Vardhaman Mahaveer Open University, Kota

Transcription

1 MA/MS MT-9 Vrdh Mhvr Uivrsiy, Ko rl Trsfors d rl Equio

2 MA/MS MT-9 Vrdh Mhvr Uivrsiy, Ko rl Trsfors d rl Equio Ui No. Ui N P No. Ui - l Trsfor - 4 Ui - Th vrs l Trsfor 4-8 Ui -3 Soluio of rdiry Diffril Equios wih Cos d 8-6 Vril Coffiis d h Soluio of Boudry Vlu Prols y l Trsfor Ui -4 ourir Trsfor 7-39 Ui -5 Mlli Trsfor 4-63 Ui -6 Th fii Hkl Trsfor Ui -7 Aliio of ourir d fii Hkl Trsfor o h 9- Soluio of Sil Boudry Prols Ui -8 ir rl Equio 3-47 Ui -9 Soluio of Grl rl Equios wih Sil Ty of Krls d y rl Trsfor Mhod Ui - Soluio of rl Equio of Sod Kid y Sussiv 79-3 Aroiio d Susiuio Ui - rl Equios wih Syri Krls Ui - Clssil rdhol Thory

3 Chir Prof. (Dr.) Nrsh Ddhih Vi-Chllor Vrdh Mhvr Uivrsiy, Ko Cours Dvlo Coi Co-ordior/Covr d Mrs Suj Covr Co-ordior Prof. D.S. Chuh Dr. Aurdh Shr Dr of Mhis Assis Profssor Uivrsiy of jsh, Jiur Dr of Boy, V.M..U., Ko Mrs :. Prof. V.P. S 4. Prof. S.P. Goyl 7. Dr. Prsh Vys E Vi-Chllor Erius Siis (CS) Assis Profssor Jiwji Uivrsiy, Dr of Mhis Dr of Mhis Gwlior (MP) Uivrsiy of jsh, Jiur Uivrsiy of jsh, Jiur. Prof. S.C. jvshi 5. Dr. A.K. Mhur 8. Dr. Vilsh Soi Dr of Mhis Assoi Prof. (ird) urr siu of E. & Th. Dr of Mhis Dr of Mhis Bhddl, or (Puj) Uivrsiy of jsh, Jiur Gov. PG Coll, Ko (j.) 3. Prof. P.K. Brj 6. Dr. K.N. Sih 9. Dr. K.K. Mishr Erius llow (UGC) Assoi Prof. (ird) urr Dr of Mhis Dr of Mhis Dr of Mhis J.N.V. Uivrsiy, Jodhur Uivrsiy of jsh, Jiur M.S.J. Coll, Bhrur (j.). Dr. K.S. Shkhw urr, Dr of Mhis Gov. Shri Kly Coll, Sikr (j.) Edii d Cours Wrii Edior Prof. S.P. Goyl Erius Siis (CS) Dr of Mhis Uivrsiy of jsh, Jiur Wrirs. Prof. Ksh Gu. Dr. Yshw Sih Dr of Mhis Dr of Mhis MNT, Jiur Sh Moi l PG Coll, Jhujhuu 3. Dr. A.K. Goyl Dr of Mhis M.S.J. Coll, Bhrur Adi d Adiisriv M Prof. (Dr.) Nrsh Ddhih Prof. M.K. Ghdoliy Mr. Yodr Goyl Vi-Chllor Diror (Adi) hr Vrdh Mhvr Uivrsiy, Vrdh Mhvr Uivrsiy, Mril Produio d Ko Ko Disriuio Dr Cours Mril Produio Mr. Yodr Goyl Assis Produio ffir Vrdh Mhvr Uivrsiy, Ko

4 PEACE Th rs ook ild rl Trsfors d rl Equio hs dsid so s o ovr h ui-wis syllus of Mhis-9 ours for M.A./M.S. (il) suds of Vrdh Mhvr Uivrsiy, Ko. lso usd for oiiv iios. Th si riils d hory hv lid i sil, ois d luid r. Adqu ur of illusriv ls d riss hv lso iludd o l h suds o rs h suj sily. Th uis hv wri y vrious rs i h fild. Th ui wrirs hv osuld vrious sdrd ooks o h suj d hy r hkful o h uhors of hs rfr ooks.

5 Ui - l Trsfor Sruur of Ui. jiv. roduio. rl Trsfor.. So or rl Trsfor.3 l Trsfor.3. Dfiiio.3. Piwis Coiuiy or Siolly Coiuous.3.3 uio of Eoil rdr.3.4 Eis odios for l Trsfor.3.5 uios of Clss A.4 So or Proris of l Trsfor.4. iriy Prory.4. Ch of Sl Prory.4.3 irs Trslio or Shifi Thor.4.4 Sod Trslio or Shifi Thor.4.5 Alr S of Sod Shifi Thor.5 Tls of l Trsfors.6 Eris ().7 l Trsfor of Drivivs.8 l Trsfor of rl.9 Muliliio d Divisio y Powrs of. Evluio of rls y Usi l Trsfor. Eris (). iil Vlu Thor.3 il Vlu Thor

6 .4 Priodi uios.5 So Sil uios.6 Sury.7 Aswrs o Slf-ri Eris.8 Eris (). jiv Th oj of his ui is o dfi l rsfor wih is is odiios d sil roris. W shll rov so ior hors rrdi is drivivs, irls, uliliio d divisio y owr of. W shll lso disuss h vluio of irls y usi his rsforio.. roduio Th Elish Eir Hvisid (85-95) usd h oriol hods i solvi hysil rols whih v irh o oriol lulus, kwo s l rsforio. Th is du o rh Mhii, Pirr d l (749-87) who usd suh rsforio i his rsrh work. Th l rsfor hods r usful s wll s ffiv i solvi diffril quios i iiil vlu rols d h id ior os Eirs d Siiss. By usi l rsforio, ri ril diffril quios rdud o ordiry diffril quios d ordiry diffril quios rdud o lri quios.. rl Trsfors Th rl rsfor of fuio f d is dfid y dfid i is dod y f ; f ; k, f d...() whr k,, iv fuio of wo vrils d, is lld h krl of h rsfor. Th oror is usully lld irl rsfor oror or sily irl rsforio. Th rsfor fuio of rfrrd o s h i of h iv oj fuio f A forul whih ivs f k is lld h ivrsio forul d is lld h rsfor vril. i.. f, f d...() is s sifi fuio, svrl irl rsfors suh s ourir Trsfors, By ki k, Mlli rsfor, Hkl Trsfor d l Trsfor hv irodud. u of his, l rsfor is os sivly sudid d usd.

7 .. So or rl Trsfors S.No. i S f r k,, for f d T, for ; N of h Noio rsfor l f; ii, for f d, for Mlli M f ; iii iv v, for si, for, for os, for i, si f d ourir Si S f ; os f d ourir Cosi C f ; i fd Col ourir f; vi, for J, for J is h Bssl uio of h firs kid of ordr..3 l Trsfor J f d Hkl H f ;.3. Dfiiio : f : f rl vlud fuio dfid ovr h irvl ], [ suh h f i.. f fuio of dfid for ll osiiv vlus of. Th h l rsfor of f, dod y f;, is dfid y f ; f d rovidd h h irl iss d fii. s h h irl ovrs for so vlus of. Th rr, is rl or ol ur u idd of. rl rl r of. f ; lrly fuio of whih is wri s f Thus f ; f 3, s.

8 To h l rsfor of f, w ulily i y d ir h rsul wih rs o for o. This orio is lld l rsforio. Hr l rsfor d oror whih rsfors f rsforio oror..3. Piwis Coiuiy or Siolly Coious Dfiiio : A fuio f is kow s h krl of h io f is lld h l is sid o iwis oiuous ovr h losd irvl, if h irvl dividd io fii ur of su irvls i (i,..., wih d ) suh h i h su irvl, (i) (ii) i f i is oiuous i h o irvl i i f d i f i i,..., h d ois of hs irvls, h rih-hd d lf-hd liis iss d fii i.. oh is d r fii. f() iur. (A siolly oiuous fuio) Hr i is osrvd h iwis oiuiy of f o losd irvl sily idis h fuio f hs fii ur of disoiuiis of h firs kid i,. El : Cosidr Hr f suirvls ; f S si ; T ; 4 is siolly oiuous fuio i [, 4] s h fuio is oiuous i h of h, j,, j d,4 d hs fii rih-hd d lf-hd liis,, 4

9 d 4 of hs suirvls..3.3 uios of Eoil rdr Dfiiio : i A fuio f f fii quiy o sid o of oil ordr s if This s h for iv osiiv ir, hr iss rl ur M suh h f M whih ilis h f M W y lso wri i s f h s. f fuio f is of oil ordr d, h f is of oil ordr. (s ; ) El : Boudd fuios suh s si,os r of oil ordr si si d os El 3 : Show h f is of oil ordr 3. Soluio : Si i i 3 3 [usi Hosil rul] is of oil ordr 3 Ai si,, h is ordr 3. El 4 : Show h f 3 is o of oil ordr. Soluio : Si i i 3 i 3 j H w o fid ur M suh h Thrfor 3 is o of oil ordr..3.4 Eis Codiios of l Trsfor 3 M Thor : A fuio f is siolly oiuous i vry fii irvl d is 5

