDZD[]\�ZD[]^

_ `�acbedgfihkjlbej m

n oqpgr8sut�suv]wxv y t{z}|iz}sut{z}su~�|e���c�]�.t�su~�t���|�t�pF�Yv wYn����*pJw8�csu~�pFt�n'|ew�x� �

� ��w]z��lvM�c�8~gs�w���@|��]w8p�z}sut����8���8~Bzl�8|izls�v]w8t�|ew8�Y~���s�z}su~�|e���c�8pJw8v]�3pJw| �

� �D�.pF�}~gsut�pFt�y�ve� ���|��8z}p��¢¡£wp ¤

¥ �D�.pF�}~gsut�pFt�y�ve� ���|��8z}p��q¦�§�v ¨

� �D�.pF�}~gsut�pFt�y�ve� ���|��8z}p��q¦���}pFp ©

ª �D�.pF�}~gsut�pFt�y�ve� ���|��8z}p�� ª v �� ©

« �D�.pF�}~gsut�pFt�y�ve� ���|��8z}p��¢¬­p�r pJw ®

¯ �D�.pF�}~gsut�pFt�y�ve� ���|��8z}p����°su�]�]z ±

� �D�.pF�}~gsut�pFt�y�ve� ���|��8z}p���²³s�w8p ´�µ

_B_ ¶�·8¸�¹�º�h»·¼)j ½.½

¾ ¬Mv ���8z}suv wt£z�v@�¿�.pF�l~gsut�p�t�n ´e´

À ¬Mv ���8z}suv wt£z�v@�¿�.pF�l~gsut�p�t�� ´g¤

Á ¬Mv ���8z}suv wt£z�v@�¿�.pF�l~gsut�p�tÂ� ´�©

à ¬Mv ���8z}suv wt£z�v@�¿�.pF�l~gsut�p�t ¥ ´�±

² ¬Mv ���8z}suv wt£z�v@�¿�.pF�l~gsut�p�t�� �M´

¡ ¬Mv ���8z}suv wt£z�v@�¿�.pF�l~gsut�p�t ª �e�

Ä ¬Mv ���8z}suv wt£z�v@�¿�.pF�l~gsut�p�t« ��®

Å ¬Mv ���8z}suv wt£z�v@�¿�.pF�l~gsut�p�t ¯ � ¨

o ¬Mv ���8z}suv wt£z�v@�¿�.pF�l~gsut�p�t�� �e®

Page 2: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

����� [����� ����������� � \�����^���X¿Z X��R^e[ � [���^e[��! �#"%$'&)( ^*�+ �^-, � ^]\/. XDZ � $'$ \)Z�.0�! �\)^ � � Z�.21354�6�798:8<;>=?8<@BA5C?DFE98<G#H?I�J!7K8'LNMO79PRQHTSK;U@9=?8VP#8<WXEKC5J!;>HT@YIZH?P[C?@YC?G!G!8<\:]^DF_`H?I/ab;U@KS98Vc�8<@KS98<@BJ�c^C5P!J!;FMVDF8<G�;F@YCA?HTDUEK\'8edfJ!C5g?8:J!798)IhH?P+\`i j�kmlon[k d'p9qr 798VP!8'J!798 n[k C5P!8s;U@KS98Vc�8<@KS98<@BJtH?I�d/uvC?@KSxw l�yTz5{ H?Ps3 z5{ u|C?MVM}H?P+SK;F@9=`J~H r 798VJ!798VP�J!7K80c^C5P!J!;FMVDF8<G7^C<A?8)8<@98VP!=T;F8<G r 7^;FM+7�C5P!8)G!\0C?DFD^H?P�DFC5P!=?8 r 798<@�M}HT\'c^C5P!8<S r ;>J!7�J!798<;>P�P!8<G~J�\0C?G!G�8<@98VP+=?_?4�LN79H r J!7KC5JJ!7K8�cKP!8<G!G+E9P!8);F@`G!EKMO7�C?@�C?G!G!8<\:]^DF_�;FG#=T;>A?8<@`]X_

� l w j d p��<�y 4[L979H r J!7KC5J�J!798���E^M}J!EKC5J!;>HT@KG�;F@YJ!798�cKP!8<G!G+E9P!8eC?@^SYJ~H?J!C?D/8<@98VP!=?_�H?I�C���EK;FSY;F@xC�M}HT@BJ!C?;F@98VP)H?I��K�N8<SA?HTDUEK\'8eds�KG!C5J!;FG~Ih_�J!798�P!8<DFC5J!;FHT@��� j � �e� l��B�t�)�m� � �e�� ����� u

c^P!H�AN;FS98<S�J!7KC5J�J!798)��EK;US�;UG�;F@�J!798VP+\0C?Dv8<WBEK;UDF;>]KP+;UEK\`4{ 4[��HT@KG!;FS98VP�C)��EK;US';F@sCeM}HT@�J!C?;F@K8VP r 7KHTG~8 r C?DFDFG�C5P+8#�^8}�9;>]^D>8�G~HeJ!7KC5J�;FJ!G/A?HTDFEK\08t\0C�_'J!C5g?8tHT@�C?@�_'HT@K8H?I�C0G!8VJ#H?I A5C?DFE98<G)¡¢dK£*¤ �*¥ ]KJ!C?;F@�J!798�c^P!H?]^C5]^;FDU;>J¦_)§ kZ¨ £�H?I ��@KSK;U@9=eJ!7K8)��EK;FS�J~H0]�8�;F@�J!7K8�8<;>=?8<@KG~J!C5J~8© r ;FJ!7�A?HTDFE^\'8�d^£ �^ª S98<@BJ!;>IZ_`J!7K8eJ r H`«mC5=?POC?@9=?8'\eEKDFJ!;>c^DF;>8VPOG�C5cKc�8<C5P+;F@9=s;U@�_?HTE9P[P!8<G!E^D>JV��=T;FA?8<@�J!7KC5JJ!7K8:\'8<C?@`8<@98VP+=?_�C?@^S�\'8<C?@`A?HTDFE^\'8)H?I|J!798�C?G!G~8<\e]^D>_�C5P!8 j C?@KS dsu9P!8<G~c�8<M}J!;>A?8<D>_?4¬ 4 ª I/J!7980­�;>]^]^G#8<@BJ~P!H?cX_?�mC?G[=T;>A?8<@®]B_�8<WX@�¯±°:4³²T´�IhH?P)C?@®C?G+G~8<\:]�D>_�;F@®C?@Y8<@^G~8<\:]�D>8?��;FGtG!c*8<MV;UC?DF;FG~8<S`J~HJ!7K8:MVC?G~8�H?I�C?@�;FG!HTDFC5J~8<S�G~_NG~J~8<\`�^G+79H r J!7^C5J�;>J�P!8<SKE^M}8<G�J~H0J!798�EKG+EKC?D�µ�HTD>J~¶<\0C?@K@�IZH?P+\��9AX;>¶?4>�· l¸� DF@#¹�ur 798VP!8:¹�;FG#J!798�G~J!C5J!;FG~J!;UMVC?D r 8<;F=T7�J#H?I�C�=T;>A?8<@Y\0C?M}P!HTG~J!C5J~8?4ºX4�°oG~_NG~J~8<\»;F@'J!798�MVC?@KHT@K;FMVC?D98<@KG!8<\:]^DF8#7KC?G�C?MVM}8<G!G�J~HeG!J!C5J~8<G r ;FJ!7�8<@98VP!=?_�8<;>=?8<@BA¢C?DFEK8<G j#k � LN7KH r J!7KC5J;FJ!G�cKP!H?]^C5]�;FDF;>J¼_0H?I HNMVMVE9cX_X;U@9=0C0G~J!C5J~8`½R© � MVC?@�]�8 r PO;>J~J~8<@`C?G§ kvl�¾V¿BÀÂÁ p^Ã�ÄUÅ~ur 798VP!8)Æ l 3 z5�B� C?@KS�Çf;FG�J!798:Èt8<DF\079HTDFJ~¶)IZP!8V8)8<@K8VP!=?_?4É 8VPO;>IZ_�J!7KC5J#J!7K8:\'8<C?@`8<@98VP+=?_�H?IÊJ!798:C?G+G~8<\:]�D>_�MVC?@�]*8 r P+;>J~J~8<@�C?GjËl �� Æ ¯hÆmÇe´OuC?@^S�IhE9P+J!798VP�G!79H r J!7KC5J�J!798)=?8<@98VP+C?D/¯hÌsÍÎ3�´¦ÏÐH?P+S98VPt\'HT\'8<@BJ�H?I J!798)S98VAN;FC5J!;>HT@�IhP!HT\Ñ;>J!G�\08<C?@`A¢C?DUE98G+C5J!;FGÒ�^8<G � ¯ jÔÓ j ´¼Õ?Ö � � l-Ó �� Æ � ¯ jÔÓ j ´¼Õ � Í�Ì � ¯ jÔÓ j ´¼Õ p�� � � j� Æ �µ�_`G~c�8<MV;FC?DF;UG!;F@9=eJ!7K;FG�=?8<@98VPOC?D�IZH?PO\`�9H?]KJ!C?;F@`J!7K8)P!8<G!EKD>J

× � l 3�B� ��Ø � je� � Ó � j � �}Ù �

y

Page 3: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

1 ��ZD[ � X�.��� /� Z���� � �¿Z�\�[���^������� �[�� � [���XDZ�^ � Z�. � ��[��! �#"®$'& \)Z�X �-\)Z �354�°o\0C5=T@K8VJ!;FM�C?G!G!8<\:]^DF_�C5J�C)�K�N8<S'J~8<\'c�8VP+C5J!E9P+8�6 ;FG�G!E9]��~8<M}J/J~H:C?@�C5cKc^DF;>8<S'\sC5=T@98VJ!;FM���8<DFS0µ[4 ª I*J!7K8@K8VJ[\0C5=T@K8VJ!;FM�\'HT\'8<@BJtH?I/J!7K8�C?G!G!8<\:]^DF_�;F@�G!J!C5J~8�½ © � ;UG�� k �*;F@YJ!798�SK;>P!8<M}J!;FHT@�H?I/J!798�C5cKc^DU;>8<S`�^8<DUS��J!7K8<@`J!798:C?G!G!HXMV;FC5J~8<S�8<@98VP!=?_�8<;>=?8<@BA¢C?DFEK8:;FG�=T;>A?8<@�]B_j#kml�¾Vk ¯ n l ��´ Ó � k±n ur 798VP!8 ¾Vk ¯ n l ��´�;UG�J!798�8<@98VP!=?_�C5P+;FG+;F@9=�IZP+HT\ÑJ!798:\�E9J!EKC?D�;F@BJ~8VP+C?M}J!;>HT@�H?IÊDUC5J~J!;FM}8:G~c^;U@KGV4L979H r J!7^C5J���EKM}J!EKC5J!;FHT@KG�H?IÊJ!798:\0C5=T@98VJ!;UM�\'HT\'8<@BJ�C5]�HTE9J#;>J!G�\'8<C?@�A¢C?DUE98�C5P!8�=T;>A?8<@�]B_� � � l � � � � Ó � ��� � l¸�B� ��� ur 798VP!8 � � ;UG�J!798�;FG~H?J!7K8VP+\0C?Dv\0C5=T@K8VJ!;FM)G!EKG!M}8Vc^J!;>]^;FDF;FJ¦_sS98}�*@98<S�]B_��� l Ø � � �Ë� z � n:Ù � �y 4[L979H r J!7^C5J#J!798:;FG!H?J!798VP+\0C?D�M}HT\'c^P!8<G!G!;>]�;FDF;>J¼_'H?I|C?@�;FS98<C?D�=TC?G�;UG�=T;>A?8<@`]X_���� l 3Ì �B�r 798VP!8)Ì ;FG�J!798e@BEK\e]�8VP#S98<@KG!;FJ¦_Ø�� H?J~8?i�;F@�=?8<@98VP+C?DmJ!798�;FG~H?J!798VPO\0C?DvM}HT\'cKP+8<G!G!;>]^;UDF;>J¦_'H?I�C���E^;FS�;FG�=T;>A?8<@`]X_

� � l�Ó 3d � � d� � � � � Ù{ 4 ª I^J!7K8#@BEK\e]�8VP|H?I^c^C5P!J!;UMVD>8<G�a ;F@�C?@'C?G!G!8<\:]^DF_���EKM}J!E^C5J~8<G/C5]�HTE9J�G~HT\'8#\'8<C?@�A5C?DFE98 a �TJ!798�c^C5P!J!;>J!;FHT@I±EK@KM}J!;FHT@`\0C<_�]�8 r P+;>J~J~8<@`C?G � l�� kh¨ ¾ p ¿ Ã^Ä Ö ¿"! ur 798VP!8$# ;FG#J!798:M+7K8<\0;FMVC?Dvc�H?J~8<@BJ!;FC?DvC?@KS�J!798�H?J!798VP[G~_N\:]�HTDFG#7^C<A?8:J!7K8<;>P�EKG!E^C?D�\'8<C?@K;F@9=94 É 8VP+;>Ih_�]X_S^;>P!8<M}J#SK;&%�8VP!8<@BJ!;FC5J!;>HT@�J!7KC5J�J!798:\'8<C?@NϦG+WBEKC5P+8���EKM}J!EKC5J!;>HT@�;F@�c^C5P!J!;FMVDF8�@XEK\e]�8VP�;FG�=T;>A?8<@`]X_*i

� a � ��Ó � a � � l ¯ �B� ´ �s� � DU@('� # �*) � ¨ � �¬ 4[L979H r J!7KC5J�J!7K8#\'8<C?@NϦG!WXEKC5P!8���E^M}J!EKC5J!;>HT@';F@�c^C5P!J!;UMVD>8�@XEK\e]�8VP/;F@�C[��EK;FS�\0C<_�]�8�8}�XcKP+8<G!G~8<S';F@�J~8VP+\0GH?I|J!798�;UG~H?J!798VP+\0C?DmM}HT\'cKP!8<G!G+;>]^;FDF;FJ¦_ � � ��6#7BE^GVi� ��,+� l � ¯±a Ó � a � ´ � �� a �r 798VP!8 � +� ;FG�J!798:M}HT\'c^P!8<G!G!;>]�;FDF;>J¼_'H?I|C?@�;FS98<C?D�=TC?GV4Ø � H?J~8?i�J!798)IZHTDUD>H r ;U@9=�P!8<DFC5J!;>HT@KG+7K;>c^G�\0C�_�]�8�IhHTEK@KS`798<D>c^IhEKDÐi

� d l¸�B� DF@ � u�m� �� # � � ¨ � l Ì l a d uC?@^S � � l-Ó 3a � � d� # � � ¨ ur 798VP!8-# ;FG�J!7K8�M+798<\0;UMVC?D�c�H?J~8<@�J!;FC?D Ù {

Page 4: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

ºX4[LNg?8VJ!M+7�J!798�;FG~H?J!798VP+\sGvIhH?PÊC���EK;FS);U@�J!798 � dÎc^DFC?@K8/IZH?P � ;F@�J!798�@98<;>=T7B]�HTE9P+79HXHXS�H?I ��� 4/��HT\0\'8<@BJmHT@J!7K8�]�8<7KC<AN;>HTE9P�H?IÊJ!798:;FG!H?J!798VP+\0C?D�M}HT\'c^P!8<G!G!;>]�;FDF;>J¼_sC?G � � ��� �^C?@KS�C?DFG~H'HT@�J!7K8)P!8<DFC5J~8<S`]�8<7KC�AX;>HTEKPH?I|J!798�SK8<@KG!;>J¼_s��E^M}J!EKC5J!;>HT@KGV4L979H r J!7^C5J�C)SK;>A?8VP+=?8<@�J�G+EKG!M}8VcKJ!;>]�;FDF;>J¼_:;FG/\0C5J!7K8<\0C5J!;FMVC?DFD>_�P!8<DFC5J~8<S0J~HeC?@0;F@^M}P!8<C?G~8#;F@'J!7K8#M}H?P!P!8<DFC5J!;>HT@DF8<@9=?J!7�H?I S98<@^G!;>J¦_BϦS98<@KG+;>J¦_s��EKM}J!EKC5J!;>HT@^G�C?@^S`M}HT\0\'8<@BJ�HT@�J!798)8}�Xc�8VP+;U\'8<@�J!C?D�\0C?@^;>IZ8<G!J!C5J!;>HT@�H?ImJ!7K;UG8 %�8<M}JV4Ø � H?J~8?i�J!798�S98<@KG!;FJ¦_BϦS98<@KG!;>J¼_�M}H?P!P+8<DFC5J!;>HT@��s¯ � Ó ��� ´�;FG#S98}�*@98<S�]B_*i�0¯ � Ó � � ´�� � ¡<Ì�¯ � ´ Ó � Ì�¯ � ´ � ¤?¡<Ì�¯ � � ´ Ó � Ì�¯ � � ´ � ¤ � u

r 798VP!8)Ì�¯ � ´�;FG�J!798:DFHXMVC?D�@XEK\e]*8VP#SK8<@KG!;>J¼_ Ù 4W ��\ � /� ^ \)^-��X � W &��#$ [­\ � � Z�\

354[L979H r J!7^C5J#J!798:@XEK\e]*8VP���;FG#C��^�X8<S`c�HT;F@BJ�H?I|J!798�S9_N@KC?\0;FMVC?D�G~_NG~J~8<\� ¯hÌ�Ío3�´ l � ¯~¯ � ¯hÌ ´~´Ou

;FI���G!C5J!;FGÒ��8<G�J!798)8<WXEKC5J!;>HT@�i � l � ¯��9´ �Èt8<@KM}8:G+79H r i� |�� 6#7KC5J�� l�{ ;UG�J!798��K�N8<S`c�HT;F@BJ�H?I|J!798�SK_X@KC?\s;FMVC?D�G~_NG~J~8<\

� ¯hÌ�Í�3�´ l�y � ¯hÌ ´ Ó { �� ��� 6�7KC5J�J!798:S9_N@KC?\0;UMVC?D�G~_XG!J~8<\ � ¯hÌ�Í 3�´ l � � ¯��9´vÍ��HT@^D>_�7^C?G#C��K�N8<S�c�HT;F@�J#C5Jt;F@N��@K;FJ¦_s;>I � l 354y 4 ¥ ]KJ!C?;F@`C?@KS`A?8VPO;>IZ_�J!798)J r Hs�K�X8<S�c*HT;U@�J!G�H?I|J!798�S9_N@KC?\0;UMVC?D�G~_XG!J~8<\

� ¯hÌ�Ío3�´ l Ø � ¯hÌ ´vÍ ¬ Ù � ¯hÌ ´mÍ y ���C?DFMVEKDFC5J~8 � ¯hÌ ´|J~H�IhHTE9P~Ï��^=TE9P+8�C?MVMVE9P+C?M}_0IhH?P�;F@K;>J!;FC?DNA5C?DFE98<G � ¯ ��´ l-Ó 3 � �935u Ó � ����� C?@KS Óty � ¬ J!C5gX;F@K=A5C?DFE98<G#H?IÊÌ%E9c�J~H0Ì l�� C?@KS�M}HT\s\'8<@�JtHT@`J!798)P!8<G+EKD>J!GV4{ 4�6�798 �uve� sut{zls�~³p��.�|iz}suv]w H?I|c�H?c^EKDFC5J!;FHT@�=?P+H r J!7�J!C5g?8<G�J!798)IhH?P+\

� ¯hÌ�Í�3�´ l ¯Ò3�Í � ´ � ¯hÌ ´ Ó � � � ¯hÌ ´r 798VP!8 � ;FG'J!7K8�=?P!H r J!7 P+C5J~8 ¯ � l ]^;>P!J!7^G! �S98<C5J!7KGO´'C?@KS"� l �

z$# � r 798VP!8 # ;FG'J!7K8�\0C¢�9;F\eE^\c�H?c^E^DFC5J!;>HT@ r 7K;FM+7ÎJ!798®8<@�AN;>P!HT@^\'8<@�J�MVC?@�G!E9cKc�H?P!JV4 ¥ ]KJ!C?;F@ÎJ!798®J r H �K�X8<S9ÏÐc*HT;U@�J0A5C?DFE98<G�H?I�J!7K8G!_XG~J~8<\ C?@KS`]KPO;>8}�^_sM}HT\0\08<@�JtHT@`J!798<;>P�c^7B_NG!;FMVC?D�G!;>=T@^; ��MVC?@KM}8?4¬ 4&%KH?PtC?@B_�G~_NG~J~8<\ r 7K;FM+7�EK@^S98VP!=?HX8<G�C'G!;F@9=TD>8�c^7^C?G~8�J~P+C?@KG!;>J!;FHT@��NJ r H0H?I J!7K8��K�X8<S�c�HT;F@BJ!G�\0C<_`;F\s\'8}ÏS^;FC5J~8<D>_�]�80;FSK8<@�J!; ��8<S®C?G�J!7980D>H r ÏÐJ~8<\'c�8VP+C5J!E9P!80C?@KS%7^;>=T7NÏÐJ~8<\'c�8VP+C5J!E9P+8e�^�X8<Sxc�HT;F@BJ!GV4('�;FG!MVEKG!G�J!7K8c�7�_NG!;FMVC?D�G!;F=T@K; ��MVC?@KM}8)H?IÊJ!798<G~8:c*HT;U@�J!G�C?@KS�8}�Xc^DUC?;F@ r 7B_�J!798V_�C5P!8:C5J~J~POC?M}J!;>A?8?4ÈtH r r HTEKDFS�_?HTE9P�M}HT@KMVDFEKG+;>HT@KG']�8�C %�8<M}J~8<S ;FI r 8YMO79HTG~8�J~H�C5cKc^D>_�J!798<G~8®C5P!=TEK\'8<@BJ!G0J~H J!798�HT@98}ÏS^;F\'8<@KG!;FHT@KC?D ª G!;U@9='\'HXSK8<D�C?G�C0G~c�8<MV;FC?D�MVC?G!8�)ºX4�° G~_XG!J~8<\ 7^C?G�8<@98VP+=?_sD>8VA?8<DFG j#kml �Nu+* � u+* � u+*-, �V�V� r ;>J!7�S98V=?8<@98VPOC?MV;>8<G/. kml 35u y u y u<3 �V�V�5� 6#798tG!_XG~J~8<\;UG);F@x8<WBE^;FDF;>]KPO;FEK\ r ;>J!7%C�J!7K8VP+\0C?D|P!8<G~8VP+A?HT;>P�C5J�J~8<\'c�8VP+C5J!E9P+8 � uvG!EKM+7xJ!7^C5J ¾ p ¿1032 4 3�IhH?P6587 ¬ �9 H?P+g'HTE9J/J!798#c�C5P!J!;>J!;>HT@'I±EK@KM}J!;FHT@��?J!798�\'8<C?@s8<@98VP!=?_'C?@KS'J!7K8t\'8<C?@0G+WBEKC5P+8<S'8<@98VP!=?_'H?I�J!798�G!_XG~J~8<\�4� 4 µ[4�°[@�8<@98VP+=?_�D>8VA?8<D r ;FJ!7�C0S98V=?8<@98VPOC?M}_`H?I�J r H�;F\'c�DF;>8<G�J r H�G!J!C5J~8<G r ;>J!7`J!798:G+C?\'8:8<@98VP+=?_?4�6�798c�C5P!J!;>J!;>HT@�I±EK@KM}J!;>HT@`;UG#C0G!EK\�HRA?8VP;:=<?>�<A@B: �^@9H?J!CD@=E�@=CF:O4¬

