Amn Halloc Mah Ecss / 7 E-mal : amn@shhs bpag : shhs/amn MATH EXERCISES GRADIENT DIVERGENCE CURL DEL NABLA OERATOR LALACIAN OERATOR CONTINUITY AND NAVIER-STOKES EQUATIONS VECTOR RODUCTS I and hn scala o do podc co o s podc In som boos s also consdd o podc dnd b GRADIENT DIVERGENCE CURL DEL NABLA OERATOR LALACIAN OERATOR GRADIENT L b a scala ld Th gadn s h co ld dnd b gad DIVERGENCE L R Q b a co ld connosl dnabl h spc o and Thn h dgnc o s h scala ld dnd b R Q d CURL Th cl o s h co ld dnd b Q R Q R R Q cl o Q R Q R cl DEL NABLA OERATOR Th co dnal opao s calld dl o nabla
Amn Halloc Mah Ecss Ug can dno gad d and cl as blo: gad d cl No ha s no h sam as Q R LALACIAN OERATOR Th Laplacan opao Δ s dnd o a scala ld U b U U U Δ U U and o a co ld Q R b Δ Δ ΔQ ΔR Som omlas o pola and clndcal coodnas ola coodnas dm ansomaon: aa lmn: da d d sandad bass: [Rma : No ha a dpnd on hn mo om pon o pon hs s h ason h hs bass s calld local bass n som boos ] I n Casan cood and ϑ h sam co n clndcal coodnas hn ϑ [Rma : V can d hs omlas b calclang h componns o n h dcons o and Ths / 7
Amn Halloc Mah Ecss / 7 smlal ϑ ϑ ] Clndcal coodnas : ansomaon: olm lmn: d d d dv ϑ ϑ sandad bass: I n Casan cood and ϑ h sam co n clndcal coodnas hn ha ollong co componns laonshp: ϑ [Rma : o ampl can g n h ollong a: ] scala ld: gadn: gad laplacan: Δ co ld: dgnc: d o cl: cl
Amn Halloc Mah Ecss EXERCISES nd a d b gad d and c cl nd gad d cl Whch on o h ollong ncons a b ln c p sass h Laplac qaon Δ? nd Δ 5 W h gnal anspo qaon U S φ ho g opaos d Δ cl o gad H U ncons S a al ncons o and 6 Whch on an o h ollong ncons a b c sass h qaon U S? H 5 U and S 7 nd hch on an o h ollong ncons a b 5 c 5 sass h qaon d U d gad S h U and S 8 5 / 7
Amn Halloc Mah Ecss 8 am 8 A W h gnal anspo qaon U S q ho g opaos d Δ cl o gad H U ncons S a al ncons o and B L U nd S n h qaon q no ha h ncon sass h qaon 9 Q6 am 8 Consd h ollong qaon U U 6 q L consan U nd h consan n h qaon q no ha h ncon sass h qaon I possbl nd o h gn paal das and a and b and c and d and 5 Hn: Ncssa condon: I has conns das hn h md das o shold b qal Ths * s h ncssa condon o h snc o a ncon ha has h gn das Dmn h al o a o hch h ssm o paal dnal qaons a and has solons Thn nd pondng o hs al o a 5 / 7
Amn Halloc Mah Ecss I possbl nd o h gn paal das and a and b and c and d and Hn: Ncssa condon: I has connos das hn h md das o shold b qal Ths Con : Con : Con : a h ncssa condon o h snc o a ncon ha has h gn das Dmn h als o a and b o hch h ssm o paal dnal qaons a and b has solons Thn nd pondng o hs als o a and b W consd an ncompssbl dns cons sad sa aabls do no dpnd on m sohmal Nonan lo h a gn loc ld V Us h ollong qaons conn and Na Sos qaons o nd n psson o pss as a ncon o and h consan μ consan g g g g and g h g 98m / s g Incompssbl conn qaon: q Na Sos qaons: componn: 6 / 7