10 of oil ordr ' ', s h f ; iss. Proof :. Th f; fd fd fd iss. Si f o is siolly oiuous i vry fii irvl Now f d f d...(3), h irl f d f d [si f is of oil ordr i.. f M ] M. d M d M if M illy w f d if. Th r H M f d d s sll s w ls y hoosi suffiily lr for iss d ordily Thus f; iss for. f d lso iss for. rks : Th odiios iv i Thor r suffii u o ssry for h is of h l rsfor. f hs odiios r sisfid, h l rsfor us is. f h ov odiios r o sisfid h h l rsfor y or y o is. Thr r fuios whos l rsfor iss v wh h odiios of hor (o) r o sisfid. 6

11 or l f is o siolly oiuous i vry fii irvl i h r. Si h rih-hd lii, i f h i h ho hs o fii rih hd lii f h is o siolly oiuous i h r. Also f wh or oil ordr, f M, or wih M d,,, 3... M is of oil ordr s Bu ; d u u du u u du u u du (ui u ) for H G K J Thus w hv show h l Trsfor of iss for v if is o siolly oiuous i h r d is of oil ordr s..3.5 uios of Clss A Dfiiio : A fuio whih is siolly oiuous i vry fii irvl d is of oil ordr ' ' s is kow s fuio of lss A. 7

12 .4 So or Proris of l Trsfors W ssu, ulss ohrwis sd, h ll fuios sisfy h odiios of hor, so h hir l rsfor is..4. iriy Prory Thor : f f; d f rsivly d if C, C r y wo oss, h Proof : By dfiiio, w hv ; h l rsfor of h fuios f, f C f C f ; C f ; C f ; C f C f ; C f C f d.4. Ch of Sl Prory C f d C f d C f ; C f Thor 3 : f f ; f h f Proof : f ; f d H G K J f d f H G K J.4.3 irs Trslio or Shifi Thor f ; H G K J (ui Thor 4 : f f ; f, h f ; f whr is rl or ol ur. f f f Proof : W hv is h l Trsfor of f, h f or f ; f d ) is h l Trsfor of 8

13 f d f; f f d KJ f.4.4 Sod Trslio or Shifi Thor Thor 5 : f f ; f d fuio f ; h ; f Proof : W hv ; d is dfid s d d.. d f d fd f d (ui ) f; f.4.5 Alr S of Sod Shifi Thor f f is h l rsfor of f d, h whr u u ; if rsfor of f u Proof : W hv = ; if f. u ; f u d 9 is h l f is h Hvisid ui s fuio dfid s

14 Now usi h dfiiio of ui s fuio, w hv ; f u H h rsul. f u d f u d f.. d f.. d f d fd f d f.5 Tl of l Trsfors S.No. f f; f (ui d d ).,.,! 3.,, N 4., if, 5. si 6. os 7. sih 8. osh,,,,

15 Th followi rsuls oid y usi firs shifi rory ; si os d sih osh El 5 : id h l rsfor of : i 3 si5 4os7 9 5 ii iii 3 h iv osh 4 osh v si4 Soluio : W hv 3 i si5 4 os 7 9 5; ; si 5; 4 os 7; 9 3 ; 5 ; ! ii osh 4; osh8 ; ; osh 8; 64 iii h ; ; h

16 3 3 ; 3 ; 3 ; ; 3 ; 3 ; 3 ; ! 3! iv N M osh ; ; KJ Q P 3 4 ; ; ; ; j j N M Q P v si 4 ; Si ; 3 i 4. ; 3 4i i 4i 3 48 i (y firs shifi hor) h h h

17 os4 i si 4 ; h 3 h 64 i 4 3 Now qui h iiry rs o oh h sids, w 64 si 4; 4 3 Ai lyi h firs shifi hor h h si 4; 4 3 El 6 : Show h f ; is Soluio : W hv f ; f d h h h h KJ KJ h h whr f f d f d f d d KJ El 7 : id h l Trsfor of. u 3 is ui s fuio Soluio : Si u u u u h d S T,,, KJ KJ usi sod shifi hor, whr u 3 3

18 u 3 ; 3 u 3 ; 6 3 u 3 ; 9 u 3 ; Slf ri Eris - r ; r... lsi ; q... ; r... r 3 ; 6 3 ; 9 3 ; f f f ;, h fill-i h lks i h followi :. f. 3. r f 4. f u ; Th oil ordr of 3 is ST o UVW ; ;....6 Eris (). Prov h h followi fuio : f S T ; ; ; 3 is siolly oiuous i, 3.. Show h h fuio f is of oil ordr ' ' if, N. 3. Show h h fuio f 4. Prov h f ; 4 KJ 3 is of oil ordr 4. f K J, whr f ; f d d r oss.

19 5. id h l Trsfor of : i si 4 ii As As. 3 iii si 3 As. iv 3 sih 5osh As. 6 h 9h 5 3 v 3 os 6 5si 6 As. 6. Evlu f ; whr f S T os ; si ; As. h 7. S d rov h rslio roris of h l rsfor. Also oi h l rsfor of h followi fuios : i ii iii f f f S T S T S T si, 3 3, 3 j T ; T ; T j os 3 ; 3 ; 3 8. id h l Trsfor of : i si5 os3 As. ii osh os As. As. As. As. H 3 K. ; T h; T. ; iii As

20 iv os As. v sih os As. 9. id h l Trsfor of : h i osh 3 As. ii os 3 As.. f f ; f, show h i sih f ; f f ii osh f ; f f Usi sod shifi hor, fid h l rsfor of : h i u ii u As. As. 3.7 l Trsfor of Drivivs Thor 6 : i f is oiuous for N ii f is of oil ordr ' ' for N iii f is siolly oiuous for N iv f; f Th f ; f f for...(4) Proof : W hv f ; f d 6

21 Usi irio y rs, w hv Si f Thor 7 : i f fd is of oil ordr ' ' i f for, hrfor Thus f ; f f, f f h f f for... f r oiuous for ordr ' ' for N. ii f is siolly oiuous for 7 N. N d r of oil Th f ; f f f... f f...(5) whr f f ; Proof : W shll rov h hor y usi hil iduio. By h hor 6 w hv f ; f ; f H h hor is ru for. h hor ru for ( fid osiiv ir) for...(6) Th f ; f f f... f f Now f d d f ; ; f ; f f f f f f f... f f f... f f f Thrfor h hor is ru for

22 H y h riil of hil iduio, h hor 7 is ru for ll N..8 l Trsfor of rls Thor 8 : f f ; f, h Proof : G f u du; f f udu G f d G ; ; ; N M ; f h Si G G G f f u du or f u du rk : Th rliio of ov rsul is u u... f u du... d u ; f whr f f ;.9 Muliliio d Divisio y Powrs of Thor 9 : f f ; f, h d f; d f Proof : W hv f f ; f d...(7) Diffrii h ov quio (7) o oh sids wih rs o, d lyi ii s rul for diffriio udr h si of irio, w hv d d f d d f d f d 8

23 Thus f ; d d f...(8) whih rovs h h hor is ru for. To slish h hor 9, w us riil of hil iduio. h hor ru for d f; d f r 9 ( fid osiiv ir), h d or f d d f...(9) Diffrii oh sids of (9) wih rs o ' r f d f d d ' d lyi ii s rul, w d or f ; f...() d Thrfor h rsul () is ru for. H y riil of hil iduio, h hor 9 is ru for ll osiiv irs. rk : ii s rul for diffriio udr h si of irio f f f, d, r oiuous fuios of d, h d d f f, d, d Thor : f f ; f, h f iss. Proof : G f, so h f G, whr, r oss idd of. N M Q ; P Tki l Trsfor of oh h sids d usi Thor 9. W hv or f ; d G ; d G ; f d d G ; Now iri oh h sids wih rs o fro o, w hv r G ; f d f u du or i G ; G ; f u du f u du, rovidd h h irl

24 or G ; f u du i G ; i G d or f ; f u du. Evluio of rls y usi l Trsfors l Trsfor usd for vluio of irls s show low: r f f ; f i.. f d f Tki h lii s d ssui h h rsuli irrl is ovr. W hv f d f El 8 : Usi h driviv forul, show h i os ; Soluio : i ii os ; h f os si, f os, f d f ; ; so h f Si f f f f...() hrfor os ; os ;. Silifyi, w os ; ii Hr l f os os si, f si d f os Also f f Susiui hs vlus i (), w si os ; os ;

25 h os ;. or or os ; h El 9 : Evlu l Trsfor of h followi fuios : i si os si os ii si Soluio : (i) si os ; si ; os ; d d si ; os ; si ; u du h h h h u du u 3 h 3 (ii) f os h f ; ; os ; H G KJ os u ; u u K J h du lou lo u N M lo Q u P u N M h lo f u jq P

26 w hv lo N M H P K Q os ; lo lo P lo h...() h u ou lou h lo u os u u du ; lo lo du u u du h h u lo u u u u lo u u h u lo u u u lo u u lo u K J u iu lo K J lo u u KJ K J UVW iu... lo u u 4 u o ST lo El : id l Trsfor of h fuio si d h oi h l Trsfor of os Soluio : si ! 5! 7! si ; ! 5! 7!