Page 5: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

�X4 9 798<@`C�c^C5P+J!;FMVD>8 r ;>J!7�G~c^;F@ �� ;FG�c^DFC?M}8<S`;U@�C�\0C5=T@K8VJ!;FM[�^8<DFS n �9;>J!G�8<@98VP!=?_�D>8VA?8<D�;FG�G~c^DF;FJ�;F@BJ~H�� # nC?@^S�;>J�7^C?G'C®\sC5=T@98VJ!;FM�\'HT\'8<@BJ�� #˯hP!8<G~c�8<M}J!;>A?8<D>_K´eC?D>HT@9=xJ!798`SK;>P!8<M}J!;FHT@ H?ItJ!798`\0C5=T@98VJ!;FM��^8<DUS�4L9E9cKc�HTG~8[J!7KC5J#C?@`C?G!G~8<\e]^D>_�H?IÊa G!EKM+7�c�C5P!J!;FMVD>8<G�HT@`C'DFC5J~J!;UM}8�;FG�c^DFC?M}8<S`;F@�C'\0C5=T@98VJ!;UM[�^8<DFS n C?@^S;UG�g?8VcKJ�C5J�C�J~8<\'c�8VP+C5J!E9P!8 � 4 %|;F@KS®J!7980;F@BJ~8VP+@KC?DÊ8<@98VP+=?_?�vJ!798'8<@BJ~P!H?cB_?� J!7980G~c�8<MV; ��M�798<C5J�C?@KSxJ!7K8J~H?J!C?D \0C5=T@98VJ!;FM:\'HT\'8<@BJ#H?I|J!7K;FG�C?G!G~8<\e]^D>_?4� �\ � /��^]\*^���X � W &��#$ []\ ����� X354�°�DF;U@98<C5P#M+7KC?;U@ ª G!;U@9=0\'HNS98<D�7KC?G#C0È[C?\0;FDFJ~HT@K;FC?@�=T;>A?8<@�]X_

l-Ó p��� k

� �k · k · k Ö � �

L979H r J!7^C5J#J!798�c^C5P+J!;>J!;>HT@�I±EK@KM}J!;>HT@�J!C5g?8<GtJ!798)IhH?P+\`i' l¸y

p���k � M}HTG!7 �

k ur 798VP!8 � k � � kÐz5�B� ;UG�J!798:M}HTE9c^DU;F@9=�c^C5P+C?\'8VJ~8VP�4¥ ]KJ!C?;F@�J!798)G~c^;F@9ϦG~c^;F@�M}H?P!P!8<DFC5J!;>HT@�I±EK@KM}J!;FHT@�;U@�J~8VPO\0G�H?ImJ!798�c^C5P+J!;>J!;>HT@�IhE^@KM}J!;>HT@�C?@KS�G!79H r J!7KC5J�IZH?PE^@K;>IhH?P+\�;F@�J~8VPOC?M}J!;>HT@�G~J~P+8<@9=?J!7�;>J�J!C5g?8<GtJ!798)IhH?P+\`i

� Õ ¯ � ´�� � · Õ · Õ5Ö� � l J!C?@K7 � �r 798VP!8 � ;FG�J!798�M}HTEKc^DF;F@9='M}HT@KG!J!C?@�J#IhH?P�J!7K8�\'HXSK8<D�4��HT\0\'8<@BJ:HT@ J!7K80c�HTG!G!;>]^;UDF;>J¦_`H?I�D>HT@K=5ÏÐP+C?@9=?8sH?POS98VP:C5cKc�8<C5P+;F@K=�;F@%J!798sMVC?G!8<GVi�¯±C�´ � 7 ����C?@KSίh]�´��l �N4y 4[L979H r J!7KC5J)J!7K8�J~POC?@KG~Ih8VP�\sC5J~P+; �®\08VJ!79HXSx\0C�_Y]�8�8}�XJ~8<@^S98<S%J~H�J!7980MVC?G~8'H?I�C?@ ª G!;U@9=�PO;F@9=�D>HXMVC5J~8<S;U@`C0M}HT@KG~J!C?@BJ#�^8<DUS n C?@^S`798<@KM}8�H?]KJ!C?;U@`J!798)IhP!8V8�8<@98VP!=?_�c�8VP�DFC5J~J!;FM}8eG!;>J~8�;F@�J!798)IhH?P+\`i� l�Ó

�Ó 3Æ DF@ Ø M}HTG!7�Æ n ��¯ ¾ � ¿�� G!;F@^7 � Æ n Í ¾ � ¿�� ´��� Ù u

r 798VP!8�C?DFD�J!798eG~_X\e]�HTDFG�7KC�A?8:J!798<;>P#E^G!EKC?D�\'8<C?@K;U@9=94Ø È[;F@BJVi 9 798<@�=?8<@K8VP+C?DF;>¶<;F@K='J!798�J~P+C?@KG!IZ8VP�\sC5J~P+; �`J~H0J!798�MVC?G~8:H?I�@9HT@NÏжV8VP+H08}�XJ~8VPO@KC?D��^8<DUS`P!8<\'8<\e]�8VPJ!7^C5J�;>J#\�EKG~J�P!8<\0C?;U@�G~_N\0\'8VJ~P+;FM�;F@`;FJ!G#;F@KSK;UM}8<GV4 Ù¥ ]KJ!C?;F@xC?@®8}�XcKP+8<G!G!;>HT@®IhH?P[J!7K8�G!c*8<MV;>��M:\0C5=T@K8VJ!;>¶<C5J!;>HT@�� l � · � �mC?@KSxSK;FG!MVE^G!G�;>J!G�S98Vc�8<@KSK8<@KM}8eHT@J!7K8�8}�XJ~8VPO@KC?D��^8<DFS n 4/M}HT\0\08<@�JtHT@`_?HTE9P#P!8<G+EKD>J!GV4{ 4�6�798��#­»P!8<MVEKP+G!;>HT@YP!8<DFC5J!;>HT@^GtIhH?P)C�HT@98}ϦSK;F\'8<@^G!;>HT@KC?D ª G!;U@9=�\'HNS98<D|;F@®C?@®8}�XJ~8VP+@^C?DÊ�^8<DFS n \0C�_�]�8r P+;>J~J~8<@�C?G��� �l � ¯Ò3�Í��K´ �¯ � Í��9´}¯Ò3�Í � �K´ � � � l ��¯ � Í��9´¯Ò3�Í � �K´ ur 798VP!8 � l�¾ p! ��"$# � C?@^S%� l¸¾ p'& "$# � 4 É 8VPO;>IZ_�J!798�8}�9;FG~J~8<@^M}8�H?IÊ�K�X8<S�c*HT;U@�J!G�C?G�IZHTDFDFH r GVi�¯ �)( u*� ( ´ l¯ �Nu ��´+� ¯ � ( u*� ( ´ l ¯ �Nu<3�´�C?@KS�CsDF;F@98�H?IÊ�K�N8<S`c�HT;F@�J!G � ( l 3)IhH?P��-,.� ( ,-3 � '�;FG!MVEKG!G#J!798)c^7B_XG!;UMVC?DG+;>=T@K; ��MVC?@^M}8�H?IKJ!798<G!8�c�HT;F@BJ!GmC?@^S�G!g?8VJ!M+7�J!798�G~_NG~J~8<\-c*HT;U@�Jm�^H r G|;F@eJ!798�J r H5ϦSK;F\'8<@^G!;>HT@KC?DBc^C5P+C?\'8VJ~8VPG!c^C?M}8�]�HTEK@KS98<S�]X_ �/, � ,Ô3)C?@KS �-,0�1, 3 �¬ 4 É 8VPO;>IZ_xJ!7KC5J�G!EK\0\s;F@9=�H�A?8VP�C?D>J~8VP+@KC5J~8�G~c^;F@^G�HT@ C�G+WBEKC5P+8sDFC5J~J!;FM}8�P!8<G!EKD>J!G:;F@%C�@98 r G!WXEKC5P!8sDFC5J~J!;UM}8P+H?J!C5J~8<S�J!79P+HTE9=T7 ¬ º32�P!8<DFC5J!;FA?8�J~H0J!798)H?P+;F=T;F@KC?D�DFC5J~J!;FM}8eC?@KS r ;>J!7`C0G+MVC?D>8�I±C?M}J~H?P�H?I�� l54 y �°[DFG~H`A?8VPO;>IZ_YJ!7KC5J)J!798'8 %�8<M}J�H?I�G!EKM+7�C`S98<MV;F\0C5J!;>HT@xHT@ J!798 ª G+;F@9=�\'HXSK8<D�;FG�J~H�M+7KC?@K=?80J!798'H?P+;>=T;U@KC?Dc�C5P!J!;>J!;>HT@oI±EK@KM}J!;>HT@�;U@�A?HTD>AN;F@9= HT@KDF_ c^C?;>POG�H?I'@98<C5P+8<G~J�@98<;>=T7B]�HTE9P+G�J~H C�IhH?P+\ ;F@BA?HTD>AN;F@9= @98<C5P!8<G~JÒÏ@K8<;>=T7�]�HTE9POGV�9@98}�XJÒÏÐJ~H5Ϧ@K8<C5P!8<G~Jt@K8<;>=T7�]�HTE9POG�C?@KS`J!798)cKP+HXSKE^M}J�H?I|IhHTE9P#G~c^;U@KG�J!C5g?8<@76 P!HTE^@KS`C0G!WXEKC5P!838>4

º

Page 6: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

�\ � /��^]\*^���X � W &��#$ []\ ���o&�� \*\354 ¥ @�J!798e]^C?G!;FG�H?I�J!798 9 8<;UG!G�\'HTD>8<MVEKDFC5P��^8<DFS`J!7K8VH?P!_?��G!79H r J!7KC5JtJ!798 ª G!;F@K=s\'HNS98<DmMVC?@K@KH?J�8}�97K;>]^;>JtCG!c*HT@BJ!C?@98VHTEKG \0C5=T@98VJ!;FG+C5J!;>HT@)IhH?PmJ~8<\'c�8VP+C5J!E9P+8<G�=?P!8<C5J~8VPÊJ!7KC?@ �olÎ� � � r 798VP!8�J!798���E9P+;>8|J~8<\'c�8VP+C5J!E9P+8;UG�=T;>A?8<@`]X_ � ��l � z5� 4y 4�°[@�;FG!HTDFC5J~8<S�c^C5P+C?DFDF8<D^c^DUC5J~8�MVC5c^C?MV;>J~H?P�7KC?G�C�c*H?J~8<@BJ!;FC?D�SK; %*8VP!8<@^M}8�d ]�8VJ r 8V8<@�;>J!G�8<D>8<M}J~P+HXS98<G<� r 7K;FMO7C5P+8�G+;>J!EKC5J~8<S C5J � l � � � 6#798�G~c^C?M}8�]�8VJ r 8V8<@ J!798�8<D>8<M}J~P!HNS98<G�;FG:HNMVMVE9c^;F8<S�]X_�C?@ ;>HT@K;FM�G~HTDFE9J!;FHT@r 7K;FM+7Î7KC?G�C�SK;>8<DF8<M}J~P+;FM`M}HT@KG~J!C?@BJ�H?I�EK@K;>J¼_?4 ¥ ]KJ!C?;F@ÎC?@ 8}�NcKP!8<G!G+;>HT@�IhH?P�Ì�¯ � ´Ou�J!798®G~c^C?M}8xM+7KC5P!=?8S^;FG~J~P+;>]�E9J!;>HT@�� r 7K;FMO7Y8}�9;FG~J!G)C5IhJ~8VP�J!7980G!_XG~J~8<\ 7^C?G�P!8<C?M+798<S J!798VP+\0C?D|8<WXEK;FDF;F]KP+;FEK\�4 %9H?P:J!798'G!C5g?80H?IG+;F\'c^DF;UMV;>J¦_Y_?HTE \0C�_ C?G!G!EK\080J!7KC5J:J!7K8sc�H?J~8<@�J!;UC?D�SK;&%�8VP!8<@KM}8s]*8VJ r 8V8<@ J!798�c^DFC5J~8<G:;UG:G~H�G+\0C?DFD�J!7KC5J¾ d 4 �B� �Ø È[;F@BJVi/J!C5g?8 � l �0C?G#C0c^DFC?@98)H?I|G~_N\0\'8VJ~P!_ r 798VP!8?i¯±C�´�Ì Ö ¯ ��´ l Ì p ¯ ��´ l Ì + ¯±G!C�_9´¯h]�´�J!798)c�H?J~8<@�J!;UC?D�8<WBE^C?DFG�¶V8VP!H94 Ù{ 4��[G!;F@K=�J!798t« C?@KSKC?E0\0HXS98<D^IZH?P�c^7KC?G~8�J~P+C?@KG!;>J!;FHT@KG�H?]KJ!C?;F@sA¢C?DUE98<G/IhH?P�J!798[M}P+;>J!;FMVC?DK8}�Xc�HT@98<@BJ!G/Æ/uÒw�C?@^S� 4

� �\ � /��^]\*^���X � W &��#$ []\ � � X � �354��[G~80J!7K8s7K;>=T7NÏÐJ~8<\0c*8VPOC5J!E9P!8�8}�Xc�C?@KG!;>HT@%J~HYG!79H r J!7KC5J�J!7980c�C5P!J!;>J!;>HT@xI±EK@KM}J!;>HT@xH?I�J!798®¯hH?c*8<@*´ ª G!;U@9=DU;F@98<C5P#M+7^C?;F@`J!C5g?8<GtJ!798)IZH?PO\ ' Îl�y M}HTG!7 p�� � uC?@^S`8}�Xc�DFC?;F@�J!798�SK; %*8VP!8<@^M}8<G�]�8VJ r 8V8<@�J!7K;FG�C?@^S`J!798�P!8<G+EKD>J�IZH?P�J!798 ª G!;F@9=�P+;F@K=94y 4��[G~8eJ!798:\'8VJ!79HNS`H?I/7K;F=T7NÏÐJ~8<\'c�8VP+C5J!E9P!8�8}�Nc^C?@KG!;FHT@`J~HsH?]KJ!C?;U@�J!798:G!c^;F@NϦG~c�;F@�M}H?P+P!8<DFC5J!;>HT@ � · Õ ·�� � H?IJ!7K8 ª G+;F@9='\'HNS98<D�C?G � · � · Õ � l '�p�� M}HTG+7�� � y ��

� � � Õ ¯ � ´� 5ur 798VP!8 � ;FG[J!7980@BE^\:]�8VP[H?I�@98<C5P!8<G!J�@98<;>=T7B]*HTEKPtc^C?;FP+GV��� ��J!C?@K7 � C?@KS � � Õ ¯ � ´t;FG[J!798'@XEK\:]�8VP�H?I=?POC5c^7KG�H?I � DF;F@98<GV� r ;FJ!7�8VA?8<@�A?8VP!J!;FM}8<G�8}�NM}8Vc^JtC5J�G!;FJ~8<G � C?@KS`Ì�4L979H r J!7^C5J#IZH?P�J!798:MVC?G~8�H?I|J!798 ª G!;F@K=0DF;F@98<C5P�M+7^C?;F@��KJ!7K;UG�P!8<SKEKM}8<G�J~H� · � · Õ � l �� Õ p � ³uC?@^S�S9P+C r J!798:M}H?P!P+8<G~c�HT@KSK;F@9=�=?P+C5c^7v4{ 4 ª I/HT@98'MVC?@xC?G!G!E^\'8eJ!7^C5J[J!7K8�;U@�J~8VP!c�C5P!J!;FMVD>8ec*H?J~8<@BJ!;FC?D��|¯ � ´#;FG[DFC5P!=?8�¯±C?@KS®c�HTG!;>J!;FA?8R´�HT@®J!798�G!MVC?D>8�H?I�B� uBIhH?P �� �� �XC?@^S�;FG�G!\0C?DFD^IZH?P � 7 � u r 798VP+8 � ;FG�J!798[\0HTD>8<MVEKDFC5P�SK;FC?\'8VJ~8VP��BJ!798<@�;>J�\0C<_0]�8[G!79H r @J!7^C5J#J!798:G~8<M}HT@^S`AX;FP+;FC?D�M}HB8���MV;>8<@�J[MVC?@`]�8 r P+;>J~J~8<@`C?Gn � l�n Ó � z5�B� �¥ ]KJ!C?;F@�8}�Nc^DF;UMV;>J#8}�NcKP!8<G!G!;FHT@KG#IhH?PtJ!7K8�M}HT@KG!J!C?@�J!G � C?@KS n �*C?@^SYG!7KH r J!7KC5JtJ!7K8eP!8<G+EKD>J!;F@9='8<WXEKC5J!;>HT@H?I�G~J!C5J~8�\0C<_�]�8�P!8<SKEKM}8<S�J~H0J!798 É C?@�SK8VP 9 C?C?D 8 Gt8<WBEKC5J!;FHT@�4¬ 4 ª I�C'=TC?GtH?I/;F@�J~8VPOC?M}J!;F@9=0c^C5P+J!;FMVD>8<G#;FG�\'HXSK8<DFD>8<S`C?G�7^C5P+S�G~c�798VP!8<G�H?I|P+C?SK;UEKG �^�*G!79H r J!7KC5JtJ!798:G~8<M}HT@KSAN;>P+;UC?D*M}HX8���MV;>8<@BJ�J!C5g?8<G�J!798�IhH?P+\`i n � l y

� � ,{ �­);>A?8<@�J!7KC5J�J!798)J!7K;>P+S�AN;>P+;UC?D*M}HX8���MV;>8<@BJ�\0C�_�]*8 r P+;>J~J~8<@`C?GVin , l 3{�� � , � � � , � � � ¯!½ � ½ ´ � ¯!½ � � ½ ´ � ¯!½ � Ó � � ½ ´�u

Page 7: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

r 798VP!8 � ¯ � ´ l�¾ p ¿��*À Å Ó 35�^G!7KH r J!7KC5J�J!7K;FG#;FG�P+8<DFC5J~8<S`J~H0J!798eG~8<M}HT@KS`AN;>P+;FC?D�M}HX8���MV;>8<@BJ n � ]B_n , l º² n �� uIhH?P�C0G!_XG~J~8<\ H?IÊ7^C5P+S�G~c^7K8VP!8<GV4ºX4�° =TC?G�M}HT@KG!;FG!J!;F@9=�H?I a MVDFC?G+G!;FMVC?D*c�HT;F@�J�c^C5P!J!;FMVDF8<G�H?IÊ\0C?G!G � HXMVMVEKc^;>8<G�C�A?HTDFEK\'8�dËC5J�J~8<\'c�8VP+C5J!E9P+8� 4 ª IÊJ!798)c^C5P+J!;FMVD>8<G#;F@BJ~8VP+C?M}J#J!7KP!HTE9=T7`C'J r H5ÏÐ]�HNS9_�c*H?J~8<@BJ!;FC?D�H?I|J!798�IhH?P+\`i

�|¯ � k � ´ l �� Õk � u

r 798VP!8 � ;FG�CxM}HT@KG~J!C?@BJV� � k � l ½ � k|Ó � � ½mC?@KS�Ìo;FG�c*HTG+;>J!;>A?8?��G!79H r J!7KC5J�J!798�MVC?@KHT@K;FMVC?D�c^C5P!J!;>J!;FHT@I±EK@KM}J!;FHT@`;FG#C079HT\0H?=?8<@98VHTEKG#I±EK@KM}J!;FHT@��9;F@�J!798:G~8<@^G~8'�¯�� � u���p , " Õ de´ l � , À �� p �� Å '�¯ � uOd�´�u

r 798VP!8���;FG#C?@�C5P!]�;>J~P+C5P!_�G!MVC?DF;F@9='I±C?M}J~H?P<4�X4��P!H�A?8eJ!798�;FS98<@BJ!;>J¼_ �� �

¾ p ¿ l-Ó � ¿+ ¾ p À ¿ p�� Å � � � ¾ p�� � �*uJ~H`G~8<M}HT@^SxH?P+S98VP�;F@�Æ�]X_�8<WXEKC5J!;F@9=�M}HX8���MV;>8<@BJ!G);F@�J!7K8�7K;F=T7NÏÐJ~8<\'c�8VP+C5J!E9P!8�¯±G!\0C?DFD ÆÊ´�8}�Nc^C?@KG!;>HT@YH?I8<C?MO7YG+;FS98�H?I|J!798)P!8<DFC5J!;>HT@v� r 798VP!8 ;FG#C?@`H?c�8VP+C5J~H?P�4� 4[L979H r J!7^C5J�C5J�7K;>=T7�J~8<\0c*8VPOC5J!E9P!8<GV�KJ!7K8�798<C5J�MVC5c^C?MV;>J¼_`H?IÊCsWBEKC?@BJ!EK\ C?G+G~8<\:]�D>_�MVC?@�]*8 r P+;>J~J~8<@�C?G× � l 3�B� �� � � ¯ � ´�

� ¯Ò3�´Ó Ø � � ¯ ´ Ù �Ø � � ¯Ò3�´ Ù � Í �K¯hÆÊ´���u

r 798VP!8)Æ l 3��RgX6)4²X4[L979H r J!7^C5J#J!798:EKG!8)H?I J!798 É C?@�S98VP 9 C?C?DUG#8<WXEKC5J!;>HT@��� � Í �d � � ¯Ðd Ó �V´ l a �B� uJ~H SK8<G!M}P+;>]�8sc�7KC?G~8�J~P+C?@KG!;FJ!;>HT@KG�;F@�C®��EK;FS�G~_NG~J~8<\�D>8<C?SKG�J~H%J!798�IhHTDFD>H r ;F@9=®A¢C?DFEK8<GeIhH?P'J!798`M}P+;>J!;UMVC?Dc�C5P+C?\'8VJ~8VP+GVi � ��l � z5y � � � u�d ��l¸{ ��u�a �B� ��l ²1� z5y � � �Èt8<@KM}8:G+79H r J!7KC5J#J!798 É C?@YS98VP 9 C?C?DFG�8<WBEKC5J!;FHT@�\0C<_�]�8 r P+;FJ~J~8<@`;F@`J!798�EK@^;>A?8VP+G!C?D�IhH?P+\���§�Í {�� � � ¯ {��� Ó 3�´ l ² �� ur 798VP!8 �§ l � z � � u �� l d z d � u �� lÎ��z���� �

� 4�µ�_ P+8}ÏÐ8}�XcKP+8<G!G!;F@9=YJ!798 É C?@�SK8VP 9 C?C?DUGe8<WXEKC5J!;>HT@ ;F@�J~8VP+\0G�H?I#J!798�P!8<SKE^M}8<S�A5C5P+;FC5]^DF8<G�� � l ¯ � Ó� � ´ z � � u � l ¯Ðd Ó d � ´ z d � C?@^S�� ��l ¯ � Óx� � ´ z�� � �9H?]KJ!C?;F@`A5C?DFE98<G�IhH?P�J!798�M}P+;>J!;UMVC?D�8}�Xc�HT@98<@BJ!G�w®C?@^S� 4���HT\0\'8<@BJ�HT@�J!798)A5C?DFE98<G r 7K;FM+7�_?HTE�H?]KJ!C?;F@�4