Amn Halloc Mah Ecss g μ q componn: g μ q componn: g μ q a V b V c V 5 am 9 A Consd h ollong qaon U U 6 6 8 L consan U nd h consan n h qaon q no ha h ncon sass h qaon q B W consd an ncompssbl dns cons sad sa aabls do no dpnd on m sohmal Nonan lo h a gn loc ld V Us h ollong qaons conn and Na Sos qaons o nd n psson o pss as a ncon o and h consan μ consan g g g g and g g h g 98m / s and V 6 6 am 9 W consd an ncompssbl dns cons sad sa aabls do no dpnd on m sohmal Nonan lo h a gn loc ld V Us h ollong qaons conn and Na Sos qaons o nd s paam a and hn n psson o pss as a ncon o and h consan μ consan g g g g and g g h g 98m / s and V 5 a 7 / 7
Amn Halloc Mah Ecss 8 / 7 7 Consd sad ncompssbl sohmal lamna saona Nonan lo n a long ond pp n h -dcon h consan ccla s-scon o ads R m Us h conn and h Na-Sos qaons n clndcal coodnas o nd h loc ld V and h pss ld h ld lo sass h ollong condons: c All paal das h spc o m a Sad lo c μ g/m s and g/m c A Consan pss gadn / /5 a/m s appld n h hoonal as -as n o noaon: / /5 c Th lo s paalll o h as ha s and c W assm ha h lo s asmmc Th loc dos no dpnd on ha s c5 Bonda cond No-slp bonda condon V ld V all : I hn c6 Bonda condon : has mamm a ha s --------------------------------------------------------------------------------------------- Th conn and h Na-Sos qaons o an ncompssbl sohmal Nonan lo dns cons s μ cons h a loc ld V n Clndcal coodnas : Incompssbl conn qaon q a Na-Sos qaons n Clndcal coodnas: -componn: g μ q b -componn: g μ q c
Amn Halloc Mah Ecss -componn: g μ q d 8 Eam Mach qson A pons W consd an ncompssbl dns cons sad sa aabls do no dpnd on m sohmal Nonan lo h a gn loc ld V c b a Us h ollong qaons conn and Na Sos qaons h consan μ consan g g g g and g g g 98m / s o nd: paams a b and c n psson o pss as a ncon o and Th GRADIENT VECTOR h chang o aabls and bass Th gadn co o h ncon s dnd as gad * I chang aabls o and plac bass cos h n lnal ndpndn cos hn can pss h sam gadn co gad n ms o aabls and cos W smpl calcla h das and n n aabls and pss as a lna combnaons o Thn sbs hos als no * S h ollong ampl 7 W consd a scala ld gn n clndcal coodnas h and bass cos a nd h psson o h gadn gad n clndcal coodnas ha s n ms o and a p h sam bass 9 / 7
Amn Halloc Mah Ecss b c am ; Q5 B pons d hs s on sd as a local bass o clndcal coodnas ANSWERS AND SOLUTIONS: Solon: Q R a Snc d ha d Ans a d b Snc gad ha o d gad d Ans b gad d d c cl Q R Ans c cl Solon: cl Ths d cl and ho gad d cl Ans: gad d o / 7
Amn Halloc Mah Ecss Δ Ans: Th ncon ln sass h Laplac qaon Ans: Δ d cl gad Δ 6 5 Solon: d U d gad Sφ d d S 6 Whch on an o h ollong ncons a b c sass h qaon U S? H 5 U and S φ S φ Solon : Th qaon U S can b n as d U d gad S 5 5 5 d d 5 5 5 q a L V calcla h das o and sbs n h l hand sd LHS and gh hand sd o h qaon q / 7
Amn Halloc Mah Ecss LHS: 5 5 5 RHS 6 Whnc LHS RHS Ths h ncon s no a solon o h qaon b LHS RHS Whnc LHS RHS and h ncon s no a solon o h qaon c L Thn LHS 6 RHS 7 Ths LHS RHS and h ncon s no a solon o h qaon Ans: Non o h ncons sass h qaon 7 Ans: ncon 5 sass h qaon 8 am 98 A W h gnal anspo qaon U S q ho g opaos d Δ cl o gad H U ncons S a al ncons o and B L U nd S n h qaon q no ha h ncon sass h qaon Solon: A U S d U d gad S d d S / 7
