Mathematics: Lecture 1 Differential Equations:

Transcription

1 Mthemtics: Lectue Dieetil Equtios: Dieetil Equtios A dieetil equtio is equtio tht ivolves oe o moe deivtives o dieetils. Dieetil equtios e clssiied b:. Te: Odi o til.. Ode: The ode o dieetil equtio is the highest ode deivtive tht occus i the equtio.. Degee: The eoet o the highest owe o the highest ode deivtive. A dieetil equtio is odi D.Eqs. i the ukow uctio deeds o ol oe ideedet vible. I the ukow uctio deeds o two o moe ideedet vible the D.Eqs. is til D.Eqs.. is til D.Eqs.. E: E: E: d 5 d d Ł d ł Ł d d ł st ode-st degee d d si 5 d d 5 d ode-d degee d ode-st degee Eecise: Fid the ode d degee o these dieetil equtios.. d cos d s:st ode-st degee. d d s:st ode-st degee. d d. 5. Solutio The solutio o the dieetil equtio i the ukow uctio d the ideedet vible is uctio tht stisies the dieetil equtio. E: Show tht c si c cos is solutio o '' sol: c si c cos ' c cos. - c si. ''- c si - c cos - c si - c cos c si c cos \ is solutio --

2 Mthemtics: Lectue Dieetil Equtios: Note: The solutio i emle bove is clled geel solutio sice it's coti bit costt c d c i.e. the geel solutio o dieetil equtio is the set o ll solutios d the ticul solutio is oe o these solutios. Eecise:. Show tht e -e - is solutio to ''-. Detemie whethe e - e - is solutio o '''. Detemie whethe - is solutio o ' - Odi Dieetil Equtios: Odi Dieetil Equtios e equtios ivolve deivtives. A. Fist Ode D.Eqs. - Vible Seble. - Homogeeous. - Lie. - Ect. - Vible Seble: A ist ode D.Eq. c be solved b itegtio i it is ossible to collect ll tems with d d ll tems with d tht is i it is ossible to wite the D.Eq. i the om the the geel solutio is: d g d d g d c whee c is bit costt. E.: Sol.: Solve d e d d e e d d e d e - e d e d - e - -d e d - e - e c E.: d Solve d Sol.: d d --

3 Mthemtics: Lectue Dieetil Equtios: t t - - d - d - l c d E.: Solve - L d d du d du Sol.: ut - u -.. d d d d du du u d d u - Ø / -/ ø \ Œ du d º u - u œ ß [ l u - - l u ] c u - l c u u - c e u Eecise: Sete the vibles d solve.. -d d s: -c. de - d s: e e c d. si cosh d s: sih -cosc. e d d s: e - l c d 5. s: c d - Homogeeous: Some times D.Eq. which vibles c't be seted c be tsomed b chge o vibles ito equtio which vibles c be seted. This is the cse with equtio tht c be ut ito om: d d Such equtio is clled homogeous. Put u u du u d d du u d becomes d d u --

4 Mthemtics: Lectue Dieetil Equtios: E.: Sol.: Solve d d d d du u u d u homo. Put du d u u - u u d d du u d du d u du u d u l c l c E.: Solve the homogeous D.Eq d d Sol.: d d du d d ut u u d d d du u u l - l u c c c d u Eecise: Show tht the ollowig dieetil equtios e homogeous d solve.. d d s: c. d -d s: l - c. d - d e s: l e - c - Lie d The equtio o the om Q whee P d Q e uctios o ol o d d costt is clled lie i d. d Fid itegtig cto I.. e the the geel solutio is Pd I.. I.. Q. d --

5 Mthemtics: Lectue Dieetil Equtios: E.: Solve d d - P - Q. e - d -l e I.. e e Solutio is e d e c E.: Solve P Q d I.. e e d. d Solutio is e e d e e c - ce is the solutio Eecise:. e d s: e - ce -. d si s: c-cos d. c d d d s: - Ect The equtio M d N d is sid to be ect i Geel Solutio is c Md tems i N do ot cotis d M N E.: Show tht the ollowig D.Eq. e ect D.Eq. d d -5-

6 Mthemtics: Lectue كلام راهزا دعاسم سردم Dieetil Equtios: -6- N M N M \ The D.Eq. is ect. b cos ] si cos [ d d cos si cos si N M - - \ the D.Eq. is ect. E.: Is the D.Eq. d d - ect o ot? Sol. d d - thed.eq.is ect \ N M N M Q E.: Solve the ect D.Eqs. i E. bove d d Sol. c d c the solutio is d E.: Solve d d Sol. thed.eq.is ect \ N M

7 Mthemtics: Lectue Dieetil Equtios: c Md tems i N do ot cotis d the solutio is d c d Eecise:. e de -d s: ce -. ttdsec sec td s: c t-lcossec. d -d s: - /c Poblems: Solve the ollowig dieetil equtios: - l d d - e d - e - d - d - d - d cos d 5- l - l d l - l d 6- d - - d 7- cos d si - cos d 8- d d 9- l e d d - e d e d Reeeces: - Clculus & Altic Geomet Thoms. - Clculus Hwd Ato. ا. رياض احمد عزت - Advced Mthemtics o Egieeig Studies - Mode Itoductio Dieetil Equtios Schum's Outlie Seies. -7-

8 Mthemtics: Lectue Dieetil Equtios: -8-

9 Mthemtics: Lectue Dieetil Equtios: B. Secod Ode Dieetil Equtios: The secod ode lie dieetil equtios with costt coeiciet hs the gel om is: b c F whee b d c e costts. I F the is clled homogeous. I F the is clled o homogeous. E: ''- 'si is lie d ode homo. ''-' si is o lie d ode o homo. '''l Homogeeous. b Nohomogeeous. - Udetemit coeiciets. - Vitio o metes. The Secod ode lie homogeous D.Eq. with costt coeiciet: The geel om is b c whee b d c e costts. The geel solutio Put 'D d ''D i eq. D is oeto D bdc D bd c usig D-oeto ow substitute D b d leve the b c is clled chcteistic equtio o the dieetil equtio d the solutio o this equtio the oots give the solutio o the dieetil equtio whee -9-

10 Mthemtics: Lectue Dieetil Equtios: b m b c Thee e two vlues o : - el equl d ot equl. - comle. Cse : I b c the d e distict d el oots d the geel solutio is c e ce Cse : I b c the d the geel solutio is: c c e Cse : I b c the the oots e two comle cojugte oots α ± iβ i d the geel solutio is: e α c cos β c si β E.: Solve Solutio: the geel solutio is c e c e E.: Solve 6 9 Solutio: c c e --