Page 8: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

� ��\ � /��^]\*^���X � W &��#$ []\ ��� \��¿\�Z354��P!H�A?8eJ!798�M}P+;>J!;UMVC?D�8}�Xc�HT@98<@BJ#;F@98<WXEKC?DF;FJ¦_���Í y Æ`Í w�� yØ È[;F@BJVi/_?HTE�\sC<_`C?G!G!EK\08)J!798)P!8<DFC5J!;>HT@^G!7K;>c��� ¯ × & Ó ×�� ´ lÎ� ¯ � � z � � ´ � & ur 798VP!8 × & C?@KS × � C5P+8�J!798�G~c�8<MV; ��M)7K8<C5J!G�C5J�M}HT@KG!J!C?@�J���8<DFS`C?@KS�\0C5=T@K8VJ!;FG!C5J!;>HT@`P+8<G~c�8<M}J!;>A?8<D>_?4y 4�°[SKSK;FJ!;>HT@KC?D^M}PO;>J!;FMVC?D�8}�Xc�HT@98<@BJ!G�MVC?@�]�8[H?]KJ!C?;F@98<S�;>I r 8)SK;&%�8VP!8<@�J!;UC5J~8tJ!798�­�;>]^]^G�IhP!8V8[8<@98VP!=?_�P!8Vc�8<C5JÒÏ8<S^D>_ r ;>J!7�P!8<G~c�8<M}J�J~H0J!798:8}�XJ~8VP+@^C?Dv\0C5=T@98VJ!;FM)�^8<DFS`µ[�KJ!7XEKGVi¯ � � z � n ´ � l � À Å�� �� ��� � À p��¼Å!ur 798VP!8:J!7K8 � C5P!8egN@9H r @YC?G[J!798��5=TC5c%8}�Nc*HT@K8<@�J!G���C?@KS � � ;FGtJ!798:P!8<S^EKM}8<S�J~8<\0c*8VPOC5J!E9P!8?4 ¥ @�J!7K8]�C?G!;FG�H?I�J!7K8 9 ;FS9HT\bG!MVC?DF;F@9=Y7�_Xc�H?J!798<G!;FGV�ÊG!79H r J!7KC5J�J!798�=TC5c 8}�Nc�HT@98<@�J!G�C5P!8�C?DFD/8<WXEKC?D�C?@KS�=T;FA?8J!7K8<;>P#A5C?DFE98�;F@�J~8VP+\sG�H?I J!7K8�c^C5P+C?\'8VJ~8VPOG#H?IÊJ!798 9 ;FS9HT\ G!MVC?DU;F@9=�J~P+C?@KG~IhH?P+\0C5J!;FHT@�4{ 4 ª I r 8eS98<@9H?J~8�J!798:H?P+S98VP�c�C5P+C?\'8VJ~8VP�¯hH?P[G~c�8<MV; ��M)\0C5=T@K8VJ!;FG!C5J!;>HT@�´�]B_ � �^G!79H r J!7KC5J#J!7K8:\'8<C?@NÏ��^8<DUSG!HTDFE9J!;>HT@�H?I|J!798 ª G!;F@9=0\0HXS98<D�MVC?@�]*8 r P+;>J~J~8<@�C?G� l J!C?@K7�� �3�Í � � Í�����ur 798VP!8 � � ;FG�J!798�P!8<S^EKM}8<S�J~8<\'c�8VP+C5J!E9P!8eC?@KS � l Æ n ;FG�J!798:P!8<SKEKM}8<S�8}�NJ~8VP+@KC?D \0C5=T@98VJ!;FM[�^8<DUS�4µ�_�M}HT@KG+;FS98VP+;F@K=-� IZH?P[J~8<\'c�8VP+C5J!E9P+8:MVD>HTG~8eJ~H � � C?@KS�IZH?Pt¶V8VP!Hs8}�NJ~8VP+@KC?Dm�^8<DFS���G!79H r J!7KC5J�J!7K8eC?G!G~H5ÏMV;UC5J~8<S�M}P+;>J!;UMVC?D�8}�Xc�HT@98<@BJ#J!C5g?8<G#J!7K8�A¢C?DUE98)Æ l �� 4Ø È[;F@BJVi/J!798�IhHTDFD>H r ;F@9=�8}�Nc^C?@KG!;>HT@

J!C?@K7 � l � Ó 3{ � , Í �K¯ ��� ´OuIhH?P�G!\sC?DFD�A¢C?DUE98<G�H?I � �KG!7KHTEKDFS�]�8�798<D>cKI±EKD�4 Ù¬ 4 ¥ ]KJ!C?;F@�C?@�8}�NcKP!8<G+G!;>HT@�IZH?P�J!798)\'8<C?@�8<@98VP!=?_ j H?ImJ!798 ª G!;F@9=�\'HXSK8<D r 7K8<@sJ!798)C5cKc�DF;>8<S'��8<DFS�;UG�¶V8VP!H9�E^G!;F@9=�J!7980G!;F\0c^D>8<G~J�\'8<C?@NÏ��^8<DUS%C5cKcKP!H��N;F\sC5J!;>HT@�4eÈt8<@KM}8sG+79H r J!7KC5J:J!798'G~c�8<MV; ��M�798<C5J × & 7KC?G)J!7K8]�8<7KC�AX;FHTE9P<i × & l �Nu IZH?P � 7 � � �l { a �Kz5y u IhH?P � � � �

9 7KC5J�;UG�J!798)A5C?DFE98)H?I|J!798�C?G!G~HNMV;FC5J~8<SYM}P+;>J!;FMVC?D�8}�Nc�HT@98<@�J���)ºX4�µ�_®M}HT@KG!;FS98VPO;F@9=sJ!7K8�]�8<7KC�AX;FHTE9P�H?I�J!798�H?P+S98VP)c^C5POC?\'8VJ~8VP�C5J)J~8<\'c�8VP+C5J!E9P!8<G �!EKG~J�C5]�H�A?8'J!798'M}P+;>J!;UMVC?Dc�HT;F@BJV�TG!79H r J!7KC5J/J!7K8#M}P+;>J!;FMVC?DX8}�Nc*HT@K8<@�J|wÊ� r 7K;FMO7';FG�C?G!G~HNMV;FC5J~8<S r ;>J!7�J!798#;FG~H?J!7K8VP+\0C?DNG!EKG!M}8Vc^J!;>]^;FDF;FJ¦_?�J!C5g?8<GtJ!798�A5C?DFE98�w l-Ó 3)C?MVM}H?P+S^;F@9=0J~Hs\08<C?@NÏ��^8<DFS�J!798VH?P+_?4°[DFG~H9� ]B_xM}HT@KG!;US98VP+;F@9=�J!798'8 %�8<M}J:H?I�C?@%8}�NJ~8VP+@KC?DFD>_xC5cKc^DU;>8<Sx\0C5=T@98VJ!;FM��^8<DUS%C5J �»lË� � �mG!7KH r J!7KC5JJ!7K8�8}�Nc�HT@98<@BJ[C?G!G!HXMV;FC5J~8<S r ;FJ!7�J!798�M}P+;>J!;UMVC?D ;FG!H?J!798VP+\ J!C5g?8<G�J!798�A¢C?DFEK8 � l�{ 4 Ø ª @®J!798�DFC5J~J~8VP�MVC?G~8?�J!7K8:;FS98<@BJ!;>J¦_ J!C?@K7v¯ � Í��9´ l J!C?@^7 � Í J!C?@^7 �3�Í�J!C?@K7 � J!C?@K7 � u\sC<_�]�8�798<D>cKI±EKD�4 Ù�X4[L979H r J!7KC5JsJ!798`H?cKJ!;F\0C?D�IhP!8V8`8<@98VP!=?_�H?I�C?@ ª G!;U@9=%\'HXSK8<D�� r 7K;UM+7 M}H?P!P!8<G~c�HT@KS^G�J~H%J!798�\'8<C?@NÏ��^8<DUSJ!7K8VH?P!_?��\0C�_�]*8 r PO;>J~J~8<@�;F@�J!798)IhH?P+\`i

Ç l-Ó a Æ DF@ Ø y M}HTG!7 ¯hÆ n à ´ Ù Í ay��� ¯ n à Ó�n ´ � ur 798VP!8/Æ l 3 z5�B� � n ;FG�J!798�8}�XJ~8VPO@KC?DFD>_�C5cKc^DF;F8<S)\0C5=T@98VJ!;FM/�^8<DFS�� n à ;FGvJ!798�8 %*8<M}J!;>A?8��^8<DFS�8}�Xc�8VP+;>8<@^M}8<S]X_�8<C?M+7YG~c^;F@�� � ;UG�J!798�M}HXH?P+SK;F@KC5J!;FHT@`@BEK\e]�8VP#C?@KS � ;FG�J!7K8:;F@BJ~8VP+C?M}J!;>HT@�G~J~P+8<@9=?J!7�4

²

Page 9: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

� 4[��HT@KG!;FS98VP�C?@ ª G!;U@9=e\'HNS98<D r 798VP!8[J!798[8}�XJ~8VP+@^C?DK�^8<DUS n[k S98Vc�8<@KSKG�HT@�J!798tc�HTG!;FJ!;>HT@0H?IvJ!7K8[DFC5J~J!;UM}8[G!;FJ~8©+4�LN79H r J!7KC5J#J!798eM}HT@KSK;>J!;>HT@�IhH?P#J!798)IhP!8V8�8<@K8VP!=?_�J~H0]�8�Cs\0;F@K;F\�EK\ J!C5g?8<GtJ!798)IZH?PO\`in À k Åà Ó�n[kvl�� ���� � · � � � u

r 798VP!8�C?DFD�J!798eG~_X\e]�HTDFG�7KC<A?8eJ!798<;>P#EKG+EKC?D�\'8<C?@K;F@K=�C?@^S`J!798�@9H?J!C5J!;>HT@Y;F@KSK;UMVC5J~8<G�J!7KC5J#J!798�G!E^\ÑH�A?8VP5®;FG:P!8<G~J~P+;UM}J~8<S J~HY@K8<C5P!8<G~Je@K8<;>=T7�]�HTE9POG�H?I�©!4`°tDFG!H�G!79H r J!7KC5JeJ!7980H?cKJ!;U\0C?D/IhP!8V808<@98VP+=?_%J!C5g?8<GeJ!7K8IhH?P+\`iÇ l Ó aÆ DF@�� y M}HTG+7��+Æ n À k ÅÃ�� Í 3y � k � n À k Åà Ó�n[k � � · k � � �

²X4�°-=?8<@98VPOC?DF;>¶V8<S ª G!;F@9='\'HNS98<D�7KC?G#J!798:EKG!E^C?D�ÈtC?\0;UD>J~HT@K;FC?@�]^E9J�8<C?MO7YG!c^;F@�A¢C5PO;FC5]^D>8�J!C5g?8<GtJ!798�A5C?DFE98<GVi· kvl-Ó � u Ó � Í 35u �>�>�>� u � Ó 35u � ur 798VP!8 � \sC<_ ]�8�8<;FJ!798VP�C?@ ;F@BJ~8V=?8VP�H?P�C�7KC?D>IZÏÐHXS^S ;F@BJ~8V=?8VP<4 �[G!;F@9=�\'8<C?@NÏ��^8<DUS J!798VH?P!_ ��@KS J!7K8M}PO;>J!;FMVC?D9J~8<\'c�8VP+C5J!EKP!8�H?I�J!7K;FG/G~_NG~J~8<\`�XC?@KS'J!798<@sEKG~8#J!7K;UG|P!8<G!EKD>J�J~H:P!8<M}H�A?8VP�J!798�M}PO;>J!;FMVC?DNJ~8<\'c�8VP+C5J!E9P+8M}H?P+P!8<G~c�HT@KSK;F@K=�J~HsJ!798�G~J!C?@KSKC5POS`J r H5ϦG~J!C5J~8 ª G!;U@9='\'HXSK8<D�4Ø � H?J~8?i�_?HTE�\0C<_`C?G!G+EK\'8)J!798�P+8<DFC5J!;>HT@KG!7^;>cvi��� p �

¾� � l G+;F@K7���� � Í �� � � G!;F@^7 Ø � z5yRÙ � Ù� 4�6�798�È�8<;UG~8<@�]�8VP!=0\0HXS98<D�IhH?P#Ih8VP!P!HT\0C5=T@K8VJ!;FG!\ ;FG#=T;>A?8<@`]X_�J!798eÈtC?\0;UD>J~HT@K;FC?@�i

l-Ó

��� kh¨ ��� ¬ k � ¬ � Ó � k � � ¬ k u

r 798VP!8 ¬ k ;FG�C�J!7KP!8V8}ϦSK;F\'8<@^G!;>HT@KC?D EK@K;>JtA?8<M}J~H?P:C?@KS � ;UG[C�EK@^;>IZH?PO\ 8}�NJ~8VP+@KC?D�\0C5=T@98VJ!;UM:��8<DFS�4 �tG!8\08<C?@`�^8<DFS�J!7K8VH?P!_�J~HsH?]KJ!C?;F@�C?@`8}�NcKP!8<G+G!;>HT@`IhH?P#J!798�M}P+;FJ!;FMVC?D�J~8<\'c�8VP+C5J!E9P+8 � � 4Ø È[;F@BJVi�6ÊC5g?8:J!798)8}�NJ~8VP+@KC?D���8<DFS�J~Hs]�8);F@�J!798��� SK;>P!8<M}J!;FHT@�4 Ù3 �N4�6�798�ÈtC?\s;FD>J~HT@K;FC?@�H?I|C0M}8VP!J!C?;U@�\'HNS98<DvG~_NG~J~8<\ ;FG#=T;>A?8<@`]X_l Ó

�a � k � ·k · � Ó�n � k · k u

r ;>J!7 · k l ��354�LN79H r J!7KC5JtJ!798:G~_NG~J~8<\ EK@KS98VP!=?HX8<G#Csc^7KC?G~8�J~POC?@KG!;>J!;>HT@�;F@�C?DFDvS^;F\'8<@KG!;FHT@KG�C?@KS`�*@KSJ!7K8:M}P+;>J!;FMVC?D�J~8<\0c*8VPOC5J!E9P!8�;F@�J!798:J!798VP+\'HNS9_N@KC?\0;FM�DF;F\s;>J�a ��� 4��HT\0\'8<@BJ#HT@`J!798:;U\'c^DF;FMVC5J!;FHT@KG�H?I|J!7K;UG�P!8<G!EKD>J�IhH?P�J!798 ª G!;F@K=�\0HXS98<DÐ4� ��\ � /��^]\*^���X � W &��#$ []\ � �� � & [354�6�798��#­»P!8<MVEKP+G!;>HT@YP!8<DFC5J!;>HT@^GtIhH?P)C�HT@98}ϦSK;F\'8<@^G!;>HT@KC?D ª G!;U@9=�\'HNS98<D|;F@®C?@®8}�XJ~8VP+@^C?DÊ�^8<DFS n \0C�_�]�8r P+;>J~J~8<@�C?G��

� �l � ¯Ò3�Í��K´ �¯ � Í��9´}¯Ò3�Í � �K´ � � � l ��¯ � Í��9´¯Ò3�Í � �K´ ur 798VP!8 � l�¾ p! ��"$# � C?@^S%� l¸¾ p'& "$# � 4 É 8VPO;>IZ_�J!798�8}�9;FG~J~8<@^M}8�H?IÊ�K�X8<S�c*HT;U@�J!G�C?G�IZHTDFDFH r GVi�¯ �)( u*� ( ´ l¯ �Nu ��´+� ¯ � ( u*� ( ´ l ¯ �Nu<3�´�C?@KS�CsDF;F@98�H?IÊ�K�N8<S`c�HT;F@�J!G � ( l 3)IhH?P��-,.� ( ,-3 � '�;FG!MVEKG!G#J!798)c^7B_XG!;UMVC?DG+;>=T@K; ��MVC?@^M}8�H?IKJ!798<G!8�c�HT;F@BJ!GmC?@^S�G!g?8VJ!M+7�J!798�G~_NG~J~8<\-c*HT;U@�Jm�^H r G|;F@eJ!798�J r H5ϦSK;F\'8<@^G!;>HT@KC?DBc^C5P+C?\'8VJ~8VPG!c^C?M}8�]*HTE^@KS98<S�]X_ �/, � , 3�C?@KS��-,0�1, 3 � µ�_:DF;F@K8<C5P+;FG!;F@K=�C5]�HTE9J J!7K8�Ih8VP!P!HT\0C5=T@K8VJ!;FM��K�X8<S�c�HT;F@BJV�H?]^J!C?;F@`J!798�\0C5J~PO; ��H?I|J!798 �#­�6¸C?@^S�G!79H r J!7KC5J#J!798�C?G!G!HXMV;FC5J~8<SYM}P+;>J!;FMVC?D�;U@KSK;FM}8<G�C5P!8 � � l�y u*� � l 3IhH?P�G!MVC?DU;F@9='I±C?M}J~H?P � loy 4

Page 10: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

y 4�6�798'ÈtC?\0;UD>J~HT@K;FC?@YIZH?P)J!7K8�HT@K8}ϦSK;F\'8<@KG+;>HT@KC?D ª G!;U@9=�\'HNS98<D|;F@xC?@®8}�XJ~8VPO@KC?DÊ�^8<DFS n \sC<_®]*8 r P+;FJ~J~8<@C?Gl-Ó � �� k � � · k · � Ó Ó Æ n � k · k*Ó � k × ur 798VP!8�� ;FG�C�M}HT@KG~J!C?@BJV�N]^C?M!gX=?P!HTEK@^SsJ~8VP+\�4�LN79H r J!7KC5J�J!798�c^C5P+J!;>J!;>HT@0I±EK@KM}J!;FHT@'IZH?P�J!798[G~_XG!J~8<\»\0C�_]�8 r PO;>J~J~8<@ C?G�C`c^P!HXS^EKM}J�H?I�J~8VPO\0GV�m8<C?MO7 H?I r 7K;UM+7%S98Vc�8<@KSKG�HT@ HT@KD>_�HT@K80H?I�J!798'8VA?8<@NϦ@XEK\:]�8VP!8<SG!c^;F@KGV4�È�8<@^M}8?��]X_`c^C5P!J!;FC?DmG!EK\0\0C5J!;FHT@`H�A?8VP�J!798e8VA?8<@NϦ@BE^\:]�8VP!8<SYG~c^;F@KG<�9H?]KJ!C?;F@�J!7K8:P!8<MVE9POG!;>HT@`P!8<DUC¢ÏJ!;FHT@KG�IhH?P � � u n � C?@KS × � 4{ 4[��HT@KG!;FS98VP J!798�J r H5ϦS^;F\'8<@KG!;FHT@KC?D ª G!;F@9=[\'HNS98<D�EK@^S98VPmS98<MV;F\sC5J!;>HT@:HT@�C[G!WBE^C5P!8�DFC5J~J!;UM}8?4 ª I r 8�HT@KD>_�J!C5g?8;U@�J~H�C?MVM}HTEK@BJ[J!7K8�M}HTE9c�DF;F@9=sM}HT@^G~J!C?@�J!G � ¯±@98<C5P!8<G~JÒϦ@K8<;>=T7�]�HTE9P�;F@BJ~8VP+C?M}J!;>HT@KG ´#C?@KS # ¯±@98}�NJÒϦ@98<C5P!8<G~JÒÏ@K8<;>=T7�]�HTE9P#;U@�J~8VP+C?M}J!;FHT@KGO´O�KJ!798)P+8<MVE9P+G!;>HT@�P+8<DFC5J!;>HT@KG�C5P!8:=T;>A?8<@�]X_� � l�y � � Í #

�# � l � � �

%|;F@^S�J!798:�K�X8<SYc�HT;F@BJ!G#IhH?P�J!798<G~8�P!8<@9H?PO\0C?DF;>¶<C5J!;>HT@�=?P!HTE9cY8<WBEKC5J!;FHT@KGtC?@^SY;FSK8<@�J!;>Ih_�J!798eM}P+;FJ!;FMVC?DvHT@98?4« ;F@98<C5P+;UG!;F@9=�J!798 �#­�6¸C5]�HTE9J�J!7^;FG�c�HT;F@BJV�XH?]^J!C?;F@�C'A5C?DFE98)IhH?P#J!798�M}P+;FJ!;FMVC?D�8}�Nc*HT@K8<@�J � 4¬ 4&'�;FG+MVEKG!G0]�HT@KS c*8VPOM}HTDFC5J!;>HT@ HT@ C J r H5ϦS^;F\'8<@KG!;FHT@KC?D#G!WXEKC5P!8®DFC5J~J!;FM}8YC?@^S���S9P+C r ;F@9=�C?@KC?DFH?=T;>8<G r ;FJ!7J!7K8 ª G!;U@9=`\'HXSK8<D r 798VP!80C5cKc^P!H?cKP+;FC5J~8?�v;U@�J~P!HNSKEKM}8�J!7980M}HT@^M}8VcKJ!G�H?I������ <���+>�C���� ��->���� C���<������� ����-@=C3>�<�� ��CD@���� <�� �vC?@KS!�"����<���+>1C�@�#�$ �� @�� <±4L979H r J!7^C5J#J!798:M}P+;FJ!;FMVC?D�cKP!H?]^C5]�;FDF;>J¼_0IZH?P�J!7K;FG�cKP!H?]�D>8<\Ñ;FG|§ �/l �� 4°tcKc^DF_'J!798�P!8<@9H?P+\0C?DU;>¶<C5J!;>HT@�=?P!HTE9c�J~H']�HT@KS�c*8VPOM}HTDFC5J!;>HT@��9EKG!;U@9=eC�G!MO798<\'8�;F@ r 7K;FMO7�C?D>J~8VP+@KC5J~8)G!;>J~8<GC5P+8�P!8<\'HRA?8<S�IhP!HT\ÑJ!798:DUC5J~J!;FM}8?4�LN79H r J!7KC5J#J!7K8 ��­Ë8<WBE^C5J!;>HT@�J!C5g?8<GtJ!798)IhH?P+\§ � l�y § � Ó § 5u

r 798VP!8v§ � ;UGvJ!798�cKP!H?]^C5]^;UDF;>J¦_tH?I9J r H)G!;>J~8<Gv]�8<;F@K=#M}HT@K@98<M}J~8<S�HT@:J!7K8�@98 r DFC5J~J!;FM}8�C?@^S�§';FGvJ!798�C?@KC?D>H?=?HTEKGWXEKC?@BJ!;>J¦_oIhH?P�J!798 HTDUS¸DFC5J~J!;FM}8?4 É 8VP+;>Ih_ÎJ!7KC5J�J!798%�K�N8<Soc�HT;F@�J!G�C5P!8`§ ( l �Nu±§ ( l 35��C?@KS § ( l¯ 4 º Ó 3�´ z5y % � � � y ��C?@KS�S^;FG!MVEKG!G�J!798e@KC5J!E9P!8)H?I|J!798<G~8��^�X8<S`c�HT;F@BJ!GV4µ�_®DF;F@98<C5P+;F¶<;F@9='J!798/�#­»J~P+C?@KG~IhH?P+\0C5J!;>HT@xC5]�HTE9J�C0�^�X8<SYc�HT;F@�JV��G!7KH r J!7KC5J[J!798�P!8<D>8VA5C?@�J[8<;>=?8<@BA¢C?DFEK8H?I|J!798)J~POC?@KG~IhH?P+\0C5J!;>HT@�\sC5J~P+; �`;FG�=T;>A?8<@�]�E� � l ¬ § ( ¯Ò3 Ó § ( � ´ ��[G!;F@K='J!7K;FG�P!8<G!E^D>JV�9H?]KJ!C?;F@`C0@XEK\'8VPO;FMVC?D�A¢C?DUE98)IZH?P�J!798:M}P+;>J!;UMVC?D�8}�Xc�HT@98<@BJ � 4

� ��\ � /��^]\*^���X � W &��#$ []\ �'& � Z�\354�° c^C5P!J!;UMVEKDFC5P`\'HNS98<D[IZH?P`J!798 M}PO;>J!;FMVC?D�]�8<7KC<AN;>HTE9P�H?I�G~c^;F@KG�HT@�C � ϦSK;F\'8<@KG+;>HT@KC?D[DFC5J~J!;FM}8 D>8<C?SKG�J~HP+8<@9H?P+\0C?DF;F¶<C5J!;>HT@�=?P!HTE9c�8<WXEKC5J!;>HT@^G�H?I|J!798)IZH?PO\