Amn Halloc Mah Ecss / 7 S q B W sbs U and n h qaon q and g S φ 8 S 8 6 8 Consqnl 8 8 S 9 Q6 am 8 Consd h ollong qaon 6 U U q L consan U nd h consan n h qaon q no ha h ncon sass h qaon Solon: 6 U U 6 U cl d gad d U d c U cl ha U cl d 6 d d 6 q W sbs U and n h qaon q and g 6 φ No ha s a consan 8 6 6 Ans: I possbl nd o h gn paal das and
Amn Halloc Mah Ecss a and b and c and d and 5 Hn: Ncssa condon: I has connos das hn h md das o shold b qal * s h ncssa condon o h snc o a ncon ha has h gn das Ans: a C b C c C d No solon c h condon * s no llld 5 Solon a Snc and h das a connos h condon * s llld and can nd o h gn das In od o nd nga h spc o h s o h qaons q q and g d C Ths C W ha ngad h spc o ho h consan sll dpnd on No o nd C dna and sbs n q and g: C C C C C nall sbsng C C n ha C h C s a consan / 7
Amn Halloc Mah Ecss Ans: om a a Thn o a ha C Ans: a C b C c C d No solon c h condon Con s no llld Solon a a and Snc h condons Con a llld and can nd o h gn das In od o nd nga h spc o h s o h qaons q q q and g d C Ths C W ha ngad h spc o ho h consan sll dpnd on and No o nd C dna and sbs n q and g: C C C C C W ha ngad h spc o ho h consan sll dpnd on and Ths C No sbsng n q ha 5 / 7
Amn Halloc Mah Ecss C C C C C nall sbsng C C n ha C h C s a consan Ans: om a a b b b b Ths all h condons a llld a and b o hs als o a and b g C Calclaon o h pss ld o a non loc ld o an ncompssbl sad sa sohmal Nonan lo Ans: a g 8 8 C 7 7 b g C c g C Solon a W sbs n q and g no ha al das h spc o a : Conn qaon: q dncall llld Na Sos qaons: componn: 6 / 7
Amn Halloc Mah Ecss 6 q componn: 6 q componn: g q No q gs 8 C * Sbson n q mpls C 6 C 8 C Hnc om * ha 8 8 C ** No sbs ** n q and g g C g C g C h C s a consan nall sbsng C g C n ** ha 8 8 g C h C s a consan 5 Solon A: U U 6 6 8 d U d gad d cl U 6 6 8 c cl U ha d cl U d d 6 6 8 6 6 8 q W sbs U and n h qaon q and g 8 8 φ No ha s a consan 6 6 6 8 6 6 6 8 6 5 6 6 8 7 / 7
Amn Halloc Mah Ecss Ans A: 5 Solon B: W sbs 6 n q and g no ha al das h spc o a : Conn qaon: q dncall llld Na Sos qaons: componn: 6 q componn: 8 q componn: g q No q gs 8 C * Sbson n q mpls C 8 C 8 C Hnc om * ha 8 8 C ** No sbs ** n q and g g C g C g C h C s a consan nall sbsng C g C n ** ha 8 8 g C Ans B: 8 8 g C h C s a consan 6 Solon V 5 a s sbs 5 a n q and g no ha al das h spc o a : Conn qaon: a a No ha V 5 Ug h Na Sos qaons g: 8 / 7
Amn Halloc Mah Ecss componn: 9 6 q componn: 5 q componn: g q 9 No q gs 6 C * Sbson n q mpls C 5 C 5 C Hnc om * ha 9 6 5 C ** W sbs ** n q and g g C g C g C h C s a consan nall sbsng C n ** ha 9 6 5 g C Ans : 9 6 5 g C h C s a consan Q7 Consd sad ncompssbl sohmal lamna saona Nonan lo n a long ond pp n h -dcon h consan ccla s-scon o ads R m Us h conn and h Na-Sos qaons n clndcal coodnas o nd h loc ld V and h pss ld h ld lo sass h ollong condons: c All paal das h spc o m a Sad lo c μ g/m s and g/m c A Consan pss gadn / /5 a/m s appld n h hoonal as -as n o noaon: / /5 c Th lo s paalll o h as ha s and 9 / 7