11 Mthemtics: Lectue Dieetil Equtios: E.: Solve Solutio: b c b ±... -± - -± i - ± i α β e c cos c si Eecise: solve. ''-'5 s:c e / c e 5/. ''-'-5 s:c e 5 c e -/. '' s:c cos/ c si/. ''-8'6 s:c e c e 5. ''9 s:c cos c si --

12 Mthemtics: Lectue Dieetil Equtios: b The Secod ode lie o homogeous D.Eq. with costt coeiciet: The geel om is: b c F whee b d c e costts. The geel solutio I h is the solutio o the homo. D.Eq. b c the the geel solutio o eq. is: h h comlemet uctio oticul itegl i is homo. ii use the tble h Methods o idig : Thee e two methods: Udetemied coeiciets: I this method om o F i eq. s ollows: deeds o the oots d o chcteistic equtio d o the F Choice o M.R. k k k k L k th degee olomil k si k cos ke β β o ce β c cos c si β m iβ Note: Fo eeted tem oot multil b. E.: Use the tble to wite F k k k k --

13 Mthemtics: Lectue Dieetil Equtios: - - F e k c ce F cos k β c cos c si F e k c k k k ce 5 F cos si c cos c si 6 F si cos c cos c si A cos B si E.: Solve. Solutio: '' '- the ch. Eq. -- h c e c e is olomil o secod degee the k k k k k Substitutio gives... k k k k k k k coe. o : k k coe. o : k k k cost : k k k k - - g h c e c e --

14 Mthemtics: Lectue Dieetil Equtios: E.: e Solutio: e... h c e c e Put ce ce 9ce... Substitute I 9ce -ce -ce e 9c-c-c c c I e g h ce ce e... قاعدة التعديل Modiictio ule F اذا كان - اذا كان - b اذا كان وكان احد جذري المعادلة القياسية يضرب F وكان احد جذري المعادلة القياسية يضرب F وكان جذري المعادلة القياسية يضرب السابق في السابق في السابق في ke ke k. m iβ α k cos β F k si β اذا كان وكان يضرب السابق في E.:Solve '' si Solutio: '' - ± i α β h c cosc si c cosc si ' -c sic cosc cosc si --

15 Mthemtics: Lectue Dieetil Equtios: '' -c cos-c si-c sic cos-c sic cos Substitutio gives -c sic cossi -c c -/ c c g c cos c si cos Eecise: Fid the geel solutio ''9 - ''-'-si ''-5e - '''e - 5 ''-'

16 Mthemtics: Lectue Dieetil Equtios: - Vitio o metes. Let h c u c u be the homogeous solutio o b c F d the ticul solutio hs the om uv u v whee v d v e ukow uctios o which must be detemied ist solve the ollowig lie equtios o v' d v' : v' u v' u v' u' v' u' F which c be solved with esect to v' d v' b Gmm ule s ollows D d u u v u u D D D v F D D u u D u u F b itegtio o v' d v' with esect to we c id v d v. E.: Solve e. u h u e e c e v u v u c e u e u e v u v u v u v u F hece v e v e v e v e e Solvig this sstem b Cme ule gives e e e 5 e D e D e D e e e e e e e e v' e e v' e 5 e e v v e e e e -6-

17 Mthemtics: Lectue Dieetil Equtios: e e the geel solutio e e is : c e e c e e E.: solve ''sec Solutio: '' - ± i α β h c cos c si u cos u si sec v u v u v cos v si the v cos v si v si v cos sec cos si D cos si si cos si D si sec si sec cos cos D cos sec si sec cos t t v t v v' v d si d l cos cos l cos cos si g c cos c si l cos cos si Eecise. ''-' e l e. ''-' 5. ''si -7-

18 Mthemtics: Lectue Dieetil Equtios: Poblems: Fid the geel solutio - e l - sit 5cost - - e - - e 5-6- sec Reeeces: - clculus & Altic Geomet Thoms. - Clculus Hwd Ato. ا. رياض احمد عزت - Advced Mthemtics o Egieeig Studies - Mode Itoductio Dieetil Equtios Schum's Outlie Seies. -8-

19 Mthemtics: Lectue Dieetil Equtios: C. Highe ode Dieetil Equtios: How to id oots o equtio: Let K be eq. o degee we deote this eq. b the: is oot o the eq. i. is eeted oot o the eq. i. I is oot o the eq. the must be cto o. I is oot o the eq. the divides. E.: Fid ll oots o 8 Solutio: 8 : m m m m 6 m 9 m is oot o the eq. Thee e two methods to ctoize : log divisio & st divisio. Fist method: Fst divisio The oots e Secod method: log divisio m ± 6 m 6 ± 9 8 m 9 ± 8

20 Mthemtics: Lectue Dieetil Equtios: Highe ode lie Dieetil Equtios: The geel om with costt coeiciet is: K F... I F the is clled homogeous othewise is clled ohomogeous. The geel solutio The methods o solvig secod ode homogeous D.Eqs. with costt coeiciets c be eteded to solve highe ode homogeous d ohomogeous D.Eq. with costt coeiciets. Homogeous: the chcteistic equtio o th ode homogeous D. Eq.: K is: Let... K be the oots o chcteistic equtio the: I... e ll distict the the solutio is: c e c e K c e h I eeted m times the h m c e c e... c e m will coti the tems: I some o oots e comle α m iβ the h will coti c cos β c si β e α E.: solve ''' -''' Solutio: e ll distict h h c e c e c c e - - c e c e

21 Mthemtics: Lectue كلام راهزا دعاسم سردم Dieetil Equtios: E.: -'''''-' m - m m m e c e c e c c e c c c e c h h E.: cto is - oot is cto is - oot β α i m e c c e c c e si cos h b Nohomogeous: the geel om o th ode ohomogeous dieetil equtio is:... F K The geel solutio is h g

22 Mthemtics: Lectue Dieetil Equtios: Methods o idig : Udetemied coeiciets We c eted the methods o solvig secod ode o homogeous D.Eqs. with costt coeiciets to solve highe ode ohomogeous D.Eq. with costt coeiciets. E.: -8''6-8si Solutio: g h -8'' ± h c e c e c e - c e - let AcosBsi ' -AsiBcos '' -Acos-Bsi ''' Asi-Bcos AcosBsi AcosBsi8Acos8Bsi6Acos6Bsi-8si 5Acos5Bsi-8si 5A A 5B-8 B-8/5 - g ce c e ce c e si 5 Vitio o metes I this method the ticul solutio hs the om v u v u v u Whee u u u e tke om h c u c u c u. To id v v v we must solve the ollowig lie eqs. Fo v' v' v' : v u v u v u - v u v u v u v u v u v u v u M v u v u

