web.math.ku.dkweb.math.ku.dk/~hugger/courses/3na/linerrslides.pdf0 # 1(2/ 3 4 & 3 5 6 7 8 9: ; < 2 =...

19

Transcript of web.math.ku.dkweb.math.ku.dk/~hugger/courses/3na/linerrslides.pdf0 # 1(2/ 3 4 & 3 5 6 7 8 9: ; < 2 =...

Page 1: web.math.ku.dkweb.math.ku.dk/~hugger/courses/3na/linerrslides.pdf0 # 1(2/ 3 4 & 3 5 6 7 8 9: ; < 2 = 4 > ?@ 6 7 8 " a5 b5c= 4 < 8 d 4 e f g hi 0 j *lk-k- m 0 n 3 o p*& 6 q

����������� ������������������� �������! �"�#�$%�&�'�)(+*,�-�/. �0�#�1( 2/���3�4 �&�3���5�

��67 8 � 9:��;�<��� 2 �=��4 ����� 2 �>�?�@�

6����7 8 � ����"���A�5� �B�5C��=��4 �<� 8 D 4 ���E� 8 F ���G�

HI���0� J *LK-K-���M��0�N�3�O�P*&��� 6���� Q �&� 8 ��4 �&�3�)�I�-� D �R���;�I�S�&�

T ���=UI���V�'�<�XW ��6 Y%�5Z;��� 8 �LK-���@ F []\_^�`0a�a Y J F �;�-4 �R�!b

c 2 \ 4 �;��ed 8 *LK-K-���gf 4 ��� 8@h bi* h (+b1j

kml?nolqp>rts>u%nwvyxzx!{|r�}~lg� �B� �'�1� ���u�l �ipy����nwu �'pg}Bvzrtpz�'�As���l!��s�l kmlB�nolqp�rts!u/�'{�vz��s���������1�'��� ���y�y���3�_l!vg��l!�_�¡ ml>�£¢_�¤vy¥

¦

Page 2: web.math.ku.dkweb.math.ku.dk/~hugger/courses/3na/linerrslides.pdf0 # 1(2/ 3 4 & 3 5 6 7 8 9: ; < 2 = 4 > ?@ 6 7 8 " a5 b5c= 4 < 8 d 4 e f g hi 0 j *lk-k- m 0 n 3 o p*& 6 q

� ���������� � ��� ����� � �����������

9 ��L( ��� � ! "�# c%$ j & ')( c%$ j+* '-, c/.$ j dc ` j0 1 �2�3� & 4 �� $ 5�2�3� & a �+� 6 $ 7 562��� �6�8 & 9;: �5� 6 $ : <

$6 $ 7

F ����)� 8 �=�P� C;�+*��L(��i� W

6 $ :

= ��*�4 ����� C;�+*��L(��i� W

> ���@?�� � 4+A9�: �i�?� B�� 4 ���,� 8 ���I�5*IK 8@C ��� ��� �=�0 � *����ED+*I� �#�+� *&�3���5� ��� c ` j h

F ��G �)� � 8 � HJI KMLON H P�PRQSP T+UMV LWNYX�Q V

Z � � & � � � [ �]\ � ! " � c%$ j ^ '`_ c .$ j 5c ^ j 8 ���?� � \ � �"� � ���5���3��-*I�5*1�S Z;���1�S�P �"�#� Z;�5�¤W��I� \4 �t���2a �;�<��� �;�=��4 ����� ��Z-Zi�B�cb��4 �&�3�)�+� ��� �2�3� h

Page 3: web.math.ku.dkweb.math.ku.dk/~hugger/courses/3na/linerrslides.pdf0 # 1(2/ 3 4 & 3 5 6 7 8 9: ; < 2 = 4 > ?@ 6 7 8 " a5 b5c= 4 < 8 d 4 e f g hi 0 j *lk-k- m 0 n 3 o p*& 6 q

� � � �c � ����� � ?��������

� c �2�3� 5�� j & c� j 5 ��� ! )#

� c � \ � 5� j & c� j 5 ��� ! � # ^ #7 8 � ���5���3��-*I�5*1� ���!�?��� ZR�?�5C-�"��4 d