Amn Halloc Mah Ecss / 7 c W assm ha h lo s asmmc Th loc dos no dpnd on ha s c5 Bonda cond No-slp bonda condon V ld V all : I hn c6 Bonda condon : has mamm a ha s Th conn and h Na-Sos qaons o an ncompssbl sohmal Nonan lo dns cons s μ cons h a loc ld V n Clndcal coodnas : ---------------------------------------------------------------- SOLUTION Incompssbl conn qaon q a Na-Sos qaons n Clndcal coodnas: -componn: g μ q b -componn: g μ q c -componn: g μ q d
Amn Halloc Mah Ecss W choos as a cal as an a n a hoonal plan and h lo s paalll h h -as W dno loc co V h and a -componn - componn and -componn n clndcal coodnas Accodng o h assmpons ha and dos no dpnd on Snc s h cal as ha ha co g-g h g98 m/s hch n clndcal coodnas gs g g g g and g No sbs / /5 a/m μg /ms n h conn and Na- Sos qaons: Snc and accodng o c conn qaon n clndcal coodnas gs Ths lls s ha s no a ncon o hmo c loc dos no dpnd on assmpon c concld ha dpnds onl on To smpl noaon dno * No sbs g g g g and g / 7
Amn Halloc Mah Ecss / /5 a/m μg /ms n h Na-Sos qaons: Th -componn o h Na-Sos qaon gs: g q -c Th -componn o h Na-Sos qaon: g q -c Th Z-componn o h Na-Sos qaon h and q -c 5 5 gs: Sp W nd h pss In od o nd h pss sol q -c q -c and h qaon s 5 ha g g 5 om hs qaons g g C 5 Sp W nd h loc componn W sol q -c h bondas c5 and c6: q -c 5 c5 / 7
Amn Halloc Mah Ecss c6 d Rma: Tchncall can nsad d aabl om q -c ha 5 c s no a ncon o onl on C sbson and c6 C C sbson and c5 C Ths and V Ans : g C 5 V 8 W consd an ncompssbl dns cons sad sa aabls do no dpnd on m sohmal Nonan lo h a gn loc ld V c b a Us h ollong qaons conn and Na Sos qaons h consan μ consan g g g g and g g g 98m / s o nd: paams a b and c n psson o pss as a ncon o and / 7
Amn Halloc Mah Ecss / 7 Incompssbl conn qaon: q Na Sos qaons: componn: g μ q componn: g μ q componn: g μ q ------------------------------------------------------------- W sbs a b c n q and g no ha al das h spc o a : Conn qaon: a a q Ths b c Na Sos qaons: componn: q componn: q componn: q Th ssm q q q s solabl onl md das a qal: : c c Con
Amn Halloc Mah Ecss 5 / 7 : b bc Con : b b Con Ths c and b W sol smpld qaons and g C g 6 8 5 Ans C g 6 8 5 9 W consd a scala ld gn n clndcal coodnas h and bass cos a nd h psson o h gadn gad n clndcal coodnas ha s n ms o and a p h sam bass b c am ; Q5 B pons d hs s on sd as a local bass o clndcal coodnas Solon: In aabls ha gad q o clndcal coodnas ha
Amn Halloc Mah Ecss 6 / 7 s h das and n coodnas h aabl s n boh cood ssms Solng h ollong ssm o and g ** W sbs h das ** n q and g *** gad To sol poblms a b c and d ms pss as a lna combnaons o and sbs hm no *** a om *** c ha mmdal gad b om nd q b Thn p om q b no *** and g gad and a collcng componns o gad c om ha
Amn Halloc Mah Ecss 7 / 7 q c ng om q c no *** gs gad d W can sol d n h sam mann as n abc b hs m can s collc ms and g h sl: gad Ans: a gad b gad c gad d gad