23 Mthemtics: Lectue Dieetil Equtios: E: solve ''''sec Solutio: Let '''' - ±i h c c cosc si u u cos u si sec v v cos v si v v si v cos v v cos v si sec D cos si cos si cos si si cos cos si D si cos sec si cos sec sec cos si D sec si cos si sec cos si cos sec D cos si cos sec si cos sec si sec t D v sec v D D v v D v D D t v sec d lsec t d t d l sect- cos-l cos si l cos g c c cosc si l sect- cos - l cos si 5

24 Mthemtics: Lectue Dieetil Equtios: Eecise: Solve '''-6'''-8 '''- 5 - ''' '''-6'''6 5 8''''''6 6 -'' Poblems: Fid the geel solutio '''-6'''-8 8''6 '''-'e -6 5 '''-' 6 Reeeces: - Clculus & Altic Geomet Thoms. - Clculus Hwd Ato. ا. رياض احمد عزت - Advced Mthemtics o Egieeig Studies - Mode Itoductio Dieetil Equtios Schum's Outlie Seies. 6

25 Mthemtics: Lectue كلام راهزا دعاسم سردم Mtices: Mtices: Whe sstem o equtios hs moe th two equtios it is diicult to discuss them without usig mtices d vectos. The size o the mti is descibed b the umbe o its ow d colums. A mti o ows d m colums is sid to be m mti. [ ] m.... j... i ij m m m m A L M L L Tes o mtices: Sque mti: It is mti whose umbe o ows e equl to the umbe o colums m. Fo emle: 5 A 8 B Digol mti: It is sque mti which ll its elemets e zeo ecet the elemets o the mi digol. Fo emle: 9 A Idetit mti: It is digol mti whose elemets o the mi digol e equl to d it is deoted b I. Fo emle: I I Tsose mti: Tsose o A is deoted b T A mes tht wite the ows o A s colums i A t. Fo emle: T A 7 A

26 Mthemtics: Lectue Mtices: Mti dditio d multilictio I A ] d B [ b ] d both A & B e m mtices the [ ij ij A B [ ij ] [ bij ] [ ij bij ] E.: 5 7 Fo scl umbe c we c multil A b c s ollows: E.: ca c[ij ] [c ij ] 6 A mti with ol oe colum i size is clled colum vecto d oe o ol oe ow m i size is clled ow vecto. Mtices multilictio Let A be k mti d B be k m mti the CAB is m mti whee the elemet i the i th ow d j th colum o AB is the sum cij i b j ib j... ibj ikbkj i... m d j.... k E. 5-7 Suose A B the - 6 AB Detemits With ech sque mti A we ssocite umbe deta o ij clled the detemit o A clculted om the eties o A s ollows: Fo det

27 Mthemtics: Lectue Mtices: Fo det Mios To ech elemet o mti thee coesods mti tht is obtied b deletig the ow d colum o tht elemet. The detemit o the mti is clled the mio o tht elemet. Fo mti o dimesio we deie det whee d is the mio o is the mio o. is the mio o E.: Fid the detemit o ech mti 5 5 b 6 6 E.5: Fid the detemit o A whee: 5 A Sol.: B choosig the ist colum we get 5 6 det A

28 Mthemtics: Lectue كلام راهزا دعاسم سردم Mtices: E.6: Evlute the detemit o A i: A Solutio: B choosig the secod ow we get det A Note tht 6 is the sme vlue tht ws obtied o this detemit i Emle bove. Note: I mti A is tigul eithe ue o lowe its detemit is just the oduct o the digol elemets: Liel Deedet d Liel Ideedet Deiitio: the vectos v v v m e liel deedet i v v v m d i v v v m the v v v m e liel ideedet. E: Let v 6 -; v 8 -; v - detemie whethe v v v e liel deedet o ot. Sol: Sice the v v v e liel ideedet E: Let v 6; v ; v detemie whethe v v v e liel deedet o ot. Sol: Sice the v v v e liel deedet

29 Mthemtics: Lectue Mtices: Eecise: Detemie whethe the give vectos e liel deedet o liel ideedet. ;- b -;-;-6 Fid detemit o the ollowig mtices b Reeeces: - Clculus & Altic Geomet Thoms. - Clculus Hwd Ato. ا. رياض احمد عزت - Advced Mthemtics o Egieeig Studies 5

30 Mthemtics: Lectue 5 كلام راهزا دعاسم سردم Mtices: Solvig sstem o lie equtios Let A be mti X colum vecto B colum vecto the the sstem o lie equtios is deoted b AXB. The ugmeted mti The solutio to sstem o lie equtios such s 6 5 Deeds o the coeiciets o d d the costts o the ight-hd side o the equtio. The mti o coeiciets o this sstem is the mti I we iset the costts om the ight-hd side o the sstem ito the mti o coeiciets we get the mti 6 5 We use veticl lie betwee the coeiciets d the costts to eeset the equl sigs. This mti is the ugmeted mti o the sstem lso it c be witte s: 6 5 Note: Two sstems o lie equtios e equivlet i the hve the sme solutio set. Two ugmeted mtices e equivlet i the sstems the eeset e equivlet. E.: Wite the ugmeted mti o ech sstem o equtios. 5 z z z 5 b 6 z z 5 6

31 Mthemtics: Lectue 5 Mtices: We'll tke two methods to solve the sstem AXB Cme's ule The solutio to the sstem b c b c Is give b D b b D D d whee D D c b D d D c b c c Povided tht D Notes:. Cme's ule woks o sstems tht hve ectl oe solutio.. Cme's ule gives us ecise omul o idig the solutio to ideedet sstem.. Note tht D is the detemit mde u o the oigil coeiciets o d. D is used i the deomito o both d. D is obtied b elcig the ist o colum o D b the costts c d c. D is oud b elcig the secod o colum o D b the costts c d c. E.: Use Cme's ule to solve the sstem: Sol.: Fist id the detemits D D d D : D 7 D D B Cme's ule we hve D D 7 d D 7 D 7 7 Check i the oigil equtios. The solutio set is. 7 7

32 Mthemtics: Lectue 5 Mtices: E.: Solve the sstem: 9 5 Sol.: Cme's ule does ot wok becuse D 6 6 Becuse Cme's ule ils to solve the sstem we l the dditio method: 9 5 Becuse this lst sttemet is lse the solutio set is emt. The oigil equtios e icosistet. E.: Solve the sstem: Sol.: Cme's ule does ot l becuse 5 D 6 Multil Eq. b - d dd it to Eq. 6 6 Becuse the lst sttemet is idetit the equtios e deedet. The solutio set is{ 5 7}. E.: Use Cme's ule to solve the sstem: 5 Sol.: Fist wite the equtios i stdd om A B C 5 Fid D D d D : D 9 5 D -5-5 D 5 5 Usig Cme's ule we get D 5 D d D 5 D Becuse stisies both o the oigil equtios the solutio se is { }.