� c Z � � 5�� j & c� j [ � c � \ � 5�� j 5 ��� ! )#� ���=�;�>bL�� � � � 8 ��K-�5�i���t�<�XW

� c Z � � 5� j & a 5 ��� ! � #D � � )# ^ � � ^ )# � h � h�Z �"� 4 ���?� �S�;���P�¤W � 8 ��� � \ � T �-�=����� � ��� Y � �-M�� 2 h

�>� p

Page 4: web.math.ku.dkweb.math.ku.dk/~hugger/courses/3na/linerrslides.pdf0 # 1(2/ 3 4 & 3 5 6 7 8 9: ; < 2 = 4 > ?@ 6 7 8 " a5 b5c= 4 < 8 d 4 e f g hi 0 j *lk-k- m 0 n 3 o p*& 6 q

�� �M�������S� ����� � � � � � � � ��� � � ��

Q �I���1�1d 9 ��L( C;�5*-�L(��i�oW ���5�I( �<�3���5�0� �5� ��� � �� \�����!�,�-� �1(�K���� �E6 ��� �����=�;4 �;�&� � C3W B�� Z-�t�O�_�3��IK � 8 � *�4 Z0� C h

M (-�I� d D ��L��� � # ������ �"� �S�+���3���*��5*1� �I���?�;��� ��� �� \�����!�,�-�@�1(�K-� �����S

7 8 � H�*-4 Z 7 ���!4 � :c�� j& B��m*�4 ��6 �;�>�?�@� = ��*�4 ���-� ���5�I( �<�3���5�0� C& 6 Z � ��� �

628 � �6 Z �3��� �6�8 �� �! " �������1(

& [ 6�� \ �#� �628 � �

6�� \ �$� �628 �� �! "bi�I�1 �

�"� ���54 Z�*&�G��C-�"� h

� � K h % *I�&���3���5�@d 9:��I( = ��*�4 ���-� �S�+�L( h

2�D�*;�OU h % *������3���+� d 9:��L( � Z-���<�_�3��LK ��6 � *-4 ��6= ��*�4 ���-� �S�+�L( h

&

Page 5: web.math.ku.dkweb.math.ku.dk/~hugger/courses/3na/linerrslides.pdf0 # 1(2/ 3 4 & 3 5 6 7 8 9: ; < 2 = 4 > ?@ 6 7 8 " a5 b5c= 4 < 8 d 4 e f g hi 0 j *lk-k- m 0 n 3 o p*& 6 q

� �;���;�g�-� = ��*�4 ����� Y �+�L(��O�3���+� D Z��t�<�_�3��LK �B�5C \�=��4

9:��L( c � � � 5�� �3� j ! " :� c � : 5 6 $ � j dc�� j6 Z �3�����628� �! " �-�����1(

& � �3������ �! "a �L(� :� �! "b,���0 � �

� �3������ �! "a �L(5 �5� � ��� 5

� & 5�;5 ! � � 5 & ` 5 < < < 5� c � : & a �� � � ���������� & a 8 ���L��� � ����P�B��(+*I�S�1(@j h

" :� c � : 5 6 $ � j & � c � 5�� j d� & c � � 5 � � j 5 � & c�� � 5�� � j �5� � ��� 8 �;�B�

� � � � � � ` �5� � ��� 5 � � � � � � `^ �5� �`_��� c � : j 5

� � � � � � a �5� ����� 5 � � � � � � a �5� �-,��� c � : j ! � � 5 & ` 5 < < < 5� � c � � ��� � ������� & 6 0 �! 6 $ j <

� _��� c � : j & ��" ! � ��� d � : & a �c�# j� ,��� c � : j & ��" ! � ��� d � : �& a �

� _��� c � : j �"� �i�<� 8 ��� ��� �]��6 � ��� �#� � �3� � ,(

�@� �(�� �#���?�P��� Z;� �e�&� �#�P� �#� � �3� � a c �� $ ( j h

%

Page 6: web.math.ku.dkweb.math.ku.dk/~hugger/courses/3na/linerrslides.pdf0 # 1(2/ 3 4 & 3 5 6 7 8 9: ; < 2 = 4 > ?@ 6 7 8 " a5 b5c= 4 < 8 d 4 e f g hi 0 j *lk-k- m 0 n 3 o p*& 6 q