27 H G S K J 3 H G 4 T KJ! 4 3! 4 H G KJ 3... U V W 3 4 os N l f si, so h f, f Now usi h forul f ; f ; f w hv or N M N M Q os ; P si ; os ; Q P H G K J H G 4 K J 3 El : f f ; f, h 4 f d f u du, ssui h h irls ovr d h rov h si d. Soluio : W hv, y Thor f f ; d fudu Tki d ssui oh h irls ovr, w f N, l f si, so h d f u du...(3) si ; f By usi (3), w 3

28 si d El : Prov h i si h du u u si d si ii d ; lo 4 4 lo5 4 d ddu h KJ Soluio : or os Si si, hrfor rodi s i El 9 ii si ; lo 4 si d 4 w fid h 4...(4) KJ lo 4 KJ Tki lii, w hv si d 4 lo5...(5) Ai lyi h hor i quio (4) w hv si ; lo 4 u 4 u KJ du h u 4 ou lou h lo h u 4 lo u 4 du lo u du 4 4 du u u du S H G K J 4 u N Mou lou 4h ulo u u 4 T 4 N M 4 4 iulo K J lo 4 4 u u u UV WQ P KJ H G K J Q P 4

29 si N M Tki lii, w Si d si d iu u Slf-ri Eris lo N M 4 d lo 4 KJ H G K J 4 Q P KJ H G K J 4 Q P i u u K J u ST...U lo 4 4 6! V u u W 4 4 KJ 4h i lo i lo i lo lo i i d i Assui h odiios of vlidiy, fill i h lks i h followi : o. f ;.... f ; f u du r { } 4. ; r... S H G T K J 5. d d 6. f, h ;... U f; V W... o f ;... 5

30 7. f ST si ; o UVW. Eris (). Giv S T U V, h ST si UVW ;... 3 ; W, show h S T ; { h } r u. Vrify dirly h u u du ; ; 3. Evlu l Trsfor of h followi fuios : U V W i si As. h ii 3 os As. iii 3 hsi 3 As. iv si 4 As. v osh As h o o h 3 4. id h l Trsfor of si. Dos h rsfor of os si d. As. o H G K J 5. Show h S T 6. Evlu ST As. lo UVW osh ; lo KJ U V W 6 KJ ; d h ddu h, dos o iss 3 6 d is? Also rov h lo.

31 7. Usi l Trsfor hiqu, vlu h followi irls : i 3 si d As. ii si d 7 As. 8. f f ; f, fid h l rsfors of f 9. f f vlidiy of rsuls. S T G,, Prov h f d d ; G ;. Prov h. Show h ST S T UVW os os ; lo U u du; V u W lo. Giv h f T,, id i f ; ii As. S r KJ KJ 4 d r f ; r r r Dos h rsul f ; f ; f hold for his s? Eli. i ii Th rsul dos o hold s f is disouiuous. iil Vlu Thor Thor : f suos h f f, si odiios of oiuous for ll d of oil ordrr s. Also is of lss A, h i f i f ; Proof : By Thor 6, w hv f ; f ; f

32 or f d f ; f...(6) Si f is siolly oiuous d of oil ordr, w hv i f d Now ki lii s i (6), w fid h i f ; f or f i f ;...(7) Si f is oiuous, w hv f i f ro quio (7), w i f i f ;...(8).3 il Vlu Thor Thor : f f is of lss A, h oiuous for ll d of oil ordr s d if i f i f ; Proof : By Thor 6, w hv f ; f ; f...(9) Tki lii s i (9), w hv i f d i f; f or f d i f ; f or f i f ; f or i f f i f ; f or i f i f ; 8

33 H h fil vlu hor is vrifid..4 Priodi uios Dfiiio : A fuio f f T f is sid o riodi if hr iss rl ur T suh h f T is h slls osiiv ur for whih suh rlio is sisfid, h T is lld h riod of h fuio. or l, h sils riodi fuios r si d os hvi riod. Thir rirols os d s r lso riodi wih riod d d o r riodi wih riod Thor 3 : f f,. f u T f u h f ; Proof : W hv is riodi fuio wih riod T i.. fu T fu, T T f ; f d f d T T 3T T T f d f d f d... T T T u T f d f u T du u T f u T du... (Pui u T, u T,. i h d d 3rd... irls rsivly) T T T u T u f u du f u du T u f u du... T h T T... u f u du T T T T u, T f u du f d (Usi h rlio; r r..., r ) r.5 So Sil uios A. Th G uio : Th fuio is dfid y h 9

34 u u du, B. Th s d osi rls : Th s d osi irls dod y S rsivly r dfid y h quios d i si S i os C u u du u u du C. Th Error uio d Colry Error uio : i Th rror fuio, dod y rf, is dfid y ii u rf du Th olry rror fuio, dod y rf, is dfid y u rf rf du u du i d Ci D. Th Ui S uio (or Hvisid s Ui uio) : Th ui s fuio, dod y U is dfid y U S T ; ; E. Th Ui uls uio or Dir Dl uio : Cosidr h fuio S T,, whr. Th rh of his fuio is show i h dir low Th Dir s dl fuio or ui iuls fuio is dod y i d is dfid s iv y i. 3

35 . Bssl uio : Bssl s fuio of ordr is dfid y J N 4 M r r. r! r K J r G. urr Polyoil : urr olyoil is dfid y h d!. d.,,,... H. Th Eoil rl : Th oil irl fuio Ei is dfid s u E u du i. Hyrori uio : Th fuio, ; ; kow s Guss s hyrori fuio or sily hyrori fuio is dfid s, ; ;! J. B uio : W dfi h B fuio B, y h irl whr, B, d El 3 : Prov h U ;, whr U is h Hvisid s ui s fuio Soluio : By h dfiiio of Hvisid s ui s fuio, W hv U S T,, u ; U d U d U d d Q P 3

36 El 4 : id ; whr is dfid i.5 d h show h i ;. Soluio : W hv S T,, Th ; d d d d d. ; d Thus i i ; i (y Hosil s ul) El 5 : Evlu Si; Soluio : W kow h, whr S i si u u du. i si S u u du KJ u u 4 u 6... du 3! 5! 7! ! 5 5! 7 7! Tki l rsfor of h sid, w Si;!. 3!!. 5!

37 H G KJ El 6 : Prov h C ; Soluio : f so h y ii s rul f i h osu u du; u u Ci os du os du u u os or f os f ; os ; d or d f ; or lo h d d f f whr f f ; KJ or d d f f is os ri oh sids wih rs o ' h ', w f lo A...() Bu fro h fil vlu hor i f i f ro () s, w hv A A h or f lo 33

38 h lo or f f ; Ci ; lo El 7 : Prov h Ei ; Soluio : W hv E i E u u du u u du ; i Q P N M v Q P v dv ui u v v ST U VW v dv d v. d dv v v. v dv { } (hi h ordr of irio) v v KJ dv lov lo v i K J E ; lo lo v El 8 : Prov h J; d h ddu h i J; ii J ; h 3 34

39 d iii J ; iv Soluio : W kow h J Dduios : J J d 3 v J 4 d 3 5. r r! r r H G K J r! r r r K J J; ; ; ; ; r! 4! 6! H G 3. K J H G. 4 KJ i Si f KJ ; H G K J f J;. S T h, whr f f ; H G K J d ii J d J ; ; U V W h H G 3 KJ... 35

40 d N M d Q P h h iii Si f f 3 ;, whr f f ; J ; J ; r iv w hv J ; J d v ui, w hv ro dduio ii, w hv J d J d 3 ui 3 d 4, w Slf-ri Eris - 3 J 4 d ill i h lks i h followi. f J ;. f f r, h i J ;... r ii J r iii J r S ;... ;... T si,, h h f ;... r h h 36

41 3. f f f ;, h { d i } 4. f rf ; r S T, h rf ;... { o } rf ;... { o } rf ; whr.6 Sury 3 { d i } ; r... is h Dir dl fuio. u f u u U V du; W... his ui you sudid ior irl rsfor kow s l rsfor, wih is odiios. You lso sudid so si roris d rsuls ivi h l rsfor of drivivs, irls, uliliio d divisio y owrs of ' '. A ur of rols o l rsfor r lso solvd o flii h udrsdi of his rsfor..7 Aswrs o Slf-ri Eriss Eris -. K J. f 3. f lo r K J 4. f , Eris -. f f... f. 3. f 4. d d! 5. d d f f h 37

42 6... d d KJ h 7.. f Eris - 3. f lo h 3 H G K J 4. d i h Eris (). f f. f f,, fid f. d f f h h As.: riodi fuio wih riod 4, whr f, S h rov h f T 3 6, 4 3. Vrify h iiil vlu hor for i 3 os ii 4. Vrify h fil vlu hor for i 3 ii 5. Show h 3 ; h 3 si i J u osu du si ii d i h os Jd uisi u du os iu [Hi. irs vlu J u ;, d h or rl d iiry rs of h rsul