33 Mthemtics: Lectue 5 Mtices: The Gussi Elimitio method Whe we solve sigle equtio we wite simle d simle equivlet equtios to get equtio whose solutio is obvious. I the Gussi elimitio method we wite simle d simle equivlet ugmeted mtices util we get ugmeted mti i which the solutio to the coesodig sstem is obvious. Becuse ech ow o ugmeted mti eesets equtio we c eom the ow oetios o the ugmeted mti. Elemet Row Oetio:. Costuct the ugmeted mti A:B.. Itechge two ows R i R j.. Multil ow b costt dieet om zeo R i kr i. Add costt multil o ow to othe ow R i R i kr j E.: Use Gussi elimitio method to solve the sstem two equtios i two vibles: Sol.: Stt with the ugmeted mti: R -R R 7 R R 7 R R R This ugmeted mti eesets the sstem d. So the solutio. set to the sstem is { } E.: Use Gussi elimitio method to solve the sstem thee equtios i thee vibles: z z 6 z Sol.: 6

34 Mthemtics: Lectue 5 كلام راهزا دعاسم سردم Mtices: 5 R 6 R R R R R -R -R 5 6 R - R 5 6 R R R R R -R 6 5 R - R 5 R R R This ugmeted mti eesets the sstem d z. So the solutio set to the sstem is { }. E.: Solve the sstem Sol.: R 7 R R R coesods to the equtio 7. So the equtios e icosistet d thee is o solutio to the sstem. E.: Solve the sstem 6 Sol.: 6 R R - R I the R o the ugmeted mti we hve the equtio. So the equtios e deedet. Fo odeed i tht stisies the ist equtio stisies both equtios. The solutio set is { } Eecises: Solve the ollowig sstems: 6 z z z

35 Mthemtics: Lectue 5 كلام راهزا دعاسم سردم Mtices: 6 Mti Ivese The mti A hs ivese deoted b A - i A whee A.A - I. We'll tke two methods to id A - whee A is mti. B Guss elimitio methodusig ow oetios:. Costuct the ugmet mti A:I. Use ow oetio util we hve I:A - E: Use Row oetio to id A - i A R R 7 R R R 7 7 R R R R R A E: Fid A - i A R R R R -R R R -R 7 R R 7

36 Mthemtics: Lectue 5 كلام راهزا دعاسم سردم Mtices: 7 R R R R -R R 7 R R 5 7 R R R R R 7R A B Cocto Method Usig detemit o the mti The cocto o the elemet ij o the mti A ij is deied b c ij - ij A ij whee A ij is the detemit o the mti tht emis whe the ow i d the colum j e deleted. To id the ivese o mti whose detemit is ot zeo - costuct the mti o coctos o A co A c ij - Costuct the tsosed mti o coctos clled the djoi o A dj A co A T - the A - det A dj A - to check ou swe A.A - I o A -.A I E.: Use detemit to id A - whee A A - A dj A 7 8 A CoA C - C - - C - - C - AdjA T

37 Mthemtics: Lectue 5 كلام راهزا دعاسم سردم Mtices: A E: Fid A - i A Solutio: A co A c c c c c c c c c 7 8 t co AdjA A

38 Mthemtics: Lectue 5 كلام راهزا دعاسم سردم Mtices: 9 Poblems: wite the ugmet mti to the ollowig sstems the id the solutio: z z z b 8 z z z c 5 Fid the ivese o ech ollowig mti 5 5 b Reeeces: - Clculus & Altic Geomet Thoms. - Clculus Hwd Ato. - Advced Mthemtics o Egieeig Studies تزع دمحا ضاير.ا

39 Mthemtics: Lectue 6 Vectos: Vecto: A vecto is mti tht hs ol oe ow the we cll the mti ow vecto o ol oe colum the we cll it colum vecto. A ow vecto is o the om: [... ] A colum vecto is o the om: b b b M b m A qutit such s oce dislcemet o velocit is clled vecto d is eeseted b diected lie segmet A vecto i the le is diected lie segmet. The diected lie segmet AB hs iitil oit A d temil oit B; its legth is deoted b AB. Two vectos e equl i the hve the sme legth d diectio. Comoet om I v is two dimesiol vecto i the le equl to the vecto with iitil oit t the oigi d temil oit v the the Comoet om o v is: v v v v I v is thee dimesiol vecto i the le equl to the vecto with iitil oit t the oigi d temil oit v v the the Comoet om o v is: v v v v v

40 Mthemtics: Lectue 6 Vectos: The umbes v v d v e clled the comoets o v. Give the oits P d Q the stdd ositio vecto v v v equl to PQ is v v z z z z The mgitude o legth o the vecto v PQ is the oegtive umbe v v v v z z The ol vecto with legth is the zeo vecto o. This vecto is lso the ol vecto with o seciic diectio. E.: Fid comoet om d b legth o the vecto with iitil oit P d temil oit Q 5 Solutio: v The comoet om o PQ is v - - b The legth o mgitude o v PQ is v 9 Vecto Additio d Multilictio o vecto b scl Additio: Let u u u d v v v be vectos with k scl. u u v u v u v u v v

41 Mthemtics: Lectue 6 Vectos: Scl multilictio: ku ku ku ku I the legth o ku is the bsolute vlue o the scl k times the legth o u. The vecto u u hs the sme legth s u but oits i the oosite diectio. I u u u d v v v u v u v u v u u v v Note tht u v v u d the dieece u v s the sum u v E.: Let u d v 7 id u v b u v c u Solutio: u v 6 7 b u v 5 - c u Poeties o vecto oetios: Let u v d w be vectos d d b be scls. u v v u u v w u v w u u u u 5 u 6 u u