§ � l � � §�Í × ¯ � � Ó � 0 ´)(�Í+*0¯ §�(T´+�( � l � 0 (�u

r 798VP!8t§xC?@KS,(sC5P+8eJ!798�M}HTE9c^DU;F@9=sM}HT@KG!J!C?@�J!GV� × ;FG[C�G~_NG~J~8<\ M}HT@^G~J!C?@�J r 7K;FM+7Y;FGtc*HTG+;>J!;>A?8eC?@^S�P!8<C?DÐ�* l ¬ Ó � ��C?@KS ��;FG�J!798�EKG!E^C?D�¯±DF8<@9=?J!7�´�G!MVC?DF;U@9=xIhC?M}J~H?P<4�µ�_ DF;F@98<C5P+;UG!;F@9=�C5]�HTE9J�J!798��^�X8<S c�HT;F@BJ¯ § ( u-( ( ´ l ¯ �Nu ��´O�mH?]^J!C?;F@xJ!798'\0C5J~P+; �YH?I�J!798-�#­�6ËC?@KS%G+79H r J!7KC5J�J!798'C?G!G!HXMV;FC5J~8<S�M}P+;>J!;FMVC?D|;F@KS^;FM}8<GC5P+8 � � l�y C?@KS%� � l ¬ Ó � 4µ�P+;F8}�^_�S^;FG!MVEKG!G�J!798e@KC5J!E9P!8)H?IÊJ!7K;FG��^�X8<S�c�HT;F@BJV4­);>A?8<@�J!7^C5JtJ!798�M}P+;>J!;FMVC?D ;F@KSK;UM}8<G�IZH?PtJ!7K;FGt\'HXSK8<DmC5P!8eJ!798eG!C?\08eC?G[J!79HTG~8eIZH?P[J!798e\08<C?@NÏ��^8<DFS�J!798VH?P!_H?I|J!798 ª G!;F@K=0\'HXSK8<D��9M}HT\0\'8<@BJ#]KP+;>8}��_sHT@`J!798:A¢C?DF;USK;>J¦_�H?P#H?J!798VP r ;FG~8)H?I|\'8<C?@NÏ��^8<DFS�J!798VH?P!_?43 �

Page 11: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

y 4�° c^C5P!J!;UMVEKDFC5P`\'HNS98<D[IZH?P`J!798 M}PO;>J!;FMVC?D�]�8<7KC<AN;>HTE9P�H?I�G~c^;F@KG�HT@�C � ϦSK;F\'8<@KG+;>HT@KC?D[DFC5J~J!;FM}8 D>8<C?SKG�J~HP+8<@9H?P+\0C?DF;F¶<C5J!;>HT@�=?P!HTE9c�8<WXEKC5J!;>HT@^G�H?I|J!798)IZH?PO\� #�¯ �V´� DF@��

l¸y #�¯ �}´mÍ�� � � ¯ �V´3 � � �Ó #�¯ �V´ �m¯ �V´3 � � � �

� �m¯ �V´� DF@��

l *�� ¯ �}´ Ó { � � ¯ �V´3 � � � ur 798VP!8 #�C?@KS �®C5P!8eJ!7K8eM}HTE9c^DU;F@9=sM}HT@KG!J!C?@�J!GV� � ;FGtC�G!_XG~J~8<\ M}HT@KG~J!C?@BJ r 7K;FMO7�;FG�c�HTG!;>J!;>A?8:C?@^S�P!8<C?DÐ�* l ¬ Ó � �XC?@^S ��;FG�J!798tE^G!EKC?DKG~c�C5J!;FC?DKP!8<G+MVC?DF;F@9=)I±C?M}J~H?P<4�6#7K8<G~8#8<WXEKC5J!;>HT@KG�C5P!8�A5C?DF;FS0IZH?P�G!\0C?DFD^A¢C?DFEK8<GH?I�*<4�­�;>A?8<@�J!7^C5J#J!798�M}P+;>J~8VPO;>HT@�IZH?PtC��K�N8<S`c�HT;F@�J�¯ # ( u�� ( ´�;FG

� # ( ¯ �V´� DF@��

l � � ( ¯ �V´� DF@6�

l �NuA?8VPO;>IZ_�J!7KC5J�C'�K�X8<S�c�HT;F@BJ:¯hJ~H0H?P+SK8VP6*O´�;UG�=T;>A?8<@�]X_

� ( l 3 � � � *{ � C?@KS � ( l-Ó � � *� �µ�_)DF;F@K8<C5P+;FG!;F@K=�C5]�HTE9JvJ!798|�^�X8<S)c�HT;F@BJV��H?]^J!C?;F@)J!798/\0C5J~PO; �[H?INJ!798 ��­)6 C?@KS�G!7KH r J!7^C5JvJ!798�C?G!G!HXMV;FC5J~8<SM}PO;>J!;FMVC?Dv;F@^SK;FM}8<G�C5P!8 � � l�y�Ó * z5{ C?@KS � � l�Ó *V4µ�P+;F8}�^_�SK;UG!MVEKG!G�J!798)@KC5J!E9P+8)H?I J!798[�K�N8<S�c�HT;F@BJ r ;>J!7�c^C5P!J!;FMVEKDUC5P�P+8VIZ8VP!8<@^M}8)J~H'J!798�SK;F\'8<@^G!;>HT@KC?DF;FJ¦_'H?IJ!7K8:DFC5J~J!;FM}8?4L979H r J!7^C5J�C?@`8}�NcKP!8<G!G!;FHT@`IZH?P�J!798�M}P+;>J!;FMVC?Dv8}�Xc�HT@98<@BJ � MVC?@�]�8 r P+;FJ~J~8<@�C?GVi

�l 3y Í *3 y Í���¯ * � ´ �

����� [����������� � �� �� � X " ��[���XDZ�^ []X ��\ � /� ^ \)^ �354�6�798#C?@KC?D>H?=TE98�H?I�8<WX@�¯±µ[4F3?3�´�IhH?P/cKP+8<G!G!E9P!8�;F@�J!7K8�MVC?@9HT@K;FMVC?DK8<@KG~8<\e]^D>8 r 798<@'8}�NJ~8<@KS98<S0J~H�J!798#=?P+C?@^SMVC?@KHT@K;FMVC?D�8<@KG!8<\:]^DF8�;FG � l-Ó�� kZ¨ � � j�kZ¨ � d � § kh¨ �­);>A?8<@�i j�kZ¨ lon[k d p9qJ!7K8VP!8VIZH?P+8

� l-Ó�� kh¨ ¯ Ó wv´ ntk d p9q?p�� § kh¨ Îl w � kh¨ ntk d p9qd � §�� ¨ l wd � kh¨ j#kh¨ § kh¨ l w jd �y 4 ��� j�� � �e� l � ¯ jÔÓ�j ´}¯ � Ó � ´ � l � j�� �e� Ó � j � � �e� �� H r ¯±µ[4F3 { ´�=T;>A?8<G � k�l�Ó � j�k¦z � d/u7K8<@KM}8 � j�� �e� l�� k § khj�k � j#k� d l-Ó 3� � k ¾ p^Ã�Ä "$# � �<j#k �j#k� d �

��D>8<C5P+D>_ r 8:@KH r¸r C?@�JtJ~H0=?8VJ#J!7K;UG#;F@BJ~H'J!798)IZH?PO\`i3?3

Page 12: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

� j � � �e� c�DFEKG 6³G~HT\'8VJ!7^;F@9='8<DFG~838>4� H?J~8?i j�k±¾ p^Ã^Ä "$# � l ¯ÐÍ �B� � ´ �� � ¾ p^Ã^Ä "$# � l��B� �®�� � ¯ � § k ´ �L9E9]^G~J!;>J!EKJ~8�;U@�J~H0J!7K8)IZH?P+\�EKDFC�IhH?P � j�� �e� J~H0H?]^J!C?;F@�i� j � ��� l-Ó 3� �B� �Y�� � � � k § k � j#kÐz � d l �B� �

� �� � ¡ � � �e� ¤Tu¯hP+8<\'8<\:]�8VP � kml-Ó � j#kÐz � d�´O4� H r SK;&%�8VP!8<@BJ!;FC5J~8�J!798)cKP+HXSKE^M}J � � �e� r ;>J!7`P!8<G~c�8<M}J�J~H � C?@KS�EKG~8)J!7K8�;FS98<@BJ!;>J¦_*i� p�� � � z � ��l � DF@ � z � � ur ;>J!7�J!798�P!8<G+EKD>JVi � j�� �e� Ó � �e� � j � l ��� j�� � �e� l��B� � � � �e� z � � �{ 4��E9J � � l � à C?@KS � � l �^£�i�J!798<@`8<WX@�¯±°:4 y?{ ´�]�8<M}HT\'8<G

§ kh¨ £ l � p�� ¾ p À ��� Ã^Ä�� � Ö � � � � Å uC?@^S�¯±°:4 y ¬ ´O��¯±°e4 y ºT´�MVC?@�]�8�=?8<@98VP+C?DU;FG~8<S�C?G#C5c^cKP!H?cKP+;UC5J~8?4/6#798�M}HT@KG~J~POC?;F@�J!G�C5P!8�=T;>A?8<@�]B_jËl�� kh¨ £ § kh¨ £ j#kh¨ £�� d l�� kh¨ £ § kZ¨ £9d^£ �6 H�;FS98<@BJ!;>Ih_`J!798'\eE^D>J!;>c^DF;F8VP+G � à u�� £ r 8�M}HT\0c^C5P!8�J!798'\0C?M}P!HTG!M}H?c�;FM�8}�Xc^P!8<G!G!;>HT@YIhH?P � j ¯±M}HT\e]^;F@98<S3VG!J�C?@KS y @KS�DFC r G#H?I J!798VPO\'HXSK_X@KC?\s;FMVGO´ r ;>J!7�J!798e\0;FM}P!HTG!M}H?c�;FM�cKP!8<SK;UM}J!;>HT@�4

%9P!HT\ J!798VP+\'HNS9_N@KC?\0;FMVG �jËlÎ�

� · Ó � � doÍ � q � q ��� q �%9P!HT\Ñ8<WX@m¯ n � �T´ �

jËl�� kh¨ £ j#kh¨ £ � § kh¨ £�Í � kh¨ £ § kh¨ £ � j�kh¨ £ �°[DFG~H9� � · l � à � kh¨ £ j�kZ¨ £ � § kh¨ £)Í �K£ � kh¨ £ d^£ � § kh¨ £ �� @�J~P+H?cB_`;FG�A5C5P+;>8<S�C5JtM}HT@KG~J!C?@BJ[d�u�G~H0J!798�M}HT@KG!J~P+C?;F@BJ#HT@ df=T;>A?8<G

� d l�� kh¨ £ d^£ � § kh¨ £*uJ!7XEKG

� · l � à � kh¨ £ j#kh¨ £ � § kh¨ £�Í �^£ � dsuC?@^S�G~H � kZ¨ £ j#kh¨ £ � § kh¨ £ l � ·� Ã

Ó �K£� à � d'u7K8<@KM}8�jËl �� Ã

Ó �K£� à � d Í�q

�� � kh¨ £ § kh¨ £ �j#kh¨ £� � q ��� q�� �� ���HT\'c^C5P+;FG!HT@�H?IÊJ!7K8/6³\0;FM}P!H 8KC?@KS76³\0C?M}P!H 8^IhH?P+\0G�H?I � j D>8<C?S^G�J~H� à l 3 z�� � �^£ l � z�� �

3 y

Page 13: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

¬ 4�µ�_�8<WXEKC?D >������ �����cKP!H?]^C5]�;FDF;>J!;>8<G § k�l 3¹ uIhH?P�C?DFDv©!� r 798VP+8)¹¸;FG�J!798�@XEK\e]*8VP�H?I�\0;UM}P!HTG~J!C5J~8<GV4�6�798e­�;>]K]�G�8<@�J~P!H?cX_`;FG�=T;>A?8<@�]B_· l-Ót� � k § k DF@ʧ k �� A?8VP!_`\'8<\e]*8VP�H?I J!798eG!EK\0\0C5J!;FHT@`;FG�J!798:G!C?\08:C?@KS�J!798VP!8eC5P!8e¹�\0;UM}P!HTG~J!C5J~8<GV�^7K8<@KM}8

· l-Ó[��� ¹�� 3¹ � � DF@ 3¹ � l�Ót� ¯ Ó DF@#¹�´ l¸� DF@�¹�uC?G�P!8<WXEK;>P!8<S�4ºX4&%KP!HT\ ¯±µ[4F3<²T´ r 8:7^C<A?8 j�l-Ó[� �� ¯Ò3 z�� ´ DF@ � l-Ó � DF@ �� Æ uIhH?P#Æ l 3 z5�B� �%KP!HT\ J!798VPO\'HXSK_X@KC?\s;FMVGV�NIZP+8V8�8<@98VP!=?_�Ç l jÔÓ%� · C?@^S

· l-Ót� � k § k DF@ʧ kvl�� DF@ � Í j�z�� �6�7BEKG<i Ç l-Ót�B� DU@ � l-Ó Æ�p��KDF@ � �ª J#IhHTDFD>H r G�J!7KC5J � l¸¾ p ¿5ÁC?@^S�� § kvl�¾V¿BÀÂÁ p^Ã�ÄUÅ �C?@^S�G~H9i jËl-Ó �� Æ ¯ Ó Æ Çe´ l �� Æ ¯hÆmÇe´ �� H r � ¯ jÔÓ j ´¼Õ?Ö � � l�� k ¯ j#k�Ó j ´¼Õ5Ö � § kl � k ¯ j k Ó j ´ Õ5Ö � ¾ ¿�À Á p^Ã^ÄZÅ l-Ó � ¯ j k Ó j ´ Õ �� Æ ¾ ¿�À Á p^Ã^ÄUÅl Ó �� Æ � k ¯ j#k�Ó j ´¼Õ ¾ ¿�À Á p^Ã^ÄUÅ Í � k ¾ ¿BÀÂÁ p^Ã�ÄUÅ �� Æ ¯ j�k*Ó j ´¼Õl Ó �� Æ � ¯ jÔÓ j ´ Õ � Í�Ì � k ¾V¿BÀÂÁ p^Ã�ÄUÅ+¯ jÔÓ j ´ Õ p�� � j� Æl Ó �� Æ � ¯ jÔÓ j ´ Õ � Í�Ì � ¯ jÔÓ j ´ Õ p�� � � j� Æ �%KH?P�J!798:MVC?G!8�Ì l 3)i ��� j � � l � ¯ jÔÓ j ´ � �� jÔÓ j � l � ��®C5gN;F@9=0J!798�MO7KC?@9=?8eH?I A5C5P+;FC5]^DF8�C?@^S�S9HT;F@9='J!7K8�SK;&%�8VP!8<@�J!;UC5J!;>HT@`C5J�M}HT@KG~J!C?@BJ�d/u� j� Æ � �B�t� × � �

3 {

Page 14: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

� � X " ��[�� X¿Z�^ []X �\ � /��^]\*^�1354[­);>A?8<@ j#kml�¾<k ¯ n l ��´ Ó � k�n u ¯��¢´J!7K8�cKP!H?]^C5]�;FDF;>J¼_0H?IÊC?G+G~8<\:]�D>_�]�8<;F@K=�;U@`G~J!C5J~8�½ © � ;FG

§ k�l ¾� p^Ã�Ä "$# � �¥ ]BAN;>HTEKG!DF_sIhP!HT\�8<WBE^C5J!;>HT@ ¯��¢´O�KJ!7K8:;F@KG~J!C?@BJ!C?@98VHTEKG�\0C5=T@98VJ!;FM�\0HT\'8<@�Jt;FG�=T;>A?8<@�]X_� kml-Ó � j#kÐz � n �

6�7BEKG�\'8<C?@`A5C?DFE98)H?I|\0C5=T@98VJ!;FM�\0HT\'8<@�Jt;FG�=T;>A?8<@`]X_*i� �Ë� l�� k � k § k�l�Ó�� k � j k� n � ¾ p^à Ä������ l-Ót�B� � DF@ �� n �

ª @BJ~P!HXS^EKM}8?i��� l � � ���� n � � l�� k � k � §

k� nl Ó 3�B� � k � k § k � j k� n Ó � DF@ �� n � k � k § kl 3�B� ¡ � � � � Ó � �Ë� � ¤TuJ!7K8VP!8VIZH?P+8 � � � l � � � � Ó � �Ë� � l��B� � � �y 4

� � l-Ó 3d � � d� � � � �%KH?PtC?@�;US98<C?D�=TC?GV� � d l a �B�¯hH?P � l � �B�ol Ì �B� ´J!7XEKG d l a �B�#z �C?@^S�G~H� +� l-Ó 3d � Ó a �B�� � � l 3d � 3� � � � d l 3� l 3Ì �B� ¯±EKG!;U@9='C5]*HRA?8:8<WXEKC5J!;>HT@�;U@�]KP+C?M!g?8VJ!G ´

7K8<@KM}8 � +� l 3Ì �B� uC?G�P!8<WXEK;>P!8<S�4{ 4(�Y8<C?@NϦG!WXEKC5P!8���EKM}J!EKC5J!;>HT@vi� ¯±a Ó � a � ´ � � l � a � Ó y a � a � Í � a � � 7l � a � � Ó y � a � � Í � a � �l � a � � Ó � a � �

� � a � � Ó a � �3 ¬

Page 15: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

­);>A?8<@�i � l � kZ¨ ¾ p ¿ Ã^Ä Ö ! ¿¯ �B� ´ � � � DF@ �� # � ) � ¨ � l ¯ �B� ´ � �� # � 3� � �� # � � ¨ �l ¯ �B� ´ �� # �� 3� � kh¨ ¾ p ¿ Ã�Ä Ö ¿"!

��a���

l �B� �� Ó 3� � � �� # � � kh¨ a ¾ p ¿ Ã�Ä Ö ¿ !

Í 3� � kh¨ Æma � ¾ p ¿ Ã^Ä Ö ¿ ! ��l Ø Ó � a � � Í � a � � Ùl � a � � Ó � a � � uC?G�P!8<WXEK;>P!8<S�4¬ 4&%KP!HT\ J!798:cKP!8VAN;>HTEKG�WXE98<G~J!;>HT@�i

� ¯±a Ó � a � ´ � � l ¯ �B� ´ ����� � DF@ �� # � ) � uOd �� H r � ;FG�c^C5P!J!;>J!;FHT@�IhEK@^M}J!;>HT@�IZH?P�J!798�­[P+C?@^SY��C?@9HT@K;UMVC?D�8<@KG~8<\e]^D>8?�K798<@KM}8 r 8�7KC�A?8?i� d l¸�B� DF@ � �L9E9]^G~J!;>J!EKJ!;F@9=�IZH?P�DU@ � u r 8:7KC�A?8?i¯ �B� ´ �0� � DF@ �� # � ) � ¨ � l ¯ �B� ´ � � � �� # � � � d�B� � ) � ¨ �l ¯ �B� ´ � d�B� � � �� # � ) � ¨ � l �B� d �� # � � �� # � � uOd �

­);>A?8<@�i � � �� # � � ¨ � l a d l Ì�u� �B� d �� # Ì ´ � l��B� d � Ì� # � l��B� d ��� Ì� d � �m� d� # � l �B� d �Ô�� d � a d ) � � d� # � l

Ót�B� d ad � � � d� # � l Ót�B� ad �m� d� # � l�Ót�B� Ì �v� d� # � �­);>A?8<@�it¯ � d z � # ´ l�Ó a � �� Í'¯ �B� Ì ´ a � � l � � � z � +� � a �Èt8<@KM}8 � ¯±a Ó ax´ � �a l � � z � +� �

� A¢C?DUEKC5J!;>HT@`H?I �VÕ ! � � � �tG!8:MVC?DFMVEKDFE^G�;US98<@�J!;FJ¦_*i� ���� � � l � ���� � � � Í� � �� � � �

� � �� � � �3<º

Page 16: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

� � � Ì� # � � l � � Ì� # � Í � � Ì� a � ! � � a� # � � �« C?G~J#J~8VP+\ ;FG�¶V8VP!HsC?G�Ì l a z d»S9HX8<G!@ 8 JtS98Vc�8<@KS�HT@�a �6�798VP!8VIhH?P!8 �m� Ì� # � � l �Ê� Ì� # � u7K8<@KM}8 � � Ì� # � � ¨ � lÓ ad � � � d� # � uC?G�P!8<WXEK;>P!8<S�4ºX4�°�J ��� u � � � �� d l �NuJ!7K8VP!8VIZH?P+8?i � � ��3 z � �� d � � �6�7BEKG�C5J ��� u � � u � � � � u�C?@KS�G!H � ¯ � ax´ � � ��� ���HT@KG!;FS98VPtP!8<DFC5J!;>HT@^G!7K;>cY]*8VJ r 8V8<@xS98<@^G!;>J¦_��*EKM}J!EKC5J!;>HT@KG[C?@KS®S98<@KG!;FJ¦_BϦS98<@KG!;>J¼_�M}H?P!P!8<DFC5J!;FHT@ �s¯ � Ó � � ´ �'[8}��@^;F@9=0a l�� �

� � Ì�¯ � ´Ou r 8e7KC<A?8� ¯±a Ó ax´ � � l � � � � ¡<Ì�¯ � ´ Ó � Ì�¯ � ´ � ¤X½ � � � � ¡<Ì�¯ � � ´ Ó � Ì�¯ � � ´ � ¤ � u

C?@^S`IZP+HT\ÑJ!798:S98}�*@K;>J!;>HT@�H?I��� ¯±a Ó a%´ � � l � � � � � � � �s¯ � Ó � � ´l � � � � � � � �s¯ � Ó � � ´l � � � � � � � � �s¯ � � � ´ IhP!HT\ G~c^C5J!;UC?Dv79HT\'H?=?8<@98<;FJ¦_l d � � � � � �s¯ � � � ´ �

��HT\:]^;U@K;F@9= r ;>J!7�P!8<G+EKD>J�H?IÊcKP!8VAN;>HTEKG�WXE98<G~J!;>HT@�i� � z � +� l Ì|p�� � � � �s¯ � ´ �� � X " ��[�� X¿Z�^ []X �\ � /��^]\*^ W354

� ¯hÌ�Ío3�´ l � Ø � ¯hÌ ´ Ù ���HT@KSK;>J!;>HT@�IhH?P�C��K�N8<S`c�HT;F@BJ�;FG � ¯hÌ�Ío3�´ l � ¯hÌ ´ l �J!7K8VP!8VIZH?P+8:G!E9]^G!J!;>J!E9J!;F@9=�C5]�H�A?8eIZH?P � ¯hÌ�Ío3�´Ou�C?@^S � ¯hÌ ´�=T;FA?8<G

� l � ¯��9´OuC?G�P!8<WXEK;>P!8<S�4¯±C�´ � ¯hÌ�Ío3�´ l¸y � ¯hÌ ´ Ó {

3 �

Page 17: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

J!7K8VP!8VIZH?P+8 � Ø � ¯hÌ ´ Ù�l�y � ¯hÌ ´ Ó�{Èt8<@KM}8;� l � ¯��9´ � � loy � Ó { C?@KS � l�{ �¯h]�´ � ¯hÌ�Ío3�´ l � � ¯hÌ ´vÍ��� l � � ¯hÌ ´vÍ��� ¯��9´ l � �)Í��

� l � ��Í ��IhH?PtC��^�X8<S`c�HT;F@BJ��¯Ò3 Ó � ´ l �� l �3 Ó � �%KH?P � l 35u � l � �y 4[­);>A?8<@�i � ¯hÌ�Ío3�´ l Ø � ¯hÌ ´mÍ ¬ Ù � ¯hÌ ´mÍ y

� ¯ � ¯hÌ ´~´ l Ø � ¯hÌ ´vÍ ¬ Ù � ¯hÌ ´vÍ y� ¯��N´ l ¯���Í ¬ ´ ��Í y� H r ��� l � ¯��9´ IZH?P�C'�K�X8<S�c*HT;U@�JV�KG~H