43 hus oid] 6. Show h d d J ; id h l Trsfor of h followi fuios : i Jd i ii Si iv v Ei As.: i iii v Ei lo 3 lo 4 3 h 4 6 h 3 ii 3 8. id l rsfor of : i N M u u As.: i lo KJ du iv h KJ 39 iii 3 vi vi d i ii rf u du ii 9. f f ; H G, f. id i KJ S H G U N T KJ V As.: M W u As.: i h ii lo h Ci Ei3 5 lo 3 5P iii iii K J ;. d f, fid f ; d ii os h d P Q os lo ; lo. id h l Trsfor of h riodi fuio hvi riod k d dfid y f k

44 As.: k k k hh G K J. Prov h J ; d ddu h i h J 3. Prov h for ; S T ii J; 4 h 3 h J ;, 4. Prov h if,, J d i; h H ddu h i u d i ; ; ii u J u f u du; f ; { u J u f u du } fu 5. Prov h { d i } J os d lo 6. Usi l rsfor, vlu As.: 7. f, J d 4 d! d. h, h rov h ;, ;. U V W 8. i rf ;. H ddu h vlu of rf As.: 4 rf 4, rf

45 9. Evlu J ; for. Prov h 3 S T r d 3 h 3 u fu U du V u f. id h l Trsfor of h fuio N M As.: As.: P] Q, 3 k 4 k k k. Show h for usi h rsul Show h ; W lo whr f f ; J;, ; ;, ; ; K J d ddu h r h { } J ; r i J ; ii J d r J ; iii iv J d { } 3 h j v J j S U ; T V W KJ 4

46 Sruur of h Ui. jiv. roduio. vrs l Trsfor.. Dfiiio.. Null uio..3 Uiquss Ui - Th vrs l Trsfor.3 vrs l Trsfor of Elry uios.4 So or Proris.4. iriy Prory.4. Ch of Sl Prory.4.3 irs Shifi or Trslio Prory.4.4 Sod Shifi Prory.5 Us of Pril rios.6 Eris ().7 vrs l Trsfor of Drivivs.8 vrs l Trsfor of rls.9 Muliliio d Divisio y Powrs of. Covoluio Thor.. Covoluio of Two uios.. Covoluio Thor. Eris (). Dirihl s Codios.3 ourir rl Thor.4 Th Col vrsio orul.5 Th Browih Coour.6 Us of sidu Thor i oii vrs l Trsfor.7 vrs l Trsfor of uios wih Brh Pois.8 vrs l Trsfor of uios wih fiily My Siulriis.9 Sury. Aswrs o Slf-ri Eriss. Eris () 4

47 . jiv Th oj of his ui is o dfi ivrs l rsfor wih is sil roris. W shll rov so ior hors rrdi is drivivs, irls, uliliio d divisio y owrs of. W shll lso disuss h ovoluio hor d ol ivrsio forul for l rsfor.. roduio h ls ui w sudid h l rsfor d is roris. his ui w dfi h ivrs l rsfor d slish vrious roris d rsuls ssoid wih ivrs l Trsfor.. Th vrs l Trsfor.. Dfiiio : f f, is h l rsfor of fuio f ; is lld h ivrs l rsfor of h fuio f d is wri s f f i.. f f h f is lld h ivrs l rsforio oror. El : ;.. Null uio f N N M!! ; fuio of suh h Nd, El : Th N is lld Null fuio. Th fuio f is Null fuio...3 Uiquss S T,,, ohrwis W kow h N; N 43

48 El 3 : ; urhr, if f ; f, h f N f Cosquly if f f h f f N whih ilis h, w hv wo diffr fuios, wih h s l rsfor. Th wo diffr fuios d f for S f hv h s l Trsfor i.., T, ohrwis,. H h ivrs l rsfor of fuio is uiqu if w do o llow Null fuios. This is idid i rh s hor iv low : rh s Thor : f irvl is uiqu. f f d f N d of oil ordr for N h iwis oious i vry fii, h h ivrs l rsfor of rk : Throuh his ui, w shll ssu suh uiquss ulss ohrwis sd..3 vrs l Trsfor of Cri Elry uios ro h dfiiio.. of ivrs l rsfor d l rsfor of so lry fuios iod i h ls ui, w oi (i) ;! ; N M Q P! (ii) ; ; 44 whr N. N M Q P (iii) ; ; if. (iv) si ; ; if (whr y rl or ol ur) si

49 (v) os ; ; os (vi) sih ; sih (vii) os h ; ; osh (viii) J ; ; J M P.4 So or Proris.4. iriy Prory : Thor : for ll i,, 3,..., if f d i r oss, h M N P Q i r l rsfors of h fuios fi f f... f f f... f Proof : By iriy rory of l rsfor, w hv f f... f f f... f f f... f f f... f or f f f f f f or f f f.4. Ch of Sl of Prory : (y dfiiio of ivrs l rsfor) H G K J Thor 3 : f f ; f, h f f ;, Proof : Si f f d f f d; 45

50 H f f vri, w hv T H G K J H G K JU W u S V f u du (ui u) f; f ; H G K J.4.3 irs Shifi or Trslio Prory : Thor 4 : f f ; f, h Proof : By dfiiio, w hv f ; f f ; f f d f f d f ; H, f ; f rk : Th rsul of his hor is lso rssil s f ; f ;.4.4 Sod Shifi Prory : Thor 5 : f f ; f d Proof : S T, f, h f ; or f ; f U whr U is h wll kow ui s fuio. By dfiiio, w hv ; d 46

51 d d. d f d f u du H ; f u or f ; (ui u).5 Us of Pril rios f f is of h for, whr d h r olyoils i, h rk f io h ril frios d iul r y r. El 4 : Soluio : El 5 : Soluio : Evlu S T By h liriy rory, w hv S T U V 5 S W 5 T W T 3 6 U V id w ST U VW 47 S S U V U V W U V W S 5 T W T si5 os 6. 4! ; w Th usi sod shifi hor w U V W r [Usi f ; f ] f f os h w f w (sy) os S T h w if if

52 El 6 : Soluio : f S T Si osh w U U V W U S V T h ; W h ; os, h fid 9 9 S T U V W h ; os El 7 : li y, w hv y Thor 3, S T U V W H G K J h ; os U S V T W H G 9 K J 9 h ; 9 os 3 Prov h H ddu h vlu of P Q P os ; ;, whr. Soluio : Si... 3! 3! KJ ! 3! S T U V W S T U V W S T ; ; ;! U V W S T 3 U V W ; ;

53 S T U V S T d i d i d i 4 6 U V W W 4 6 os ;...!!! Now l f So h Pui k El 8 : (i) S T S T U V. Th y hor 3, w hv k k k k k k k W os os ; k ; U V W os, w fid h S T Soluio : (i) (ii) U V W k os ; Evlu h ivrs l rsfor of 3 7 (ii) Si S T U V W S T S N MT ; ; ; S T S T U 4 3 S T ; ; j U V S W T U V W S 3 T 7 U V jw Q P U V W U V W U V W V W ; ; 3 T 34 j 4 j S U V W

54 El 9 : S T id U U S H V T K W S 3 H 3 T K 3 3 os si H G 3 KJ 3 T S T 3 os 3 si ; ; 3 3 V S W 3 os T V S si ; 3 W, S T h 3 T S 3 os si VU 3 ; U V W U V W U W U U V W H oi h ivrs l rsfor of Soluio : W kow h J ; h h 5 3. Now diffrii wih rs o ' ', w S d d J d ; d T h W h U V 3 or d d J J ; ; or h P Q h 3 ; J P J J J 3 5

55 Dduio : P Q o P 3 3 5h ; ; 4 4 h 3 ; J J. El : Us ril frios o fid h ivrs l Trsfor of Soluio : h h h N M N T W ; ; ; 4 T 4 4 S S T 4 U V U V M S T S S W ; ; T U V S U V WQ P U V W W ; ; T U V W Q P os si os si 4 N M KJ os si KJ Q P 4 os sih si osh 5

56 Slf-ri Eris - ill i h lks : S T S T S T S T ST U V W ; U V ; W... U V W ;... U V 5 4 ; W... UVW 5. os ;....6 Eris (). id vrs l rsfor of : h (i) (iii) (ii) (iv) KJ As. (i) 3os 4 si 4 osh 4 6sih Show h 3 6 (ii) 5 os3 6si3 4 (iii) si (iv) (i) (ii) si ; 3! 5! 7! ST U VW S U ; T V W J d i 5...