= �,��� d c�� j 8 �1� � 8 � *-�;�ED+*�� �#�5� *��3���+�c � & 5 ;5 ! � � 5 & ` 5 < < < 5� j

� �3��� � &���� � � � ��� ���� � � � : �5� � ,��� c � : j

` � ^ �5� � _��� c � : jc�� j

� �3����� &�� � a �+� � ,��� c � : j� � ���������� � �+� � _��� c � : j

c�� j� �M 6 � _��� �� � ��� c 4 ���N� ��6 ���;�ij � 8 ���

� � ����� & � � � � ������ � � � : �-�L( � �3����� & a�hM 6 � _��� � � ��� c 8 ��Z-Z;�;�1�S C�*�� �#�;�"(��54 j � 8 ���

� � ����� & ,(

�-�L( � �3����� & � � � � ������ �c � & 5 j h

Page 7: web.math.ku.dkweb.math.ku.dk/~hugger/courses/3na/linerrslides.pdf0 # 1(2/ 3 4 & 3 5 6 7 8 9: ; < 2 = 4 > ?@ 6 7 8 " a5 b5c= 4 < 8 d 4 e f g hi 0 j *lk-k- m 0 n 3 o p*& 6 q

�+� � ���� ��?������)��� � ��� ?)� � �������� � �

� �J� � � � � ���J�

9 �e�I( Z ! " � c%$ j 5 c � 5�� j ! " :� c � : 5 6 $ � j dc�� jZ � � \ � ! "�# c%$ j 5

0 1 Z & 4 [ 0 1 � \ � �� 0 �Z & a �+� 6 0 � * 6 $ 7

6 Z �6�8 � & 9;: [ 62� \ �#� �

6�8 � �5� 6 0 � * 6 $ :6 Z �6�8 � & � � � : � � � �+� 6 0 � 6 $ 5

& ` 5 < < < 5�

� � � �J����� � P Q� ����� ����� � K�� N�� � U V�X��)U�� ��Q��� U NYX Q V Z & Z �3��� � & � �3��� � & � �3�

!

Page 8: web.math.ku.dkweb.math.ku.dk/~hugger/courses/3na/linerrslides.pdf0 # 1(2/ 3 4 & 3 5 6 7 8 9: ; < 2 = 4 > ?@ 6 7 8 " a5 b5c= 4 < 8 d 4 e f g hi 0 j *lk-k- m 0 n 3 o p*& 6 q

�� �M��� � �)� � � �������`��� � ����������� �� �J� � � � � ����� ����� � ��� � � �;� ��� � ?�� � �������)�

`�h Q *����3�Z��¤W F 2 �<� 8 �����N� 6_*-�L�'�3���5� h^�h M �����1K+�g�&��� ��UI��� �5��� �;�=��4 ����� h� h /�R� �3�t���P������1K+�����3���5� *1�'��LK C;�+*��I(@�R�oW ���5�I( �<�3���5�0� h

9:��L( Z ! " � c%$ j 5 c � 5�� j ! " :� c � : 5 6 $ � j dc�� jZ � � \ � ! "�# c%$ j 5 Z�� ��� & a 5c � � [ � 7� j c Z 5 � j [ � �� c � 5���5�� j & c� � � � :� ��� � � � ����������� � ��� � � ����� �"! � � #%$& �('! �*) � �+ , �.- �0/ � � 1 2�3 4 - 5 3 46/�7�8 , � / � � 1 2:9 /;7�8 < + , �.->= �./ ��@?, �.- �./ � � A 1 2CB AD�E F - ,

FHG , I F ->= � ,FHG ,

/ , 7KJ� �, � / � � A 1 2 B AD�L M � / , 7KJ� �, �ON �0P �RQS�./ � � TU2WV P ,YX � I Q , I F - = � ,

FZG ,/ , 7KJ

[ \�]>^@_.]a` b c dfehgZikj;l m�nYo prqKs t�pZj urvZwkxyu(j sze|{i}u~t�w�e>v � � � R��� P � P R��� Q � Q R�Z� �

Page 9: web.math.ku.dkweb.math.ku.dk/~hugger/courses/3na/linerrslides.pdf0 # 1(2/ 3 4 & 3 5 6 7 8 9: ; < 2 = 4 > ?@ 6 7 8 " a5 b5c= 4 < 8 d 4 e f g hi 0 j *lk-k- m 0 n 3 o p*& 6 q