� l ¯��)Í ¬ ´ ��Í y � � l-Ó 35u Ó[y �Èt8<@KM}8� ¯ ��´ l Ó 3 � �93 DF;F\Õ���� � ¯hÌ ´

l-Óty�

� ¯ ��´ l Ó � ����� DF;F\Õ���� � ¯hÌ ´l � �

� ¯ ��´ l Óty � ¬ DF;F\Õ���� � ¯hÌ ´l�Óty �

LN8V8�LX8<M}J!;>HT@ 354 �X4 { IZH?P�J!798�P!8<G~J�H?I|J!7K;UG�G~HTDFE9J!;>HT@v4{ 4�«mH?=T;FG~J!;FM�8<WXEKC5J!;>HT@Y;FG � ¯hÌ�Ío3�´ l ¯Ò3�Í � ´ � ¯hÌ ´ Ó � � � ¯hÌ ´OuJ!7XEKGVi � l ¯Ò3�Í � ´ � Ó �B� � C?@KS�G~H � l �*¯ � Ó �B�9´ �µ�E9J � l �z$# u7K8<@KM}8 � l ��¯ � Ó � �# ´OuC?@^S�G~H � l �*¯Ò3 Ó �# ´ � �°[G ���l �Nu-� l �'H?P # �;Ð4 8?4 ª IÊc�H?c^EKDFC5J!;FHT@�¶V8VP+H9�^;>J�P!8<\0C?;U@KG�¶V8VP!H94 9 798VP!8<C?Gt;>I ;FJ�=?P!H r G<� # ;FG#J!798�DF;F\0;FJV4

3 �

Page 18: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

¬ 4[��HT@KG!;FS98VP'G~c�;F@KGeHT@ C%DFC5J~J!;FM}8?4 %9H?P � 7 ��� u|J!798VP+8�;FG'SK;FG~H?POS98VPeHT@ G!MVC?D>8<G 7�� r 798VP+8��x;FG���@^;>J~8?4��HTC5P+G~8}ÏÐ=?P+C?;F@^;F@9=�8 %*8<M}J!;>A?8<DF_YP!8<S^EKM}8<G���C?@KS®J!7980G!_XG~J~8<\ ��H r G�EK@KSK8VP �#­�6�J~H`M}HT\'c�D>8VJ~80SK;FG~H?POS98VP<�r 7K;FM+7 ;FG�J!7K8�7K;>=T79ÏÐJ~8<\'c�8VP+C5J!E9P!8'�^�X8<S�c*HT;U@�JV4`°[MVM}H?P+SK;F@9=TDF_®J!7K8�7K;>=T79ÏÐJ~8<\'c�8VP+C5J!E9P!8'�^�X8<S�c*HT;U@�J:;UG> < <�� > � <�� E�@��%KH?P �

� � uvJ!7980P!8VA?8VPOG~8sC5P!=TEK\08<@�J:7KHTDFSKG)C?@KS%J!7K8sD>H r ÏÐJ~8<\'c�8VP+C5J!E9P!8'�^�X8<Sxc�HT;F@BJ�Ï r 7K;UM+7%M}H?P!P!8}ÏG!c*HT@^SKG�J~H0c�8VP!Ih8<M}J�H?P+S98VP�Ï�;UG�C?DFG~H0C5J~J~P+C?M}J!;>A?8?4ª @�c^C5P+C?\'8VJ~8VP#G~c�C?M}8tJ!798VP+8[\�EKG~J�]�8tC18 r C5J~8VPOG!798<S 8>� r 7K;UM+7�G~8Vc^C5P+C5J~8<G�J!798�J~P+C"�~8<M}J~H?P+;>8<G�J~H�J!798�D>H r Ï �C?@^S%7K;>=T7NÏ � �K�N8<Sxc�HT;F@�J!G<4:6#7K;FG[;FG�J!798'M}P+;FJ!;FMVC?D|G!E9P!I±C?M}80C?@KSxM}HT@�J!C?;U@KG)J!798�� � # @�xH?P �"����<���+>1C��K�N8<Sc�HT;F@BJV4%KH?P#HT@98}ϦSK;F\08<@KG!;>HT@KC?D ª G!;F@9=9�XJ!798)G!;>J!EKC5J!;FHT@�;UG�G!g?8 r 8<S���C?G�J!798 �ol �e�K�N8<S�c*HT;U@�J�;FG�C?DFG~H�J!798)M}P+;>J!;UMVC?D�^�X8<S`c�HT;F@BJV4�%9H?P � 7 �Nu �#­�6o��H r ;UG�J~H0J!798)�K�N8<S`c�HT;F@BJ�C5J � ��� �ºX4�|C5P!J!;FJ!;>HT@�G!EK\ � l 3�Í y?¾ p ¿10 � Í y?¾ p ¿$0 � Í ¾ ¿$0 ��Y8<C?@�8<@K8VP!=?_

� j � l � y * � ¾ p ¿10 � Í y * � ¾ p ¿10 � Í *-, ¾ p ¿10 �� ��Y8<C?@NϦG!WXEKC5P!8:8<@98VP!=?_� je� � l � y * � � ¾ p ¿$0 � Í y * �� ¾ ¿$0 � Í * �, ¾ p ¿$0 �� �

�X4�°[G#J!7K8)c^C5P!J!;FMVD>8<G�C5P!8)HT@�C'DFC5J~J!;UM}8�J!798V_`\0C<_�]�8�J~P!8<C5J~8<S®C?G#SK;FG~J!;U@9=TEK;FG!7^C5]^D>8?4 Èt8<@KM}8?i��� k l ¯ � � ´ ur 798VP!8 � � ;FG�J!7K8�c^C5P!J!;>J!;FHT@�IhEK@^M}J!;>HT@�IZH?PtC?@�_�HT@98�G~c�;F@�4 � H r� � l ��¾ p 0 2+"$# �l ¾ p ! & "$# � Í ¾ Ö ! & "$# �l y M}HTG!7 ¯ # n'z5�B� ´ �6�7BEKG ��� k l Ø y M}HTG!7 ¯ # n0z5�B� ´ Ù �Èt8<DF\07KHTD>J~¶)IZP+8V8�8<@98VP!=?_�Ç Ç l-Ót�B� DU@ ��� k l-Ó a �B� DU@ Ø y M}HTG!7 ¯ # � z5�B� ´ Ù �ª @BJ~8VP+@KC?D�8<@98VP+=?_�� l � j �

�l � j � l �B� �Y�� � DF@ ��� k l Ó a # n J!C?@K7e¯ # n0z5�B� ´ �� @�J~P+H?cB_ · l ¯ jÔÓ Çe´ z5�B�l a � ¡�DF@ Ø y M}HTG!7(# n'z5�B�ÎÓ # n0z5�B� J!C?@K7:¯ # n0z5�B� ´ Ù ¤ ��®C5=T@98VJ!;FMe\'HT\'8<@BJ �� l a # l a � k # k § k

l a � # ¾ ! & "$# � Ó # ¾ p ! & "$# � ¾ !&"$# � Í ¾ ! & "$# � l a ! J!C?@K7 # n0z5�B� �

LNc*8<MV;>��M�798<C5J × & × & l � � j� � )&l a � ¯ # n0z5�B� ´ � z M}HTG!7 � ¯ # n'z5�B� ´ �

3<²

Page 19: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

� � X " ��[���X¿Z�^ [­X ��\ � /��^]\*^ �354&%|;>POG~J�c^C5P!J#M}HRA?8VP!8<SY;U@`°�cKc�8<@KS^; ���[4'[8}��@K8?i � Õ ¯ � ´ l � · Õ · Õ?Ö� � �� ;FG�SK;FG~J!C?@KM}8)]�8VJ r 8V8<@YG!;>J~8<GV�K\'8<C?G+E9P!8<S�;F@`E^@K;>J!G�H?I|DFC5J~J!;FM}8:M}HT@KG!J!C?@�J&� �

� Õ ¯ � ´ l �0p�� �� ��· Õ · Õ?Ö� ¾ ����� �Ä� �� Ä � Ä � Ä � � � � k � �

k�B� ���8}Ï r P+;>J~8 � � Õ ¯ � ´ l � � �� · Õ · Õ?Ö�

¾ � Ä � Ä � Ä� � �

��HT@KG!;FS98VP#@K8<C5P!8<G~J�@98<;F=T7�]�HTE9P�MVC?G~8?i � l 3 �� � Õ ¯Ò3�´ l �

� ��· Õ · Õ?Ö � ¾ � Ä � Ä � Ä � � l �� � Õ

�� ��

¾ � Ä � Ä � Ä � �l �� � ' C?@^S�;F@KSKE^M}J!;>A?8<D>_ ' � Õ ¯ � ´ l �� � Õ

�� � Õ?Ö � �V�V��� � Õ?Ö� p�� �

�Èt8<@KM}8?i y

� p��k � M}HTG!7 � k � Õ ¯Ò3�´ l�y

� p �k � M}HTG!7 � k �`G!;U@K7 � Õ uJ!7K8VP!8VIZH?P+8 � Õ ¯Ò3�´ l J!C?@K7 � Õ u� Õ ¯ � ´ l � k � J!C?@K7 � Õ5Ö k p�� ��[@K;>IhH?P+\Ñ;F@BJ~8VP+C?M}J!;>HT@ � � Õ ¯ � ´ l J!C?@K7 � ���HT@KG!;FS98VPtDF;F\0;>J � � � � � 7 �Nu�J!C?@^7 � 3:C?@KS8� Õ � �sC?G � � � � Èt8<@KM}8eIZH?P ��l �Nu � � �C?@^S�� Õ � 3 �y 4[­�8<@98VP+C?DF;>¶V8eÈtC?\0;UD>J~HT@K;FC?@�i

l-Ó p��� k

� � ·k · k Ö � �

l-Ó p��� k � � ·

k · k Ö � Ó�n � k � ·

k �­�8<@98VP+C?DF;>¶V8:J~P+C?@KG~Ih8VP�IhE^@KM}J!;>HT@�i

� ¯ · k u · k Ö � ´ l¸¾ p ¿ À � Ä ¨ � Ä� � Å l�¾ Ö ¿3� � Ä � Ä � � Ö ¿ & À � Ä Ö � Ä� � Å " � �«m8VJ#Æ � � � u^C?@KS�Æ n � � �6 P+C?@KG!IZ8VP�\sC5J~P+; �¦ � � � Ö�Ö � Ö p� p Ö � p�p )

l � ¾ Ö�� ¾ p ¾ p ¾

p � )r 798VP!8�\0C5J~P+;>��8<D>8<\08<@�J!GtC5P!8)=?8<@98VP+C5J~8<SYIZP+HT\`i���� � � ¯ · k�l �'35u · k Ö � l ��3�´ �� ;>=?8<@BA¢C?DFEK8<GVi � � l�¾ M}HTG+7&� � � ¾ � G!;F@^7 � ��Í ¾ p � � � " � �� l � Ö Í � p3 �

Page 20: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

C?@^S�C?G�a ��� u9HT@KDF_�DFC5P+=?8VP#8<;>=?8<@BA¢C?DFEK8�� Ö ;FG�P+8<D>8VA¢C?@BJV4%KP!8V8:8<@98VP+=?_�c�8VP�DFC5J~J!;UM}8:G!;>J~8?i � l-Ó �¿ DF@� Ö u^C?@KS�G~H9i� l-Ó � Ó Æ p�� DF@��FM}HTG!76� � � ¾ � G!;F@K7 � ��Í ¾ p � � � " � � �

�l � � l-Ó Æ � �� �l G!;F@K7 � � G!;U@K7��*M}HTG!7&� � ¾ � G!;F@^7 � ��Í ¾ p � � p�� " ��FM}HTG!7&� � � ¾ � G!;U@K7 � �|Í ¾ p � � � " � � �

°[G�� � �Nu � � �[IhH?P�C?DFD � � � �NM}8Vc^J ��l � r 798<@eHT@98�MVC?@'7KC<A?8�Ct��@K;>J~8�c�8VP+\0C?@98<@BJÊ\sC5=T@98VJ!;>¶<C5J!;>HT@�4{ 4� �l � ¯Ò3�Í��K´ �¯ � Í��9´}¯Ò3�Í � �K´ ¯Ò3�´+� � � l ��¯ � Í��K´¯Ò3�Í � �K´ ¯ y ´ �

¯ � �Tu*� �¢´ l ¯ �Nu<3�´ � WX@v¯Ò3�´ � � l � ��33 l � 8<WX@m¯ y ´ � 3 l 3T¯ �tÍ�3�´3�Í � l 3¯ � �Tu*� �¢´ l ¯ �Nu ��´ � WX@v¯Ò3�´ � � l � ��33 l � 8<WX@m¯ y ´ � � l � � �3 l �

� �l 35u � ,0� � ,Ô3 � WX@v¯Ò3�´ � 3 l 3T¯Ò3�Í��9´ �¯Ò3�Í��K´ � l 3 �'� 8<WX@m¯ y ´ � � � l � �

�l 3�� �ol � � � l �Nu n�l � u � l 35u n l �NuJ!7K8VP!8VIZH?P+8�IZ8VP+P!HT\0C5=T@98VJ!;FM)�K�X8<S�c*HT;U@�J�;FG�¯ �Nu<3�´ �¬ 4 � C?M+7YG~c^;F@�HT@�CsG!WBE^C5P!8�DFC5J~J!;FM}8:;U@�J~8VP+C?M}J!G r ;>J!7�J!798 ¬ @98<C5P!8<G~J�@K8<;>=T7�]�HTE9POG�HT@�C0G!WXEKC5P!8)P!HTEK@^S`;>JV4

'[8<@9H?J~8�C?D>J~8VP+@KC5J~8eG~c^;F@KG�HT@�DFC5J~J!;FM}8�]X_ � k uKP+8<\0C?;F@KS98VP�]X_ � k i¡ · k ¤ l ¡ � k ¤�ÍΡ � k ¤ ��|C5P!J!;FJ!;>HT@�G!EK\ MVC?@`]�8)8}�XcKP+8<G!G~8<S�;F@�J~8VPO\0G#DF;>g?8�� Ä � �V�V�

¾ Ä À � � Ö � � Ö � Ö ��� Å �V�V�

C?@^S�;>I r 8:S9H'J!7K8:G!EK\�HRA?8VP � kml �'3� l �� � Ä � �V�V� �

¾À � � Ö � �~Ö � !Ö � +Å�Í ¾ p À � � Ö � �~Ö � !Ö � � �V�V�

r 798VP!8 � � Ó � C5P!8:J!798�@98<C5P!8<G~Jt@98<;>=T7B]*HTEKP#G~c^;F@KG�J~H08<C?MO7 � k �;Ð4 8?4/c^C5P!J!;>J!;FHT@�IhEK@^M}J!;>HT@`;FG�J!798�c^P!HXS^EKM}J�H?I|\sC?@�_�G!;F\0;UDFC5P�J~8VP+\0G�J~H0J!7^C5JV4� H r P!8}ϦDFC5]�8<D�J!798�P+8<\0C?;F@K;F@K=�G~c�;F@KG · k ��¥ E9P�P!8<@KH?P+\0C?DF;>¶<C5J!;FHT@`M}HT@KSK;>J!;>HT@`;UGVi� ¾ À � � Ö � �ÒÖ � !Ö � � Å Í ¾ p À � � Ö � � Ö � ~Ö � � Å l � ¾ ��� �� À � � � �ÒÖ � � � ~Ö � � � Ö � � � � Å Ö � À � � � !Ö � � � � Å Ö � � � � � � � ¤ �6�7K;FG�\�EKG~J#79HTDFS�IhH?P�C?DFDvc*HTG+G!;>]^D>8tA¢C?DUE98<G�H?I · � u · � u · , C?@KS · 8?4 =94 >1C C · k l Í�3�H?P >1C C l�Ó 3�=T;>A?8<GVi¾ Í ¾ p! l � ¾ � � Ö � �ÒÖ y �

Page 21: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

C?@^S`H?J!798VP�c�HTG!G+;>]^D>8�M}HT\:]�;F@KC5J!;>HT@KG<i y:l � ¾ p � � Ö � � Ö �¾ � Í ¾ p � l � ¾ p �yel � ¾ p � �ÒÖ �� �<A@ · k C5P!8:G!c^;F@KG�HT@`J!798 � @�� DUC5J~J!;FM}8?4�Èt8<@KM}8:@98<C5P!8<G!JÒϦ@98<;>=T7B]*HTEKP#J~8VP+\0G#DU;>g?8 · � · � r ;FDUD�C5cKc�8<C5P�;U@ �� @�H?J!798VP�G!WXEKC5P!8[]KP+C?M+g?8VJV4�6#7K;UG/=T;>A?8<G�CeIhC?M}J~H?P�H?I y G~H�J!7KC5J�J~H?J!C?D�M}HT@�J~PO;>]^E9J!;>HT@sIZP+HT\ · � C?@KS · � ;UG¾

� � � � � �ª @�C?DFDÐ�XIhH?PtabG~c^;U@KG�HT@`H?P+;>=T;F@^C?D�DFC5J~J!;FM}8� l � ¯ � ´ " � � ������� 8}�Nc)¡ � �

� � k � � · k · � ¤�Í � � �� � kh¨ ��� � · k · � Í � , � k � ��� ·k · � · # ·� u

r 798VP!8?i � � l 3¬ DF@�M}HTG+7m¯ ¬ � ´+� � � l 3² DF@�M}HTG!7 ¯ ¬ � ´� , l 3² DF@�M}HTG!7m¯ ¬ � ´ Ó 3y DF@�M}HTG!7 ¯ y � ´ �& � X " ��[�� X¿Z�^ []X �\ � /��^]\*^ 354&%KH?P�J!7K;FG#_?HTE�@98V8<S�J~H�S9P+C r J!798)=?P+C5c^7KG�C?G#SK;FG!MVE^G!G~8<S`;F@�LX8<M}J!;FHT@ { 4F354³ºX4y 4�6ÊC5g?8 � l ��HT@%J!7980M}8<@BJ~P+C?D�c^DUC?@98?� r ;>J!7xÌ Ö ¯ ��´ l Ì p ¯ ��´ l Ì + ¯±G!C�_9´O4�­�8<@98VP+C?DF;FG!8`¯ { 4 y?y ´O��¯ { 4 y ²T´Oi��P+G~J�Ì�¯ � ´ l Ì + ¾ p �� "$# � C?@KS�G~8<M}HT@KS�¯hIZH?P�c*HTG+;>J!;>A?8)C?@KS�@98V=TC5J!;FA?8eM+7^C5P!=?8<GO´ r 8�7KC�A?8¾ Ì�¯ � ´ l¸¾ ¯hÌ Ö Ó Ì p ´ l�¾ Ì Ö ¯ ��´ ¾ p �� "$# � Ó ¾ Ì p ¯ ��´ ¾ �� "$# �l-Óty?¾ Ì + G!;F@K7�¡ ¾ � z5�B� ¤ �6ÊC5g?8��ÊHT;FG+G~HT@ 8 G�8<WXEKC5J!;>HT@�C?G

�� ����� l ¬ � ¾ Ì�¯ � ´ l ² � ¾ Ì + G+;F@K7*¡ ¾ � z5�B� ¤

%² � ¾ � Ì ��B� C?G ¾ � 4 �B� �C?@^S`IZP+HT\ ¯ { 4 { 3�´ l y �� � � r 798VP!8 � l�� ¬ � ¾ � Ì +�B� � � " � �

6�798�G~HTDFE9J!;FHT@�H?IÊJ!7^;FG�SK;&%�8VP!8<@BJ!;FC?D�8<WBE^C5J!;>HT@�MVC?@�]�8 r P+;FJ~J~8<@�C?G� l � G!;F@K7�¯ 4 y � z � ´�Í n M}HTG+7m¯ 4 y � z � ´ �

9 ;>J!7`]�HTEK@^SKC5P!_sM}HT@KS^;>J!;>HT@KG � l �'C5J � l �N��;>J#;FG�8<C?G!;FDF_�G~8V8<@�J!7KC5J r 8:\eE^G~J#7KC<A?8 n�l � � °tDUG~H9�� l ��3 z5y d C5J �

l� ��uG!H

� l d y 3G!;F@K7v¯ 4 y � z � ´ uC?@^S� l d y � G!;F@K7�¯ 4 y � z � ´G!;F@^7�¯ 4 y � z � ´ �L9E9]^G~J!;>J!EKJ~8[]�C?M!g`;F@BJ~H0J!798)8}�NcKP!8<G!G!;FHT@�IZH?P#J!7K8:G~c^C?M}8:MO7KC5P!=?8?i

Ì�¯ � ´ %Ó[y?¾ � Ì + ��B� l Ót¾ � Ì +�B� d G!;F@K7�¯ 4 y � z � ´G!;U@K7�¯ 4 y � z � ´ �y 3

Page 22: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

{ 4 � WBE^;FDF;>]KPO;FEK\ \0C5=T@K8VJ!;>¶<C5J!;>HT@�M}H?P+P!8<G~c�HT@KSKG�J~Hs\s;F@K;F\�EK\ IZP+8V8)8<@98VP!=?_?4/%KP!HT\ LX8<M}J!;FHT@ { 4 { 4 y 4� Ç� � l y � � � Í ¬ � � , l �l y � � + ¯ � Ó%� � ´ � Í ¬ � � , u

7K8<@KM}8?i � l �)H?P � � �˯ � Ó%� � ´J!7XEKG � �˯ � Ó%� � ´+� " � � �N� " �� uC?@^S�G~H Æ l 3 z5y �6 H0H?]KJ!C?;U@�w®C?@KS � C?S^S�C0\0C5=T@98VJ!;UM)J~8VP+\ SKEK8)J~H08}�XJ~8VPO@KC?D��^8<DFS n iÇ l Ç + Í � � + ¯ � Ó%��� ´ � � Í � � Ó�n �J!7K8VP!8VIZH?P+8 � Ç� � l�Ó�n Í y � � + ¯ �ÎÓ%� � ´ � Í ¬ � � , l � �

%KH?PtM}P+;>J!;UMVC?Dv;FG~H?J!798VPO\`� �olÎ� � C?@KS`G~H n � � , uKJ!7XEKGVi � l�{ �� H r � l � �� n � �J~H�SK;&%�8VP!8<@�J!;UC5J~8�]�H?J!7`G!;FS98<G�H?IÊ8<WXEKC5J!;>HT@`IhH?P#8<WXEK;FDF;>]^P+;FEK\»\0C5=T@98VJ!;>¶<C5J!;>HT@ r 4 P<4 J n i3 l�y � � + ¯ � Ó%� � ´ � �� n � � Í�3 y � � �0� �� n � �C?@^S � l � y � � � � � � Ío3 y � � � � p�� � w l 3

� � X " ��[�� X¿Z�^ []X �\ � /��^]\*^ �354 ª @ ��7KC5cKJ~8VP %9HTE9P r 8'S98VP+;>A?8<SYJ!798�P!8<G!EKD>JtIZH?P�J!798�MVD>HTG~8<S ª G!;F@9=�M+7KC?;F@xC?G[8<WX@ ¯ ¬ 4 y ��´ r 7K;FM+7®\0C�_�]�8r P+;>J~J~8<@�;F@�J!7K8)IZH?P+\�i � l�y M}HTG+7 � �¢3�Í�J!C?@^7 � � �ª @'J!798#MVC?G!8#H?I�J!798�H?c�8<@'M+7KC?;F@v�TJ!798#HT@KDF_:=?P+C5c^7 r ;>J!7'C?@'8VA?8<@0@XEK\e]�8VP�H?I*A?8VP+J!;FM}8<G�;FG|J!7K8�¶V8VP!H5ÏÐH?P+S98VP<4Èt8<@KM}8�J!7K8:C5]�H�A?8:P+8<G!EKD>J�]�8<M}HT\'8<G ' Îl�y M}HTG!7 p�� � �� H?J~8'J!7KC5J�;F@YJ!798'MVC?G~8�H?I�C?@®H?c*8<@xMO7KC?;F@ r 8'7KC�A?8 � l a Ó 3�C?G[J!798�@BEK\e]�8VPtH?I�c^C?;>P+G�H?I�DFC5J~J!;UM}8G+;>J~8<GV4y 4�µ�_`S98}��@^;>J!;>HT@�i

� · �[· Õ � l � p�� �� � � · �[· Õ

¾ p ¿ l �'p�� �� � � · �[· Õ �

� kh¨ � � ¾ � Ä � 2 �6�798�C5P!=TEK\08<@�J/c^P!HXM}8V8<S^G�C?G/IhH?P�J!7K8#c^C5P!J!;>J!;FHT@'IhE^@KM}J!;>HT@��TC?@KS�EKG!;F@K=[J!7K8�;FS98<@BJ!;>J¦_�=T;>A?8<@�C?G/8<WX@Y¯ ¬ 4 ¬ ´Oi� · � · Õ � l � p�� M}HTG!7 � � �

� � � �� kh¨ � � · � · Õ ¯Ò3�Í � · k · � ´y?y

Page 23: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

m−2 m−1 m n n+1 n+3

%|;>=TE9P+8035i��r 798VP!8 � �oa + H?I�@98<C5P!8<G~JÒϦ@K8<;>=T7�]�HTE9P�c�C?;>P+GV4� H r ;F@�J~P+HXSKE^M}8eDFC5J~J!;FM}8'=?POC5c^7KG�;F@YHT@98}ÏÐJ~H5ÏÐHT@98sM}H?P!P+8<G~c�HT@KS98<@KM}8 r ;>J!7®J!798�J~8VP+\0G�;F@®J!798�8}�Nc^C?@KG!;FHT@�4°[G�]�8VIZH?P+8?�BP!8<WXEK;>P+8(@ E1@����0G~c^;U@s;F@�CecKP!HNSKEKM}J�;FG�J~H�]�8[P+C?;FG~8<S�J~H'C?@ @=E1@���c�H r 8VP�J~H'7KC<A?8�C�@9HT@NÏжV8VP!HM}HT@BJ~P+;>]�E9J!;>HT@�4���� <)C?G�7KC<A?8�I±C?M}J~H?P · � · Õ u IhH?PeJ!7K8<G~8�J r HxG!;>J~8<G r 8�P!8<WXEK;>P+8sC?@ � ��c�H r 8VPeH?I#J!7K8G!c^;F@�4Èt8<@KM}8 r 8eP!8Vc^DFC?M}8 .�¯ � ´�;F@`J!7K8�8}�Xc�C?@KG!;>HT@`IhH?P�J!7K8:c^C5P!J!;FJ!;>HT@`I±EK@KM}J!;>HT@`]X_ � � Õ ¯ � ´Ou r 798VP!8 � � Õ ¯ � ´�;UGJ!7K8:G!C?\'8:C?G .*¯ � ´�8}�NM}8Vc^JtC5J�G!;FJ~8<G � C?@KS`Ì ���HT@KG!;FS98VP#CsDF;F@98<C5P ª G!;F@9=0MO7KC?;F@�4 � l a Ó 3 �%KH?P#C?@sH?c�8<@�M+7KC?;F@�J!798VP+8tC5P!8 �� �MVD>HTG!8<Ss=?P+C5c�7KG�C?@^S�798<@^M}8tJ!798[HT@KD>_0M}HT@BJ~P+;>]�E9J!;>HT@0J~H�J!798[c^C5P!J!;>J!;FHT@I±EK@KM}J!;FHT@ r HTEKDUS�]*8)J!7K8�¶V8VP!H5ÏÐH?P+S98VPt=?P+C5c^7�4J!7K8VP!8VIZH?P+8 � l�y M}HTG+7 p�� � �¯Ð��HT\'c^C5P!8�J~HsJ!798)P!8<G!EKDFJ#;F@�LX8<M}J!;>HT@ ¬ 4 y IZH?PtJ!798 � C $: @ � ª G!;F@9=�P+;U@9=94 ´µ�8<MVC?EKG!8®H?I:J!798YI±C?M}J~H?P · Õ · � u�J!798 � + J~8VPO\ S9HX8<G!@ 8 J�M}HT@BJ~P+;>]^E9J~8YJ~H J!798®M}H?P+P!8<DFC5J!;>HT@�4�6#798YHT@KD>_@KHT@NÏÐA¢C?@^;FG!7K;F@K=�M}HT@BJ~P+;>]^EKJ!;>HT@`;FG�IhP!HT\ÑJ!798�J~8VPO\ G!79H r @�;F@�J!798��^=TE9P!8?4� �<A@¯Ò3�´#C?DFD�M}HT@K@98<M}J~8<S�A?8VP!J!;FM}8<G�8}�NM}8VcKJ � C?@KS`Ì%C5P!8�8VA?8<@¯ y ´�A?8VP!J!;FM}8<G � C?@KS`Ì C5P!8)HNSKS�46�7K;FG�J~8VP+\ 7^C?G�Ì Ó � A?8VP!J!;FM}8<GtC?@KS�;FG�J!7K8VP!8VIZH?P+8�H?I|H?P+S98VP�½ Ì Ó � ½ �

� · � · Õ � l � p�� M}HTG+7 � � y �� � � � Õ ¯ � ´� l � y M}HTG+7 p�� � � p�� � y M}HTG+7 p�� � � � Õ p � J!798VP!8VIhH?P!8 � · � · Õ � l � Õ p � l J!C?@K7 Õ p � � �{ 4�6�798 É C?@�S98VP 9 C?C?DFG#8<WXEKC5J!;>HT@`J!C5g?8<GtJ!798�IhH?P+\� � Í �d � � ¯Ðd Ó �V´ l a �B� u

r 798VP!8 � z d � P!8VcKP+8<G~8<@�J!G|J!7K8�8 %*8<M}J/H?I*\eEKJ!EKC?D9C5J~J~P+C?M}J!;>HT@s]*8VJ r 8V8<@s\'HTD>8<MVEKD>8<G|C?@KS ��;FGÊJ!798 6 8}�9MVDFEKSK8<SA?HTDUEK\'838^SKEK8�J~H0J!798:��@K;>J~8�G!;F¶V8�H?I|J!798:\'HTD>8<MVE^D>8<GV4�6�798)8<WBE^C5J!;>HT@�;FG#]^C?G~8<S�HT@YC0\'HNS98<D r 7K8VP!8 �|¯ � ´;UGtJ!C5g?8<@�C?G)M}H?P!P!8<G~c�HT@KSK;U@9=sJ~H�C 6³7KC5P+S%G!c^798VP!838*c�H?J~8<@�J!;FC?DÊIhH?P � , � u*]�E9J[;UG r 8<C5gND>_YC5J~J~P+C?M}J!;FA?80IZH?P� 7 � �%KP!HT\ 8<WX@KG:¯ ¬ 4 � { ´�C?@KS�¯ ¬ 4³º?²T´ r 8:7^C<A?8n � l�Ó 3y�� � l-Ó 3y�� � � � ¾ p À Å "$# � Ó 3 uC?@^S��9HT@�J!798)]�C?G!;FG�H?I|HTE9P�C?G!G+EK\'cKJ!;>HT@^G�C5]�HTE9J �ʯ � ´Ou r 8:\0C�_�\sC5g?8:J!798�G!;U\'c^DF; �*MVC5J!;>HT@¾ p À Å "$# � % � IZH?P �� �� �

% 3 Ó � z5�B� u IhH?P � 7 � �y?{

Page 24: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

6�798<@��KSK;>AN;FSK;U@9=:J!798)POC?@9=?8�H?I�;F@BJ~8V=?P+C5J!;>HT@�;F@BJ~H'J r Hsc�C5P!J!GV� r 8)H?]KJ!C?;F@n � l 3y�� �+ ¬ � � � � � Í 3y � �� ¬ � � � �ʯ � ´�B� � �

lon Ó � z5�B� ur 798VP!8 n l y

� � ,{ l ¬ � + ur 798VP!8 � + �-A?HTDFEK\08)H?I|C'\0HTD>8<MVEKD>8?�KC?@KS� l-Óty

� � �� �� �|¯ � ´ � � �� H r �^IhP!HT\ ¯ ¬ 4 � y ´#C?@^S`J!798<G~8�P+8<G!EKD>J!GV�NJ!798:cKP!8<G!G!EKP!8);FG�=T;>A?8<@�]X_

� l a �B�d � 3�Í a d � n-Ó ��B�s� ���«m8VJ�EKG#SK8}��@98�J!798 É C?@YS98VP 9 C?C?DFGtM}HT@KG~J!C?@BJ!G�C?GVi

� l ¬ a � + ur 7K;FM+7`;UG�J!798)J~H?J!C?D 8}�NMVDFE^S98<S�A?HTDFEK\'8)H?I|a \'HTD>8<MVEKDF8<GV�KC?@KS� l a � � u

r 7K;FM+7Y;FGtJ!798eJ~H?J!C?DÊ8 %�8<M}J�H?I�;F@BJ~8VP+C?M}J!;>HT@KGt]�8VJ r 8V8<@®C?DUDvc�HTG!G!;>]�D>8)c^C?;>P+G<4�6#798<@YJ!798:8<WXEKC5J!;>HT@®IhH?PtJ!7K8c^P!8<G!G!E9P+8[]�8<M}HT\'8<G � l a �B�d Í a �B�d � �d Ó �d � �H?P�i � Í �d � %a �B�d � 33 Ó � z d ur 798VP!8xJ!798 ;F@BJ~8VP!cKP!8VJ!C5J!;FHT@ ;U@ÎJ~8VPO\0G`H?IeC���P+G~JÒÏÐH?P+S98VP�J~P+E^@KMVC5J!;>HT@oH?I�J!798x]^;F@9HT\0;UC?D�8}�Nc^C?@KG!;FHT@ ;UG�!EKG~J!;>�^8<S�IZH?P�G!\0C?DFDKA¢C?DFEK8<G�H?I � z d � 6�798<@0\�EKD>J!;>c^DF_X;F@K=tC?M}P!HTG!G<�XC?@KS�MVC?@KM}8<DFDF;U@9=)C?G�C5cKc^P!H?cKP+;FC5J~8?��_X;>8<DUSKG� � Í �d � � ¯Ðd Ó �V´ l a �B� u

C?G�P!8<WXEK;>P!8<S�4¬ 4&%KP!HT\ ¯ ¬ 4 � { ´�C?@KS�¯ ¬ 4³º?²T´O� n � l-Ó 3y�� � , � � ¯ � ´ �6�798�7KC5P+S�G!c^798VP!8�c�H?J~8<@�J!;UC?DvG!C5J!;FGÒ��8<G

� ¯ � ´ l � � � �l � � 7 � �Èt8<@KM}8 � ¯ � ´�G!C5J!;UGÒ�^8<GVi

� ¯ � ´ l-Ó 3 � �'�l � � 7 � �6�7BEKG#J!798�;F@BJ~8V=?P+C?D�]�8<M}HT\'8<GVi n � l � Ó �� � Ó 3 � ¬ � � �+ �

�� �l y

� � ,{ uy ¬

Page 25: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

C?G�P!8<WXEK;>P!8<S�4%KH?P n ,5�KG~J!C5P!J#]X_`S9HT;F@9=�J!798e;F@�J~8V=?POC?D r ;>J!7�P!8<G!c*8<M}J�J~H � � �KJ!7XEKGVi

¯ � ´ l � � , � � � ¯ � � ´ � ¯!½ � Ó � � ½ ´6�798�;F@BJ~8V=?P+C?@KS�;FG Ó 3 � Ó 3 l 3#IhH?P � � D>_X;U@9= r ;>J!7K;U@�CeG~c^798VP+8�H?I�P+C?SK;FEKG ��M}8<@BJ~P!8<S�HT@0J!798tH?P+;>=T;F@0C?@^SC?DUG~H r ;>J!7K;F@�C�G~c^798VP!8tH?ImP+C?S^;FEKG �0M}8<@�J~P+8<S�HT@ � � l � 4/6#7XEKG ¯ � ´�;FG�J!798�A?HTDFEK\'8[H?I G~c�C?M}8[HNMVMVE9c^;F8<S]X_�J!798�J r H�H�A?8VPODFC5cKc^;F@K=�G~c^798VP!8<G<��C?@KS0]X_'G~_N\0\'8VJ~P!_'M}HT@KG!;UG~J!G/H?I�J r H�G~c^798VP+;UMVC?D)8 MVC5c^G�8>�XHT@s8<C?MO7�G!;US98H?I�C'c�DFC?@98�J!79P!HTE9=T7 � � l �

z5y 4«m8VJ��®]�8sC�M}HXH?P+SK;U@KC5J~8�D>_N;F@9=`]�8VJ r 8V8<@ �z5y C?@KS �^��M}H?P!P+8<G~c�HT@KSK;F@9=�J~H®HT@98�7^C?D>I�H?I�J!7K8sA?HTDFEK\08sH?I;U@�J~8VP+G!8<M}J!;>HT@�4�6#7K8<@`;>J#;FG�8<C?G!;FDF_�G~8V8<@�J!7KC5J

¯ � ´ l�y � � " � � ¯�� � Ó � � ´ � �

l�� ¬ � ,{ Ó � � � Í � ,3 y ) �

%|;F@^C?DFD>_?� n , l 3{ ¬ � � �+ �� ¯ � ´ � � l º � � ���3<² l º² n �� u

C?G�P!8<WXEK;>P!8<S�4ºX4�6�798tP+8<DFC5J!;>HT@KG!7^;>cs7KC?G�]�8V8<@ r PO;>J~J~8<@sJ~H'\0C5g?8�J!798[cKP!H?]�D>8<\fD>HXH?g0\'H?P!8�SK; ��MVEKD>JV4/LXJ!C5P!J�]B_s;U@�A?8VP!J!;U@9=;FJVi� , À �� p �� Å 'e¯�� � u���p , " Õ d�´ l '�¯ � uOd�´ �

6�798)P+;>=T7BJ#7KC?@KS`G!;US98�;FG�HTE9P�G!J!C5P!J!;F@9='c�HT;F@BJVi�IZP+HT\Ñ8<WB@^G:¯ ¬ 4 { 3�´O�|¯ ¬ 4 {?y ´O��¯ ¬ 4 {?{ ´�C?@^S�¯ ¬ 4 { ¬ ´ r 8�7KC�A?8'�¯ � uOd�´ l 3

a�� ¯ y � � �B� ´ �� � � � 8}�Nc �� Ó ��B� � k � 3½ � k�Ó � � ½ Õ �� u

r 798VP!8 r 8�7^C<A?8�G!E9]^G~J!;FJ!E9J~8<S%J!798�=T;>A?8<@�c�H?J~8<@BJ!;FC?D�4�«mHBH?g C5J�J!798�µ�HTD>J~¶<\sC?@K@ 8}�Nc�HT@98<@�J:�^P+G~JV4 9 8r C?@BJ � � � � �KG~H r 8�\0C5g?8eJ!798�M+7^C?@9=?8:H?IÊA¢C5PO;FC5]^D>8<GVi� kvl ��� " Õ � k u

C?@^S`J!798�8}�Nc�HT@98<@�J�]�8<M}HT\'8<G<i��B� � k � 3¯�� � " Õ ½ � k*Ó � � ½~´ Õ l-Ó �� � � � k � 3½ � k�Ó � � ½ Õ �� H r D>HXH?g�C5J#J!7K8�H?J!798VPtM}HT@^G~8<WBEK8<@KM}8<G#H?IÊJ!798:MO7KC?@9=?8�H?I|A5C5P+;FC5]^D>8?i

� �k l � , " Õ � � k

d l��� �k�l � , " Õ � � � kvl � , " Õ d �

d � l � p , " Õ dy º

Page 26: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

� H r ��;>=T@9H?PO;F@9=�J!798:M}HT@KG~J!C?@BJ�c^P!8VIhC?M}J~H?P�¯ � a � ´ p�� � r 7K;FM+7 r ;FDFD�MVC?@KM}8<DvC?M}P!HTG+GV� r 8�7KC<A?8?i'�¯ � uOd�´ l ¯ y � � �B� ´ �� � �� � �

� 8}�Xc �� Ó �� � � � k � 3½ � k*Ó � � ½ Õ ��

l ¯ y � � � � � ´ �� � , À �� p �� Å � � � 8}�Nc �� Ó �� � � � k � 3½ � k�Ó � � ½ Õ ��l � , À �� p �� '�¯�� � u�� p , " Õ d�´�u

C?G�P!8<WXEK;>P!8<S�4�X4 �� �

¾ p ¿ l-Ó � ¿+ ¾ p À ¿ p�� Å � � � ¾ p�� � � �� H?J~8:J!7^C5J#J!798�A5C5P+;FC5]^DF8 ��;FG�@98<M}8<G+G!C5P+;FD>_ ��¯hÆÊ´Ou^C?@^S`8}�Xc�C?@KS�]�H?J!7�G!;FS98<G�C?G�c�H r 8VP�G~8VP+;>8<G�;F@�Æ �# � � · � l �� � Ø 3Ó Æ Í�¯hÆ ´ � z5y �¢Í �V�V� Ùl Ó Æ � � � Í Æ �y

�� � � Í Æ �y

� � � Í �V�V�<�� � � · � l Ó � ¿+ Ø 3 Ó ¯hÆ Ó �K´ Í 3y

�¯hÆ Ó �9´ � � Í �V�V� Ù

� � � Ø 3Ó� Í � �y

�� Í �V�V� Ù � �l Ó � ¿+ � � � � ��Í �

¿+ �¦¯hÆ Ó �K´ �

��� Í � � � � � � �)Í �V�V�l Ó � ¿+ � � � � ��Í �¿+ Æ � � � � �

Ó � ¿+ � � ��� � ��Í �

¿+ � � � � � �:Í �V�V�l Ó Æ � � � ÍΡ<Æ � � � �

Ó Æ �y � � � ¤�Í Æ �y � � � Í �V�V�l Ó Æ � � � Í Æ �y � � � Í Æ �y � � � Í �V�V�l # � � · �� 4&%KH?P�J!798:MVC?@KHT@K;FMVC?D�8<@KG!8<\:]^DF8�;F@�J!798)8<@98VP!=?_�P!8Vc^P!8<G~8<@BJ!C5J!;>HT@�� r 8:7KC�A?8?i

× � l�� j� � ) � u jËl��B�t�Y�� � DF@ �r 798VP!8 � l �

�¾ p ¿ l ��� 3 Ó Æ Í Æ �y

�� Í �K¯hÆ , ´ �l �Z�

� 3 Ó Æ � � Í Æ �y � � � Í �K¯hÆ , ´ �l �� 3 � 3 Ó Æ

�� � � 3 ��

�y � � ��� 3 Í �K¯hÆ , ´ � �

y�

Page 27: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

6�798<@ j l �B�t�®�� � DF@*¡ � � 3 � 3 Ó Æ�� �� 3 Í Æ �y � � ��

� 3 Í �K¯hÆ , ´ � ¤l �B� �®�� � ¡�DF@ � � 3�Í DU@ � 3 Ó Æ � � �� 3 Í Æ �y � � ��

� 3 Í �K¯hÆ , ´ � ¤l �B� �®�� � DF@ Ø 3 Ó � Í �K¯hÆ , ´ Ùr 798VP!8 � l Æ � � �

� 3 Í Æ �y � � ��� 3 �� �Xc�C?@KS`DF@��F3 Ó � Í �K¯hÆ , ´���J~H��K¯hÆ , ´Ou^S^;&%*8VP+8<@�J!;FC5J~8s¯hJ!7K8)Æ ´ r 4 P<4 J � C?@KS�P!8<WXEK;>P+8<S�P+8<G!EKD>J�IhHTDFD>H r GV4²X4 � � Í �d � � ¯Ðd Ó �V´ l a �B�

��P+;>J!;FMVC?D�c�HT;F@BJ � ;F@N�^8<M}J!;FHT@�H?I|;UG~H?J!798VP+\0G�J!7K8VP!8VIZH?P+8?�� �� d � � ���l � � �� d � ) � ���

l �=T;FA?8<G�M}HT@KSK;FJ!;>HT@KGV4��8}ϦC5P!P+C?@9=?8'd ��� 8<WBEKC5J!;FHT@�i � l a �B�d Ó �

Ó �d � u ¯��¢´� �� d � � lÓ a �B�¯Ðd Ó �V´ � Í

y �d , l ��IZH?P ��lÎ��� uOd l d � �6�7BEKG<i y �d ,� l a �B� �¯Ðd ��Ó �V´ � �y ��¯Ðd �/Ó �V´ �d ,� l a �B� � � ¯�� �¢´� � �� d � ) � l

y a �B�¯Ðd Ó �}´ , Ó �1�d ¬ l ��IZH?P ��lÎ��� uOd l d � �C?@^S { �d � l a �B� �¯Ðd ��Ó �V´ , u7K8<@KM}8 { ��¯Ðd �/Ó �V´ ,d � l a �B� � � ¯�� � �¢´� WBE^C5J~8:«mÈ)L�H?I�¯�� �¢´�C?@KS�¯�� � �¢´Oi y � ¯Ðd ��Ó �V´ �d ,� l�{ � ¯Ðd �/Ó �}´ ,d �� d ��l¸{ � �L9E9]^G~J!;>J!EKJ!;F@9=�IZH?P[d � ;F@BJ~HY¯�� �¢´ � a �B� ��l ²1� z5y � �)C?@KS`]�H?J!7�d � C?@KS � � ;F@ ¯��¢´ � � ��l � z5y � � � �LN8VJ �§ l � z � � u �� l d z d � C?@KS �� lÎ��z�� � 4Èt8<@KM}8)c^EKJ � l � � �§vu�d l d � ��sC?@KS ��lÎ��� �� ;F@�8<WBEKC5J!;FHT@ ¯��¢´�C?@KS�G!E9]^G~J!;FJ!E9J~8tIhH?P � � uOd � C?@KS � � uXJ~H=?8VJVi � �y � � � �§'Í ��� � � � � � ¯ { � �� Ó �V´ l a � �� ²1�y � � 3a �y �

Page 28: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

� �y � � � ���§'Í{�� � � � { � � �� Ó 3{*� l ²1� ��y � � uC?@^S�G~H � �§�Í {�� � � ¯ �� Ó 3 z5{ ´ l ² ��{ �

� 4&%KP!HT\ J!798:cKP!8VAN;>HTEKG�WXE98<G~J!;>HT@�i ���§�Í {�� � � ¯ {��� Ó 3�´ l ² �� �LN8VJ��§ l �§ Ó 3 l ¯ � Ó � � ´ z � � 798<@KM}8 �§ l §�Ío3� l �� Ó 3 l ¯Ðd Ó d � ´ z d � 7K8<@KM}8 �� l dÎÍ�3

� l �� Ó 3 l ¯ � Ó%� � ´ z�� � 798<@KM}8 �� l � � Ío3� � §�Ío3�Í {

¯Ò3�Í �N´ � � Ø { ¯Ò3�Í �N´ Ó 3 Ù�l ²9¯Ò3�Í � � ´ ��®EKD>J!;Fc^D>_0J!79P+HTE9=T7`]X_x¯Ò3�Í �N´ � u^\�EKD>J!;>c�D>_'HTE9J�C?@KS�P!8}ϦC5P!P+C?@9=?8?iy § � 3�Í � �y Í ¬ � � Í { � ,y ) l-Ót{ � , Í�² � � ¯Ò3�Í y �)Í � � ´H?P § l-Ó {y � , Í � � ¯ ¬ Ó � ��Í � � � Í �V�V� ´vÍ �V�V� u7K8<@KM}8 w � Ó d � p��� � l Í � � d� � � � l � �d � � � §� � � � l � �d � ¯ Ó � � � ´C?@^S�G~H � � � � p��� J!798VP!8VIhH?P!8�w l 3 �� � ��P+;>J!;FMVC?D�;FG~H?J!7K8VP+\ �olÎ� � u � ��l �'J!798VP!8VIhH?P!8�§ � � , 798<@KM}8 � l¸{ �� �<A@#6#798<G~8�C5P!8�\'8<C?@NÏ��^8<DFSeA5C?DFE98<G|C?@KS É C?@'S98VP 9 C?C?DUG�;FG @KH r C?SKC<_NG�P!8}Ϧ;F@BJ~8VP!cKP+8VJ~8<SeC?G�C[\'8<C?@NÏ��^8<DUSJ!7K8VH?P!_?4

� � X " ��[�� X¿Z�^ []X �\ � /��^]\*^ �354[­);>A?8<@ ��� ¯ × & Ó ×�� ´ lÎ�Ô�m� �� � � �

&J!7K8VP!8VIZH?P+8× & l

� � � � � � &��� Í × �� H r × � \eE^G~J�]�8)c*HTG+;>J!;>A?8);�4 8?4 × � � �J!7K8VP!8VIZH?P+8× & �

� � � � � � &��� ¯��¢´ �'[8}��@^;>J!;>HT@KG�H?I�M}P+;>J!;UMVC?D�8}�Xc�HT@98<@BJ!GVi

× & � ½ � � ½ p £ u ��� �f½ � � ½ p9q u� �˯ Ó � � ´ ¿ � � �� �Î�

&�˯ Ó � � ´ ¿ p�� �

y ²

Page 29: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

� H r ¯ Ó � � ´ � l ½ � � ½ �6�7BEKG�;F@98<WXEKC?DF;>J¼_Y¯��¢´�]�8<M}HT\'8<G ½ � � ½³p £ �-½ � � ½ � À¿ p��¼Å Ö q �Èt8<@KM}8 Ó ��, y ¯hÆ Ó 3�´vÍ wÊuJ!7XEKG ��Í y Æ Ó y Í�w � �NuC?@^S�G~H ��Í y Æ�Í w�� y �y 4�6�798 9 ;US9HT\ G!MVC?DF;U@9=�P!8<DFC5J!;>HT@�;UGVi�s¯�� q � � u���� n ´ l � �0¯ � � u n ´ �

'�;&%�8VP!8<@BJ!;FC5J~8 � J!;F\08<G�HT@`]�H?J!7�G!;FSK8<G�C?@KS`P!8}ϦC5P!POC?@9=?8?i� � ¯ � � u n ´ l � � � p�� � À � Å ¯Ò3�´ q ¯ � � u���� n �

Èt8<@KM}8:;FJ�IZHTDFDFH r G�J!7^C5JVi� À � Å ¯ � �~n ´� À � p��¼Å ¯ � � u n ´ l � � � p�� � À � Å ¯�� q � � u�� � n ´� À � p��¼Å � p�� � À � p��¼Å ¯ � q � � u�� � n ´l � � � À � Å ¯ � q � � u�� � n ´� À � p��¼Å ¯�� q � � u�� � n ´ �� H r # � � · � l � � p�� �� ¯hIhP!HT\ S98}��@^;>J!;>HT@�´O4°[DFG~H0MO79HXHTG~8 � l � p�� " � u9J!7BE^GVi

� p�� �� l � � p�� " � � � � � ¯Ò35u�� � n ´� À � p��¼Å ¯Ò35u�� � n ´l � p � " �J!798VP+8VIZH?P!8 � � l § z � u�;U@KS98Vc�8<@KS98<@BJ�H?I � �{ 4�6�798�\'8<C?@�\sC5=T@98VJ!;>¶<C5J!;>HT@ � l � · � + C?@KS`IhP!HT\ \'8<C?@���8<DFS�J!798VH?P!_*i� · � + l J!C?@K7m¯hÆ n Í y�� � Æ � · � + ´ �Èt8<@KM}8:;U\0\'8<SK;FC5J~8<DF_ r 8:MVC?@ r PO;>J~8?i� l J!C?@K7m¯hÆ � ��� Í �}´ r 798VP!8&� � Æ n �� H r \'8<C?@`��8<DFS�J!798VH?P!_�=T;>A?8<G � Æ � � l 3�H?P � � l 3 z Æ � �Èt8<@KM}8 � l J!C?@K7�� Æ �Æ � Í���� l J!C?@K7�� � � �� Í���� l J!C?@K7 Ø � z ¯Ò3�Í � � ´vÍ�� Ù �

LN8VJ n l �'C?@KS`8}�Nc^C?@KS`IhH?P � � ��� �K;F@ r 7K;FM+7`MVC?G!8�� � ;FG�G!\0C?DFDÐi� l �3�Í � � Ó 3{ � ,¯Ò3�Í � � ´ , uJ!7XEKG � � 3 Ó 33�Í � �5� l-Ó 3{ � ,¯Ò3�Í � � ´ , uy �

Page 30: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

7K8<@KM}8 � l �H?P � � l-Ót{ � � ¯Ò3�Í � � ´ ,¯Ò3�Í � � ´ lÔÓ[{ � � ¯Ò3�Í � � ´ � �6ÊC5gN;F@9='J!798:@KHT@�J~P+;FAX;FC?D�MVC?G~8?� � � ½ Ó { � � ½ � " � uC?@^S`]B_�M}HT\'c^C5PO;FG~HT@ r ;FJ!7�J!798)8<WBE^C5J!;>HT@ r 7^;FM+7`S98}�*@98<G�J!798:M}P+;FJ!;FMVC?D�8}�Xc�HT@98<@BJViÆ l 3 z5y �

¬ 4&%KH?P n l �Nu r 8�7KC�A?8 l Ó� � � kh¨ ��� · k · � r 798VP!8 � � kh¨ ��� ;FG:G!E^\ HRA?8VP0@98<C5P!8<G~J�@98<;>=T7B]�HTE9P+GV4�6�798\08<C?@�8<@98VP!=?_�H?I|J!798�G~_NG~J~8<\ ;UG�=T;>A?8<@�]X_jËl � %� l-Ó

��� kh¨ ��� � · k · � � �ª @�D>H r 8<G!JÒÏÐH?P+S98VPt\08<C?@��^8<DFS`C5cKc^P!H<�9;F\0C5J!;>HT@v�KG~c^;F@KG�C5P!8e;F@KS98Vc�8<@KS98<@BJ�C?@KS� · k · � � l � · k � � · � � �Èt8<@KM}8 jËlÔÓ

��� kh¨ ��� � · k � � · � � l-Ó

�� a y � �

r 798VP!8 � l � · � �-H?POS98VP#c^C5P+C?\08VJ~8VP<4 %9P!HT\ J!798)J!798VP+\0HXS9_N@KC?\0;UM[SK8}��@K;>J!;>HT@�H?IÊJ!798�798<C5J�MVC5c�C?MV;>J¦_?�r 8:7KC�A?8 × & l � j� � )&l-Ó[y

�� a y � � �

�� l-Ó

�� a � � �

�� �

%KH?P� 7 ��� i � l � J!798VP!8VIhH?P!8 × & l ����,��� i � l ¯ Ót{ � � ´ � " � �Èt8<@KM}8 � �� � l 3y ¯ Ót{ � � ´ p�� " � � Ó � � �� � l Ót{y � p�� � � �� � l

Ót{y � p�� � p��� uC?@^S�G~H � j� � )&l {y � � a � � p�� � p��� l {y � � a

�l {y a � C?G ���l��B� � �

Èt8<@KM}8 × & ;FG#SK;FG+M}HT@�J!;F@XE9HTEKG�C5J �olÎ� � C?@KS � l � �ºX4&%KP!HT\ J!798eS98}��@K;>J!;FHT@�H?IÊJ!798:G+EKG!M}8VcKJ!;>]�;FDF;>J¼_?� r 8:7KC�A?8� � l � �� n l Æ � �� � uC?@^S�C?DFG~H � l J!C?@K7 � �3�Í � � Í � � %

�3�Í � � Í�� IhH?P � 7 � � �� H r � Ó �3�Í � � l �Ru{ �

Page 31: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

J~HsJ!7K;FG�H?P+SK8VP#H?IÊC5c^cKP!H<�9;F\0C5J!;FHT@`C?@KS��9P!8}ϦC5P!POC?@9=T;F@9=9� r 8�7KC�A?8?i� l � 3�Í � �� � � � �Èt8<@KM}8 � � � � �� � � 3� � C?G � � � �C?@^S�G~H � � � � p��� u w l-Ó 3 ���HT@KG!;FS98VP�J!7K8�8 %*8<M}JtH?IÊC?@�8}�XJ~8VPO@KC?DFD>_�;F\'c�HTG~8<S��^8<DFS`C5J �olÎ� � u r 798VP+8 � �/l �Nu�C?@KS�G~H�3�Í � ��l 3 ��[G~8�J!7K8:;FS98<@BJ!;>J¦_*i� l J!C?@K7m¯ � Í��V´ l ¯hJ!C?@K7 � Í�J!C?@^7&�V´}¯Ò3�Í�J!C?@K7 � J!C?@K7&�V´

%� � Ó � ,{ Í�� Ó � ,{ ) ¯Ò3�Í J!C?@K7 � J!C?@K7&�V´

��C?@KM}8<D�J!798�I±C?M}J~H?P�H?I � HT@`]�H?J!7�G!;FSK8<G�C?@KS`P!8<C5P!POC?@9=?8?�KJ~H0H?]KJ!C?;U@�i� � � ,{ Í � ,{ Ó � � Ó � ,{ Í�� Ó � ,{ ) � � � Ó � � ,{ Ó � � ,{ Í �V�V� )

J!7K8VP!8VIZH?P+8 � � � , z �[IhH?P�G!\sC?DFD �Ru � i^7K8<@KM}8 � l�{ �ª I r 8�G~8VJ6� � � , HT@`J!798�PO;>=T7�J#7^C?@KS�G!;FSK8?� r 8:MVC?@�A?8VP+;>Ih_�C?DUD�J~8VP+\0G#H?I|H?P+SK8VP�7K;>=T798VP�J!7^C?@ ��� � ,��C5P+8:@98V=TD>8<M}J~8<S�4�X4&%KP!HT\ 8<WX@�¯ � 4 �5y ´O�*J!798:\0C¢�9;F\�EK\�IZP+8V8�8<@98VP!=?_�;FGVi

Ç l Ç + Ó 3y a � � � · � �+ Í n � a � · � +r 798VP!8 n � lon à Ó�n �/Ç + l

Ó aÆ DU@ Ø y M}HTG!7e¯±Æ n à ´ Ù � � · � + l J!C?@K7:¯hÆ n à ´ �L9E9]^G~J!;>J!EKJ!;F@9=�C?G�C5cKcKP+H?cKP+;FC5J~8?iÇ l Ó aÆ DU@ Ø y M}HTG!7e¯hÆ n à ´ Ù^Ó 3y a � � J!C?@^7 � ¯hÆ n à ´vÍ a�¯ n Ã Ó n ´BJ!C?@^7:¯hÆ n à ´ �6�798)A¢C5PO;FC5J!;>HT@KC?D�cKP+HXM}8<SKEKP!8)=T;>A?8<GVi

¯ n à Ó�n ´ l � � � · � + l � � J!C?@K7:¯�Æ n à ´ ���8 r P+;>J~8)J!7^;FG�C?G J!C?@^7:¯hÆ n à ´ l

n à Ó�n��C?@^S�G!E9]^G!J!;>J!E9J~8);F@BJ~H0J!798)8}�NcKP!8<G!G!;FHT@�IZH?P�Çfi

Ç l Ó aÆ DF@ Ø y M}HTG+7e¯hÆ n à ´ ÙKÓ 3y a � � ¯ n à Ó�n ´ �¬ � � � � Í a�¯ n Ã Ó n ´ ¯n à Ó�n ´�

�l Ó aÆ DF@ Ø y M}HTG+7e¯hÆ n à ´ Ù Í ay��� ¯ n Ã Ó n ´ � uC?G�P!8<WXEK;>P!8<S�4

{ 3

Page 32: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

� 4�Èt8VP!8 r 8�P!8Vc�8<C5J�J!798�cKP!8VAN;>HTEKG�MVC?DFMVEKDFC5J!;>HT@ r ;>J!7�J!798�=?8<@98VP+C?DF;>¶<C5J!;FHT@ J!7^C5J�J!7K8�8}�NJ~8VP+@KC?D�\0C5=T@98VJ!;FM��8<DFS`;FG�@9H0D>HT@K=?8VP�C'M}HT@KG~J!C?@BJ�]^EKJ�A5C5P+;>8<G�IhP!HT\ÑHT@K8�DFC5J~J!;FM}8:G!;FJ~8)J~H0J!798�@98}�NJ#C?@KS`;FG#SK8<@9H?J~8<S`]X_ n[k �IhH?P#J!798:©ÐJ!7�DFC5J~J!;FM}8eG!;>J~8?4ª J)IhHTDFD>H r G)J!7KC5J)J!79808 %�8<M}J!;>A?8'��8<DFS n à � n À k Åà C?@KS J!7980\'HTD>8<MVEKDUC5P��^8<DFS n � � n �k C?DFG!H94�LXHT\'8'H?IJ!7K8�g?8V_�P!8<DFC5J!;>HT@KG+7K;>c^G�C?DFG~H'=?8<@K8VP+C?DF;>¶V8�G~J~POC?;>=T7�J~IhH?P r C5P+SKD>_?�9J!7XEKGViÇ + l-Ó Æ p�� � k DF@�� y M}HTG!7 �+Æ n À k ÅÃ�� u ¯Ò3�´C?@^S � · k � + l J!C?@K7 �OÆ n À k Åà � � ¯ y ´6�798<@��9IhP!HT\ ¯Ò3�´�C?@KS ¯ y ´O� r 8:\0C<_�8<C?G!;UD>_sH?]KJ!C?;U@�i� Ç +� n À k ÅÃ

l � k � · k � + � ¯ { ´� 8}�NJ r 8:J!C5g?8:8<WX@�¯ � 4 �?²T´ r ;>J!7�J!798�8<WXEKC?DF;>J¼_�C?@^S�=?8<@98VP+C?DF;>¶V8:J~H0J!798:@9HT@XEK@K;FIZH?P+\�MVC?G~8?�^J!7BEKG<i

Ç l Ç + Ó � kh¨ � �k � � · k · � � + Í � k n �k � · k � + u ¯ ¬ ´

C?@^S�� r ;>J!7`C?@�H?]BAN;>HTEKG�=?8<@98VPOC?DF;>¶<C5J!;>HT@`J~H�J!798�;F@K7KHT\'H?=?8<@98VHTEKG#MVC?G!8?�Ç l Ç + Ó � kh¨ � �k � � · k · � � + Í � k � n À k ÅÃ Ó ntk � � · k � + � ¯�ºT´

°[G/]�8VIhH?P!8?��HT@sJ!798tE^@9c�8VP!J!E9P!]�8<S�\'HXSK8<D�� r 8#J~P!8<C5J�J!798�G~c�;F@KG�C?G�;U@KS98Vc�8<@KS98<@BJV4 ��8<\0;F@KS^;F@9=[HTE9P+G~8<D>A?8<GH?I|J!798)c^P!H?c�8VP!J!;>8<G�H?I|J!798�S9HTE9]�D>8)G!EK\`� r 8:\0C�_ r P+;>J~8?i� kh¨ � �k � � · k · � � l � y � k � � � � � · k � + � · � � + u ¯ �T´

r 798VP!8 � 5 � S98<@9H?J~8<G 6³G!EK\�H�A?8VP[J!798:@K8<C5P!8<G~J#@98<;>=T7B]�HTE9P+G�H?IÊ8<C?M+7Y© 8>4� H r r 8tMVC5P!P!_'HTEKJ�J!798#A5C5P+;FC5J!;FHT@��BSK; %*8VP!8<@BJ!;FC5J!;F@K=�ÇÔC?G�=T;>A?8<@s]X_�¯�ºT´/C?@^SY¯ �T´O� r ;>J!70P+8<G~c�8<M}J/J~H n Àk Åà �J!7XEKGVi � Ç� n À k ÅÃ

l � Ç +� n À k ÅÃÓ � y � k � � � � � � ·

k � +� n À k Åà � · � � + Í � y � k � � ��� � · k � + � � · � � +� n À k ÅÃÍ � n À k ÅÃ Ó ntk � � � · k � +� n À k ÅÃ Í � k � · k � + � ¯ � ´6 r H0c�HT;F@BJ!G�G!79HTE^DFS�@9H r ]*8�@KH?J~8<S�i¯±C�´ %KP!HT\ ¯ { ´O� r 8:G!8V8�J!7KC5J#J!798��^P+G~J�J~8VP+\ HT@�J!798)P+;F=T7�J#7KC?@^S�G!;FS98)MVC?@KM}8<DUG#J!798:DUC?G~J�J~8VP+\`� �!EKG~J#C?G;U@�J!798:79HT\'H?=?8<@K8VHTEKG�MVC?G~8?4¯h]*´�6�798:G~8<M}HT@KS�J~8VP+\ ;F@�A?HTDFAX;F@K='J!798:S9HTE9]�D>8�G!EK\ÑA5C?@K;FG+798<G�]�8<MVC?EKG~8 � � · � � + z � n À k Åà l �0IhH?P 5 �l ©!�C?@^S 5s;UG w8p�r pF� 8<WBEKC?DvJ~H0©!46�798<@��TG~8VJ~J!;F@K= � Ç z � n À k Åà l �N��C?@KS�8<WXEKC5J!;F@9=)M}HX8���MV;>8<@BJ!G�H?I � � · k � + z � n À k Åà � r 8�H?]^J!C?;F@�J!798�M}HT@KSK;>J!;FHT@IhH?P�C?@�8}�XJ~P!8<\�EK\ C?G<i n À k Åà Ó�n[k�l

�� ���� � · � � + � ¯�²T´

9 8�\0C<_�I±E9P!J!798VP r P+;>J~8�J!7K;FG#M}HT@^SK;>J!;>HT@`;F@�J!7K8�EKG~8VI±EKD�IZH?PO\`in À k Åà Ó�n[kml�� ���� J!C?@K7 �OÆ n

À � ÅÃ � u ¯ � ´{?y

Page 33: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

r 798VP!8 r 8:7KC<A?8�G!E9]^G!J!;>J!E9J~8<S�IZH?P � · k � + IZP!HT\ ¯ y ´O49 8`\0C�_ M}HT@KMVDUEKS98s]X_ r PO;>J!;F@9=�SKH r @�J!798�H?cKJ!;F\0C?D�IhH?P+\bH?I#J!798�IZP!8V8�8<@98VP!=?_?4 %KP!HT\2¯�ºT´:C?@^So¯ �T´O�C?DFHT@9= r ;>J!7 ¯Ò3�´�C?@KS�¯ y ´O�^J!7^;FG�;FGVi

Ç l Ó 3Æ � k DU@�� y M}HTG+7��OÆ n À k Åà � Ó � y � k � � ��� J!C?@^7 �+Æ nÀ k ÅÃ�� J!C?@K7��!Æ n À � Åà �

Í � k � n À k Åà Ó�n k � J!C?@K7 �OÆ n Àk Åà � � ¯Ò3 ��´

6�798<@�G!E9]�G~J!;>J!E9J!;F@K=eIhP!HT\ ¯ � ´�;U@�J~H'J!798e\0;FSKSKDF8tJ~8VP+\ r 8:7^C<A?8Ç l-Ó 3Æ � k DF@�� y M}HTG!7��+Æ n À k Åà � Í 3y � k � n À k Åà Ó�n k � J!C?@K7 �+Æ n À

k Åà � � ¯Ò3?3�´� H?J~8:J!7^C5J#J!7K;FG�P!8<G+EKD>J�;FG wv�z CsG!;F\'c^DF8t=?8<@98VPOC?DF;>¶<C5J!;>HT@`H?I|J!798)P+8<G!EKD>J�H?IÊJ!798�cKP+8VAX;>HTE^G�cKP!H?]^D>8<\�4²X4(�Y8<C?@NÏ��^8<DUS�C5cKcKP!H��N;U\0C5J!;>HT@�i

l-Ó ¯ n Í � ��� ´

�k � ·

kr 798VP!8

l � · k � �6�798�c�C5P!J!;>J!;>HT@eIhE^@KM}J!;>HT@eIhC?M}J~H?P+;F¶V8<G�;F@�J~H)C[cKP!HNSKEKM}J H?I^a G!;U@9=TD>8}ϦG~c^;U@�c^C5P!J!;FJ!;>HT@:I±EK@KM}J!;FHT@KGV�¢8<C?M+7'=T;>A?8<@]X_*i

' � l ��� p �¾ � l G!;F@^7 Ø ¯ � Í �� ´ � ÙG!;F@K7 Ø � z5yRÙ u

r 798VP!8�l Æ�¯ n Í � ��� ´ �

¥ ]KJ!C?;F@`C0MVD>HTG!8<S`8<WBE^C5J!;>HT@�IhH?P � ]X_ r H?P!gN;F@9='HTE9J�

l � · � l 3' � ��� p � ·¾ � l � DF@(' ���� �

°�J n�l �N��

l ¯ � Í 3y ´XM}H?J!7 � � � Í 3y � Æ � ��� � Ó 3y M}H?J!7 � 3y Æ � ��� � �µ�_'M}HT@KG!;US98VP+;F@9=[J!798#IhH?P+\ËH?I*J!798�G!HTDFE9J!;>HT@��TM}HT@^MVDFEKS98�J!7KC5J�J!798VP!8#C5P!8�@KHT@NÏжV8VP!H:G~HTDUE9J!;>HT@KG|IhH?P � r 798VP+8J!7K8:G!D>H?c�8�H?I|J!7K;FG�I±EK@KM}J!;>HT@`C5J � l �';FG�=?P!8<C5J~8VPtJ!7KC?@�EK@^;>J¦_*i

� ¯ � Ío3�´{ Æ � � 7 3H?P �

� � � � ¯ � Ío3�´{ �

�� �{?{

Page 34: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

%KH?P�J!798 ª G+;F@9=�\0HXS98<DÐ�*J!C5g?8 � l �� 4;'[8<@9H?J!;F@9= ª G!;F@9=�A¢C?DFEK8<Gt]X_YC?@xHRA?8VP!]^C5P<�vSK8}��@98 · k�l y · k u�G!EKM+7J!7^C5J · kml ��3)C?@KS�

l � · k � l¸y � uC?DFHT@9= r ;>J!7 � l � z ¬ ��J~H0=T;>A?8�J!798�M}H?P+P!8<M}Jt;F@BJ~8VP+C?M}J!;>HT@�ÈtC?\0;UD>J~HT@K;FC?@�4 9 ;>J!7`J!798<G!8:M+7KC?@K=?8<G r 8)��@KS

l J!C?@K7 ¯hÆ � � � ´�uC?@^S � � l �

�z5� �

� 4�6ÊC5g?8?i�l� �� � � 7 �

� ¬ � � l � �� �l�Ó

( ���� k ¬ k � �� Ó � � k ¬ k � �� �

Èt8VP!8 (®;FG�J!798�M}HXH?P+SK;U@KC5J!;>HT@�@XEK\:]�8VP�H?I[J!798�DFC5J~J!;FM}83�#@9H?PO\0C?DFD>_ r 8�EKG!8 � ]^E9J�J!7KC5J r HTEKDFS MVC?EKG~8M}HT@KIhEKG+;>HT@ r ;>J!7�J!798eM}HBH?P+S^;F@KC5J~8)H?I|J!798�G!C?\'8e@KC?\'8?46�798)G!MVC?DFC5P�cKP!HNSKEKM}J�cKP!H �~8<M}J!G�HTE9J�J!798�M}HT\'c�HT@98<@BJ�H?I ¬ k ;F@�J!798 �� SK;>P+8<M}J!;>HT@�C?@^S r 8)S98<@9H?J~8�J!7K;UG�]X_· �k 4l-Ó ¯ � Í ( ��� ´ � k · �k �

' Îl ' � ur 798VP!8?i

' � l�y � ���+ G!;F@ � � � ¾V¿BÀ � Ö�� � � Å����� �®C5g?8:J!7K8:M+7KC?@K=?8�H?I|A¢C5P+;UC5]^D>8<GVi

M}HTG � l #�u G!;F@ � � � l-Ó� #�u

' � l Óty� �%p��� � # ¾ ¿BÀ � Ö�� � � Å !l y

� � �p�� � #¾ ¿�À

� �� � � Š!l y

�Æ�¯ � Í ( � � ´ � ¾<¿�À � Ö�� � � Å Ó ¾ p ¿BÀ � Ö�� � � Å l ¬ �Æ�¯ � Í ( � � ´ G!;F@K7 Ø Æ�¯ � Í ( � � ´ Ù ��Y8<C?@�A5C?DFE98)H?I�G~c�;F@�i { ¬