57 3. Evlu h ivrs l rsfor of : (i) (iv) 6 4 (ii) (v) 5 (iii) 3 9 h (vi) 4 9 (vii) 3 (viii) As.: (i) 6 os4 si 4. (ii) (iii) J (v) (vi) (vii) S T S T 4, si 3, (iv) J 3 6, J 3 6 U, U 4 (viii) os5u 4 5 K J 3 4 3! 4! Q P 4. Show h As.: 5. f 6. Show h 4 6 os ;! 4! 6! ST U VW Q... rf ; P, id ; M ;. P S T rf H G J KJ U V N Q 3 ; W !! 3! 3 7. id fuios whos l rsfors r : (i), (ii) 5 4 (iii) 3 53

58 As.: (ii) (iii) S T S T 6 (i) S T 3 5, osh, or osh U, 5 5 or U 5 6, 5, or U, o 8. Us Pril frios o fid ivrs l rsfor of h followi fuios : (i) (ii) (iii) 3 h (iv) 4 h 4 (v) h (vi) 3 h 5h As.: (i) (ii) (iii) os si (iv) si os si (v) (vi) si si 3.7 vrs l Trsfor of Drivivs Thor 6 : f f ; f, h Proof : Si, w hv N M d f d f ; ; f,,, 3,... d f d f f ; 54

59 N M d f d f ; ; f.8 vrs l Trsfor of rls Thor 7 : f f ; f, h f f u du ; Proof : ro Thor of ui, w hv f ; { } f u du f udu (rovidd h is f.9 Muliliio d Divisio y Powrs of Thor 8 : f f ; f d f, h f f ; f Proof : ro Thor 6 of ui, w hv ;, h f f ; f ; is h dir dl fuio or ui iuls fuio. 55 T U V iss) W f ; f f f f H f f rk. f f or f f f whr rk. Grliio o f ; is ossil, i.. f d ; f f d rovidd h f f f f... Thor 9 : f f ;. f f is siolly oiuous d of oil or- iss, h for >, w hv dr d suh h i f h

60 Proof : G ST f ; U VW f u du fudu G f d G ; f ; Th G ; G ; G G ; or f G G H f N M Q f v N M Q P ; v ; P G f u du Thor. f ; f, h Th G Proof : G f u dudv Si G G f u dudv fudu d G f ; r ; G; Now G ; G ; G G G ; or f G or or f N M v f Q P ; G f u dudv Th ov rsul y lso wri s N M f rl, w hv N M Q ; P Q f d f ; P... f d 56

61 . Covoluio Thor.. Covoluio of Two uios : f d wo fuios of lss A, h h ovoluio of h wo fuios d dod y f is dfid y h rlio : f f u udu...() Dfiiio : f Th quio () wri s f f u u du Th ovoluio f.. Th Covoluio Thor : Thor : f is lso kow s lu or rsul of f d. d wo fuios of lss A d l f f ;. Th f. ; fu udu f. Proof : Hr w shll rov h...() ; d f u u du; f....(3) H f u u du u { } H ; f u u du d Th rio of irio is oudd y h urvs,, u d u. Thus is h hlf of h firs qudr. u u=...(4) =u u iur. W ovr i y horiol sri whih srs fro u o. or his sri u vris fro o. 57

62 H ; f u du u d H ; f u u u u u u v f udu vdv u v u f u du u d ;. ; or f H or f f u u du f rk : Th ovoluio hor rwri s : f u udu; f; ; El : id h ivrs l rsfor of (ui u v ) (i) h (ii) (iii) lo KJ or lo Soluio : (i) Si d d KJ d (iv) o KJ h ; si d d N M KJ Q P h ; ; N M d KJ Q P d si h ; (Usi Thor 6) 58

63 (ii) P Q o P h ; ; P Q h ; P (y firs shifi hor) (iii) f si KJ lo f ; f lo lo lo N M ;. os os ; f os N M os lo ; KJ Q P o or f ; ; KJ ; ; f Bu f f whih ilis h or f (iv) f f or f ; ; ; 59 h (rodi s ov)

64 or f si or f o ; si El : id h ivrs l Trsfor of (i) (ii) 3 h (iii) lo Soluio : (i) Si (ii) (iv) 4 d N M d KJ ; ; Q P h ; h ; si ; udu si os h u du h ; os si u u du 3 h ; si os lo lo lo u u u u du u u (iii) f 6

65 or f f ; or f ; f ; or (iv) W hv lo ; lo ; Q u u u P ; ; 4 4 N M El 3 : Clul (i) (iii) 4 Q P N M du 4 u u 4 ; 4 H G 4 rf ; d i du d ; (ii) K J lo ; (iv) (usi Thor 9) (ui 4u ) ; h h ; 6

66 (v) Soluio : (i) W hv N M h 3 ; d i d i d i Q P N M Q P ; ; ; d i (ii) (iii) Q S T. rfd i rf ; d i d i M P N Q N M Q rf d i S rf ; T P M P P ; ; ; M H K f log J lo j lo M N U V W U V W f f ; P Q P f ; J f ; J N M 6

67 or N H KQ J G M J P lo ; (iv) H Si h P Q P h ; os h P Q N M P M P ; ; ; h h M h (v) Si J ; M P M N Q N os r U os ru S T, os, os os, Diffrii oh sids wih rs o ' ', w S T d d J d ; d d d J ; or J ; h 3 P Q j h 3 h 3 U V W J J P 63

68 El 4 : Aly ovoluio hor o rov h Soluio : Si S T 4 rf ; d i U V W S T ; ; U V W 4. d ; ST U VW By ovoluio hor, w El 5 : Evlu 4u. ;.. du 4 u. rf 4 ; d i siuos udu. os Soluio : f siu u du d f ; siu os u du; f d rf 64 d (u 4u du d ) si ;. os ; (y ovoluio hor). h ; h i

69 N M K J f si d d KJ ; Q P ;KJ El 6 : Aly ovoluio hor o rov h B, u u du H dudu h si os B, whr B, is lld B fuio. Soluio : f u u du By ovoluio hor, w hv f ; u u du; ;. ;. N M.,, f ; ; Now ki, w hv u u du B, Dduio : Pui u si, du si os d N M 65

70 B, si os. si os d B, si os d Thus, w h rquird rsul. Slf-ri Eris - ill i h lks :- r. f ;.... f u du; S T U V W ; S h ovoluio hor for l Trsfor.. Eris (). id h ivrs l Trsfor of (i) lo (iv) As. (i) (iv) J u du (ii) (v) os os 3 66 o S T U V W ;... (iii) (vi) lo (ii) (v). By ki us of ovoluio hor, fid (i) ; (ii) 3 9 As. (i) (ii) (iii) 8 si os h (iii) P Q (vi) h ; P (iii) 3 KJ os si h 4h ; KJ

71 3. Us ovoluio hor o fid (i) (iii) As. (iii) h ; (ii) 4h ; h ; (iv) h ; 3 (i) si si os si 4. Prov h J u u du J 5. Evlu : (iv) (ii) si os os si Q P (i) As. (i) os d (ii) d. Dirihl s Codiios f f (ii) sisfis h followi odiios (iii) (iii) Q P si d (i) (ii) is dfid i h irvl d f oh r iwis oiuous i i.. f is riodi wih riod. f f (iii) f f Th ov odiios r suffii (u o ssry) odiios for h ovr of ourir Sris..3 ourir rl Thor Thor : f d if fd ovrs (or f oiuiy of f, sisfy h Dirihl s odiios i vry fii irvl is soluly irl i, r i h h oi of f dv v u f u os v u du...(6) 67

72 f ' ' is oi of disoiuiy, h h.h.s. of (6) is rld y h vlu of f h oi of disoiuiy. 68 i.. f f Th ov odiios r suffii u o ssry. Si w kow h siv ur is lwys odd fuio of v, hrfor, w hv ro (6) d (7), w r v u dv f u si v u du...(7) i v u f dv f u du v u i v i v u or f dv f u du v u This rsul is kow s h ol for of ourir irl..4 Th Col vrsio orul Thor 3 : f f...(8) hs oiuous driviv d is of oil ordr for lr f h f f is iv y osiiv vlus of, whr d if f d f i f ; i i Proof : f hs oiuous driviv d if rrsd y h ourir s irl suh h i v i vu dv u du Now l us k S T f d,...(9) f,, h d is soluly ovr for H fro (), w hv for N M r i v u i vu f f u du dv d is soluly ovr, h, r y...()

73 rk : u rk : f f N M iv i i i i iv u f u du dv i v f f f iv dv i i f d f d, h ov roof, w lso ssu h d (ui iv, so h dv ) i u f u is soluly irl i, f u du ovrs, so h ourir s is irl hor lid. Th irio i (9) is o rford lo li, i.. i h ol l whr u iv. Th rl ur is hos so h h li lis o h rih of ll h siulriis (ols, rh ois or ssil siulriis)..5 Th Browih Coour C Th irl (9) i Thor 3 lso vlud y osidri h oour irl f d C v D B T Q E u T A G iur. whr C is h oour, s show i h i... Th oour C is kow s Browih oour d is dod y (i) li AB (ii) r BDEGA of irl of rdius wih r h orii. Also l h r BDEGA dod y, h w hv C it it f d f d f d 69

74 or it f d f d f d i it i i C i T or s N T d usi h irl of h quio (9), w hv f im f d f dp i i...() C.6 Us of sidu Thor i oii vrs l Trsfor Thor 4 : Suos h oly siulriis of f r ols whih ll li o h lf of h li for so rl os. Also suos h f Q k, whr k d M r oss, suh h i f d, h h ivrs l rsfor of f iv y f is su of rsidus of f ll h ols of f...() i T Proof : f d f d f d i i T i C whr C is h Browih oour d is h irulr r BDEGA Now y Cuhy s rsidu hor, w hv i f d su of rsidus of f C...(3) ll ols of f isid C...(4) Usi (4) i h quio (3), w hv i i T i T f d sidus isid C i f d. Tki h lii s T (or s sidus isid C f El 7 : Us ol ivrsio forul o oi h ivrs l rsfor of 7 T ), w fid h i f d d U S i V f f d i i T, W f su of rsidus of f ll h ols of.