� � ^���� ��� �y_���� �k^���� � � _.^�� � ]a` �� ^@_ �U\y] � _ _.^ _ ����� � � � ����� ����� ��� �"!#�$�k^��%�

& '�'"(*),+ -$.,/ 02143,560�/8797:1;7 /=<?>@5A-B(C1 )=D 18E,/F7 560G0�/80H/JIK /F)L-MD +"(NEO/LD

P (Q),' - R� R S�T6U ! V W - �FX AD�E Y Z"[U�\ Z VU + < ]@? V U - � [ / V < ] � U P R� [ Q R� [ / V Y U_^ I ] � V U / V` / R � T U ! V Y # $& U '! [ ) Va 1b-c/ W ] � U P R� [ Q � [ / V Y d ef g $ ] �ih,_j �ON j U - R� [ / Vk .,/87:/

] �ih, � N U - [ / V Y T 2WV F - NFZG N

ll /nmm A 1 2 7KJ [U�\6\ Vo R J , [ p Y \ [cqMqrqM[ts

uwv S�T6U ! V [ � TFU ! Vyxz $ U ! V�{ | - R #�} U ! V W;- X AD�E Y Z%~� .,/8) U�\ Z V x + U - R� [ / V Y ^ U / V I ] � U / V �

Page 10: web.math.ku.dkweb.math.ku.dk/~hugger/courses/3na/linerrslides.pdf0 # 1(2/ 3 4 & 3 5 6 7 8 9: ; < 2 = 4 > ?@ 6 7 8 " a5 b5c= 4 < 8 d 4 e f g hi 0 j *lk-k- m 0 n 3 o p*& 6 q

� � �4� � ���y_��� ` � _ _.^@_ � �B�$� ` � �$�k^��\ � � (2D�3 7:/ -B(��A/ -$7 ( 560"56)O' -c/ADi- D��b5,3 / (*) U�� V W

� $ � � � �

Y � $� qU�\�� V� �F/F7 - 5F(*) D -c1 Eb5@7 (�5���0H/AD v 1%7 -$.,/ /F797 1%7 v >6)O3J-B( 1?) � �F/87 - 58(Q)=D -c1 -$.,/ �81 >�),'%5#7�� 3 1?)O'"( -B(C1 )=D �

� � � 1 0H1 3,560�(��L/ 5�),' D 1 0 E,/ -$.F(2D + 021��b560 �#7:1��602/ K>=D / Di- 56)O';5@7 ' � 5A-$7 (�� ����0�( - -B(*)O+ 5�>O+ K /8)A-c/=' ���� K 141 -$. /F)8(*)O+ �u -c/F7 5L-B(NEO/ �#7:1 3A/LDcD W U � .F(*)�� Y � ! 7 Di- V

" (NE,/F) $# }&% v 1%7 ' Y \ [ � [Mqrqrq '61U�\�( V) � � � �

� #+* % #+* %

Y � $ < � $ #,*.- $ %� q

� #+* %0/ $#+* % U D K 1"1 -$.,/8)F(*),+ V q

1

Page 11: web.math.ku.dkweb.math.ku.dk/~hugger/courses/3na/linerrslides.pdf0 # 1(2/ 3 4 & 3 5 6 7 8 9: ; < 2 = 4 > ?@ 6 7 8 " a5 b5c= 4 < 8 d 4 e f g hi 0 j *lk-k- m 0 n 3 o p*& 6 q

� � ] _ � �$�k^�� � � � ^ _��� \ ` � ^@_� � �4� � ���y_��� ` � _ _.^@_ � �B�$� ` � �$�k^��

Z � u )8( -B( 560�Di-c/ �5 � � 1 K ��>L-c/ �� # } % -$.,/ u )F( -B(�5�0 � 797:1;7 � D -B( I

K 5A-c/ � � )�0 � 5 �@7 ( 1;7 ( �@),1 k ) (*) v 1;7 K 5L-B(C1 )K 5�� �F/ > D /=' �

\ � P 1%7 ( v 7:1 K \ >6)A-B(*0"D 1 K / Di-c1����8(Q),+ 387 (N-c/87 (�5(2D D 5A-B(2D ! /=' ' 15 � � 1 K ��>L-c/ /AD -B( K 5A-c/LD �P #,* % 56)O' �Q #,* % 1 v -$.,/

a /8> K 5�)6) D���0�( - -B(*)O+ v >6)O3J-B( 1?) D � '6/ �F/8)O'OI(*)O+ 1 ) �� #,*.- $ % �

� � � 1 K ��>L-c/ �� #+* % -$. / � 5 k � 797 1%7 � D -B( K 5A-c/' / �F/8)O'�(*)O+ 1 ) �P #,* % 5�)O' �Q #,* % �3 � � 1 K ��>L-c/ �� #,* % -$. / � K 141 -$. /F) / ' � 7 7:1%7

� D -B( K 5A-c/ �& " � � u � � \ � " /8),/87 560%/ < >F(*0�( �@7 (*> K

�#7:1 3A/ ' >�7:/ v 1%7 7:/ 3A18E,/87�� 1 v /F7 7:1%7 /AD -B( K 5A-c1;7

� �

Page 12: web.math.ku.dkweb.math.ku.dk/~hugger/courses/3na/linerrslides.pdf0 # 1(2/ 3 4 & 3 5 6 7 8 9: ; < 2 = 4 > ?@ 6 7 8 " a5 b5c= 4 < 8 d 4 e f g hi 0 j *lk-k- m 0 n 3 o p*& 6 q

� � � ] � ����� � ��� ��� � ]%!�^��H] _� ^ � � \y] ] _ _.^ _� � ���H] � � ���y_.^��%� ` ���$�k^��%� �P � ��� �Q^ � P R� ����� Q � �

P (*),' �� R �S U ! V � S �� & U ! [ ) V WU�\�� VU + , < ] ?, V U �� [ / V Y � , U / V I ] �, U / V I ]��, U �P [ �Q [ / Vp Y \ [rqrqMqr[_s [ ` / R �� U ! V � � � � � U ! V q[ \�]>^@_.]a` � � szehi}uht�wke>v t0e m����o j! ~w sKt s w#" qrv%$e>v>i'& w(" �P qrv)$ �Q q dfj szj;ikj+*�t0j,$ t0e s q~t�w�s,"-& t�pZj��< >F(*0�( �@7 (*> K � 1?)O'"( -B(C1 )

� , U / V I ] �, U / V I ]��, U �P [ �Q [ / V Y Z"[` / R �. U�/ 0 V [ p Y \ [cqMqrqr[_s [U�\+1 V

2 3)4+564 783)4 ���:9<;>=>? @BA �DCFEHGJIKGJL:MNGOE)P QRE%S<;UTVGW@XS+?Y 564 $ 4:Z\[]4,^ _a`

b. c / 0ed f gih j bkmlacon dqpsr t ucav,w dc8x 0 y z�{0 d c}|F~ h�d f � � | j b��licon dqp r ti�f ��� Y [ g v ��� bk l con dqp r t ~ � f v�~��!����~o� �

���

Page 13: web.math.ku.dkweb.math.ku.dk/~hugger/courses/3na/linerrslides.pdf0 # 1(2/ 3 4 & 3 5 6 7 8 9: ; < 2 = 4 > ?@ 6 7 8 " a5 b5c= 4 < 8 d 4 e f g hi 0 j *lk-k- m 0 n 3 o p*& 6 q

������� ��� � � ������������� ��� ������ ���� �!�! " ����!�#�$��� % ��!�#'&(�)&*��!$+,.- &/���0! ����$�1�$2$&3 4�5&*�6!�+ 78 ��� 8 l:9<;

cav>= dQDGJS@? c bA ~ bB d j b� Cl cED C ~GF�n H d I � Cl cED C ~GF\n H d uzH0 c bA ~ bB ~ h�d f c8x 0 y z {0 d c 78 ~ h:d y J 0 c h�d�y z C0 c h:d

� f v ~�������~8� ~ ��h j bk Cl con d I k HLK ClM @ T!= u N O = =+CFEBGOIKGJL�M GJE%P ;>@XS@? G TVGW@XS cav 1 d Gs? ? 9>TQPGs?SR�=>? G A