42 Mthemtics: Lectue 6 Vectos: 7 bu b u 8 u v u v 9 b u u bu Uit vectos A vecto v o legth is clled uit vecto. The stdd uit vectos e: i j k v v v v v v v v v v v i v j v k We cll the scl o umbe v the i-comoet o the vecto v v the j-comoet o the vecto v d v the k-comoet. I comoet om P d P is z z P P i j z z k I v the v u is uit vecto i the diectio o v clled the diectio o the v ozeo vecto v. E.: Fid uit vecto u i the diectio o the vecto P d P. Solutio P P i - j - k i j - k P - 9 P P P u P P i j k i j k

43 Mthemtics: Lectue 6 Vectos: The uit vecto u is the diectio o P P. Midoit o lie segmet The Midoit M o lie segmet joiig oits P d P is the oit z z z z E.: The midoit o the segmet joiig P d P 7 is 7 5 Poduct o vectos u & v e vectos Thee e two kids o multilictio o two vectos: - The scl oduct dot oduct u.v. The esult is scl. - The vecto oduct coss oduct u v. The esult is vecto. The dot oduct I this sectio we show how to clculte esil the gle betwee two vectos diectl om thei comoets. The dot oduct is lso clled ie o scl oducts becuse the oduct esults i scl ot vecto. De.: The dot oduct u v u u v o vectos u u u d v v v is: v uv uv 5 u v

44 Mthemtics: Lectue 6 كلام راهزا دعاسم سردم Vectos: 6 Note: E.: 7 5 scl j i j i b 6 5 scl k j i k j i Agle betwee two vectos The gle θ betwee two ozeo vectos u u u u d v v v v is give b cosθ v u v u v u v u cos θ whee θ θ E.: Fid the gle betwee two vectos i sce 9 8 cos 9 8 cos cos - θ θ θ v u v u. j k k j j i k k j j i i k j i k j i u v

45 Mthemtics: Lectue 6 Vectos: E.: Fid the gle θ i the tigle ACB detemied b the vetices A B5 d C5 B5 C5 CA 5 d CB - CA CB 5 - CA 5 9 A CB θ cos 9 Othogol vectos Vectos u u u d v v v e othogol o eedicul i d ol i u v u v E.: u d v 6 e othogol becuse u v b u i j k d v j k e othogol becuse u v c is othogol to eve vecto u sice u u u u Poeties o the Dot oduct I u v d w e vectos d c is scl the u v v u cu v u cv c u v u v w u v u w u u u 5 u 7

46 Mthemtics: Lectue 6 Vectos: Vecto ojectio Vecto ojectio o u oto v oj v u u v oj v u v v "The vecto ojectio o u oto v" E.: Fid the vecto ojectio o u 6 i j k oto v i j k d the scl comoet o u i the diectio o v. Solutio: We id oj v u om eq.: oj u v u v u v v i j k i j k i j k v v v v We id the scl comoet o u i the diectio o v om eq.: Poblems: Let u d v 5. Fid the comoet om d b mgitude legth o the vecto.. u 5v. u v 5 5 Fid the comoet om o the vecto:. The vecto PQ whee P d Q-. b. The vecto OP whee O is the oigi d P is the midoit o segmet RS whee R - d S. c. The vecto om the oit A to the oigi. d. The sum o AB d CD whee A - B C - d D 8

47 Mthemtics: Lectue 6 Vectos: Let v u d w s i the igue: id u v b u v w c u v d d u w w u v Fid the vectos whose legths d diectios e give. T to do the clcultio without witig: Legth Diectio. i b. - k c. j k d. 7 i j k Fid the diectio o P P d b the midoit o lie segmet P P.. P 5 d P 5 b. P d P 6 Fid v u v u the cosie o the gle betwee v d u the scl comoet o u i the diectio o v d the vecto oj v u. v i j 5 k u i j 5 k b v i k u 5 i j 5 5 c v i j u i j k d v 5 i j u i 7 j e v u 9

48 Mthemtics: Lectue 6 Vectos: 7 Fid the gles betwee the vectos: u i j k v i k b u i 7 j v i j k c u i j k v i j k 8 Fid the mesues o the gles betwee the digols o the ectgle whose vetices e A B C d D Reeeces: - Advced Egieeig Mthemtics Ewi Keszic- 8 th Editio. - Clculus Hwd Ato. ا. رياض احمد عزت - Advced Mthemtics o Egieeig Studies

49 Mthemtics: Lectue 7 Vectos: The Coss oduct vecto. The coss oduct is lso clled vecto oduct becuse the oduct esults De.:The coss oduct to the le. u v u v siθ uit vecto oml eedicul Note: The vecto u v is othogol to both u d v u v Pllel vectos Nozeo vectos u d v e llel i d ol i u v. Poeties o the coss oduct I u v d w e vectos d s e scls the u sv s v u u v w u v u w v w u v u w u v u u v 5 u - v u Notes: i j j i k j k k j i k i i k j i i j j k k

50 Mthemtics: Lectue 7 Vectos: Clcultig Coss oduct usig detemits I u u i u j k d v v i v j k the u v u v i u v u v j k u v E.: Fid u v d v u i u i j k d v i j k Solutio u v i j k i - - j - k -i - 6j k v u - u v i 6j k E.: Fid vecto eedicul to the le o P Q d R. Solutio The vecto to both vectos. PQ PR is eedicul to the le becuse it is eedicul PQ i j k i j k PR i j k - i j k i j k - - PQ PR i - j k - - 6i 6k E.: Fid uit vecto eedicul to the le o P Q d R. Solutio Sice PQ PR is eedicul to the le its diectio is uit vecto eedicul to the le

51 Mthemtics: Lectue 7 Vectos: PQ PR 6i 6k i PQ PR 6 k Clcultig the Tile scl oduct volume: lso clled Bo oduct u v w u v w u v w u v w E.: Fid the volume o the bo detemied b u i j k v i k d w 7 j k. Solutio u v w - 7 The volume is u v w uits cubed. Lies d Ples i Sce I the le lie is detemied b oit d umbe givig the sloe o the lie. I sce lie is detemied b oit d vecto givig the diectio o the lie. Equtios o lie Suose tht L is lie i sce ssig though oit P z llel to vecto v v i v j k. The L is the set o ll oits P z o v which P P is llel to v. L v