75 . Soluio : i, w hv i i i i i j j 3 j 8 3, for 8, for i i i d i d C ( C i Browih oour) ol). su of rsidus of ols (sil ol) d ( doul Now h rsidu sil ol is iv y i S T U V W 4 d h rsidu h doul ol is iv y d i! d S T 7 U V P W Q S T i d P d h i 4 U V W

76 El 8 : id Soluio : Hr f Si f P Q h P 4 4, usi ol ivrsio forul. h i i i i 6 4 for i, hrfor Su of rsidus of i i i h i i i i i C i i Now, rsidu ol of ordr i d i d i d i d i i i 4 i ols. i i i i i i d d i d i whih r doul ols. i i i 3 i 4 i i 4 i Ad h rsidu ol of ordr 4 i i 4 i i is 7

77 i i P i i i i P h Q i i i i h h i 4 4 os si si os El 9 : Us ol ivrsio forul o oi ivrs l Trsfor of h. Soluio : or i, w hv f 8, hrfor 3 i i i i i h i i i C d d Now, sidu sil ol sidu sil ol = su of rsidus of sil ols, i i 73 i i i d i. S T V i i W i i i S T V W i i i i i i 4i i Siilrly rsidu sil ol i is i i 4i U U

78 h P Q i i i i i P i 4 4 i KJ i i i i si os h KJ.7 vrs l Trsfor of uios wih Brh Pois f f hs rh ois h Browih oour C is suily odifid.. if f hs oly o rh oi, h h oour iv i h fi..3 usd. Hr BDE d NA r h rs of h irl of rdius ' ' wih r whil HK is h r of sll irl rd of rdius. This rodur udrsood y h followi l. D El : Evlu iur.3 y h us of ol ivrsio forul. Soluio : Usi ol ivrsio forul, w hv (y Thor 3 d quio (9)) f i i i Bu h oi is rh oi of d i. d...(5) i i. Thrfor w osidr h oour C s show i h ov i..3 i.. Browih oour whih is idd h oi y s of irl of sll rdius wih r. 74

79 . d d i i i C AB BDE d i E d i HK d i K d i NA d...(6) Si h siulriy of h ird is o isid C, h irl o h lf of (6) vishs y Cuhy s hor. Also h ird sisfis h odiio of Thor 4 i.. i f d j so h o ki h lii s, h irls lo BDE d NA ds o ro. ilis h f i i i d i i AB i d d i d E HK K d...(7) Alo E, u i i so h u i u d s os fro o, u os fro o. H w hv d E d ui u u du Siilrly lo K, u i, u i i u os fro o. Thrfor d s os fro o, u d d K ui u u du Also lo HK, HK i d, w hv i i d i i i i d i i d Now susiui hs vlus i h quio (7), w 75

80 f N M N M ui u ui u i i i du du i d i u u i i i i i i j d i u i u i u i i i Bu i d. d f H G or f rf Si d i u u d i du i d u i i si u d i si u du u KJ H G u rf u si u du u KJ du i d si d Q P Q P d d os d 4. os d 4 d 4 d KJ rf H G K J El : id Soluio : Si f. P Q P d Also Si f d. 76

81 d Thrfor f d ST H G K J 4 d d d UVW vrs l Trsfor of uios wih fiily My Siulriis his s, w hv o hoos h rdius of Browih oour of h urvd orio suh h hr is oly fii ur of h siulriis isid i d h urvd orio dos o ss hrouh y siulriy. Thrfor h rquird ivrs l rsfor oid y ki rori lii s d his will lr y h followi l. osh u El : id osh N M, whr u Soluio : irs of ll, w hv o fid ou h siulriis of f u osh osh, u...(8) 4 d i d i 4 d i d i u! u 4!...! 4!... 4 u! u 4!...! 4!... h...(9) Bu y isio, i rs h is rh oi du o h rs of 77 d i i h quio (8). Bu i is o so, hrfor i is vid fro (9) h hr is sil ol. So h fuio f hs ifiily y ols whih oid y h roo of h quio j, osh or i k i, k,,... or k i or k K J K J

82 D B E C A iur.4 whih r h sil ols. H f hs sil ols d K J,,, 3,... whr Thrfor, h rquird ivrs l rsfor oid y usi h Browih oour. Th li AB is suh h ll h ols li o h lf of i. Ai w hv o hoos h Browih oour so h h urvd orio BDEA is r of irl wih r h orii '' d rdius whr is osiiv ir. This ilis h h oour dos o ss hrouh y of h ols. Now o fid h rsidus of f osh S T K J osh u osh u sidu is i osh Ai rsidu i U V W,,, 3,... is iv y h ols. W hv osh u S U i T V W S U T V W S osh osh T f C is h oour of fi..4, h i U S V T dsih id iw i osh u 4 S T U V W i osh u j os u K J U V W 78

83 i C 4 osh u d osh j osh u K J Now, ki h lii s d h irl roud ds o ro, w fid h.9 Sury Q osh u P 4 osh j osh u. his ui you sudid ior rsuls for ivrs l rsfor. Vrious hods for h vluio of ivrs l rsfor wr lid d illusrd wih h hl of solvd d usolvd rols. W lso disussd ol ivrsio forul d ivrs l rsfor of ri fuios wr oid y usi his forul.. Aswrs o Slf-ri Eriss.!. Eris os4 si 6 os 3. Jd i 4. J 5. J Eris - K J si. f f. Eris (). id h ivrs l Trsfor of h of h followi usi ol ivrsio forul : (i) (ii). (iii) (v) 3 4 (iv) 4 (vi) h 3 h 3 5h (vii) whr d r y osiiv oss. 79

84 As. (i) os (ii) (iii) (v) (vii) 3 4 (iv) 35 4 si os 4 (vi). Us h ol ivrsio forul o vlu : (i) (iii) As. (i) N M sih u u osh 3 os, (ii) 8 u (ii) 4 (iii) 3. id 4 h 6 6 As. 4. Evlu 3 N M Q P osh u 3 osh os si N M osh u osh u si os u os y h ol ivrsio forul., u u, u As. 6 u h os os 3 Q P 8

85 5. id As. rf 6. id. H ddu h KJ As. os h sih 7. Evlu sih As. N M rf k P Q rf P y h ol ivrsio forul.,, K J rf KJ H G k KJ 8

86 Ui - 3 Soluio of rdiry Diffril Equios wih Cos d Vril Coffiis d h Soluio of Boudry Vlu Prols y l Trsfor Sruur of h Ui 3. jiv 3. roduio 3. Soluio of rdiry ir Diffril Equios wih Cos Coffiis 3.3 Soluio of rdiry Diffril Equios wih Vril Coffiis 3.4 Eris 3 () 3.5 Pril diffril Equios d Boudry vlu Prols 3.6 Soluio of Boudry Vlu Prol 3.7 H Coduio Equio 3.8 Wv Equio 3.9 Sury 3.9 Aswrs of Slf-ri Eris 3. Eris 3 () 3. jiv Th i oj of his ui is o iv liio of h l rsfor for fidi soluio of ordiry diffril quios wih os d vril offiis d oudry vlu rols suh s h oduio quio d wv quio. 3. roduio Th l rsfor is Mhil ool for fidi h soluio of ordiry d ril diffril quios. By h liio of Thors 7 d 9 of Ui-, h l rsfor rdus diffril quio io lri quio (whih is kow s susidiry quio i h rsford fuio). Th rquird soluio is hus oid y fidi h ivrs l rsfor of h rsford fuio. This hod is vry usful silly wh h iiil odiios i.. h vlu of h fuio d is drivivs = (sy) r iv i h rol. Th dv of his hod is h i yilds h riulr soluio dirly wihou h ssiy of firs fidi olry fuio d riulr irl d h vlui h rirry oss. 8

87 3. Soluio of rdiry ir Diffril Equios wih Cos Coffiis us osidr lir diffril quio wih os offiis d d d d f d d d d whr d f soluio......() is iv fuio of h idd vril. Suos w w of his quio sisfyi h iiil odiios,,,,..., U d V...() wh W W lso suos h hr iss rsfor of h soluio of () d of is drivivs d d d d,,.... Also l f ; f d d d 83 Now ulilyi ll rs of () y d h iri w.r. o ' ' w liis o d usi h forul for l rsfor of drivivs, w hv d d d d... d d d d d d f d d d d f d N M d N M d h... h... f Now olli h offiis of, w hv... h h... h f 3h...(3) Th quio (3) is lld h susidiry quio. Dividi y... h, w is fuio of. Now rsolvi his io ril frios d ki ivrs l rsfor w is fuio of. This will h rqurid soluio of () udr h iv odios ().