T C 0VU W b. r t con d X bk Cl con d �cavZY dc b. r t con d ;>@XS�Ti9HGJS,?�T O = =)[ T!=HS+?UGW@XS,? L]\ ^>=BM @ @FE>TQP?UG_?]= / 0 @�A 9)I I%A E%S ;eTVGW@FS,? GJS b. c / 0qd d5`M @ T!= u h a v b c bA ~ bB d A�M @XP 9%I I c S@XS c GJM G ; O I =eT d=BI =HP =HS�T =>?4d]=>? ; @FS>TeM GJL)E>T!=fe G ` = `g G T O cav>Y d =BS AN@ M ; =>?he cavZ= d Gs? d Is@FL 9)I A @�M c bA ~ bB d5`

�:i

Page 14: web.math.ku.dkweb.math.ku.dk/~hugger/courses/3na/linerrslides.pdf0 # 1(2/ 3 4 & 3 5 6 7 8 9: ; < 2 = 4 > ?@ 6 7 8 " a5 b5c= 4 < 8 d 4 e f g hi 0 j *lk-k- m 0 n 3 o p*& 6 q

� I @XL 9)I��:M @XL ` 2 � 783�����7 � 9�M T ` � @ ;9%I�:M @FL%I =BP 2 � 783 � 9�M T `

-1-0.5

00.5

11.5

2

x

-1-0.5

00.5

11.5

2

y

0

0.2

0.4

0.6

0.8

1

phi �� �� f ���G A � � � � v ~ � � � � v

� =BI ? =

� E�TQP�� � � \%M 9%P G_?

���

Page 15: web.math.ku.dkweb.math.ku.dk/~hugger/courses/3na/linerrslides.pdf0 # 1(2/ 3 4 & 3 5 6 7 8 9: ; < 2 = 4 > ?@ 6 7 8 " a5 b5c= 4 < 8 d 4 e f g hi 0 j *lk-k- m 0 n 3 o p*& 6 q

� � � �� &�� �)&*��! ����1�$� ��������� ��� � �$� � ���*&(�.�)&�!�� � ��! � �5& ��!�+

� =qT b r t con d f ? � 9)S g � �� � l 9���� �� t ~ � �� O =BM = g � �� � l 9���� �� t Gs? 9 � 9:M TVG TVGW@FS @BA v GJS�T!@ � E>TQP� � � \%M 9%P G_? A E%S ;eTVGW@FS,? c IKGOS=<9�M � 9 d M 9%S@d)= L 9 P?UGs? A�E%S<;UTVGW@XS+? @�A � =)[ T!=HS@?)=>? L \ ^>=BM @ @XE�T�?UG_?)=� d e =)[+T!=BS ?]=<? T!@ � O @XI = �]\%M��%P G_? A�E%S��UTVGW@XS+? GOS� I � P � S�T�? � � O � M � v �j b c � d5`M @ T � u N O � � C EHGJIKGOL�M GJE%P �>@XS@? G TVGW@XS cav�� d G�� � TVGJI I����TVG��SR � ? G A

T C U W b r t con d X bk Cl con d �cav�� dg G T O cav�� d � S>A ��M!� � ?�e cav>= d A ��M �XS � � \%M��)P G_?A�E%S��UTVG"�XS G�� I#�$�%�%I�T&� � ' Y)(+*-, �/. 0 Y ^ � � � 1 [32 `4 E�T ? G � � M � S�T �5�>T�� O � � GJS-67�XI86 � T O � ���)P � bA �%S ?bB TeM��7� � �Ze O � S�� �� G T O cav%� d � S A ��M!� � ?�e cavZ= d A ��M/�FS � � \%M��%P G_? A�E%S��]PTVG"�XS G�� � TVGJI I d I��FL9�)I A ��M c bA ~ bB d5`

�;:

Page 16: web.math.ku.dkweb.math.ku.dk/~hugger/courses/3na/linerrslides.pdf0 # 1(2/ 3 4 & 3 5 6 7 8 9: ; < 2 = 4 > ?@ 6 7 8 " a5 b5c= 4 < 8 d 4 e f g hi 0 j *lk-k- m 0 n 3 o p*& 6 q