52 Mthemtics: Lectue 7 Vectos: The stdd equtio o the lie though P llel to z v vi vi vk is: tv tv z z tv - t d z tv tv z tv E.: Fid the equtios o the lie though llel to v i j k. Solutio With P equl to d v v i v j k equl to z v v i j k t t z t E.: Fid the equtios o the lie though P d Q. Solutio The vecto PQ i j 7k is llel to the lie d equtio with z give t t z 7t We could hve choose Q t t z 7t A equtio o Ple i sce Suose tht le M sses though oit P d is oml to z the ozeo vecto Ai Bj Ck. The M is the set o ll oits P z o which P P is othogol to. Thus the le though P oml to Ai Bj Ck hs equtio: z A B C z z o A B Cz D whee D A B Cz

53 Mthemtics: Lectue 7 Vectos: E.: Fid equtio o the le though P 7 eedicul to 5i j k. Solutio A B C z z 5 z z 7 5 z Notice i this emle how the comoets o 5i j k become the coeiciets o d z i equtio 5 z. The vecto Ai Bj Ck is oml to the le A B Cz D. E.: Fid equtio o the le though A B d C. Solutio We id vecto oml to the le d use it with oe o the oit to wite equtio o the le. The coss oduct: i j k AB AC i j 6k is oml to the le. 6 z 6z 6 Lies o itesectio - Two lies e llel i d ol i the hve the sme diectio. - Two les e llel i d ol i thei omls e llel. - The les tht e ot llel itesect i lie. E.: Fid vecto llel to the lie o itesectio o the les 6 z 5 d z 5. 5

54 Mthemtics: Lectue 7 Vectos: Solutio The lie o itesectio o two les is eedicul to both les' oml vectos d d theeoe llel to. i.e. is vecto llel to the les' lie o itesectio. i j k 6 i j 5k 8 E.: Fid the oit whee the lie t t z t 6z 6 itesects the le Solutio The oit 8 t t t 8 t t 6 t 6 8 6t - t 6 6t 6 8t -8 t - The oit o itesectio is z t Agles betwee les The gle betwee two itesectig les is deied to be the gle detemied b thei oml vectos. E.: Fid the gle betwee the les 6 z 5 d z 5 Solutio The vectos i 6 j k d i j k θ cos e omls to the les. The gle betwee them is 6 cos

55 Mthemtics: Lectue 7 Vectos: Poblems: Sketch the coodite es d the iclude the vectos u v d u v s vectos sttig t the oigi. u i v j b. u i - k v j k c. u i - j v i j d. u i j v i j I the tigle tht detemied b the oits P Q d R id uite vecto eedicul to le PQR.. P Q d R b. P Q d R Let u 5 i j k v j 5k d w 5i j k. Which vectos i e:. Peedicul? b. Pllel? Fid equtios o the lies:. The lie though the oit P llel to the vecto i j k. b. The lie though P d Q. c. The lie though the oigi llel to the vecto j k. d. The lie though the oit llel to the lie t t z t e. The lie though llel to the z-is.. The lie though 5 eedicul to the le 7 5z g. The lie though 7 eedicul to the le z h. The lie though eedicul to the vectos u i j k d v i j 5k i. The - is. 7

56 Mthemtics: Lectue 7 Vectos: j. The z - is. 5 Fid equtios o the les:. The le though P oml to i j k b. The le though llel to the le z 7 c. The le though d d. The le though P 5 eedicul to the lie 5 t t z t e. The le though A eedicul to the vecto om the oigi to A. 6 Fid the le detemied b the itesectig lies: L: L : t t z t s s z s - t - s 7 Fid le though P eedicul to the lie o itesectio o the les z z. 8 Fid le though the oits P eedicul to the P le z 7. 9 Fid the gles betwee the les:. z b. 5 z z Fid the oit i which the lie meets the le.. t t z t z 6 b. t z t 6 z Reeeces: - Clculus & Altic Geomet Thoms. - Clculus Hwd Ato. ا. رياض احمد عزت - Advced Mthemtics o Egieeig Studies 8

57 Mthemtics: Lectue 7 Vectos: 9

58 Double Itegl : Lectue 8 Double Itegl Deiitio: let R be closed egio i the - le. I is uctio o two vibles tht is deie o the egio R the the double itegls o R is witte b R Lim A A R da اذا كانت المنحنيات بهذه الصيغة يو خذ المقطع شاقولي dd da R b d d R b اما اذا كانت المنحنيات بالشكل التالي يو خذ المقطع افقيا dd d da R c dd d c R

59 Double Itegl : Lectue 8 Emles: Evlute 8 dd ii i sketch: sice dd veticl 8 dd 8 d {[ ] [ ] } {[ 6] -[ ] } { } d d d Evlute da ove the tigul R eclosed b R i sketch: i i & i i & -

60 Double Itegl : Lectue 8 كلام راهزا دعاسم سردم R dd da [ ] [ ] { } { } d d d d Evlute d d e Revese the ode o itegtio Sice dd hoizotl om o e - e e e e d d e d d e e d d

61 Double Itegl : Lectue 8 si Evlute d d Fom let Fom ight vlue o om evese the ode si d d si d d si d si d si d - cos si dd si dd [ cos ] d [ cos ] [- si ] d 6 Wite equivlet double o itegtio evesed d d d d dd dd -

62 Double Itegl : Lectue 8 كلام راهزا دعاسم سردم 5 7 Dw the egio bouded b e si - d evlute its e. e e e e e d e dd A e e cos cos cos si si si 8 Fid the e bouded b - - d. Solutio: 6 6 d d d d dd dd A - -

63 Double Itegl : Lectue 8 Poblems dd cos dd 5 dd 7 e dd l e dd l 8 l e dd 6 cos dd siθ 8 cosθ ddθ 9 Evlute da R: st qudt bouded b & R Evlute da R: the egio bouded b & R Evlute R da R: the egio i the st qudt eclosed b: Evlute R si da R: the egio bouded b & e dd Reeeces - clculus & Altic Geomet Thoms. - Clculus Hwd Ato. ا. رياض احمد عزت - Advced Mthemtics o Egieeig Studies 6

64 Mthemtics: Lectue 9 Pol Itegl: Pol Coodites d Ghs Pol Coodite sstem Ech oit P c be ssiged ol Coodites θ whee: is the distce om the ole oigi to the oit P. is ositive i mesued om the ole log the temil side o θ d egtive i mesued log the temil side eteded though the ole. θ is the gle om the Iitil to o. The gle θ is ositive i the ottio is couteclockwise d egtive i the ottio is clockwise. Review i tigoometic uctios: si θ siθ csc θ cscθ t θ tθ cot θ cotθ cos θ cosθ sec θ secθ odd uctios eve uctios si θ cos θ sec θ t θ csc θ cot θ si si cos cos si i si si cos cos m cos cos ± si si t m t t m ± t t i i cos cos si t t t cos cos cos si

65 Mthemtics: Lectue 9 Pol Itegl: Covetig om ol to ectgul om d vice ves We hve the ollowig eltioshi betwee ectgul Coodites Ctesi d ol Coodites θ : cosθ siθ tθ dd da dd o o o cosθ si θ θ t d dθ Ctesi Coodites Pol Coodites θ Ghig ol equtios Sketch i smmetic bout -is i elcig θ b θ does ot chge the uctio. ii Smmetic bout -is i elcig θ b θ does ot chge the uctio. iii Smmetic bout the oigi i elcig b does ot chge the uctio. iv θ M

66 Mthemtics: Lectue 9 Pol Itegl: E.: Covetig equtio om Ctesi om to ol om Sice d si θ siθ siθ o siθ the gh o is the ole. becuse the ole is icluded i the gh o we c disced d kee ol siθ - siθ E : Covetig equtio om ol om to Ctesi om cosθ cosθ Multil both sides b E : Covetig equtio om ol om to Ctesi om cosθ-/ cosθ cos/ siθ si/ cosθ siθ 6 E : Covetig equtio om ol om to Ctesi om cosθ cosθ

67 Mthemtics: Lectue 9 Pol Itegl: Some imott cuves si θ } cosθ cicle - cosθ - si θ cosθ si θ codioid si θ cos θ } Leed ose si θ cos θ } Leed ose cos θ Stdd Pol Ghs Cicle θ θ θ M b si θ i elce θ b -θ si - Not smmetic bout -is ii elce θ b -θ si -θ si θ smmetic bout -is iii Not smmetic bout the oigi. θ si θ θ 6

68 Mthemtics: Lectue 9 Pol Itegl: c cosθ i elce θ b -θ cos -θ cosθ smmetic bout -is θ Cdioids cosθ Smmetic bout -is θ Rid ol sketchig E: Sketch cosθ θ vies om Cos θ vies om cos θ vies om cosθ vies om to / to to 8 to / to to - to - to to - to - to to / / to to to to 8

69 Mthemtics: Lectue 9 Pol Itegl: b si θ H.W - cosθ - si θ EX.: Fid the e o the egio eclosed b the cdioids - cosθ A cos θ ddθ cos θ dθ cosθ cos θ dθ θ siθ θ si θ cosθ dθ cosθ cosθ dθ

70 Mthemtics: Lectue 9 Pol Itegl: Poblems Covetig equtios om Ctesi om to ol om 6 b 5 c d Covetig equtio om ol om to Ctesi om siθ b cosθ siθ c d θ sketch 5 si θ b sketch 8 cos θ chge the Ctesi itegl ito equivlet ol itegl. The evlute the ol itegl b c dd dd dd 6 d dd 5 Use ol coodite dd /

71 Mthemtics: Lectue 9 Pol Itegl: 6 Fid the e o the egio R tht lies iside the cdioid cosθ d outside the cicle. 7 Fid the e o the egio R tht lies iside the cdioid cosθ d outside the cicle. 8 Fid the e o the egio R tht lies iside the cicle si θ d outside the cicle. 9 Fid the e o the egio R cut om the ist qudt b the cdioid si θ. Fid the e o the egio commo to the cosθ d cosθ. Reeeces: - clculus & Altic Geomet Thoms. - Clculus Hwd Ato. ا. رياض احمد عزت - Advced Mthemtics o Egieeig Studies

72 Mthemtics: Lectue Pol Itegl: Pol Coodites d Ghs Leves ose cos θ si θ si θ cos θ } Leed ose si θ cos θ } Leed ose cos θ - Numbe o leves: cos θ si θ I eve No. o leves b I odd No. o leves - The mjo es o the ist le : cos θ θ θ b si θ θ θ - Limit o the ist le begi d ed b θ θ cos θ θ θ θ θ si θ θ θ

73 Mthemtics: Lectue Pol Itegl: - Comlete dwig the othe leves: 6/No. o leves om mjo es. E.: cos θ - No. o leves - cos θ θ θ θ θ - cosθ θ θ 6 - this me eve le eeted. E.: si θ - No. o leves - siθ θ θ θ θ - si θ θ θ 6 - this me eve le eeted.

74 Mthemtics: Lectue Pol Itegl: E.: cos θ - No. o leves - cos θ θ θ θ θ 6 - cosθ θ θ this me eve le eeted. E.: si θ - No. o leves - siθ θ θ 6 θ θ - si θ θ θ 6 - this me eve le eeted.

75 Mthemtics: Lectue Pol Itegl: E.5: cos θ - No. o leves 8 - cos θ θ θ θ θ 8 - cosθ θ θ this me eve le eeted. 8 8 E.6: si θ E.7: cos 5θ

76 Mthemtics: Lectue Pol Itegl: E.8: si 5θ - No. o leves 5 - si5θ 5θ θ 5θ θ - si 5 θ 5θ θ this me eve le eeted E.9: cos θ - No. o leves - cos θ θ θ θ θ - cosθ θ θ 6 - this me eve le eeted. 5

77 Mthemtics: Lectue Pol Itegl: E.: si θ - No. o leves - siθ θ θ θ θ - si θ θ θ 6 - this me eve le eeted. Poblems Fid the e o the egio eclosed b the cdioid cos θ. Fid the e o the egio eclosed b the cdioid cos θ. Fid the e o the egio i the ist qudt bouded b d si θ θ Reeeces: - clculus & Altic Geomet Thoms. - Clculus Hwd Ato. ا. رياض احمد عزت - Advced Mthemtics o Egieeig Studies 6

78 Mthemtics: Lectue Pol Itegl: Ghs o some oblems: 6 Fid the e o the egio R tht lies iside the cdioid cosθ d outside the cicle. 7 Fid the e o the egio R tht lies iside the cdioid cosθ d outside the cicle. 8 Fid the e o the egio R tht lies iside the cicle si θ d outside the cicle. 7

79 Mthemtics: Lectue Pol Itegl: 9 Fid the e o the egio R cut om the ist qudt b the cdioid si θ. Fid the e o the egio commo to the cosθ d cosθ. Fid the e o the egio eclosed b the cdioid cos θ. Fid the e o the egio eclosed b the cos θ. Fid the e o the egio i the ist qudt bouded b d si θ θ 8