88 rk : How o oi susidiry quio? f h iv difril quio is wri s D D D... h f, h.h.s. of h susidiry quio is oid y rli D y d y. Th firs r of h.h.s. is h l rsfor of f whrs rii rs r rs i,,..., ulilid y so olyoil i ' '. Ths olyoils r oid y dividi... 3 sussivly y,,,..., d droi off y iv owr of. 3.3 Soluio of rdiry Diffril Equios wih Vril Coffiis Th l rsfor hiqu is lso usful i solvi h diffril quios i whih h offiis r vril. or his uros, w lwys us h rsul of hor 9 of ui -, us rssio of h for El : Soluio : W hv D y d y is ivolvd i h iv diffril quio. d Solv d 4 y y, suj o odios; 4 d y y y y 4 h y 4 3 hy y h y h y y 3 4 or y y. Tki l rsfor of oh sids, w hv h y y, y y, y y, d y y 3 h y 4 h h h y h h (rsolvi io ril frios) Tki ivrs l rsfor o oh h sids, w hv y H G KJ KJ KJ 84

89 y osh os El : Solv D 9 y Soluio : y Bu y h os, if y, y K J. 9h y os y y y is o iv, so l us ssu y y y 9 y or or y. Th ki l rsfor of h iv diffril quio, w hv h. y 4 4h 9h h h h h h 4 y h h h 4 H y y os3 si 3 os Bu w r iv h y j 5 Pui h vlu of i h ov quio, w h rquird soluio i y os3 si 3 os h os, y, dy El 3 : Solv D y 85 d wh. Soluio : Tki l rsfor of oh h sids of h iv diffril quio, w h y os y y, whr y y d d 4 4 KJ y 4 y h

90 y h h h h h 5 y y 9 P Q M N P 5 M P M h h 3 4h si si si u. si udu P Q M N (y ovoluio Thor d 5 5 si si os u os du si si si u u os si si si os si y 5 4 si si os h El 4 : D 3 D y, y, y s. Soluio : Tki l rsfor o oh h sids, w 3 h y y 3 y Bu y y is o iv, so l us ssu y A h 3 y 3 A 4 si ] or y A 3 3 A 3 K J K J K J 86 A A K J

91 K J K J A K J A K J K J K J 5 A A. 5 5 A 4 A. 5 5 A A y K J j K J Bu y s A 4 5 A 4 A 5 us ro H y is h rquird soluio. d El 5 : Solv 8 CU d fuio. Soluio : Th iv quio y wri s D 8h CU Tki l rsfor o oh h sids, w 8h C U or 8 C if, d U h d is ui s h or C 4h 4h C or 8 S T 4W Tki ivrs l rsfor o oh h sids, w 87 U V 4 h

92 S T h, y, S C if 8 os r os, os, if El 6 : Solv D 4 y f whr f T 88 y Soluio : f ; f d.. d.. d f d H G K J Tki l Trsfor of h iv diffril quio, w 4 h y f y y si d y h h 4 y y 4 hq si u P si du os 4 h P Q 4 P S T S T El 7 : y y 4y if y 3 os, 4, os si, 4 si os os, 4 4, y. Soluio : ki l Trsfor of h iv diffril quio, w hv

93 y y 4 y d or d y y y y y d 4 d y. whr y d or d d d or or d h d 4 d d 4 irio, w h lo lo 4 lo or y y y J Bu y 3 H y 3 J 4 h 4 3, y 5 El 8 : Solv y y y 89, y. Soluio : Tki l rsfor of h iv quio, w y y y y d or d y d d y y y d or d y y y d d y y y y y h d or d y d 5 d y 5 y 5 y (si y is o iv, l y )

94 h or d y 3 y d or d y 3 y d whih is lir diffril quio. 3 d d KJ KJ Now.. =...(4) Thrfor h soluio of h ov quio (4) is y. d 5 or y 5, whr is h os of irio 5 Thus y y M y 5 Buy y h H y 5 is h rquird soluio. 3.4 Eris 3 () Solv h followi diffril quios y s of h l rsfor : N Q P. [As. y s d y y ; y, y d os ] 9 9 h,,, wh. [As. y dy. D 4 y 9 y 7 d h,, wh. dy 3. D 3 D y y d h iv h y y [As. y h ] 4. D 3 D D y 8, d si ] As. y Q P y.

95 d y d y dy y 3, sisfyi h odiios, y, y, y. d d d As. y K J C C 6 C 3 d y dy y, iv h y 3, y d d K J whr C 6 7. [As. y ] d y d y f, whr y, y [As. y os si f u si u du ] h j 3 8. D D 5D y, y, y, y 8 ] h [As. y os si 9. D y si, y Dy wh. As. y si os os. D 3 3D 3D y 5 [As. y A A A3 6 h wih y y 3 h, if y Dy D y. D y. D y 9 whr A, A d A 3 r rirry oss.],. [As. y os ] wh. As. y H G K J H G K J H G KJ os Q P d y dy 3. 3 y, y d d h 4. D D y 3 ; y y, y s. As. y As. y 3 os 3 si KJ Q P 3

96 h 5. id h oudd soluio of h quio y y y, y. As. y J J 6. Solv y y y, y 7. Solv y y y 4, y 8. Solv y y y ; y d y [As. y ] d y. [As. y ] d i As. y J, y, y. [As. y 9. y y y iv h y, y. [As. y ]. y y y ]. y y y, y y,. [As. y ] if y 5, y [As. y, y y [As. y ] d y. [As. y J. D D y 3. y y y 4. y y y, y 5 ] ] 3.5 Pril Diffril Equios d Boudry Vlu Prols My rols i Physis d Eiri r ovrd y ril diffril quios ohr wih ri rsrid odiios (kow s oudry odiios) of h fuio whih ris fro h hysil siuio. Suh rols r kow s oudry vlu rols. solvi suh rols, l rsfor rovids ffiv hod of k. 3., h l rsforio ws usd o rdu ordiry diffril quios o lri quios. h s wy, ril diffril quio i wo vrils d y rdud o ordiry diffril quio i y s of l rsforio wih rs o. or his, h quio us lir d h offiis of h ukow fuio d is drivivs us idd of ' ' i.. h rs of h for Thor : f u, suil rsriios o u i u u,. r s. fuio of wo idd vrils for u,, w hv u u u,, h udr 9

97 Proof : ii iii iv i u d u d u u u u,,, u u d u d whr u, u, u d r u, u, d H G K J d u u, (y irio y rs) i u, u, u, u, T, u, T r u Assui h u, so h i ii iii T u T is of oil ordr ' ' u, T u, u,. 93 s T By h ii s rul for diffriio udr h irl si, w hv u u u u d d d d d u d, d u, d u d u, d d u d u H G N M KJ Q, V,,, r, V V P N M Q P, whr V u u u u

98 u u u u,,, whr u u, H G K J iv Ai, y h ii s rul for diffriio udr h irl si, w hv u u d d u d d, u d d u d d u d u,, d d u d 3.6 Soluio of Boudry Vlu Prols l Trsfor (wih rs o or ) i o disiol oudry vlu rol ovrs h ril diffril quio (or quios) io ordiry diffril quio. Th rquird soluio h oid y solvi his quio y h hods disussd rlir. wo disiol rols, w usully ly l rsforio wi (for l, wih rs o d wih rs o ) d h rriv ordiry diffril quio. suh s, h rquird soluio is oid y doul ivrsio. This ross is usully rfrrd o s ird l rsforio. A siilr hiqu lid o hr (or hihr) disiol rols. 3.7 H Coduio Equio Th h flow i ody of hooous ril is ovrd y h h quio or KJ u u u u k y,...(5) u whr u, y,, u, is h rur i h ody, k is h hrl oduiviy, h sifi h, h dsiy of h ril of h ody d h os, is lld h diffusiviy of h ody. Also u is kow s li oror. f hr is o flow of h i h dirio, i.. h rur i h ody is idd of, h h h quio (5) os u KJ u u...(6) y whih is lld h h quio for wo disiol flow rlll o y l. 94

99 f w osidr h h flow i lo hi r or wir of os ross-sio d hooous ril whih is lo -is d is rfly isuld lrlly, so h h h flows i h -dirio oly, u dds oly d d hrfor h h quio os u u...(7) whih is kow s o-disiol h quio. 3.8 Wv Equio Th rsvrs disl u of lsi sri is ovrd y h o-disiol wv quio u u T,...(8) whr h vril u, is h disl of y oi of h sri i. Th os T, whr T is h (os) sio i h sri d is h (os) ss r ui lh of h sri. This quio is lil o h sll rsvrs virio of u, flil sri, iiilly lod o h -is d s io oio. (s i.) y u(, ) iur 3. f u, y, is h rsvrs dil of y oi, y of r i h, y l y i, h h virios of his r, ssud sll, r ovrd y h quio u u u y...(9) KJ whih is lld h wo disiol wv quio. Siilrly u u u u y KJ u Whr u is lld h li of u, y,, is h quio for h rsvrs virios i hr disios. El 9 : id h soluio of u u, whr, ohr wih h odiios 95

100 u, 3si, u,, u,. Soluio : Tki l rsfor of oh h sids of h iv ril diffril quio d usi h iiil odiios, w oi u, u, d u, whr u, u, d or d u u, 3si d or D hu 3si...() whih is sod ordr lir diffril quio whos C.. C C d si si P D 4 h h Thrfor h rl soluio of h ov quio () is u, C C 3si 4...() To vlu C d C, w k h l rsfor of hos oudry odiios whih ivolv, w hv u, u o, d u, u, Usi hs odiios i quio (), w C C d C C Solvi h ov wo quios, w fid h C, C d Thrfor, h quio () os 96 3 u, si...() 4 Now ki ivrs l rsfor of oh sids of (), w u, 3 si 4 whih is h rquird soluio.