Page 35: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

l � DF@(' �� ¯hÆ � ´ l M}H?J!7 Ø Æ�¯ � Í ( � � ´ Ù�Ó Ø Æ�¯ � Í ( � � ´ Ù p��%KH?PtG~c�HT@BJ!C?@98VHTEKG�\sC5=T@98VJ!;>¶<C5J!;>HT@��^J!C5g?8 � l �N�^C?@KS�M}HT@KG+;FS98VP

l M}H?J!7 ¯�� � ´ Ó 3� � uIhH?P&� l Æ ( � 4��D>H?J~J!;F@9='J!7K8�J r HsG!;FSK8<G�H?I J!7K8�8<WBE^C5J!;>HT@�� r 8:G~8V8�J!7KC5Jt;>J#7KC?G#@9HT@9ÏжV8VP!HsG~HTDFEKJ!;>HT@KG r 798<@vi

�� � � M}H?J!7ʯ�� � ´ Ó 3� � �

� +l � { 7 3 �

¥ P r 798<@ � � � l

( �z5{?� �

3 �N4�6�798�G~HTDFE9J!;FHT@�J~H0J!7K;FG�cKP+H?]^D>8<\Ñ;FG�8<G+G~8<@�J!;UC?DFD>_�LN8<M}J!;>HT@ � 4³²X4� � X " ��[�� X¿Z�^ []X �\ � /��^]\*^ �354

� �l � ¯Ò3�Í��K´ �¯ � Í��9´}¯Ò3�Í � �K´ ¯Ò3�´+� � � l ��¯ � Í��K´¯Ò3�Í � �K´ ¯ y ´ �

¯ � �Tu*� �¢´ l ¯ �Nu<3�´ � WX@v¯Ò3�´ � � l � ��33 l � 8<WX@m¯ y ´ � 3 l 3T¯ �tÍ�3�´3�Í � l 3¯ � �Tu*� �¢´ l ¯ �Nu ��´ � WX@v¯Ò3�´ � � l � ��33 l � 8<WX@m¯ y ´ � � l � � �3 l �

� �l 35u � ,0� � ,Ô3 � WX@v¯Ò3�´ � 3 l 3T¯Ò3�Í��9´ �¯Ò3�Í��K´ � l 3 �'� 8<WX@m¯ y ´ � � � l � �

�l 3�� �ol � � � l �Nu n�l � u � l 35u n l �NuJ!7K8VP!8VIZH?P+8�IZ8VP+P!HT\0C5=T@98VJ!;FM)�K�X8<S�c*HT;U@�J�;FG�¯ �Nu<3�´ �« ;F@98<C5P+;UG~80¯Ò3�´�C?@KS ¯ y ´#C5]*HTEKJ � � l � l �Nu � � l � l 3 �¯Ò3�´ � � l � ¯Ò3�Ío3�´ �¯ �tÍ�3�´}¯Ò3�Í ��´ ����� � �������

l ¬ � Ó Ó Ó ¯��N´¯ y ´ � � l ��¯ �tÍ��K´3�Í � ����

+� � �

l��

LN8VJ� � Ó 3 l � � � J!7BEKG � � l � � � Í�3�Ó 3 l � � J!7XEKG �

l � ��Í�3 �C?@^S�G~H ¯ y ´ � � � � Ío3 l ¯ � �:Ío3�´ ��l¸y � � �eÍ 3�Í � ¯ � � � ´ �C?@^S � � � l¸y � � � z � l 3 l�y � � Ó Ó Ó ¯ �V´ �{ º

Page 36: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

%KP!HT\ ¯��9´�C?@KS�¯ �V´ � � �� � � ) l � ¬ �� y ) � �� � ) i � ;F=?8<@�A5C?DFE98<G � � l ¬ u�� � l�y �%KP!HT\ � kvl � � k u � l�y � r 8e7KC<A?8?i � � l�y u � � l 3 �y 4 � l��

� ���k������� ¨ ¨�

¨�����

¾�� Ä À � Ä � � Ö � Ä � � Å Ö ���� � Ä � � Ö ¿ & � Ä Ö ���� � Ä � � Ö � � � �

'[H0J!7K8�c^C5P!J!;FC?D�J~POC?M}8:]X_�G!E^\0\0;F@9=�H�A?8VP · � l ��35u · l ��3 �V�V���� � l ������ � � ¨ � ¨ ��� ����� �

k������� ¨ ¨� ������ ¾ � À � Ä � � Ö � Ä� � Å Ö ¿ & Ö ���� À � Ä � � Ö � Ä � � Å Ö � � �

Í ¾� p À � Ä � � Ö � Ä � � Å�p ¿ & Ö ���� À � Ä � � Ö � Ä� � Å Ö � � �

��8}ϦDFC5]�8<DvG~c^;U@KG�G~H'J!7KC5J#J!798V_`C5P!8eC5=TC?;F@�@XEK\:]�8VP!8<S`M}HT@^G~8<MVE9J!;>A?8<D>_*i� � l �� � � �

k � ¾ � À Ö ���� Å À � Ä Ö �� � � Å Ö ¿ & Ö � � �Í ¾

� p À p ���� Å À � Ä Ö � Ä� � űp ¿ & Ö � � � Ó Ó Ó ¯ � ´ ���8<WBE^;>P!8)J!7KC5J�@K8 r c^C5P!J!;>J!;>HT@�IhEK@^M}J!;>HT@`G!79HTEKDUS�]�8�G!C?\08�C?G#HTDFS`]�E9J r ;>J!7`@98 r M}HTEKc^DF;F@9='M}HT@KG!J!C?@�J!GVi� � l �

� � � �kо� � Ä � Ä� � Ö ¿ & � Ä Ö � � Ó Ó Ó ¯ n ´ �

¥ ]BAN;>HTEKG!DF_?�X8<WXEKC5J!;>HT@^G:¯±°�´�C?@KS�¯±µ#´�C5P!8�M}HT@^G!;FG~J~8<@BJ�;>I i¾� � Ä � Ä � � Ö ¿ & � Ä Ö � � l ¾

�À Ö ���� Å À � Ä Ö � Ä � � Å Ö ¿ & Ö � � �Í ¾

� p À p ���� Å À � Ä Ö � Ä � � Å�p ¿ & Ö � � � �� WBE^C5J!;F@9='8}�Nc*HT@K8<@�J!G�IhH?P · k u · k Ö � l ��3�_N;>8<DFS^G { 8<WBE^C5J!;>HT@KGVi· k u · k Ö � l 3)i ¾ Ö ¿ & Ö � l�¾ � Ö � ¿ & Ö � � Í ¾ p � Ö � � �· k u · k Ö � l-Ó 3�i ¾ p ¿ & Ö � l�¾ � p � ¿ & Ö � � Í ¾ p � Ö � � �· kvl-Ó · k Ö � l ��3�i ¾ p Ö � l�¾<¿ & Ö � � Í ¾ p ¿ & Ö � � �6�798<G~8�MVC?@�]�8�G~HTDFA?8<S`J~H0_N;>8<DFS�i¾<� ¿

& l�¾<� ¿

& M}HTG!7m¯ y � Í Æ n ´ z M}HTG!7 ¯ y � Ó Æ n ´ �¾

l M}HTG!7m¯ y � Í�Æ n ´XM}HTG!7ʯ y � Ó Æ n ´ z M}HTG!7 � Æ n �¾ � l�¾ � � M}HTG+7 ¯ y � Í�Æ n ´XM}HTG!7ʯ y � Ó Æ n ´XM}HTG!7 � Æ n �{ 4[­);>A?8<@ � � l¸y � � Í #

�# � l � � uC5J�J!798)�K�N8<S`c�HT;F@BJ!G r 8:7^C<A?8 � ( l¸y � ( � Í #

( �#(l � ( � � ¯Ò3 y ´

{�

Page 37: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

ª J�;FG�8<C?G+;FD>_'A?8VP+; ��8<SsJ!7KC5J�J!798t�K�N8<Ssc�HT;F@BJ!G�C5P+8�¯ � ( u # ( ´ l ¯ �Nu ��´Ou�¯ � u � ´�C?@KSx¯Ò3 z5{ u<3 z � ´ � 6#7K8��^POG~JJ r H�C5P!8�J!7K8�7K;>=T7NÏÐJ~8<\'c�8VP+C5J!EKP!8�C?@KS D>H r ÏÐJ~8<\'c�8VP+C5J!E9P+8�c�HT;F@�J!GsC?@KS C5P!8�J~PO;>AX;UC?D�4 6#798�@9HT@9ÏÐJ~P+;>AN;FC?D�^�X8<S`c�HT;F@BJ�;FG:¯ � ( u # ( ´ l ¯Ò3 z5{ u<3 z � ´�C?@^S r 8:DU;F@98<C5P+;FG!8�C5]�HTE9J�J!7^;FGV4�LX8VJVi� � l � ( Í � � � � # � l #( Í � # �� l � ( Í � � � # l #( Í � # � ¯Ò3 { ´

� WB@^G:¯Ò3 y ´�]�8<M}HT\'8?i� � � l ¬ � ( � � Í � # Í�¯ y � ( � Ó � ( Í #

( ´� # � l y � ( � � Í�¯ � ( ��Ó # ( � ´ �Èt8<@KM}8?i � � � �� # � ) l � ¬ � ( 3y � ( � ) � � �� # ) l � ¬ z5{ 3yTz5{ � ) � � �� # ) �

� 8<;F=?8<@�A5C?DFE98<G � l 3{ ¯ y � 4 3 �B´J!7K8VP!8VIZH?P+8 � � l 3 � �5y?y � � � l-Ó � � { � � �%KH?P�J!798:M}PO;>J!;FMVC?D�8}�Nc*HT@K8<@�J � r 8:7^C<A?8�l 3 z ��u r 798VP!8�� � l � � � u � l54 y �Èt8<@KM}8

�l DU@��DF@� �

l DF@v¯ 4 y ´DF@v¯Ò3 � �5y?y ´ l � � �?º y �¬ 4[��HT@KG!;FS98VP#CsDFC5J~J!;FM}8 r ;>J!7�c^P!H?]^C5]^;FDU;>J¦_)§YJ!7KC5J�C?@B_�J r H�G!;>J~8<G#C5P+8:M}HT@K@98<M}J~8<S�]B_�C']�HT@KS�46 r H�H?P)\'H?P!8�G!;>J~8<G�M}HT@K@98<M}J~8<S l MVDFEKG~J~8VP��*C?@KS®J!798�M}P+;>J!;UMVC?DvcKP!H?]�C5]^;FDF;>J¼_�§ � ;FGtJ!798�cKP!H?]^C5]�;FDF;>J¼_�H?I�CMVDUEKG~J~8VP�G~c�C?@K@K;F@9=�J!798:DUC5J~J!;FM}8?4%KH?P/§ § � �9cKP!H?]^C5]^;UDF;>J¦_sH?I|C?@B_�J r HsG!;>J~8<G�SK;FG~J!C?@KM}8 � C5c^C5P+J#]�8<;F@9='M}HT@K@98<M}J~8<SY;FG � ¯ � ´�G!EKMO7`J!7KC5J� ¯ � ´ � � C?G � � � IhH?P § § � �ª @BJ~P!HXS^EKM}8�M}H?P!P!8<DFC5J!;FHT@�D>8<@9=?J!7`]X_ r P+;>J!;F@K=�C?G!_X\'c^J~H?J!;FM�IZH?PO\`i � ¯ � ´�� � p " � C?G � � � � r 798VP+8�K¯ §�´ l M}H?P!P+8<DFC5J!;>HT@�DF8<@9=?J!7�4 'tPOC r C?@KC?D>H?=?_x]�8VJ r 8V8<@�§ ;F@%J!7K;FG:\'HXSK8<D/C?@KS J~8<\0c*8VPOC5J!E9P!8s;F@ ª G!;U@9=\0HXS98<DÐ4/°tG|§ � § � �KM}H?P!P!8<DUC5J!;>HT@�D>8<@9=?J!7 ��� �KS98}��@K8�M}P+;>J!;FMVC?D�8}�Nc�HT@98<@�J

� �˯ § Ó § � ´�� C?G § � § � ���HT@KG~J~P+EKM}J#Cs@98 r DFC5J~J!;FM}8:]B_�c^DFC?MV;F@K=�G+;>J~8<G#C5J�M}8<@BJ~P!8:H?IÊ8<C?M+7®EK@K;FJ�M}8<DUD�H?I�¯hH?PO;>=T;F@KC?DZ´v§KϦDUC5J~J!;FM}8?4��E9J�]�HT@KSKG�M}HT@K@98<M}J!;F@K=0C?@�_�J r HsG!;>J~8<G�@KH?J#M}HT@K@98<M}J~8<S`HT@�§KϦDUC5J~J!;FM}8?4 � 8 r DFC5J~J!;FM}8�;FG�;FSK8<@�J!;FMVC?D�J~H'HTDUSDUC5J~J!;FM}8�]^E9J#7^C?G�]�HT@KSKG r ;>J!7`c^P!H?]^C5]^;FDU;>J¦_

(l 3 Ó § �

��C?DFD�J!798:@98 r DFC5J~J!;FM}8)J!798 (RϦDFC5J~J!;UM}8?4ª I�§ 7 § � �*J!798<@YC5J[D>8<C?G!J�HT@98�M}HT@�J!;F@XE9HTEKGtc^C5J!7®C?M}P!HTG!GtJ!798#§^ϦDFC5J~J!;FM}8�]^E9J�M}HT@KG!J~P+EKM}J!;>HT@�P+EKD>8<G�IZH?P!]�;FSJ!7^;FG�HT@`J!798�(RϦDFC5J~J!;UM}8?4ª I*§ 7 § � J!798<@ ( (k C?@KS�M}HT@�A?8VP+G!8<D>_`;>I^§ § � J!798<@ ( 7 (

� 4�6#7K8VP!8VIZH?P+8:;>ImJ!7K8�§KϦDFC5J~J!;FM}8e;FG#M}P+;>J!;UMVC?DG!H�C?DFG~H0;UG�J!798 (RϦDFC5J~J!;UM}8?4�6#7K8VP!8VIZH?P+8 ( � l 3 Ó § � C?@KS�C?G�J!798eDFC5J~J!;FM}8<G�C5P+8:;FS98<@BJ!;FMVC?D�� ( � l § � 4�È�8<@KM}8§ ��l 3 z5y 4¯±C�´���HTC5P+G~8}ÏÐ=?P+C?;F@^;F@9=�J~P+C?@KG~IhH?P+\0C5J!;>HT@v4 �#8<\'HRA?8:C?D>J~8VP+@KC5J~8)G!;FJ~8<G�IZP!HT\�J!7K8�DUC5J~J!;FM}8)C?@KS`S9P+C r C'@K8 r]�HT@KS�;>I�J!798VP+8 r 8VP!8)C5J�D>8<C?G~J�J r H']*HT@^SKG�M}HT@K@K8<M}J!;F@9=�J!79HTG~8[c^C5P!J!;FMVE^DFC5P�G+;>J~8<G�HT@�J!7K8tHTDFS�DFC5J~J!;FM}8?4{ �

Page 38: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

¯h]*´1��8}ϦG!MVC?DF;F@K=94��#8<SKE^M}80C?DFD/D>8<@9=?J!7^G�]B_ � l 4 y 4 � 8 r DFC5J~J!;FM}8s;FG�P+H?J!C5J~8<S � z ¬ J~H�HTDFSxHT@98?ie@K8 r]�HT@KS^G�DF;>8)C?D>HT@9=0HTDUS`SK;FC5=?HT@KC?DFG<46�79P!8V8�M}HT@N�^=TEKP+C5J!;>HT@KG#MVC?@�M}HT@�J~P+;F]^E9J~8

� §|¯��N´ l § � §|¯ �}´ l § , ¯Ò3 Ó §�´� §|¯ � ´ l § � ¯Ò3 Ó §�´ �C?@^S�G~H § � l §' /Í ¬ § , ¯Ò3 Ó §�´vÍ y § � ¯Ò3 Ó §�´ � l�y § � Ó § �°�J��K�N8<S�c*HT;U@�JV�5§ � l §m�

�#­ � � § Ó�y § � ÍY§ l �H?P§|¯ § Ó 3�´}¯ § � Í®§ Ó 3�´ l � � § ( l �Nu<3 p�� ��� �� 49 8eP!8 �~8<M}J p�� � �� �^C?G|§ 7 �N4§ �l � � @KH']*HT@^SKGʧ ( l 3 � C?DUD�G!;>J~8<G�]�HT@KS98<Sv46�798VP!8VIhH?P!8:J~P+;FAX;FC?D�C?@^S�C?@KC?D>H?=?HTE^G�J~H�7K;>=T7�C?@^S�D>H r ÏÐJ~8<\'c�8VP+C5J!E9P!8eMVC?G~8<G�;F@`J!798VP+\sC?DmG~_NG~J~8<\0GV4���HT@9ÏMVDUEKS98�M}P+;>J!;UMVC?D*�K�X8<S�c*HT;U@�J�;FGʧ ��l § ( l ¯ 4 º Ó 3�´ z5y % � � � y 4°�J��K�N8<S�c*HT;U@�JV�!�#­ � ]�8<M}HT\'8<G § ( l�y § ( � Ó § ( 5�LN8VJ�§ � l § ( Í � § � �?§ l § ( Í � §m�KJ!7K8<@

� § � ÍY§ ( l�y ¯ § ( l � §�´ � Ó ¯ § ( Í � §�´ l�y § ( � ¯Ò3�Í � §§ ( ´� Ó § ( T¯Ò3�Í � §§ ( ´ C?@^S`J~H'�^P+G!J�H?P+S98VP�;F@ � §

� § � l ¬ § ( ¯Ò3 Ó § ( � ´ � § � � § � l � � ¯ § ( ´ � § �°[G#J!7^;FG�;FG�C0G!;U@9=TD>8}ÏÐc^C5P+C?\08VJ~8VP�G~c^C?M}8?�KJ!7K8VP!8:;FG�HT@KDF_�HT@98)8<;F=?8<@�A5C?DFE98?�^C?G � � ;FG�C'G!MVC?DFC5P<�� � l ¬ § ( ¯Ò3 Ó § ( � ´C?@^S�C?G�§ ( l � � � y � � � l 3 � º { 4°[G6� l54 ºX� r 8�7^C<A?8:J!7K8:M}P+;>J!;FMVC?D�;U@KS98}��� � C?G� � l DF@� �DF@�� l DF@)3 � º {DF@ 4 yC?@^S`P!8<MVC?DFDF;U@9=�J!7KC5J � l 3 z � � ��l DF@ 4 yDF@)3 � º { l � � ² y �

� � X " ��[�� X¿Z�^ []X �\ � /��^]\*^��354 ª \'c�HTG!;F@K=�J!7K8�M}HT@KSK;>J!;>HT@^G�IZH?P#J!7K8)�K�N8<S`c�HT;F@�J�HT@`J!7K8�=T;>A?8<@`8<WXEKC5J!;>HT@KG<i

� l¸y # ( Í �� � (3 � � � Í *0¯ * � ´

� l *�� ( Ó { � ( �3 � � �{ ²

Page 39: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

LNHTD>AX;U@9=�J!798:G~8<M}HT@KS�8<WBEKC5J!;FHT@��� ( l ��H?P � ( l 3 � � �{ * �

L9E9]^G~J!;>J!EKJ!;F@9=�J!798:G~8<M}HT@KS�H?IÊJ!7K8<G~8�;F@BJ~H0J!798��^P+G!J�8<WBEKC5J!;FHT@# ( l Ó

��

� *ÊÍ �K¯ * � ´�uÈt8<@KM}8�¯ # ( u�� ( ´�;FG�=T;FA?8<@`]B_ ��Ó � �� *Vu 3 � � �{ * ) uC?G�P!8<WXEK;>P!8<S�46 HsDF;U@98<C5P+;FG~8?�9D>8VJ � l � ( Í � �# l # ( Í � #C?@^S r ;>J!7`C'Ih8 r DF;U@98<G�H?I P+HTE9J!;F@98�C?D>=?8V]^P+C

� � #� DU@��

l ��y[Ó *{�� � #�Í ��

3 � � �� 3�Í *

�� � �

� � �� DU@��

l-Ó * � � �9 PO;>J!;F@9='J!7K;UG�;F@�J!798�IhH?P+\

����� � #� DF@ �� � �� DU@��

�����ln �

� � #� � )

n �l ��

y�Ó * z5{ ��

3 � � � � 3�Í 0��� Ó *

��%KP!HT\ r 7^;FM+7 r 8:M}HT@KMVDFE^S98?i

� � l¸y�Ó *{ 7 �Nu � � l�Ó * �Èt8<@KM}8�J!7^;FG#;FG�C ��s��.p�� �K�N8<S�c�HT;F@�J#C?@^S`S98VJ~8VP+\0;F@K8<G#M}P+;>J!;FMVC?D�]�8<7KC�AX;FHTE9P�IZH?P * 7 �0C?@^S � ¬ 4��P+;>J!;FMVC?D�8}�Nc�HT@98<@�J � ;FG�H?]KJ!C?;F@K8<S�IZP!HT\

�l 3� �J!7K8VP!8VIZH?P+8

�l 3y�Ó * z5{ l 3y ��3�Í *

� Í *0¯ * � ´ �l 3y Í *3 y Í *s¯ * � ´Ou

C?G�P!8<WXEK;>P!8<S�4 { �

Page 40: HJIFKLMIFNPORQ S8TFUJUFV - University of Edinburghdmccomb/exercises.pdf · t´ + ( ,(× * ¬ ¯ § (-(´ - % ¬

y 4[­);>A?8<@�i § � l � � §�Í � ¯ � � Ó � 0 ´)(#Í �K¯ § (?´�u( � l � 0 (�u* l ¬ Ó � �

%|; �N8<S`c�HT;F@BJ�¯ § ( u-( ( ´ �˯ �Nu ��´O4« ;F@98<C5P+;UG!;F@9=�C5]�HTE9J�J!798)�K�N8<S�c�HT;F@BJV�!��­)6¸J~P+C?@KG!IZH?P+\sC5J!;>HT@`\0C5J~P+; �`;FGVin �

l � � � � ¯ � � Ó � 0 ´� � � )r ;>J!7�8<;>=?8<@BA¢C?DFEK8<G�C?@KS�8<;>=?8<@BA?8<M}J~H?P+GVi

� � l � � ¾ À �¼Å l� 3+ �� � l � 0 ¾ À � Å l�Ó

�+ � �%KP!HT\ � kml � �k � � � l�y u � � l * �%KH?P#C�\s; �X8<Sx¯±M}P+;>J!;UMVC?DZ´|�^�X8<S�c�HT;F@BJ�;F@�J!7K;FG�MVC?G~8)\eE^G~J�7KC<A?8 � � l * �Ni/798<@KM}8 � 7 ¬ 4�È�8<@KM}8�=T;>A?8<@�^�X8<S c�HT;F@BJ';FG�C%M}P+;>J!;FMVC?D�c�HT;F@BJeIhH?PsDFC5J~J!;UM}8�S^;F\'8<@KG!;FHT@�=?P!8<C5J~8VP�J!7KC?@ � l ¬ 4 ­�;>A?8<@ ;F@KIZH?P+\sC5J!;>HT@;U\'c^DF;>8<GsJ!7K;FG�\'HNS98<D�;FG�;F@ G!C?\'8%EK@K;>A?8VPOG!C?DF;>J¼_ MVDFC?G!G`C?G�J!798%\'8<C?@ �^8<DUS J!798VH?P+_ H?I ª G!;F@K=94 È�8<@KM}8;U@KSK;FMVC5J~8<G�c�HTG!G+;>]^D>8tA¢C?DF;USK;>J¦_�H?I|J!798�\'8<C?@`��8<DFS�J!798VH?P!_�C5]�H�A?8 � l ¬ 4

¬ �