� &�� � � � � � GOS7� � ? G � � M � S>T �9��T�� O � � ? � � � S7� T GJS P67�XI86 � T O � ���%P � z H K � c bA ~ bB ~ h�d e� G T O cav%� d � S�� ��M!� � ?�e cavZ= d � ��M/�FS � � \%M��%P G_? ��E%S��]PTVG"�XS G�� � I#�$�%�%I P ��TeM G [ ��M �FL%I � P � ��M T O � 69�:MNG P�)L%I � � z H K � c bA ~ bB ~ h�d GJS � � ��T�� O ��� � �@? GJE�� � S �co�H~�� �-��M}M � � � �FS ?�OS@d T&� �OS�T � M6S9�� � ?4d � � �JS T O ��5�>T�� O d u

� b f 7 � �c�� ��d� ���4� �1� � � �B[���� � 2 ��� Y �\[ 2 � Y � Y�� 2 ( 2��. Y�� ���'` � . � ��� � ( � �\[ � ( � ���! #" 2%$ � � ( � � . Y [)^ �\[��'`� .

� 7& � c h�d f � ~c���v d

' , 2�� 2 7& � c h:d f c8x y z { d c 78 ~ h�d<y J c h�d y z C c h:d () 2 ( 78 _/2 *H[ 2��+� ��� 2 � ( � � * ( �,� *H[%^ � ^.- � � ( 78 ( �/8 f 78 0 1 %2 ' � (+, c8x y z { d c 2 ~ h�d f v (3 , 2 4 ' � (+, � � *5����2 � ( 2 � * � � ^��6*H[ [ ��� � 7 �98 ��: 7���<;=" 7 �98 2 � (+, 2 > *��6*5: * 2@? ' � ��A�� 2 �

� B C / � C D � E FG H D& HJI+KMLON�PI �Q� L

� 1 (+, 2@� :QR�� � S 7!T 8 2 S R (+, 2@� � 2 SVU � (�S " (WYX

Page 17: web.math.ku.dkweb.math.ku.dk/~hugger/courses/3na/linerrslides.pdf0 # 1(2/ 3 4 & 3 5 6 7 8 9: ; < 2 = 4 > ?@ 6 7 8 " a5 b5c= 4 < 8 d 4 e f g hi 0 j *lk-k- m 0 n 3 o p*& 6 q

� ���������� ) 2 ( . R,� � ���* � T ? � U :���� T R : S ' T ����<R T :�� R�� *���� *���� R : ��� R�� T : T :������+:�*5� �<R T :�� S R!�T :������+:5*��"��? 7 � S$# � % B C / � % A�� �Q� R�A��&� � ����� " ' T ���*5: * �+A T ����* �'� ��� U *�� T R : �(� � R 8 ��? () �)� � R,� �*� �Y* � T ? � U :���� T R : S ' T ��� �<R T :�� R�� *��,+� *-�.� T : �VR��+:"��� �<R T :�� S : R�� R : ��� # � %6B C / � %A��

I � / � 0 FG N L B C � / / � 0 FG N10"2I �"3 L

' �"���(� 0 ��* S ��R � � R :"��:�� S 46587:9<;)=> 7;@? 7KBA�C ' T ��� BD *

�VR�E � U �(F�A�G&� FH���JI R@K T E F�� T R : � R D"LNM�OP,Q<RS@T H I�U LVM 2(W LNM 2�KML C 4X5 7:9 ; >ZY\[�] 7;@? 7

K.A�C_^ O` �"�.a b % B c C d e % ��F S F U a Tgf U � S R�G U � T R�a # h � T ���i j R�Glk*� S PVm�m ^ Oi n S T a�?J�o�p�.a�?J�Ba�� R!� h ��F��q� f U F�� T R�a T S I(�.E R�k ��?R)I E T a T E Tsr � S ���"� � U ��G T ? T F�a aQR)I�E R�� B c E 0 R!k*�BIF-GgG S R�G U � T R�a S � R PVm�m ^ I(� S �p�B��� T k �.Gt� O

WVu

Page 18: web.math.ku.dkweb.math.ku.dk/~hugger/courses/3na/linerrslides.pdf0 # 1(2/ 3 4 & 3 5 6 7 8 9: ; < 2 = 4 > ?@ 6 7 8 " a5 b5c= 4 < 8 d 4 e f g hi 0 j *lk-k- m 0 n 3 o p*& 6 q