101 El : id h soluio of u u 3, iv h u,, u, K J d u, 3 os 5. Soluio : u, u,. Th ki l rsfor o oh sids of h iv diffril quio, w fid h or d u u, u, 3 d d u u os5 d 3 Th rl soluio of ov lir diffril quio is u, C C os5...(3) 75 To vlu C, C, w k h l rsfor of oudry odiio uvolvi, w oi d j j u, u, wh u, wh ro quio (3) w C 3 3 C d C 3 C 3 Solvi hs quios, w fid C d u, wh d C d quio (3) os 3 u, os5...(4) 75 H y ki ivrs l rsfor of h ov quio (4), w oi 75 u, 3 os5 whih is h rquird soluio. El : id h soluio of h quio u u k whih ds o ro s d whih sisfis h odiios 97

102 u f wh, d u wh, Soluio : Tki l rsfor of oh h sids of h iv quio, w hv or u u k d,, u d d u d k u u, Th soluio of h quio (4) is iv y...(4) k k u, C C...(5) whr C d C r rirry oss. Si u s u, s whih ilis h C us ro. Thrfor quio (5) ivs k u, C...(6) Bu w r lso iv h u f wh, u f wh ro (6), w hv C f H fro (6), w hv u f Alyi ol ivrsio forul, w u, i i i k k f d. El : A si-ifii rod is iiilly rur ro. A i, os rur V is lid d iid h f. id h rur y oi of h solid y i. Soluio : Th rur u, disiol h quio u u k,, wih h oudry odiios, u, u, V y oi of h rod y i is ovrd y h o...(7) Tki l rsfor of oh h sids of h quio (7), w hv 98

103 or u u k d,, u d d u d k u whos soluio is k k u, C C...(8) Si u is fii wh u is lso fii wh C, ohrwis u s Tki h l rsfor of h oudry odiio u, V, w hv V u, V d ro (8), w hv u, C H u, u, V k V N M si C V k V rf k H G KJ S T Q rf P H G K J El 3 : A ifii lo sri hvi o d is iiilly rs o h -is. Th d udros riodi rsvrs disl iv y A si,. id h disl of y oi o h sri y i. Soluio : us suos h u, is h rsvrs disl of h sri y oi y i, h h oudry vlu rol is ovrd y h quio U V W u u,,...(9) y A si w u(, ) wih h iiil d oudry odiios iur 3. 99

104 ...()...(), si,...(), M u, u, u A d h disl is fii i.. u Tki h l Trsfor of quio (9), w hv u N M u or u u u d u,,, d or A.E. is d u u or D d whih ivs KJ u u A, B Si u, is lso fii A, ohrwis u os ifii wh is fii, so u,,, u B ro (), w hv u, A si u, A...(3) i.. u A wh B A H (3) ivs u, A Tki ivrs l rsfor, w u, A

105 A si w, K J,. f f,, usi sod shifi hor KJ El 4 : A flil sri hs is d ois o h -is d. A i, h sri is iv sh dfid y si of h sri y i. K J, or d rlsd. id h disl of y oi A sri is srhd w wo fid ois, d,. f i is disld i o urv u H G si K J d rlsd fro rs i h osiio i. id h disl of y i of y oi. Soluio : Th disl u, of y oi of h sri is ovrd y h quio u u...(4) wih h oudry odiios i u, iii u, ii u, iv u, si H G K J Tki l rsfor of h ov quio (4), w u u u d u,,, d Alyi h oudry odiios, D whos rl soluio is u KJ H G si K J u, C C si H G K J M H G K J P M N P Q K J...(5)

106 H G K J u, C C si. d ii, s u, Bu fro h oudry odiios i u, w hv Alyi h ov wo oudry odiios, w H G K J C C...(6) d C C...(7) Solvi (6) d (7), C C Th h quio (5) ivs H G K J u, si. H G Tki ivrs l Trsfor of oh h sids, w H G K J H G K J u, si.os El 5 : Solv h oudry vlu rol u u,, wih h oudry odiios u, u, ;, i u,, u f Soluio : Tki l Trsfor of h iv quio u u u d u,,, d Alyi h oudry odiios, w hv u d u, d K J

107 or d u d u whos rl soluio is, u, C C...(8) Bu w r iv h u f...(9),,...(3) u, f d f d i u i u viw of (33), w C H u, C Alyi h odiio (9), w f C u, f Tki ivrs l rsfor, w whr U K J u f u, Slf-ri Eris ill i h Blks : K J is Hvisid ui s fuio. d. ki l rsfor of 8 CU d vril, i ovrs o... d soluio is S T S T U V... r. y y y,... (wih y 5, y ) u ; W... U V u ; W... whr u, K J 3 wih d w.r...., u, 5

108 5. Wri wo-disiol h oduio quio? 6. Wh is oudry vlu rol? 3.9 Sury his ui you sudid h soluio of ordiry diffril quios wih os d vril offiis d oudry vlu rols y h hod of l rsfor. This hod is illusrd wih h hl of solvd ls. 3. Aswrs o Slf-ri Eris. 4 4 h d S 8 if T os os, os, if. 3 J 4. u, 5 3. d u, whr u, u, ; d r 5. u u y k u 3. Eris 3 (). Solv u u 9 suj o h odiios : u,, u,, u, si si 5 d u, [As. u, si os6 si5 os5 ]. Solv u u 6, suj o h odiios u,, u 3,, u, d u, os 6os3 8os [As. u, os si 4 os3 si os5 si ] Solv u u, iv h u,, u 5,, u, si 4 3 [As. u, si 4 ] 4

109 4. id h oudd soluio u,,, of h oudry vlu rol. u u rovidd h u, [As. u, ] 5. id h oudd soluio of u u,, suh h u, As. u rf, H G K J 6. id h soluio of h diffusio quio, u,. u u k,, suj o h iiil d oudry odiios u,, ; H G u K K J f,, d u, s d (whr k d K r rsivly h hrl diffusiiviy d oduiviy of h ril of h iv solid). v As. u f, v dv K kv KJ Q P 4 / k 7. A si-ifii solid hs is iiil rur qul o ro. A os h flu 'A' is lid h f so h K u, A. id h rur y oi of h solid. A A k As. u k, K H G 4 K rf k 8. A r of lh ' l ' is os rur u. A, h d l is suddly iv h os rur u d h d is isuld. Assui h h surf of h r is isuld, fid h rur y oi of h r y i. N l l As. u u u u rf rf, S T V k W M S T k 9. Solv h o-disiol diffusio quio i fii diu KJ U U V WQ P u u k,, 5

110 udr h odiios for ; u, u for d u, u for,. U T KJ KJ V W y oi y i, of h si-ifii rod is iv y l l As. u u rf rf, M S k k. Th rur u, h diffril quio u u k, suj o h odiios, d i u wh ii iii u is fii wh u A wh, Usi l rsfor show h h rur h f fr i is A k.. A of lh l whih hs is d fid, is iiilly rs. A os for r ui r is lid loiudilly h fr d. id h loiudil disl y oi of h y i. As. u E l, 8 M si os P l l Q P 6

111 Sruur of h Ui 4. jiv 4. roduio 4. Dfiiio 4.3 vrsio Thors 4.3. Col ourir Trsfor 4.3. ourir Si Trsfor ourir Cosi Trsfor Ui - 4 ourir Trsfor 4.4 lioshi w ourir Trsfor d l Trsfor 4.5 So Usful suls for Dir Aliios 4.6 Elry Proris of ourir Trsfor 4.6. iriy Prory 4.6. Ch of Sl Prory Shifi Prory Modulio Thor 4.7 Eris 4 () 4.8 Th Covoluio or lu of Two uios 4.8. Covoluio Thor for ourir Trsfor 4.9 Prsvl s diy for ourir Trsfor 4. ourir Trsfor of Drivivs 4. Sury 4. Aswrs o Slf-ri Eriss 4.3 Eris 4 () 4. jiv Th oj of his ui is o dfi ol ourir, ourir si d osi rsfors d slish ivrsio hors, ovoluio hor d driviv foruls for hs rsfors. 4. roduio My lir oudry vlu d iiil vlu rols i lid his, hil hysis d iri si ffivly solvd y h us of h ourir rsfor, h ourir osi d si rsfors. 7