80 Mthemtics: Lectue Pol Itegl: 9

81 Mthemtics: Lectue Fouie seies: Fouie Seies: Ae seies o cosie d sie tems d ise i the imott cticl tsk o eesetig geel eiodic uctios. Peiodic uctios: A uctio is clled eiodic i it is deied o ll el d i thee is some ositive No. T such tht T The No. T is clled eiod o. Fouie sid I T T: eiodic No. The cos b si T T Whee & b e Fouie coeiciets d B T A d b B cos d T T A B si d T T A A B Notes: si ± ±... itege No. - cos cos o ll ± ± ±... cos - cos si - si eve odd

82 Mthemtics: Lectue كلام راهزا دعاسم سردم Fouie seies: EX.: Wite Fouie seies o - T Fist we id b & [ ] d d T B A d b cos cos cos v u d d d T T B A [ ] cos cos si si cos - d d d b B A B A si dv u si si b si d cos cos

83 Mthemtics: Lectue Fouie seies: b b b b si si b si b si si si si b si L si L si L Fouie eve & odd uctios I is eve the i b ii iii T T B B d cos d T I is odd the i ii b T B si T d De.: A uctio is eve i o ll. Fo emle A uctio is odd i o ll. Fo emle.. Notes: - I smmetic bout -is eve. cos t - I smmetic bout oigi odd. si EX.: Wite Fouie seies o T - - -

84 Mthemtics: Lectue Fouie seies: i Fom sketch smmetic bout oigi odd. b B B si d si d - - cos cos - b cos. To id b ut i eq. b b b b b 5 b si 5 si si si si si si... 5 Notes: cos cos cos si si cos cos cos si si cos cos cos cos cos cos [cos cos ] dd We c obti si si b subtctio.

85 Mthemtics: Lectue Fouie seies: si si cos cos si si si cos cos si si si cos si cos si si si subtctio EX.: Wite Fouie seies o T cos - - is eve i b This is tue i d ol i the othe itevl B d cos si [ ] d T T cos d cos cos d... [cos cos ] d si si si si si... To id ut i eq. 5

86 Mthemtics: Lectue كلام راهزا دعاسم سردم Fouie seies: 6 si cos cos d d i eq. } {si } {si {}} { } {si } 5 {si 5... cos 5 cos cos cos Poblems: Wite the Fouie seies o the ollowig uctios: si

87 Mthemtics: Lectue كلام راهزا دعاسم سردم Fouie seies: k Fid the Fouie seies o the uctio which is ssumed to hve the eiod. Fid the Fouie seies o the uctio / - / k -

88 Mthemtics: Lectue Fouie seies: 5 Fid the Fouie seies o the uctio k -k Reeeces: - Advced Egieeig Mthemtics Ewi Keszic- 8 th Editio. - Clculus Hwd Ato. ا. رياض احمد عزت - Advced Mthemtics o Egieeig Studies 8

89 Mthemtics: Lectue كلام راهزا دعاسم سردم Ptil Dieetitios: Ptil Dieetitios z o Z Z Z Z Z st til deivtives Z Z Z Z Z Z Z Z d til deivtives Z Z E. I Z id Z Z owe uctio costt l costt - d Z Z E. I Z t show tht Z Z Z. L Z

90 Mthemtics: Lectue كلام راهزا دعاسم سردم Ptil Dieetitios: Z. L Z & e equl Poeties: I g υ υ chi ule o υ υ υ υ υ υ I s h gs \ chi ule s s s Totl dieetil I... z dz d d d o dz d d d z z E. I z z z Fid d d B oet

91 Mthemtics: Lectue كلام راهزا دعاسم سردم Ptil Dieetitios: d dz z d d z d d z d d dz z d z d z d dz d d d z E. I... ct g ct Show tht c t Thee e two methods to solve this E. Fist method: Let s ct ct Eq. becomes... s g c s g c t s s s g t t... s g c t c s g c c c t s s s g c t c t s g s s s g g I eq. c t

92 Mthemtics: Lectue كلام راهزا دعاسم سردم Ptil Dieetitios: Secod method: ل ةبسنلاب ةرشابم ةلداعم قتشن t [ ]... g c t c c ct g c c ct t c ct g c ct t [ ]... g ct g ct ct g ct Fom & c t E. I z Show tht z z z... z z z... z z

93 Mthemtics: Lectue كلام راهزا دعاسم سردم Ptil Dieetitios: 5 Fom & z z z E. Eess d s i tems o s & i z s z l s s z s z s z z 8 s s z s s s z z s s s Poblems: Fid z z z - si cos - z Fid υ whe - u i υ υ υ u u I show tht 5 I d θ θ si cos show tht 6 I z z d d id z &

94 Mthemtics: Lectue Ptil Dieetitios: 7 Fid the diectiol deivtive o diectio o A i j t t i the 8 I which diectio is the diectiol deivtive o 9 The D.D. o t i the diectio towds is d the D.D. t towds is - id D.D. t towds the oigi. Reeeces: - clculus & Altic Geomet Thoms. - Clculus Hwd Ato. ا. رياض احمد عزت - Advced Mthemtics o Egieeig Studies 6

95 Mthemtics: Lectue كلام راهزا دعاسم سردم Ptil Dieetitios: Ptil Dieetitios The Gdiet & Diectiol Deivtive z uit vectos & k j i k z j i υ The Diectiol Deivtive o z t z i the diectio o k j i A gdiet & A A u u D.D. υ υ E.: Fid D.D. o z t i the diectio o 6 k j i A Sol.: Fist we id υ z k j i k z j i υ υ A A u D.D. i i k j i k j i υ υ

96 Mthemtics: Lectue Ptil Dieetitios: Mim Miim & Sddle oit will hve M m S ccodig to: to id the suggested oit b. The b is M o m ccodig to egtive o ositive. The b is sddle oit. E.: Locte Mm & S i multi b Sice -- is m E.: Locte Mm & S i

97 Mthemtics: Lectue Ptil Dieetitios: Fom I & 6 6 t 9 is sddle oit t i i is m ii i is M Poblems: Fid z - z z si - cos z

98 Mthemtics: Lectue Ptil Dieetitios: Fid υ whe u υ i u υ u υ - I show tht 5 I d cosθ siθ show tht 6 I z & z 7 Fid the diectiol deivtive o diectio o A i j dz id d t t i the 8 I which diectio is the diectiol deivtive o 9 The D.D. o t i the diectio towds is d the D.D. t towds is - id D.D. t towds the oigi. Reeeces: - clculus & Altic Geomet Thoms. - Clculus Hwd Ato. ا. رياض احمد عزت - Advced Mthemtics o Egieeig Studies