� ������� ����� ��J� �� �� BU � �� BW

� ����� Q<RS@T H I BU 2 BW 2 KML � ��������������� ��� �"!��#���%$ !'& I BU 2 BW L�)( * +,��!.-0/1��2 /1!3��*0/ �4! !.�#� ��/1�52 �5�6� � 7 � �8�9*0��7�:� �;*<( * =8�5�?>.=�� (@!�/A=6�B�?!.� � �#��� ���#� �0=82 -5��� !�&Q RS@T H I BU 2 BW 2 KML�C ( * � ���D� � �:��� ����� 7E�F2 ��� (G�H!.� !�&I���#����<�"!��#�5�%$ (J+;*��"�0K

LNM �PO � � � O@�RQSO � M E�TOJU �V �Q ���p� Q RS@T H I BU 2 BW 2+K�L

W � ! �5/1��2 ���"� +;*6�X�����"(DYZ = � 2 ���"�B�F� � B[ \L I � L � �:��� &]=0���G�B�H!��<( � ����� (^=0+J_+5!`�a� �F� ���#� �F�"�4���P�H!�� !�& b ��/1�52 ���"�c(D� -,*0/?*0�#�d_�F� � ���6(^=0/e�c( �F� !.�#� �<>.=;*6�B�H!�� &P!`� !.�#� =0�8f;�#!<� �Q<RS@T H I BU 2 BW 2 KML �g�hKi�0Kj* � !k!37 (d����=,*"�B�H!���Kl !����,�F� � K C m �F� !��#� +5!k�F�"� !'& �#*��'� �5/1��2 ���"��F2 +0/n�H�"( o \ Sqp r Bs tu 7 I � L C vdm`w K W �5�)( /)�#*D�#�6( !��0/x$����� yz(G�F2 +8/)�6{ +,��!.-0/1��2 ( � �:��� F � >3=;*6�B�?!.� �F� F=8�0f;��! � ��K

| � � �#��� } �~7��5� ��>.=���/n�F-T��*"�B�H!���YW �#� /n�F�#�#*,�N+B$0��*02 �?7 &�=8���G�B�?!.� ( ��*0� -5� ���5+8/?*��"� 7-B$ �5� � ���5� !`�~7��5� +d$0��*02 �?75(<K

W��

Page 19: web.math.ku.dkweb.math.ku.dk/~hugger/courses/3na/linerrslides.pdf0 # 1(2/ 3 4 & 3 5 6 7 8 9: ; < 2 = 4 > ?@ 6 7 8 " a5 b5c= 4 < 8 d 4 e f g hi 0 j *lk-k- m 0 n 3 o p*& 6 q

� � �p� �� �

l �F�#7 D LVM � ��� I � L�� � I��� L YI b� L� D LNM C � �F� � C �Am 2 F�� � � � � 2DJLVM � m !�� ��� 2

� ����� � (@��/)�<�G�4� 7 (^=#��� ���;*6�

D LVM I��!2��,L � I�� r���� � � L I�� � � � � ��L &P!`� I��!2��,L � � Pi � *0�#7�*T�~7 �'*T�P�n*"�B�H!��,*0/5&P!`� 2 =8/?*"�B�H!���K

i !#" =0���1&�!�� 2 -��x_ >.=,*�7.� *"�B�?� �5/1��2 ���"�c( K

i W �"( � &�=8���G�B�?!.� ( ��� $[ L I � L �<>.=,*0/ %6�5��! �F� !.�#��"!�� �#��� !�& �#*��'� ��/1�52 �5�6� Ki W ��� *8/R&�=0�#���B�H!.� ( �F� $& L I � L &�!����D�<7 �"!��"�B�F�8=#!�=<(�F� ���#� (S*02 � +5!k�F�"�c( K' !�2 +0=6� *"�B�H!��<( ( � *,� �B��� � *"� ����� -5!�=8��7`*,�%$ *0��72 !��5�F� � �F�"� *,�~7�(<K

i $W ( m (G�F�#�D� ) \ *� m �^�#���%$D� �#�����0K

i $U � +-, I/. S@T H L �10 � m 2 ! 2 b 2 3 � $2 � 3 465 � S87 K9;: