Overview. Review Elliptic and Parabolic. Review General and Hyperbolic. Review Multidimensional II. Review Multidimensional

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1 Mlil idd variabls March 9 Mlidisioal Parial Dirial Eaios arr aro Mchaical Egirig 5B iar i Egirig Aalsis March 9 Ovrviw Rviw las class haracrisics ad classiicaio o arial dirial aios Probls i or ha wo idd variabls olio b saraio o variabls Probls wih lil ohoogos bodar codiios olios or rcaglar gor owork robl or clidrical gor Rviw Gral ad rbolic Doai o ddc or Th ara i - sac whos vals ac h val o Rgio o ilc o Th ara i - sac whos vals ar acd b h val o Aras or hrbolic aios show blow -rig characrisic d/d < Iiial odiio rv Doai o ddc Righ-rig characrisic d/d > 3 Rviw Elliic ad Parabolic Iagiar characrisics or lliic aios lik alac ad Poisso s Eir solio rgio is boh doai o ddc ad rgio o ilc Parabolic aios icall ivolv i ad sac as coordias Doai o ddc a is ir doai ad < Rgio o ilc a is ir rgio ad > Rviw Mlidisioal a hav aios i hr sac disios ad i lassiicaio as lliic arabolic or hrbolic dos o al o aios wih or ha wo disios oordias ca hav lliic-lik arabolic-lik ad hrbolic-lik bhavior i lidisioal aios E. g. i is a arabolic coordia arsia lidrical Rviw Mlidisioal II alac Disio av c hr r r r z r r r r r θ z r r si coφ φ θ r φ r φ 5 6 ME 5B Egirig Aalsis

2 Mlil idd variabls March 9 ME 5B Egirig Aalsis 7 Rviw D Disio Two-disioal disio aio or Y Y X X T T s saraio o variabl aroach wih all variabls XYT 8 Rviw D Disio II Bodar codiios giv gral solio as s o all igcios si si si si Eigcio asio or dd si si 9 ook a a osa dd si si h w sbsi w ca sara h ad igraios Gral Rsl d d dd si si si si ook a cosa II Boh igrals ar civl h sa or z ad or z [ ] v odd z d z cos cos si 6 si si d d zro or v or Rsl or a osa Rlac b ad b o g odd idics ol ad di w arars ad as ollows old old si si si si 6 Disiolss Parars Modi oial arg as ollows hr bsi io ad divid b si si 6

3 Mlil idd variabls March 9 ME 5B Egirig Aalsis 3 3 Iora Parars si si 6 Rsl shows ha / is a cio o / / / ad / a sili dobl saio i his cas b sliig oial r Prodc olio a ow sara ad ss si si si si 6 si si 6 Prodc o odisioal solios 5 ozro Bodaris r-iovill igcio asios rir zro bodar codiios For ozro bodaris sli solio as i D cas v w v saisis disio aio wih zro bodar codiios w saisis alac s ad disio aio wih ozro bodar codiios saisis disio aio wih w a bodaris 6 ozro Bodaris II olv alac aio or w wih srosiio i rird olio or v is sa as rvios solio or wih zro bodaris Iiial codiio or od ro v w b sig v si si 7 ozro Bodaris III s igcio asio o dri o sais iiial codiio Ms icld w i coig siilar o cas wih o sac disio si si w si si w 8 ozro Bodaris IV a igcio asio sd or zro bodar robl givs ils al is cas whr w hav osa iiial oial Bodar oial o zro o hr sids osa oial sid o orh sid Kow alac aio solio or w dd w si si

4 Mlil idd variabls March 9 Probl ad olio ar Disio aio or olio v w whr v js od si si w Kow w ro rvios solios o sig h ad olio alac aio solio or w w w ad w si sih w sih Ms s his solio i aio w si si dd alac s aio 9 Gig h Aswr av o ror igraio or o diicl b a lo o algbra ad oial or rror Firs ar o igral or a cosa availabl ro rvios al cod ar wih D sad-sa solio rirs os work MATAB dos o giv good rsl or igral o siasihb Igraio dails a d o rsaio Gig h Aswr II alac aio solio or w si sih w sih li aio or io wo ars 6 si si dd w si si dd zro lss boh ad odd Rsl or Igraio shows or v odd odd v a v odd 3 G Disiolss olio Divid solio b ad acor o / r i oial arg 35K si si w w si si 35K w si sih sih ME 5B Egirig Aalsis

5 Mlil idd variabls March 9 Eaios or w ad Divid b ad acor o / doiaor o scod racio odd v olio / dds o / / / / ad / 5 Rsls or / olio / dds o / / / / ad / Viw solio as / coors as a cio o / ad / coordias lc / ad / Bodar b a whr / Obai o lo or a giv val o / ial los show volio / ro iiial codiios o sad sa 6 Thr-Disioal alac s cobiaio o saraio o variabls ad srosiio ar wih basic solio ha has hoogos bodar codiios a iv sracs ad z z z z z z 7 8 Thr-Disioal alac II olio o alac aio i z siilar o disio solio i av hrbolic cosi i z dircio isad o oial i dca z sih z z sih si si si si dd 9 owork or March 6 Krszig ag 586 robls 7 Two disioal wav aio i a clidr r R θ c r r r r r θ Fid rθ b saraio o variabls Priodic bodar codiio i θ dircio: r r 3 ME 5B Egirig Aalsis 5

6 Mlil idd variabls March 9 Ridr ad Addiioal Marial Midr dsda March ovrs arial o howork assigs sbid o ad icldig las Moda March ill b o os ad book icldig igral abls b o cors hars 3 o 3 show sar o howork robl or Moda hars 35 o 3 show dails o idig h coicis o char 3 or disio i wo sac disios 3 ork o owork Probl Pag 57 robl solv h wav aio or ad wih bodar ad iiial codiios show c c.. ar wih saraio o variabls rsl T X [ Asi λc B cos λc ][ si λ D cos λ ] 3 ork o owork Probl av alrad s ha bodar codiios ha giv ollowig solio c c A si B cos si oicis dd o iiial dislac ad vloci g A g si d B c si d For all B 33 ork o owork Probl olio or B A si λc si λ s cio giv or g o g A A c A c g si d. si d c ad c. si d 3 D Disio Dails olio is giv b si si w is od ro iiial codiio ad sad-sa solio w w si si dd Al o a cosa hr D Disio Dails II alac aio solio or w si sih w sih li aio or io wo ars 6 si si dd zro lss w si si dd boh ad odd sids a zro ad o sid a ME 5B Egirig Aalsis 6

7 Mlil idd variabls March 9 6 G cod Par o Igral w si si dd si sih si si dd sih si si d sih sih si d 37 Dails Al igral abl orla ro o bsih a cosb a cosh asi b sih asi bd a b bsih a cosb a cosh a si b sih a si bd a b bsih a cosb b sih cos a cosh a si b a cosh si a b sih a si bd bsih a cosb a cosh a si b sih a si bd a b 38 sih si d Dails II bsi a / ad b / - sih cos cosh si sih a si bd sih si d sih 39 Rsls or Two Igrals si si d sih si d ohrwis sih Two rsls ar lilid oghr givig a zro lss Prvios rsl: a odd igr Rviw ar o or w si si dd 6 si si dd w si sih sih odd ad ol obi Prvios Rsls 6 w dd si si zro lss boh ad odd 6 6 sih si si d sih si d sih ohrwis ME 5B Egirig Aalsis 7

8 Mlil idd variabls March 9 ME 5B Egirig Aalsis 8 3 Rsl or zro or v sih sih 6 6 v odd