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FOURIER SERIES PERIODIC FUNCTIONS A fuctio f () is sid to hv piod T if fo, f ( T) f ( ), wh T is positiv costt. Th st vu of T> is cd th piod of f (). EXAMPLES W kow tht f () = si = si ( 4 ) = Thfo th fuctio hs piod, 4, 6, tc. Howv, is th st vu d thfo is th piod of f(). Simi cos is piodic fuctio with th piod d t hs piod. DIRICHLET S CONDITIONS A fuctio f () dfid i c c c b pdd s ifiit tigoomtic o sis of th fom cos b si, povidd. f () is w dfid d sig vud, piodic d fiit i (c, c). f () is cotiuous o picwis cotiuous with fiit umb of fiit discotiuitis i (c, c).. f () hs o o fiit umb of mim o miim i (c, c). SOME BASIC TRIGONOMETRIC OBSERVATIONS: (i) cosπ = (-) (ii) siπ = (iii) cos()π= (iv) si()π= (-) EULER S FORMULAS If fuctio f () dfid i (c, c) c b pdd s th ifiit tigoomtic sis o cos b si th c f ( )cos d, c
b c f ( )si d, c [Fomus giv bov fo d b cd Eu s fomus fo Foui cofficits] DEFINITION OF FOURIER SERIES Th ifiit tigoomtic sis o cos b si is cd th Foui sis of f () i th itv c c, povidd th cofficits giv b th Eu s fomus. EVEN FUNCTION If f () = () i (-, ) such tht ( ) = (), th f () is sid to b v fuctio of i (-, ). If ( ) f ( ) ( ) i (,) i (, ) Such tht ( ) = ( ) o ( ) = ( ), th f () is sid to b v fuctio of i (-, ). EXAMPLE: = cos, = ODD FUNCTION v fuctios. If f () = () i (-, ) such tht ( ) = - (), th f () is sid to b odd fuctio of i (-, ). If ( ) f ( ) ( ) i (,) i (, ) f () is sid to b odd fuctio of i (-, ). Such tht ( ) = - ( ) o ( ) = - ( ), th EXAMPLE; = si, = odd fuctios. FOURIER SERIES OF EVEN AND ODD FUNCTIONS. Th Foui sis of v fuctio f () i (-, ) cotis o cosi tms (Costt tm icudd), i.. th Foui sis of v fuctio f () i (-, ) is giv b
PROBLEMS f () = o cos, wh f ( )cos d.. Th Foui sis of odd fuctio f () i (-, ) cotis o si tms, i.. th Foui sis of odd fuctio f () i (-, ) is giv b f () = b si, wh b f ( )si d.. Fid th Foui sis of piod fo th fuctio f () = ( ) i (, ). Dduc th sum of f () = Soutio: Lt f () = o cos b si i (, ) () d ( )cos o ( si cos si ) ( ) ( ) cos 4 = 4 ( ) d. b d ( )si = Usig ths vus i (), w hv usig Boui s fomu., ( - ) = 4 cos i (, )..() Th quid sis c b obtid b puttig = i th Foui
sis i (). = is i (, ) d is poit of cotiuit of th fuctio f () = ( ). Sum th Foui sis i () = f() i.. 4 cos = ( - ) 4 i... -... =. Fid th Foui sis of piod fo th fuctio f () = cos i < <. Soutio: o Lt f () = cos cosd o cos b si... () cos( ) cos( ) d si( ) cos( ). ( ) =, if = cos d si cos 4 b cossi d ( cos) d. si( ) cos( ). ( ) if, 4
si( ) si( ) d cos( ) si( ). ( ) =, if b cos si d si d = cos Usig ths vus i (), w gt f() = cos si si 4,,... si. Fid th Foui sis psio of f () = si i (-, ). Soutio: cos( ) si( ). ( ) if Sic f () is dfid i g of gth, w c pd f () i Foui sis of piod. Aso f ( ) = si[(-)] = -si = - f () f () is odd fuctio of i (-, ). Hc Foui sis of f () wi ot coti cosi tms. Lt f () = b si cos cos d si si.(), 5
( ) si Usig ths vus i (), w gt si si ( ) si ( ) si ( ) si si si ( ) si 4. Fid th Foui sis psio of f () = i (, ). Hc obti sis fo cosc Soutio: Though th g (, ) is smmtic bout th oigi, o odd fuctio. o Lt f () = i (, ) th gth of th g is cosd is ith v fuctio cos b si.... () ( ) ( ) ( ) sih ( ) cos si o sih 6
b si d si cos ( ) sih ( ) Usig ths vus i (), w gt ( ) ( ) sih sih ( ) sih ( ) si = cos i (, ) Sum of th Foui sis of f ( ) f (), i.., sih ( ) i.., ( ) cosch i.., ( ) cosch HALF-RANGE FOURIER SERIES [Sic = is poit of cotiuit of f()] Th Foui psio of o piodic fuctio f() dfid i th itv (, ) of gth is kow s hf g psio o hf g Foui sis.i pticu hf g psio cotis o cosi tms is kow s hf g Foui cosi sis of f() i th itv (, ).I simi w hf g Foui si sis cotis o si tms. (i) Th hf g cosi sis i (, ) is f () = o cos wh o f ( ) d. f ( )cos d. 7
(ii) Th hf g si sis i (, ) is f () = b si, wh b f ( )si d. (iii) Th hf g cosi sis i (, ) is giv b f () = o cos wh o f ( ) d. EXAMPLES f ( )cos d. (iv) Th hf g si sis i (, ) is giv b f () = b si, wh b f ( )si d.. Fid th hf-g (i) cosi sis d (ii) si sis fo f () = i (, ) Soutio: (i) To gt th hf-g cosi sis fo f () i (, ), w shoud giv v i.. put () tsio fo f () i (, ). f = = Now f () is v i (, ). i ( o f () =, ) cos.() f ( )cosd. cosd 8
si 4. ( ) 4( ) cos, si o f ( ) d d Th Foui hf-g cosi sis of is giv b ( ) 4 cos i (, ). (ii) To gt th hf-g si sis of f () i (, ), w shoud giv odd tsio fo f () i (-, ). i.. Put () f = - Now f () is odd i (-, ). i (-, ) = - i (-, ) f () = b si.() b f ( )si d ( ), 4, ( ) si d cos si cos if is odd if is v Usig this vu i(), w gt th hf-g si sis of i (, ).. Fid th hf-g si sis of f () = si i (, ). Soutio: W giv odd tsio fo f () i (-, ). i.. w put f () = -si[(-)] = si i (-, ) f () is odd i (-, ) 9
Lt ) ( f = si b d b.si si d si si cos cos si si ) (. si ) ( si ) ( si ) ( Usig this vus i (), w gt th hf-g si sis s si. ) ( si si. Fid th hf-g cosi sis of ) ( f = i (, ). Dduc th sum of 5. Soutio: Givig odd tsio fo ) ( f i (-, ), ) ( f is md odd fuctio i (-, ). Lt f() = b si.. () d b si cos v is if odd is if,, 4
Usig this vu i (), w gt = ) (, si 4,,5 i Sic th sis whos sum is quid cotis costt mutips of squs of b, w pp Psv s thom. d f b ) (. 8 8.. 6...,,5 i i 4. Epd ) ( f = - s Foui sis i - < < d usig this sis fid th.m.s. vu of ) ( f i th itv. Soutio: Th Foui sis of ) ( f i (-, -) is giv b () f = o b si cos. () )...(...... ) ( o o d d f si ) ( cos si cos )cos ( d d f cos cos 4cos.()
b f ( )si d cos si d si cos cos cos ( ) b Substitutig (), (), (4) i () w gt cos ( ) 4( ) ( ) f () = cos si W kow tht.m.s. vu of f() i (-, ) is Fom () w gt Fom () w gt Fom (4) w gt.........( 4) o b.(5) 4 4 o o...(6) 9 b 4( ) ( ) b Substitutig (6), (7) d (8) i (5) w gt 6 4 4 6 4 4 9 5. Fid th Foui sis fo f () = 4 4 4 4 9 Soutio: i. Th Foui sis of f () i (-, ) is giv b f () = Hc show tht 4( ) cos..(7).. (8) Th co-fficits,, b o
, ) 4(, o b Psv s thom is 4 5 5 4 4 5 6 9 5.,. 6 9 5.,. 4 4 ) ( o o i i b d b d f 9.,. 6 45 8 4 4 4 4 i i.., 5 = 9 4 FOURIER INTEGRAL THEOREM If ) ( f is giv fuctio dfid i (-, ) d stisfis Diicht s coditios, th ) ( )cos ( ) ( d dt t t f f At poit of discotiuit th vu of th itg o th ft of bov qutio is. ) ( ) ( f f EXAMPLES. Epss th fuctio ) ( fo fo f s Foui Itg. Hc vut d cos si d fid th vu of. si d Soutio: W kow tht th Foui Itg fomu fo ) ( f is ) ( )cos ( ) ( d dt t t f f.() H ) (t f = fo t i.., f(t) = i - < t <
Equtio () f (t) = fo t f (t) = i t d t f ( ) cos( t ) dt d si ( t ) d si ( ) si ( ) d si ( ) si ( ) d si cos f ( ) d. () This is Foui Itg of th giv fuctio. Fom () w gt Substitutig (4) i () w gt Puttig = w gt si cos d But si cos d = si d [Usig si (AB) si (A-B) = si A cos B] = f ( ).( fo f ( )..(4) fo. Fid th Foui Itg of th fuctio f ( ) Vif th psttio dict t th poit =. Soutio: Th Foui itg of f () is fo fo 4
f ( ) f ( t)cos( t ) dt d.() t f ( t)cos( t ) dt.cos( t ) dt t f ( t)cos( t ) dtd cos( t ) dtd cos t si( t d ) f () cos si d. () Puttig = i (), w gt f () d t t t Th vu of th giv fuctio t = is. Hc vifid. FOURIER SINE AND COSINE INTEGRALS Th itg of th fom () f ( ) si f ( t)si t dt d is kow s Foui si itg. Th itg of th fom f ( ) cos f ( t)cos t dt d is kow s Foui cosi itg. PROBLEMS. Usig Foui itg fomu, pov tht b ( b ) ( u u si u du )( u b ) (, b ) 5
Soutio: Th psc of si u i th itg suggsts tht th Foui si itg fomu hs b usd. Foui si itg psttio is giv b f ( ) si u f ( t)si ut dt du b t bt si u du si ut dt t si u du u si u du u u si ut u cosut bsi ut u cosut b u u bt b u ( b ) ( u. Usig Foui itg fomu, pov tht cos cos d 4 u si u du )( u b ) Soutio: Th psc of cos i th itg suggsts tht th Foui cosi itg fomu fo cos hs b usd. Foui cosi itg psttio is giv b f ( ) cos f ( t) cos t dt d cos cos d t cost cost dt cos d t cos( ) t cos( ) t dt 6
( ) cos d ( ) ( ) cos d ( ) ( )cos d. 4 t t cos( ) t ( ) si( ) t cos( ) t ( ) si( ) t HARMONIC ANALYSIS Th pocss of fidig th Foui sis fo fuctio giv b umic vu is kow s hmoic sis. I hmoic sis th Foui cofficits o,, d b of th fuctio = f () i (, ) giv b o = [m vu of i (, )] = [m vu of cos i (, )] b = [m vu of si i (, )] (i) Suppos th fuctio f () is dfid i th itv (, ), th its Foui sis is, d ow, f () = o cos b si o = [m vu of i (, )] = b = m vu of cos i (, ) m vu of si i (, ) (ii) If th hf g Foui si sis of f () i (, ) is, f () = b = b si, th m vu of si i (, ) (iii) If th hf g Foui si sis of f () i (, ) is, f () = b si, th 7
b = m vu of si i (, ) (iv) If th hf g Foui cosi sis of f () i (, ) is, f () = o cos o, th = [m vu of i (, )] = m vu of cos i (, ) (v) If th hf g Foui cosi sis of f () i (, ) is, f () = o cos, th EXAMPLES o = [m vu of i (, )] = m vu of cos i (, ).. Th foowig tb givs th vitios of piodic fuctio ov piod T. T 6 T f ().98..5. -.88 -.5.98 Show tht f () =.75.7 cos.4 T si, wh T Soutio: H th st vu is m ptitio of th fist thfo w omit tht vu d cosid th miig 6 vus. = 6. Giv....() T T T T T 5T wh tks th vus of,,,,, 6 6 T 5T 6 tks th vus,,, T, 4 5,. (B usig ()) Lt th Foui sis b of th fom f o ) cos b si, () ( wh o, 8
cos, si, b = 6 cos si cos si.98..98..5.866.65.58.5 -,5.866 -.55.99. - -. 4 -.88 -.5 -.866.44.76 5 -.5.5 -.866 -.5.65 o b 4.6.. 6.5, 6 6 si.456 Substitutig ths vus of o,, d b i (), w gt cos.7 f () =.75.7 cos.4 si. Fid th Foui sis upto th thid hmoic fo th fuctio = f () dfid i (, ) fom th tb 6 6 f ().4..6.8.5.88.9 Soutio: W c pss th giv dt i hf g Foui si sis. f ) b si b si b si.....() ( 6 4 6 = f() si si si si si si.4..5.87..9. 6.6.87.87.9.9 9.8 -.8 -.8 5 6 9
.5.87 -.87.44 -.44 5.88.5 -.87.44.76.88 8.9 si 6 Now b 4.. 4 b si 6.6. 7 b si 6.5. 75 Substitutig ths vus i (), w gt f () =.4 si. si.75 si 4..6.5. Comput th fist two hmoics of th Foui sis fo f() fom th foowig dt Soutio: H th gth of th itv is. w c pss th giv dt i hf g Foui si sis i.., 6 9 5 8 f () 54 897 785 5499 66 f ) b si b si () ( si si 54.5.87 6 897.87.87 9 785 5499.87 -.87 5 66.5 -.87 si Now b 7867. 84 6
b si 6 56.84 f () = 7867.84 si 56.84 si 4. Fid th Foui sis s f s th scod hmoic to pst th fuctio giv i th foowig dt. Soutio: 4 5 f () 9 8 4 8 6 H th gth of th itv is 6 (ot ) i.., = 6 o = Th Foui sis is f o ) cos cos b si b si..() ( cos si cos 9 9 9 si 8 9 5.7-9 5.6 4 4 -.9-4 8-8 8 4 4 8 6 - -.6 -.6 5 5-7.4 - -7.4 Now b b o 5-5 -.4-9.8 6 6 5 4.66, 6 6 cos 8. si. cos 6. 6 6 si 6.9
Substitutig ths vus of o,, b, d b i (), w gt f ( ) 4.66 8. cos 6. cos. si 6.9 si COMPLEX FORM OF FOURIER SERIES Th qutio of th fom f ( ) c is cd th comp fom o poti fom of th Foui sis of f () i (c, c). Th cofficit c is giv b c c i c f ( ) d i Wh =, th comp fom of Foui sis of f () i (c, c ) tks th fom PROBLEMS i f ( ) c, wh c c c f ( ) i d.. Fid th comp fom of th Foui sis of f () = Soutio: Sic = o =, th comp fom of th Foui sis is f ( ) c c i f ( ) i i d i i d i i i (, ).
Usig this vu i (), w gt i i i cos i si. Fid th comp fom of th Foui sis of f () = si i (, ). Soutio: H = o =. i Th comp fom of Foui sis is Usig this vu i (), w gt si f i ( ) c..() c si i 4 i d 4 4. 4 isi cos i i i (, ). Fid th comp fom of th Foui sis of f () = Soutio: Lt th comp fom of th Foui sis b f ( ) c i i (-, ). c f ( ) i d i d
i / d i i i i i ( ) ( ) i i cos isi sih ( ) i sih. i ( ) ( ) Usig this vu i (), w hv ( ) i i sih i (-, ) 4. Fid th comp fom of th Foui sis of f () = cos i (-, ), wh is ith o o itg. Soutio: H = o =. Th comp fom of Foui sis is f i ( ) c.() c cos. i Usig this vu i (), w gt i icos si d i i icos si icos si ( ) si si ( ) cos i i (-, ). 4
SHORT QUETIONS. Dtmi th vu of i th Foui sis psio of f ( ) i. As: f ( ) is odd fuctio.. Fid th oot m squ vu of f ( ) i th itv, ) (. As: RMS V of f ( ) i, ) ( is 5 5 d 4 5. Fid th cofficit b 5 of th itv (, ) As: H f ( ) si 5 Foui cosi sis is 4. If o f () = f ( )cos d 5 4 d 5 cos 5 i th Foui cosi sis of th fuctio f ( ) si 5 i cos cos(5 ) cos(5 ) 5 5, wh si 5 cos d si(5 ) si(5 ) d cos, if f ( ) d f ( ) f ( ) fo, fid th sum of th Foui 5, if sis of f () t. As: H is poit of discotiuit. Th sum of th Foui sis is qu to th vg of ight hd d ft hd imit of th giv fuctio t. 5
i.., f ( ) f ( ) cos 5 5. Fid b i th psio of f ( ) 49 s Foui sis i (, ). As: Sic b = f ( ) is v fuctio i (, ). 6. If f () is odd fuctio dfid i (-, ) wht th vus of As: = sic f () is odd fuctio. 7. Fid th Foui costts b fo As: Sic b = si i (, ). f ( ) si is v fuctio i (, ). 8. Stt Psv s idtit fo th hf-g cosi psio of f () i (, ). As: wh f ( ) d f ( ) d f ( )cos d 9. Fid th costt tm i th Foui sis psio of f ( ) i (, ). As: = sic f () is odd fuctio i (, ).. Stt Diicht s coditios fo Foui sis. As: (i) f () is dfid d sig vud cpt possib t fiit umb of poits i (, ). (ii) f () is piodic with piod. (iii) f () d f () picwis cotiuous i (, ). Th th Foui sis of f () covgs to 6
() f () if is poit of cotiuit (b) f ( ) f ( ) if is poit of discotiuit.. Wht ou m b Hmoic Asis? As: Th pocss of fidig th Foui sis fo fuctio giv b umic vu is kow s hmoic sis. I hmoic sis th Foui cofficits fuctio = f () i (, ) giv b,, d b of th o o = [m vu of i (, )] = [m vu of cos i (, )] b = [m vu of si i (, )]. I th Foui psio of th cofficit of si. As:, f ( ) i (, ). Fid th vu of b,, Sic f () is v fuctio th vu of b =. ( ) I i..,, f ( ) f ( ). Wht is th costt tm d th cofficit of cos, i th Foui psio of f ( ) As: Giv i (-7, 7)? f ( ) f ( ) ( ) f ( ) Th giv fuctio is odd fuctio. Hc d o. 4. Fid Foui si sis fo th fuctio f () = ; < <. As: Th Foui si sis of f ( ) b si.() 7
b f ( )si d cos si d b, wh ' ' is v 4, wh ' ' is odd 4 f ( ).si,,5, ( ) 5. If th Foui sis fo th fuctio cos cos4 f ( ) si..5 As: Puttig w gt f ( ) is si f..5 5.7...5 5.7 4..5 5.7 6. Dfi Root m squ vu of fuctio? Dduc tht...5 5.7 4 As: If fuctio = f () is dfid i (c, c), th d is cd th oot msqu(r.m.s.) vu of i (c, c) d is dotd b. c Thus. d c 7. If f ( ) is pssd s Foui sis i th itv (-, ), to which vu this sis covgs t =. As: Sic = is poit of cotiuit, th Foui sis covgs to th ithmtic m of f () t = - d = c c 8
f () f ( ) 4 4 i.., 4 8. If th Foui sis cospodig to f ( ) i th itv (, ) is ( cos b ( b ). si ), As: B usig Psv s idtit, without fidig th vus of,, b fid th vu of 8 ( ). b d 9. Fid th costt tm i th Foui sis cospodig to f ( ) cos pssd i th itv (, ). As: Giv f ( ) cos cos si Now cos d d LONG QUESTIONS. (i) Epss f ( ) si s Foui sis i. (ii) Show tht fo < <, vu of, dduc th vu of si si si. Usig oot m squ p. (i) Fid th Foui sis of piodicit fo f ( ) i < <. (ii) Fid th Foui sis psio of piod fo th fuctio f () which is dfid i (, ) b ms of th tb of vus giv bow. Fid th sis upto th thid hmoic. f ()..4.9.7.5.. 4 5 9
.(i) Fid th Foui sis of piodicit fo f ( ) fo < <. (ii) Show tht fo < <, 4 cos cos. Dduc tht 4 4. 4 4 5 96 4. (i) Fid th Foui sis fo th sis ( )., f ( ). Hc dduc th sum to ifiit of, (ii) Fid th comp fom of Foui sis of f ( ) ( ) i th fom sih i ( ) i ( ) d hc pov tht. sih 5. Obti th hf g cosi sis fo f ( ) i (, ). 6. Fid th Foui sis fo f ( ) cos i th itv (, ). 7. (i) Epdig ( ) s si sis i (, ) show tht. 5 (ii) Fid th Foui sis s f s th scod hmoic to pst th fuctio giv i th foowig dt. 4 5 f () 9 8 4 8 6 8. Obti th Foui sis fo f () of piod d dfid s foows L f ( ) L i ( L,) i (, L) Hc dduc tht. 5 8 9. Obti th hf g cosi sis fo f ( ) i (, ).. (i) Fid th Foui sis of f ( ) (ii) Obti th si sis fo th fuctio i (, ) i (, )
i f ( ) i. (i) Fid th Foui sis fo th fuctio i (, ) f ( ) d f ( ) f ( ) fo. i (, ) (ii) Dtmi th Foui sis fo th fuctio, f ( ) ( ),. Obti th Foui sis fo. 6 f ( ) i (, ). Dduc tht. Obti th costt tm d th fist hmoic i th Foui sis psio fo f () wh f () is giv i th foowig tb. 4 5 6 7 8 9 f () 8. 8.7 7.6 5..6 8. 6. 5. 6.4 9..4 5.7 4. (i) Epss f ( ) si s Foui sis i (, ). (ii) Obti th hf g cosi sis fo 5. Fid th hf g si sis of f ( ) cos i (, ). f ( ) ( ) i th itv < <. 6. (i) Fid th Foui sis psio of f () = i (, ) (ii) Fid th hf-g si sis of f () = si i (, ). 7. Epd f () = - vu of f () i th itv. s Foui sis i - < < d usig this sis fid th.m.s. 8. Th foowig tb givs th vitios of piodic fuctio ov piod T. T 6 T f ().98..5. -.88 -.5.98 T T 5T 6 T Show tht f () =.75.7 cos.4 si, wh T
9. Fid th Foui sis up to th thid hmoic fo th fuctio = f () dfid i (, ) fom th tb 6 6 f ().4..6.8.5.88.9 6 4 6 5 6. (i) Fid th hf-g (i) cosi sis d (ii) si sis fo f () = i (, ) (ii) Fid th comp fom of th Foui sis of f () = cos i (-, ).
INTEGRAL TRANSFORM FOURIER TRANSFORMS Th itg tsfom of fuctio f () is dfid b b f ( ). k( s, ) d wh k(s, ) is kow fuctio of s d d it is cd th k of th tsfom. Wh k(s, ) is si o cosi fuctio, w gt tsfoms cd Foui si o cosi tsfoms. COMPLEX FORM OF FOURIER INTEGRALS Th itg of th fom f ( ) i f ( t) i t dt d is kow s Comp fom of Foui Itg. FOURIER TRANSFORMS COMPLEX FOURIER TRANSFORMS Th fuctio F f ( ) ist f ( t). dt is cd th Comp Foui tsfom of f (). INVERSION FORMULA FOR THE COMPLEX FOURIER TRANSFORM is Th fuctio f ( ) F f ( ). ds is cd th ivsio fomu fo th Comp Foui tsfom of F [ f ( )] d it is dotd b F F( f ( )). FOURIER SINE TRANSFORMS Th fuctio F S f ( ) f ( t).si st dt is cd th Foui Si Tsfom of th fuctio f (). Th fuctio f ( ) FS f ( ).si s ds is cd th ivsio fomu fo th Foui si tsfom d it is dotd b F F ( f ( )). S S
FOURIER COSINE TRANSFORMS Th fuctio F C f ( ) f ( t).cosst dt is cd th Foui Cosi Tsfom of f (). Th fuctio f ( ) FC f ( ).coss ds is cd th ivsio fomu fo th Foui Cosi Tsfom d it is dotd b F F ( f ( )). PROBLEMS C C. Fid th Foui Tsfom of i f ( ) i si s s cos s Hc pov tht s s 6 cos ds. Soutio: W kow tht th Foui tsfom of f () is giv b F f ( ) f ( ). f ( ).. ( ( is d is is ). d d is d ( ) i s f ( ). ). is is d i s d is ) ( is is is. f ( ). is d is d s is is is s is i s is ( s is is ) ( is is is ) 4 coss s 4 si s s 4 (si s s s coss) B usig ivs Foui Tsfom w gt
f ( ). 4 s 4 s (si s (si s 4 s (si s s cos s).(coss s cos s) coss ds 4 s s cos s). (si s is ds i si s) ds s cos s) isi s ds Th scod itg is odd d hc its vus is o. i.., Puttig si s si s scos s f ( ) coss ds s s 4 s cos s si s s cos s coss ds s coss ds f ( ), w gt si s s cos s si s s s 4 6 cos ds f. s cos s s s 4 6 cos ds.. Fid th Foui si tsfom of, (o) si m d. Soutio: Th Foui si tsfom of f() is giv b 4 4, >. Hc vut H = F S f ( ) f ( ).si s d fo > F S.si s d s s Usig ivs Foui si tsfom w gt si b d b b f ( ) F.si s ds s
s s. s si s.si s ds s ds i.., i.., f ( ) s s.si s ds s s.si s ds Rpcig b m w gt i.., s.si ms ds s m.si m m i.., d [sic s is dumm vib, w c pc it b ]. Fid th Foui cosi tsfom of. Soutio: W kow tht f ( ) f ( ).coss d F C H f ( ). FC f ( ).coss d Lt f ( ) F( s) Th F C F( s).coss d Difftitig o both sids w..t. w gt, 4
5 si. ) ( si ). si (.cos.cos ) ( b b d b s s ds s df d s d s d s s d s ds d ds s df ) (.og ) (.og.. ) ( s s ds s s s F 4. Fid th Foui cosi tsfom of. b Soutio: W kow tht th Foui cosi tsfom of f() is ).cos ( ) ( d s f f F C H f b ) (.cos d s F b b C og ) ( og ) ( og cos.cos s b s b s s F F d s d s b c c b
s 5. Fid f (), if its si tsfom is. s Hc dduc tht th ivs si tsfom of. s Soutio: W kow tht th ivs Foui si tsfom of F S f () is giv b H F ( ) F f ( ).si s ds f S f ( s s f ( ) s.si s ds s S ) d f ( ) d d f ( ) d f ( ) s s. (si s) ds.coss s ds s s f ( ) t To fid th ivs Foui si tsfom of : s Put =, i (), w gt f ( ) t ( d ). s cosb d t.coss ds b PROPERTIES. Liit Popt If F(s) d G(s) th Foui tsfom of f () d g () spctiv th F f ( ) b g( ) F( s) b G( s) Poof: 6
F[ f ( ) b g( )] f ( ) b g( ) is d f ( ). is d b g( ). is d f ( ). is d b g( ). is d F( s) b G( s). Chg of Sc Popt s If F(s) is th Foui tsfom of f () th F f ( ) F, Poof: F f ( ) Put = Wh d = d f ( ). is d i.., d = d, d, F f ( ) F s f ( ). is d.. Shiftig Popt ( Shiftig i ) f ( ). is If F(s) is th Foui tsfom of f () th F f ( ) F( s) Poof: is F f ( ) f ( ). d i s. d Put Wh - = d = d, d, F f ( ) f ( ). is ( ). d is f ( ). is. d is f ( ). is. d is F( s) 7
4. Shiftig i spct of s Poof: If F(s) is th Foui tsfom of f () th F i f ( ) F( s ) F i f ( ) i f ( ) is d i( s ) f ( ). d F( s ) 5. Modutio Thom If F(s) is th Foui tsfom of f () th F f ( ) cos F( s ) F( s ) Poof: F f ( ) cos is f ( ).cos. d F f ( s f ( ). ) is i f ( s ) i i( s ) i( s. f ( ). d. f ( ). ) d f ( ) cos F( s ) F( s ) COROLLARIES ( i) FC f ( ) cos FC ( s ) FC ( s ) ( ii) FC f ( )si FS ( s) FS ( s) ( iii) FS f ( )cos FS ( s ) FS ( s ) ( iv) FS f ( )si FC ( s ) FC ( s ) 6. Cojugt Smmt Popt d f ( s ) f ( s ) Poof: If F(s) is th Foui tsfom of f () th F f ( ) F( s) W kow tht F( s) f ( ). is d 8
Tkig comp cojugt o both sids w gt is F( s) f ( ). d Put = - d = -d Wh, d, F( s) f ( ). is ( d) f ( ). is d f ( ). is d F f ( ) 7. Tsfom of Divtivs If F(s) is th Foui tsfom of f () d if f () is cotiuous, f () is picwis cotiuous difftib, f () d f () bsout itgb i (, ) d im f ( ), th F f ( ) is F( s) Poof: B th fist th coditios giv, F f () d F f () ist. F f ( ) f ( ) is f ( ) is d is f ( ) d, isf f ( ), b th giv coditio. is o it gtig b pts. is F( s). Th thom c b tdd s foows. If ( ) f, f, f,, f cotiuous, () f is picwis cotiuous, ( ) bsout itgb i (, ) d f, f, f,, f s f, f, f,, th, f ( ) F f ( ) ( ) ( is) F( s) 8. Divtivs of th Tsfom If F(s) is th Foui tsfom of f () th F. f ( ) ( df( s) i) ds 9
Poof: F( s) f ( ) is d df( s) ds d ds f ( ) is d i. f ( ) is d if f ( ) ( df( s) i) ds F. f ( ) Etdig, w gt, F. f ( ) ( i) d F( s) ds DEFINITION f ( u) g( u) du is cd th covoutio poduct o simp th covoutio of th fuctios f () d g () d is dotd b f ( )* g( ). 9. Covoutio Thom If F(s) d G(s) th Foui tsfom of f () d g () spctiv th th Foui tsfom of th covoutio of f() d g() is th poduct of thi Foui tsfoms. i.., F f ( ) * g( ) F( s). G( s) Poof is F f ( ) * g( ) f ( ) * g( ) d g( u) F( s). F( s). G( s) Ivtig, w gt f ( u) g( u) du f ( u) d d du, o chgig th od of it gtio. g( u) ius F( s) du, g( u). ius du is is b th shiftig popt.
) ( * ) ( ) ( ) * ( ) ( ). ( s G F s F F g f s G s F F PROBLEMS. Evut ) )( ( b d usig tsfoms. Soutio: W kow tht th Foui cosi tsfom of.. ) ( s is f Simi th Foui cosi tsfom of.. ) ( b s b is f W kow tht ) ( ). ( ) (. ) ( d g f ds g F f F C C ) ( ) )( (.,. ) ( ) ( ) )( (.,.......,. ) ( ) ( b b b d i b b b ds ds b s s b i d ds b s b s i b b b. Fid th Foui tsfom of d hc dduc tht (i) dt t t cos (ii) ) ( s s i F Soutio:
.. ) ( ) ( ) ( ). ( ) ( d d if if f H d f d f d f f F is is is is is ) ( ) (. d d is is ) ( ) ( ) ( ) ( s F is is is is is is Usig ivsio fomu, w gt ) (. ) ( cos cos si cos. ) ( o f d s s ds s ds s s i s ds s f is. cos dt s t
Puttig =, w gt, F. s d coss ds s ( o) cost dt t FINITE FOURIER TRANSFORMS If f () is fuctio dfid i th itv (, ) th th fiit Foui si tsfom of f () i < < is dfid s PROBLEMS F S f ( ) f ( ).si d Th ivs fiit Foui si tsfom of F S f () is f () d is giv b f ( ) F S f ( ) si Th fiit Foui cosi tsfom of f () i < < is dfid s F C f ( ) f ( ).cos d Th ivs fiit Foui cosi tsfom of F C f () is f () d is giv b f ( ) FC () F C f ( ) cos. Fid th fiit Foui si d cosi tsfoms of Soutio: Th fiit Foui si tsfom is f ( ) i < <. H F S f ( ) f ( ) f ( ).si d
4 ) ( ) ( cos cos cos si cos.si S d F Th fiit Foui cosi tsfom is C d f f F ).cos ( ) ( H ) ( f C d F cos. ) ( cos si cos si. Fid th fiit Foui si d cosi tsfoms of ) (, ) ( i f. Soutio: Th fiit Foui si tsfom of ) (, ) ( i f is ).si ( ) ( d f f F S H ) (, ) ( i f
F S.si d cos si cos ( ). Th fiit Foui cosi tsfom of f ( ) i (, ) is F C f ( ) f ( ).cos d H f ( ) i (, ) F C.cos d si cos cos p ( ). Fid f () if its fiit si tsfom is giv b, p itg d. Soutio: W kow tht th ivs Foui si tsfom is giv b f ( ) p ( ) H F S f () = p F S p Substitutig () i (), w gt ( ) f ( ) p p f ( ) p si p si p ( ) wh p is positiv ( 4 p ) p p si p p cos 4. If f ( p) fid F C f ( p) if < <. ( p ) Soutio: W kow tht F p cos H f ( p) ( p ) C f ( p) FC () F C f ( ) cos 5
6 Lt ) ( ) ( p f f F C.cos ) ( cos cos ) ( () ) ( C C p p p f f p f F
SHORT QUESTIONS. Stt th Foui itg thom. As: If f () is giv fuctio dfid i (-, ) d f ( ) f ( t) cos ( t ) dt d. Stt th covoutio thom of th Foui tsfom. As: If F(s) d G(s) th Foui tsfom of f () d g () spctiv th th Foui tsfom of th covoutio of f() d g() is th poduct of thi Foui tsfoms. i.., F f ( ) * g( ) F( s). G( s). Wit th Foui tsfom pi. As: F f () d F F( S) Foui tsfom pis. 4. Fid th Foui si tsfom of As: f ( ) f ( ).si s d F S f ) ( ( > )..si s d s s s s si b d 5. If th Foui tsfom of f () is F(s) th pov tht. F f ( ) is F( s) As: F f ( ) is f ( ). d Put - = d = d Wh, d, F f ( ) f ( ). is ( ). d is f ( ). is. d b b is f ( ). is. d is F( s) 6. Stt th Foui tsfoms of th divtivs of fuctio. As: ( ) F f ( ) ( is) F( s) 7. Fid th Foui si tsfom of As: f ) (. 7
H F S f ( ) f ( ).si s d fo > s.si s d s s s si b d b b s 8. Pov tht FC f ( ) FC Poof:, f ( ) f ( ).coss d Put Wh F C = d = d, d, F C f ( ) F C s i.., d = d f ( ).cos s d. 9. If F(s) is th Foui tsfom of f () th pov tht Poof: ( F( s) df( s) ds df( s) i) ds i F. f ( ) f ( ) d ds is f ( ). f ( ). Fid th Foui si tsfom of As: f ( ) f ( ).si s d F S d is is f ( ) d d if f ( ) f ( ).cos.si s d s s s F. f ( ). d si b d ( df( s) i) ds b b 8
s s. Fid Foui si tsfom of As: f ( ) f ( ).si s d F S.si s d si d,. Fid Foui cosi tsfom of As: F C f ( ) f ( ).si s d f ( ).coss d s s cosb d. If F(s) is th Foui tsfom of f () th FS f ( )cos FS ( s ) FS ( s ) Poof: F S f ( ) cos f ( ).cos.si s. d. f ( ) si( s) si( s) d b F S. f ( )cos F ( s ) F ( s ) S f ( )si( s). d. f ( ).si( s) d. S 4. If F(s) is th Foui tsfom of f () th F f ( ) cos F( s ) F( s ) Poof: F f ( ) cos is f ( ).cos. d 9
f ( ). is i i d. f ( ).. i( s ) i( s ) d f ( ). d f ( s ) f ( s ) f ( s ) f ( s ) F f ( ) cos F( s ) F( s ) 5. If F(s) is th Foui tsfom of f () th F f ( ) s F, Poof: F f ( ) is f ( ). d Put = d = d d i.., d = Wh F f ( ), d, F s f ( ). is d. f ( ). i s. d. Fid th Foui Tsfom of i f ( ) i si s Hc pov tht s cos s s cos s 6 ds.
. Fid th Foui cosi tsfom of.. Fid th Foui Tsfom of f () if f ( ),, Hc dduc tht si t t 4 dt 4. Evut d ( )( b ) usig tsfoms 5. Fid th Foui tsfom of (i) cost t dt s (ii) F i ( s ) 6. Show tht th Foui tsfom of 7.. Fid th Foui tsfom of f () if Hc dduc tht si d d hc dduc tht si 8. Fid th Foui si tsfom of si, f ( ),. Fid th Foui tsfom of (i) cos d 6 8 8 4 f ( ) is d f ( ),, d hc dduc tht (ii) si d 6 othwis. Stt d pov covoutio thom fo Foui tsfoms. 8. Fid th Foui cosi tsfom of. (i) Fid th Foui cosi tsfom of (ii) Fid th Foui si tsfom of, f ( ),
4. Fid Foui si d cosi tsfom of d hc fid th Foui si tsfom of d Foui cosi tsfom of.
PARTIAL DIFFERENTIAL EQUATIONS A pti diffti qutio is qutio ivovig fuctio of two o mo vibs d som of its pti divtivs. Thfo pti diffti qutio cotis o dpdt vib d o idpdt vib. so tht f, H wi b tk s th dpdt vib d d th idpdt vib. W wi us th foowig stdd ottios to dot th pti divtivs. p, q,, s, t Th od of pti diffti qutio is tht of th highst od divtiv occuig i it. Fomtio of pti diffti qutio: Th two mthods to fom pti diffti qutio. (i) B imitio of bit costts. Pobms (ii) B imitio of bit fuctios. Fomtio of pti diffti qutio b imitio of bit costts: ()Fom th pti diffti qutio b imitig th bit costts fom b b. Soutio: Giv b b... () H w hv two bit costts & b.
Difftitig qutio () pti with spct to d spctiv w gt p () b q. () Substitut () d () i () w gt p q p q, which is th quid pti diffti qutio. () Fom th pti diffti qutio b imitig th bit costts, b, c fom. b c Soutio: W ot tht th umb of costts is mo th th umb of idpdt vib. Hc th od of th sutig qutio wi b mo th. b c... () Difftitig () pti with spct to d th with spct to, w gt p c q b c Difftitig () pti with spct to, p Wh........()........() ( )..(4) c c Fom () d (4), c Fom (5) d (6), w gt p p........(5)........(6)., which is th quid pti diffti
qutio. () Fid th diffti qutio of sphs of th sm dius c hvig thi ct o th o-p.. Soutio: Th qutio of sph hvig its ct t,,b, tht is o th o -p d hvig its dius qu to c is ( ) ( b) c. () If d b ttd s bit costts, () psts th fmi of sphs hvig th giv popt. Difftitig () pti with spct to d th with spct to, w hv bp () d bq.() Fom (), b.(4) p q Usig (4) i (), b..(5) p Usig (4) d (5) i (), w gt q p p c. i.. p q c p, which is th quid pti diffti qutio. Pobms Fomtio of pti diffti qutio b imitio of bit fuctios: ()Fom th pti diffti qutio b imitig th bit fuctio f fom f b soutio: Giv f b i.. f b ()
Difftitig () pti with spct to d th with spct to, w gt wh p q u b f ' u. f ' ub..().() Eimitig f (u) fom () d (), w gt q b p i.. q bp () Fom th pti diffti qutio b imitig th bit fuctio, Soutio: Giv, () Lt u, v Th th giv qutio is of th fom u, v. Th imitio of fom qutio (), w gt, u u v v p p i.. q q q p i. p q i. p q () Fom th pti diffti qutio b imitig th bit fuctio f fom g f Soutio: Giv f g. () 4
Difftitig () pti with spct to, p f u. gv..() Wh u d v Difftitig () pti with spct to, u. gv ( ) q f. () Difftitig () pti with spct to d th with spct to, f u. 4 g v9.. (4) s.. (5) d f u. g v.( ) Difftitig () pti with spct to, u. g v. t f.. (6) Eimitig hv u d g v f 4 9 s = t fom (4), (5) d (6) usig dtmits, w i.. 5 5s t o 6 (4) Fom th pti diffti qutio b imitig th bit fuctio fom Soutio: Giv u u....() Wh u Difftitig pti with spct to d, w gt p u ( ) u u ( ) () q u. u..() u. u u u...(4) 5
s u u u ( ) (5) t u. u...(6) Fom (4) d (6), w gt i.. t u u = u u = Soutios of pti diffti qutios Cosid th foowig two qutios d b..() f..() Equtio () cotis bit costts d b, but qutio () cotis o o bit fuctio f. If w imit th bit costts d b fom () w gt pti diffti qutio of th fom p q. If w imit th bit fuctio f fom () w gt pti diffti qutio of th fom p q. Thfo fo giv pti diffti qutio w m hv mo th o tp of soutios. Tps of soutios: () A soutio i which th umb of bit costts is qu to th umb of idpdt vibs is cd Compt Itg (o) Compt soutio. (b) I compt itg if w giv pticu vus to th bit costts w gt 6
Pticu Itg. (c) Th qutio which dos ot hv bit costts is kow s Sigu Itg. To fid th g itg: Suppos tht,,, p, q f...() is fist od pti diffti qutio whos compt soutio is,,,, b..() Wh d b bit costts. Lt b f Th () bcoms, wh f is bit fuctio.,,,, f.() Difftitig () pti with spct to, w gt. f b.(4) Th imit of btw th two qutios () d (4), wh it ists, is cd th g itg of (). G soutio of pti diffti qutios: Pti diffti qutios, fo which th g soutio c b obtid dict, c b dividd i to th foowig th ctgois. () Equtios tht c b sovd b dict (pti) itgtio. () Lgg s i qutio of th fist od. () Li pti diffti qutios of high od with costt cofficits. Equtios tht c b sovd b dict (pti) itgtio: Pobms: u t ()Sov th qutio cos, t Aso show tht u si, wht. Soutio: u if u wh t d wh. t 7
u t Giv: cos, t Itgtig () pti with spct to,.() u t f t t si.() u Wh d t t i (), w gt t Equtio () bcoms u t si t f. Itgtig () pti with spct to t, w gt.() t u si g (4) Usig th giv coditio, m u wh t, w gt o g si g si Usig this vu i (4), th quid pticu soutio of () is u si t t Now im u si im i.. wh t si t t, u si. () Sov th qutio d cos simutous. Soutio: Giv...() cos...() Itgtig () pti with spct to, f..() Difftitig () pti with spct to, f Compig () d (4), w gt...(4) 8
f f cos si c......( 5) Thfo th quid soutio is si c, wh c is bit costt. Lgg s i qutio of th fist od: A i pti diffti qutio of th fist od, which is of th fom Pp Qq R wh P, Q, R fuctios of,, is cd Lgg s i qutio. Wokig u to sov ()To sov d P d Q Pp Qq R Pp Qq R, w fom th cospodig subsidi simutous qutios d. R ()Sovig ths qutios, w gt two idpdt soutios u d v b. ()Th th quid g soutio is f u v o u v o v u,. Soutio of th simutous qutios. Mthods of goupig: B goupig two of th tios, it m b possib to gt odi diffti qutio cotiig o two vibs, vthough P;Q;R i g, fuctios of,,. B sovig this qutio, w c gt soutio of th simutous qutios. B this mthod, w m b b to gt two idpdt soutios, b usig difft goupigs. d P d Q d R Mthods of mutipis: If w c fid st of th qutitis,m, which m b costts o fuctios of th vibs,,, such tht P mq R, th th soutio of th simutous qutio is foud out s foows. d P d Q d R d md d P mq R 9
Sic P mq R, d md d. If d md d is ct diffti of som fuctio u,,, th w gt du. Itgtig this, w gt d d d u, which is soutio of. P Q R Simi, if w c fid oth st of idpdt mutipis, m,, w c gt oth idpdt soutio v b. Pobms: ()Sov p q. Soutio: Giv: p q. This is of Lgg s tp of PDE wh P, Q, R. d d d Th subsidi qutios. Tkig fist two mmbs d d Itgtig w gt og og og c c i.. u Tkig fist d st mmbs i.. Itgtig w gt d d..() d d. c...() v. Thfo th soutio of th giv PDE is u, v i.., ()Sov th qutio p q. Soutio: Giv: p q. This is of Lgg s tp of PDE wh P, Q, R. d d d Th subsidi qutios..()
Usig th mutipis,,, ch tio i ()= d d d. d d d. Itgtig, w gt () Usig th mutipis,,, ch tio i ()= d d. d d d. Itgtig, w gt b () Thfo th g soutio of th giv qutio is, f. ()Show tht th itg sufc of th PDE p q. Which cotis th stight i, is. Soutio: Th subsidi qutios of th giv Lgg s qutio d d d. Usig th mutipis,,,. ch tio i ()= Itgtig, w gt d d d. () d d d. () d d d Usig th mutipis,, -, ch tio i () =. d d d. Itgtig, w gt b () Th quid sufc hs to pss though Usig (4) i () d (), w hv b (4) (5)
Eimitig i (5) w gt, b.(6) Substitutig fo d b fom () d () i (6), w gt sufc., which is th qutio of th quid Mthods to sov th fist od No Li pti diffti qutio: Stdd Tp : Equtio of th fom f ( p, q)...() i. th qutio cotis p d q o. Suppos tht b c...() is soutio of th qutio, b p, q b substitut th bov i (), w gt f (, b) o sovig this w c gt b, wh is kow fuctio. Usig this vu of b i (), th compt soutio of th giv pti diffti qutio is c () is compt soutio, To fid th sigu soutio, w hv to imit d c fom c Difftitig th bov with spct to d c, w gt, d =. Th st qutio is bsud. Hc th is o sigu soutio fo th qutio of Tp. Pobms: () Sov p q.
Soutio: Giv: p q.() Equtio () is of th fom f ( p, q). Assum b c.() b th soutio of qutio (). Fom () w gt p, q b. () b b.() Substitut () i () w gt This is compt soutio. To fid th g soutio: W put i.. c...(4) c f i (4), wh f is bit fuctio. f (5) Difftitig (5) pti with spct to, w gt f (6) Eimitig btw qutios (5) d (6), w gt th quid g soutio. To fid th sigu soutio: Difftit (4) pti with spct to d c, w gt, =.(which is bsud) so th is o sigu soutio. () Sov p q pq Soutio:
Giv: p q pq..() Equtio () is of th fom f ( p, q) Assum b c () b th soutio of qutio (). Fom () w gt p, q b () b b b...() Substitutig () i (), w gt c (4) This is compt soutio. To fid th g soutio: W put c f i (4), w gt f Difftitig (5) pti with spct to, w gt f..(5)..(6) Eimitig btw qutios (5) d (6), w gt th quid g soutio To fid th sigu soutio: Difftitig (4) with spct to d c., d = (which is bsud). So th is o sigu soutio. Stdd Tp : Equtios ot cotiig d picit, i.. qutios of th fom, p, q f.() Fo qutios of this tp,it is kow tht soutio wi b of th fom 4
.() Wh is th bit costt d is spcific fuctio to b foud out. Puttig d u, () bcoms u o u d u p. du d u q. du d du d du If () is to b soutio of (), th vus of p d q obtid shoud stisf (). d d i.. f,,..() du du Fom (), w gt d du,.(4) Now (4) is odi diffti qutio, which c b sovd b vib spb mthod. Th soutio of (4), which wi b of th fom g u b o g, b compt soutio of ().,, is th Th g d sigu soutio of () c b foud out b usu mthod. Pobms: ()Sov p q. Soutio: Giv: p q () Equtio () is of th fom f, p, q Assum u wh, u b soutio of (). d u d d p. p.() du du du d u d d q. q () du du du Substitutig qutio () & () i (), w gt d du d du 5
d du d du d du B vib spb mthod, d du B itgtig, w gt d du c c k c This is th compt soutio. To fid th g soutio: W put k f i (4), w gt f Difftit (5) pti with spct to, w gt.(4)..(5) f..(6) Eimitig btw qutios (4) d (5), w gt th quid g soutio. To fid th sigu soutio: Difftit (4) pti with spct to d k, w gt 6
..(7) d (which is bsud) So th is o sigu soutio. ()Sov9 q 4 p. Soutio: Giv: 9 q 4 p () Equtio () is of th fom f, p, q Assum u wh, u b soutio of (). d u d d p. p.() du du du d u d d q. q () du du du Substitutig qutio () & () i (), w gt 9 9 d du d du d du d du d du 4 4 9. d du 4 Itgtig th bov, w gt d du c u c c 7
k This is th compt soutio. To fid th g soutio: W put k f i (4), w gt..(4) f Difftit (5) pti with spct to, w gt..(5) f..(6) Eimitig btw qutios (4) d (5), w gt th quid g soutio. To fid th sigu soutio: Difftit (4) pti with spct to d k, w gt..(7) d (which is bsud) So th is o sigu soutio. Stdd Tp : Equtios of th fom f p g, q,.. () i.. Equtio which do ot coti picit d i which tms cotiig p d c b sptd fom thos cotiig q d. To fid th compt soutio of (), W ssum tht f p g, q,.wh is bit costt. Sovig f, p,w c gt p, d sovig g, q,w c gt q, Now d d d o i.. d, d, d pd qd Itgtig with spct to th cocd vibs, w gt, d, d b.(). Th compt soutio of () is giv b (), which cotis two bit costts d b. Th g d sigu soutio of () c b foud out b usu mthod. 8
Pobms: ()Sov pq. Soutio: Giv: pq p..() q Equtio () is of th fom f, p g, q Lt Simi, Assum p q (s) p p.() q q () d pd qd b soutio of () Substitut qutio () d () to th bov, w gt d d d Itgtig th bov w gt, d d c d c k..(4) This is th compt soutio. Th g d sigu soutio of () c b foud out b usu mthod. () Sov p q. Soutio: Giv: p q p q.. () Equtio () is of th fom f, p g, q Lt p q (s) 9
p p. () Simi, q q () Assum d pd qd b soutio of () Substitut qutio () d () to th bov, w gt d d d Itgtig th bov w gt, d d d c c This is th compt soutio. c (4) Th g d sigu soutio of () c b foud out b usu mthod. Stdd Tp 4: (Ciut s tp) Th qutio of th fom p q f ( p, q) () is kow s Ciut s qutio. Assum b c () b soutio of ()., b p, q b Substitut th bov i (), w gt b f (, b)..() which is th compt soutio. Pobm: () Sov p q pq Soutio: Giv: p q pq.() Equtio () is Ciut s qutio Lt b c..() b th soutio of ().
Put p, q b i (), w gt b b.() which is compt soutio. To fid th g soutio: W put b f i (), w gt f f (4) Difftit (4) pti with spct to, w gt f f f..(5) f Eimitig btw qutios (4) d (5), w gt th quid g soutio To fid sigu soutio, Difftit () pti with spct to, w gt. b b b..(6) Difftit () pti with spct to b, w gt. b..(7) b Mutipig qutio (6) d (7),w gt 4 4 b b q () Sov p q p p
q Soutio: Giv: p q p.() p Equtio () is Ciut s qutio Lt b c...() b th soutio of (). Put p, q b i (), w gt b b.() which is th compt soutio. To fid th g soutio: W put b f i (), w gt f f..(4) Difftit (4) pti with spct to, w gt f f f..(5) Eimitig btw qutios (4) d (5), w gt th quid g soutio To fid th sigu soutio: Difftit () pti with spct to, b b b b... (4) Difftit () pti with spct to b,.(5)
Substitutig qutio (4) d (5) i qutio (), w gt Equtios ducib to stdd tps-tsfomtios: m m Tp A: Equtios of th fom f p, q o f p, q, Wh m d costts, ch ot qu to. W mk th tsfomtios X Th p Y q. Y d m Y. X m. m P, wh P d X X Q, m Thfo th qutio f p, q qutio. m Th qutio f p, q, qutio. wh Q Y ducs to mp, Q, ducs to mp, Q,, f.which is tp f.which is tp. Pobm: ()Sov p 4 q. Soutio: Giv: p This c b witt s 4 q p q.
m Which is of th fom f p, q,, wh m=,=. Put X m ; Y P. p p X X Q. q q Y Y Substitutig i th giv qutio, P q. This is of th fom p, q, Lt Z f X Y f., wh u X Y d d P, Q du du Equtio bcoms, d Sovig fo, du d du d du d og og d du 8 8 du 8 X Y b 8 b is compt soutio. Th g d sigu soutio c b foud out b usu mthod. k k k k Tp B: Equtios of th fom f p, q o f p, q,, Wh k is costt, which is ot qu to -. W mk th tsfomtios Z k. Z d k Th P k p 4
Z Q k k p k k Thfo th qutio f p, q qutio. k k Th qutio f p, q,, tp 4 qutio. P Q ducs to f,, which is tp k k P Q ducs to f,,,, which is k k Pobms: 4 ()Sov: q p. 4 Soutio: Giv: q p. Th qutio c b witt s q p Which cotis p d q. Hc w mk th tsfomtio Simi Z P P p q Q Usig ths vus i (), w gt p () Z Q P 9..() As () is qutio cotiig P d Q o, soutio of () wi b of th fom Z b c.() Now P d Q b, obtid fom () stisf qutio () b 9 i.. b 9 Thfo th compt soutio of () is Z 9 c i. compt soutio of () is 9 c Sigu soutio dos ot ist. G soutio is foud out s usu. 5
6 Tp C: Equtios of th fom, q p f k k m, wh ;, k m W mk th tsfomtios, k m d Z Y X Th dx d d dz X Z P.. m p k m k. d q k Q k. Thfo th giv qutio ducs to, Q k P k m f This is of tp qutio. Pobm: ()Sov q p Soutio: Giv: q p It c b witt s q p..() which is of th fom q p k k m w mk th tsfomtios, k m d Z Y X i.., d Z Y X Th P d dx X Z dz d p.... p P, Simi, q Q, Usig ths i (),it bcoms
P Q () As () cotis o P d Q picit, soutio of th qutio wi b of th fom Z X by c.() Thfo P d Q b, obtid fom () stisf qutio () i.. b b, Thfo th compt soutio of () is Z X Y c Thfo th compt soutio of () is c Sigu soutio dos ot ist. G soutio is foud out s usu. p q Tp D: Equtio of th fom f, B puttig Z Z wh P d Q. X Y Pobms: ()Sov pq. Soutio: Giv: Rwitig (), X og, Y og d Z og th qutio ducs to f P, Q, pq..() p q..() p q As () cotis d, w mk th substitutios X og, Y og d Z og Th P d Z dx... P. dz X d 7
i.. p P Simi, q Q Usig ths i (), it bcoms PQ..() which cotis o P d Q picit. A soutio of () is of th fom Z X by c (4) Thfo P d Q b, obtid fom (4) stisf qutio () i.. b o b Thfo th compt soutio of () is Thfo th compt soutio of () is G soutio of () is obtid s usu. Z X Y c og og og c..(5) APPLICATION OF PARTIAL DIFFERENTIAL EQUATIONS I m phsic d giig pobms, w ws sk soutio of th diffti qutios, whth it is odi o pti, which stisfis som spcifid coditios cd th boud coditios. A diffti qutios togth with ths boud coditios is cd boud vu pobm. I this chpt w sh stud som of th most impott pti diffti qutios occuig i giig ppictios. O of th most fudmt commo phom tht foud i tu is th phom of wv motio. Wh sto is doppd i to poud, th sufc of wt is distubd d wvs of dispcmt tv pid outwd. Wh b o tuig fok is stuck, soud wvs popgtd fom th souc of soud. Whtv is th tu of wv phom, whth it is th dispcmt of tight sttchd stig, th dfctio of sttchd mmb, th popgtio of cuts d pottis og ctic tsmissio i, ths titis govd b pti diffti qutio, kow s th Wv Equtio. Vib Spb Soutio of th Wv Equtio t, t X. T t.. () Lt 8
b th soutio of th qutio.. () t Wh X is fuctio of o d T t is fuctio of t o. Th d T d XT d X T, Wh T d X Stisf qutio () t dt d i.., XT T X T () X T Th L.H.S of () is fuctio of o d th R.H.S is fuctio of t o. Th qu fo vus of th idpdt vib d t. This is possib o if ch is costt. d X T k, Wh k is costt. X T X kx.. (4) T k T.. (5) Th tu of th soutio of (4) d (5) dpds o th tu of vus of k. Hc th foowig th css is. Cs : k is positiv. Lt k p Th qutio (4) d (5) bcom D p X d D p T d d Wh D d D d dt Th soutios of ths qutios p p X A B d pt pt T C D Cs : k is gtiv. Lt k p Th qutio (4) d (5) bcom D p X d D p T Th soutios of ths qutios X Acos p Bsi p d T C cos pt Dsi pt 9
Cs : k. Th qutio (4) d (5) bcom d X d d T d dt Th soutios of ths qutios X A B d T Ct D Sic, t X. T is th soutio of th wv qutio, th th mthmtic possib soutios of th wv qutios p p pt pt t A B C D, (6), t Acos p Bsi pc cos pt Dsi pt. (7), t A B Ct D (8) d Pobms: ()A uifom stig is sttchd d fstd to two poits pt. Motio is sttd b dispcig th stig ito th fom of th cuv (i) k si d (ii) k d th sig it fom this positio t tim t=. Fid th dispcmt of th poit of th stig t distc fom o d t tim t. Soutio: fig. Th dispcmt, tof th poit of th stig t distc fom th ft d t tim t is giv b th qutio (fig.). () t
Sic th ds of th stig = d = fid, th do ot udgo dispcmt t tim. Hc, t, fo t. () d, t, fo t. () Sic th stig is sd fom st iiti, tht is, t t=, th iiti vocit of v poit of th stig i th -dictio is o. Hc,, fo.. (4) t Sic th stig is iiti dispcd i to th fom of th cuv f, th coodits,,stisf th qutio f, wh, is th iiti dispcmt of th poit i th -dictio. Hc, f, fo. (5) Wh i (i) d i (ii). Coditios (),(),(4) d (5) coctiv cd boud coditios of th pobm. W hv to gt th soutio of qutio (), th ppopit soutio, cosistt with th vibtio of th stig is, t Acos p Bsi p C cos pt Dsi pt (6) Wh A, B, C, D d p bit costts tht to b foud out b usig th boud coditios. Usig boud coditios () i (6), w hv A C cos pt Dsi pt fo t A Usig boud coditios () i (6), w hv Bsi p C cos pt Dsi pt fo t Bsi p Eith B o si p, t which is migss. If B=, th soutio bcoms, si p p p Wh,, Difftitig both sids of (6) pti with spct to t, w hv, t Bsi p. p C si pt Dcos pt t Wh p Usig boud coditios (4) i (7), w hv Bsi p. p. D fo As B d p, w gt D Usig ths vus of A, p, D i (6), th soutio ducs to t, t BCsi cos, wh,, (7)
Tkig BC=k, Eq.() hs ifiit m soutios giv bow. t, t k si cos t, t k si cos t, t k si cos, tc. Sic Eq.() is i, i combitio of th R.H.S mmbs of th bov soutios is th g soutio of Eq.().Thus th most g soutio of Eq.() is t, t ck si cos o t, t si cos.... (8) Wh is t to b foud out. Usig boud coditios (5) i (8), w hv si f, fo (9) If w c pss f() i sis compb with th L.H.S. sis of (9), w c gt th vus of. (i) f k si k si si 4 Usig this fom of f() i (9) d compig ik tms, w gt k k,, 4 4 4 Usig ths vus i (8), th quid soutio is k t k t, t si cos si cos 4 4 (ii) k If w pd f() s Foui hf-g si sis i,, tht is i th fom b si it is compb with th L.H.S sis of (9). Thus b f si d, b Eu s fomu k si d
k cos si cos k 4 odd is if k v is if, 8, Usig this vu of i (8), th quid soutio is cos si 8, t k t () Sov th o dimsio wv qutio,, t i t giv tht.,,,,,, b d t t t Soutio: Shiftig th oigi to th poit,, w gt Y X Wh Y d X,, th coodits of th poit (, ) with fc to th w sstm of coodit s. Th diffti qutio i th w sstm is,, t X t.. () Th boud coditios bcom, t Y (), t Y... (). t fo, X t Y. (4) d X i X b X i X b X Y,,,.(5) Sic th st boud coditio i th od sstm is i b i b,,, Th quid soutio of qutio () is cos si si 8, t X b t X Y
Sicsi, Wh is v itg, th soutio c b witt s 8b X t Y X, t si si cos Wh X, t.,,5, Chgig ov to th od sstm of coodits, th soutio bcoms 8b t, t si si cos,,5 Now si si si cos cos si si cos, Sic is odd. Th quid soutio is 8b t, t si cos cos Wh 8b d,,5,, t t cos cos t.,,5, () A tight sttchd stigs with fid d poits = d =5 is iiti t st i its quiibium positio. If it is sid to vibt b givig ch poit vocit (i) v v si d 5 (ii) v v si cos, 5 5 Fid th dispcmt of poit of th stig t subsqut tim. Soutio: Th dispcmt (, t) of poit of th stig t tim t is giv b t.. () W hv to sov qutio () stisfig th foowig boud coditios., t, fo t () 5, t, fo t.... (),, fo 5 (4) Sic th stig is i its quiibium positio iiti d so th iiti dispcmt of v poit of th stig is o., f, t fo 5.... (5) wh f v si 5 Fo (i) d f v si cos, Fo (ii) 5 5 4
Th suitb soutio of Eq (), cosistt with th vibtio of th stig, is, t Acos p Bsi pc cos pt Dsi pt... (6) Usig boud coditios () i (6), w hv A C cos pt Dsi pt fo t A Usig boud coditios () i (6), w hv B si 5 p C cos pt Dsi pt fo t Eith B o si5p If w ssum tht B=, w gt tivi soutio. si 5p 5p p 5 Wh,, Usig boud coditios (4) i (6), w hv B si p. C fo 5 As B, w gt C Usig ths vus of A, p, C i (6), th soutio ducs to t, t k si si, 5 5......( 7) wh k BD d,, Th most g soutio of Eq.() is t, t si cos 5 5.... (8) Difftitig both sids of (8) pti with spct to t, w hv t, t. si cos t 5 5 5..... (9) Usig boud coditio (5) i (9), w hv si v. Sic v,, 5 5 t (i) v v si d 5 v si si 4 5 5 v si si si. 5 5 4 5 5 Compig ik tms, w gt v v, d 5 4 5 4 5, fo,4,5,6 75v 5v, d 4 5 6 Usig ths vus i (8), th quid soutio is 5
75v 5v t, t si si si si 5 5 6 5 5 (ii) v v si cos 5 5 v si si 5 5 v si si si. 5 5 5 5 Compig ik tms, w gt v v, d, fo,4,5,6 5 4 5 4 5 5v 5v, d 4 5 Usig ths vus i (8), th quid soutio is 5v t 5v t, t si si si si 5 5 5 5 (4) A tut stig of gth, fstd t both ds, is distubd fom its positio of quiibium b imptig to ch of its poits iiti vocit of mgitud k. Fid th dispcmt fuctio, t. Soutio: Th dispcmt (, t) of poit of th stig t tim t is giv b t.. () W hv to sov qutio () stisfig th foowig boud coditios., t, fo t (), t, fo t... (),, fo... (4), k, t fo (5) Th suitb soutio of Eq (), cosistt with th vibtio of th stig, is, t Acos p Bsi pc cos pt Dsi pt (6) Usig boud coditios () i (6), w hv A C cos pt Dsi pt fo t A Usig boud coditios () i (6), w hv B si p C cos pt Dsi pt fo t Eith B o si p If w ssum tht B=, w gt tivi soutio. si p p p 6
7 Wh,, Usig boud coditios (4) i (6), w hv fo p C B. si As, C gt w B Usig ths vus of A, p, C i (6), th soutio ducs to,,,...(7)..., si si, wh t k t Th most g soutio of Eq.() is cos si, t t. (8) Difftitig both sids of (8) pti with spct to t, w hv cos si., t t t. (9) Usig boud coditio (5) i (9), w hv si, si b fo k Which is Foui hf-g si sis of. k i,. Compig ik tms, w gt fomu s Eu b d f b, si. d k si k 8 cos 4 si cos k 4 4 odd is if k v is if,, 4 4 Usig this vu of i (8), th quid soutio is 4 4 cos si 64, t k t
ONE DIMENSIONAL HEAT FLOW VARIABLE SEPARABLE SOLUTIONS OF THE HEAT EQUATION Th o dimsio ht fow qutio is u u t....() Lt u(,t) = X().T(t)....() b soutio of Eq.(), wh X() is fuctio of o d T(t) is fuctio of t o. u u dt d X Th XT d X T, wh T d X, stisf Eq.(). t dt d i.., XT X T i.., X T X T....() Th L.H.S. of () is fuctio of o d th R.H.S is fuctio of t o. Th qu fo vus of idpdt vibs d t. This is possib o if ch is costt. X T k, wh k is costt. X T X kx d T k T... (4) Th tu of th soutios of (4) d (5) dpds o th tu of th vus of k. Hc th foowig th css com ito big. Cs : k is positiv. Lt k p. Th qutios (4) d (5) bcom ( D p ) X d ( D p ) T, wh d d D d D. d dt Th soutios of ths qutios X C p C p d T C p t Cs : k is gtiv. Lt k p. Th qutios (4) d (5) bcom ( D p ) X d ( D p ) T, Th soutios of ths qutios X C cos p C si p d Cs : k= Th qutios (4) d (5) bcom d X dt d d dt Th soutios of ths qutios X C C d T C T C p t Sic u(, t) = X.T is th soutio of Eq.(), th th mthmtic possib soutios of Eq.() u(, t) u(, t) p p p t ( A B )....(6) p t ( Acos p Bsi p)... (7) 8
d u(, t) A B... (8) wh C C d C C hv b tk s A d B. PROBLEMS. Fid th tmptu distibutio i homogous b of gth which is isutd t, if th ds kpt t o tmptu d if, iiti, th tmptu is k t th ct of th b d fs uifom to o t its ds. Soutio: Figu 4. psts th gph of th iiti tmptu i th b. k Equtio of OA is d th qutio of AB is k i.., k ( ) Hc k, i u(,) k ( ), i Th tmptu distibutio u(, t) i th b is giv b u u t t...() W hv to sov Eq.() stisfig th foowig boud coditios. u (, t), fo t...() u(, t), fo t... () k, i u(,) k ( ), i... (4) As u(, t) hs to mi fiit wh t, th pop soutio of Eq.() is u(, t) Usig boud coditio () i (5), w hv p t ( Acos p Bsi p).....(5) 9
p t A., fo t A = Usig boud coditio () i (5), w hv p t Bsi p., fo t B o si p B = ds to tivi soutio. si p p o p, wh,,, Usig ths vus of A d p i (5), it ducs to u(, t) wh,,, Thfo th most g soutio of Eq.() is u(, t) t Bsi.(6) B si. t Usig boud coditio (4) i (7), w hv B si f ( ) i (, ), wh k, f ( ) k ( ), i i.(7) If th Foui hf-g si sis of f ( ) i (, ) is B si, it is compb with B si. Hc B b k si d 4k cos si 8k si Usig this vu i (7), th quid soutio is 8k t u(, t) si si. 8k u(, t) ( ) ( ) si( ). k ( )si d cos ( ) () t si 4
. Sov th o dimsio ht fow qutio u u t stisfig th foowig boud coditios. (i) u (, t), fo t u (ii) (, t), fo t ; d (iii) u(,) cos, Soutio: Th ppopit soutio of th qutio u u t () stisfig th coditio tht u wh t is u(, t) Difftitig () pti w..t., w hv u p t (, t) p( Asi p Bcos p) Usig boud coditio (i) i (), w hv p t ( Acos p Bsi p)....() p t p. B., fo t if p, u(, t) A, Usig boud coditio (ii) i (), w hv....() B = which is migss p t p. Asi p., fo t Eith A = o si p A = ds to tivi soutio. si p p o p, wh,,, Usig ths vus of B d p i (), it ducs to u(, t) wh,,, t Acos...(4) Thfo th most g soutio of Eq.() is u(, t) A cos. t Usig boud coditio (iii) i (5), w hv..(5) A cos cos i (, ) I g, w hv to pd th fuctio i th R.H.S. s Foui hf-g cosi sis i (, ) so tht it m b compd with L.H.S. sis. I this pobm, it is ot css. W c wit cos s ( cos), so tht compiso is possib. 4
Thus A cos cos Compig ik tms, w hv A, A, A A A4 Usig ths vus i (5), th quid soutio is 4 t u(, t) cos u u. Sov th qutio stisfig th foowig coditios. t (i) u is fiit wh t. u (ii) wh =, fo vus of t. (iii) u = wh =, fo vus of t. (iv) u u wh t =, fo < <. Soutio: W hv to sov th qutio u u () t stisfig th foowig boud coditios. u (, t), fo t () u(, t) =, fo t...() u(, ) u fo < <.... (4) Sic u is fiit wh t, th pop soutio of Eq.() is u(, t) Difftitig (5) pti w..t., w hv u p t (, t) p( Asi p Bcos p) Usig boud coditio (ii) i (6), w hv p t ( Acos p Bsi p).......(5) p. B. p t, fo t if p, u(, t) A,....(6) B = which is migss Usig boud coditio (iii) i (5), w hv p t Acos p., fo t Eith A = o cos p A = ds to tivi soutio. cos p p odd mutip of o ( ) ( ) p, wh,,,. 4
Usig ths vus of B d p i (5), it ducs to ( ) () t 4 u(, t) Acos......(7) wh,,,. Thfo th most g soutio of Eq.() is ( ) () t 4 u(, t) A cos......(8) Usig boud coditio (iv) i (8), w hv ( ) A cos. u i (, ).(9) Th sis i th L.H.S of (9) is ot i th fom of th Foui hf-g cosi sis of fuctio i (, ), tht is, cos. Hc, to fid A, w pocd s i th divtio of Eu s fomu fo th Foui cofficits. ( ) Mutipig both sids of (9) b cos d itgtig w..t. btw d, w gt ( ) ( ) A cos d u cos d A. A oth it gs i th L. H. S. vish ( ) ( ) si si u ( ) ( ) ( ) A. u. si ( ) 4u A ( ) ( ) Usig this vu i (8), th quid soutio is 4uo ( ) ( ) u(, t) cos.. ( ) () t 4 Sovd pobms () Fom pti diffti qutio b imitig bit costts d b fom b As: Giv b. () 4
p q b p q Substitutig () & () i (), w gt 4 4 () Fom pti diffti qutio b imitig th bit costts d b fom th qutio b cot. As: Giv: b cot.. () Pti difftitig with spct to d w gt p cot.. () b cot q... () () p cot (4) () b q cot.. (5) Substitutig (4) d (5) i () w gt 4 4 p cot q cot cot. p q t. () Fid th compt soutio of th pti diffti qutio p q 4 pq. As: Giv p q 4 pq... () Lt us ssum tht b c () b th soutio of () Pti difftitig with spct to d w gt p.. () q b Substitutig () i () w gt b 4b Fom th bov qutio w gt, 4b 6b 4 b b b 4. (4) Substitutig (5) i () w gt 4 b c b (4) Fid th PDE of ps hvig qu itcpts o th d is. As: Th qutio of such p is b. () 44
Pti difftitig () with spct to d w gt p b p b.. () q b q b.. () Fom () d (), w gt p q (5) Fid th soutio of p q. As: Th S.E is d d d Tkig fist two mmbs, w gt d d Itgtig w gt c i. u c Tkig st two mmbs, w gt d d Itgtig w gt c i. v c Th compt soutio is, (6) Fid th sigu itg of th pti diffti qutio p q p q. As: Th compt itg is b b. b b b Thfo 45
46 4 4 4 4 4 (7) Sov:. m q p As: m q p Giv.. () Lt us ssum tht c b () b th soutio of () Pti difftitig with spct to d w gt b q p.. () Substitutig () i () w gt m b This is th quid soutio. (8) Fom pti diffti qutio b imitig th bit costts d b fom. b As:. b Giv () Pti difftitig with spct to d w gt.. q b b q p p.. () Substitutig () i () w gt q p q p This is th quid PDE. (9) Fom pti diffti qutio b imitig th bit costts d b fom. b As: Giv b. () q q p b b p Substitutig () & () i (), w gt p q.
pq 4 () Fom pti diffti qutio b imit th bit fuctio f fom f. As: Giv : f.. p p f... q q f.. Fom (), w gt p f p Substitutig () i(), w gt.....().....()......() () Obti pti diffti qutio b imitig bit costts d b fom b. As: Giv b. () p p...( ) b q b q Substitutig () & () i (), w gt p q p q () Fid th compt itg of p q pq, wh p, q. As: Lt us ssum tht b c () b th soutio of th giv qutio. Pti difftitig with spct to d w gt p q b.. () Substitutig () i () w gt b b b Substitutig th bov i () w gt 47
c This givs th compt itg. () Fid th PDE of th fmi of sphs hvig thi cts o th i ==. As: Th qutio of such sph is Pti difftitig with spct to d w gt p....() q....() Fom (), p....() p Fom (), q....(4) q Fom () d (4), w gt p q p q This is th quid PDE. 48
VECTOR CALCULAS UNIT STRUCTURE 5. Objctivs 5. Itoductio 5. Vcto difftitio 5. Vcto opto 5.. Gdit 5.. Gomtic mig of gdit 5.. Divgc 5..4 Sooid fuctio 5..5 Cu 5..6 Itio fid 5.4 Poptis of gdit, divgc d cu 5.5 Lt Us Sum Up 5.6 Uit Ed Ecis 5. OBJECTIVES Aft goig though this uit, ou wi b b to L vcto difftitio. Optos, d, gd d cu. Poptis of optos 5. INTRODUCTION Vcto gb ds with dditio, subtctio d mutipictio of vt. I vcto ccuus w sh stud difftitio of vctos fuctios, gdit, divgc d cu. Vcto: Vcto is phsic qutit which quid mgitud d dictio both. Uit Vcto:
Uit Vcto is vcto which hs mgitud. Uit vctos og coodit is i d j, k spctiv. i = j = k = Sc Tip Vcto: Sc tip poduct of th vctos is dfid s. b c o b c. Gomtic mig of b c mius dgs, b d c. is voum of ppipd with cott W hv, b c = b c = c b b c = - b c Vcto Tip Poduct: Vcto tip poduct of b d c is coss poduct of d b b c o coss poduct of b d c b c =. c b. b c b c =. c b b. c Rmk : Vcto tip poduct is ot ssocitiv i g i.. b c b c c i.. Cop Vctos: Th vctos, b d c cop if b c = fo, b, c 5. VECTORS DIFFERENTIATION Lt v b vcto fuctio of sc t. Lt cospodig to th icmt t i t. v b th sm icmt i Th,
v v t t - v(t) v = t v t t - v(t) t Tkig imit t w gt, im v = im t t t dv v = im = im v t t - v(t) t v t t - v(t) dt t t t t dv = im dt t v t t - v(t) Fomus of vcto difftitio: t (i) d = k v = k dv k is costt dt dt (ii) (iii) (iv) d du dv u v = dt dt dt d dv du u. v = u. v. dt dt dt d dv du u v = u v dt dt dt (v) If v vi v j vk Th, Not: dv dv dv dv i j k dt dt dt dt If i j k th = = Emp : If t i t t - j t - t k fid d dt d d dt
Soutio: t i t t - j t - t k d dt d dt i t j j k Emp : t - k If cos wt b si wt wh w is costt show tht d d = w b d = -w dt dt Soutio: cos wt b si wt------------ (i) d dt cos wt b si wt------------ (ii) d dt cos wt b si wt -w si wt bw cos wt b = w cos wt b w si wt b b = b w cos wt b w si wt b w cos wt si wt b w w Agi difftitig q b b = = - b d dt - w cos wt - b w si wt = -w cos wt b si wt -w fom (i) Emp. Evut th foowig: i) d b c dt ii) d d d dt dt dt
Soutio: i) d b c dt d =. b c dt d =. b c b c. dt d dt dc db d =. b c b c. dt dt dt dc db d =. b. c b c. dt dt dt dc db d = b c b c dt dt dt Soutio: ii) d d d dt dt dt d d d d d d d dt dt dt dt dt dt dt = c (Fom Rsut i) d d = dt dt =. d dt d dt Emp 4. Evut th foowig: d Soutio: = b c dt dc d = b b c dt dt dc db d = b b c dt dt dt d = b c dt dc db d = b c b c dt dt dt d d Emp 5. Show tht = dt, wh dt =
Soutio : W hv d d dt dt d d - dt dt d d dt dt = L.H.S. = d d dt dt d d dt dt d dt d dt R.H.S Emp 6. If Soutio: = t i t - 5t = t i t j d dt 5t t i 6t j 5t j. Th show tht d dt = k L.H.S. i j k d dt 5t t t t 6t 5t
i - j k t 6t - t t 5t 5t 5 5 k 6t 6t 5 5 k R. H. S. Emp 7. If Soutio: mt mt = b. Show tht d mt mt = b...(i) d dt d dt d dt mt mt m - m b m m b mt mt m b mt mt m m Chck ou pogss: du () If = w u d dv = w v dt dt d Show tht u v = w u v dt () If = t i t - t j 7t k Fid d, dt dt d dt = (fom (i)) () If: = t i t j st - k, Fid d d d d,,, dt dt dt dt (4) If t = i cos t j 7si t j Fid d dt t t = (5) Show tht: mgitud of.. d dt = d dt wh = i j k d is
5. VECTOR OPERATOR Th vcto diffti opto is dfid s = i j k. 5.. Gdit: Th gdit of sc fuctio is dotd b gd o d is dfid s = i j k. qutit. 5.. Gomtic mig of gdit: Not tht gd is vcto Th gd is vcto ight gd to th sufc, whos qutio is,, = c, wh c is costt. Hc fo = i j k poit o sufc. d = i.. t is ight gs to d d d is o th tgt p to th sufc t P. d Gomtic psts vcto om to th sufc,, = costt. Emp 8: Fid gd, wh Soutio: gd = = i j k = i j k = i j k = i j k Emp 9: If i j k fid gd Soutio:
= i j k = Gd = i j k = i j k = i j. k = i j k = i jk gd = Emp : If i j k fid gd Soutio: = i j k = gd = =,, = i j k = i j k = i j k = i j k
= i j k =. i j k = = Emp : If fid gd t (, -, ) Soutio: gd = i j k - = i - j - k - = i 6 j - k - = i 6 j - - k At (, -, d ) gd = 6 i j k = 6 i j 4 k = -6 i 6j k Emp : Evut gd Soutio : Gd, wh = i j k = i j k = i. j. k. = i.. j. k. = i j k =
Emp : Fid gd Soutio: gd = = i j k Emp 4: Fid gd Soutio: gd og = i j k = i j k = i j k = i j k = i j k = og = i j k (og ) = i (og ) j (og ) k (og ) = i j k = i j k = i j k = = gd og = gd (og ) = gd (og ).. Emp 5: Show tht gd i j k wh
Soutio: t = i j k. =. gd. i j k ow i i i i simi j d k k j - - - - -. gd i j k - i j k
-. -. -. -. - Chck ou pogss: () If i j k d Show tht: ) gd og b) gd c) gd f () If f = 4 5 Fid gd t (,, -) () Show tht (4) If gd - 5 F,, Fid F t (,, ) (5) Show tht f wh i j k (6) Fid uit vcto om to th sufc t (,, ) [Hit :- Uit vcto om to sufc i.. ] 5.. Divgc: If v (,, ) = vi v j v k c b dfid d difftitd t ch poit (,, ) i gio of spc th divgc of v is dfid s div v =. v
= i j k. v i v j v k = v v v Emp 6 If F = i j k, fid F Soutio: div F =. F = i j k. i j k = = = 4 Emp 7 Show tht div = wh = i j k Soutio: div =. F = i j k. i j k = = = Emp 8 Fo = i j k show tht wh div = () Soutio: L.H.S. div =. = i j k. i j k = =
= =... = = = = = R.H.S. Emp 9 Evut div wh = i j k Soutio: W hv = i j k div =. = i j k = i j k. = - - - = - - - = - - - = - - - = = -
- = = Emp If Soutio: gd F = F 4 F = Fid div (gd F) = i j k 4 4 = i j 4 k div (gd F) 4 4 4 =. i j 4 k = 4 = 4 4 4 4 6 Emp Fid th vu of div vcto d = i j k Soutio: div = i. = i.. = i. = i. i wh is costt = i. i = i i. i = i
i. 5..4 Sooid Fuctio: A vcto fuctio F is cd Sooid if div F = t poits of th fuctio. 5..5 Cu: Th cu of vcto poit fuctio F is dfid s cu F F if Fi F j F k. cu F F F = i j k Fi F j F k = i j k F F F F F F F F F = i - j k Th cu of th i vocit of ptic of igid bod is qu to twic th gu vocit of bod. i.. if w = wi w j wk b th gu vocit of ptic of th bod with positio vcto dfid s = i j k th i vocit v = w. Hc cu v v = w i j k = w w w = i w -w j w -w k w -w
= i j k w -w w -w w -w = i w w j w -w k w w = w i w j w k = w cu v = w 5..6 Iottio fid: A vcto poit fuctio F is cd iottio if F t poits of th fuctio. Emp Fid cu (cu F ) If Soutio: Cu F = i j k = i - k = F = i - j k t (,, ) cu cu (F) = i j k i j k = i j k = i j k = j At (,, )
cu F () j 4 j Emp Fid cu V if Soutio: cu V V i j k V = i j k = i - j k i - - j - k - Emp 4 Evut cu wh if = i j Soutio: Cu = i j k i - j k i - j k k Emp 5 Evut cu Soutio: wh if = i j k = = i j k
cu = i j k i j k i k - j i ---------------------------- i ---------------------------- i j k = i j k Emp 5 If Soutio: cu F i j k F = i j k fid div (cu F ) i - j k i - - j - k - - i - div (cu F) k
i j k. - i k - = Emp 7 If F = gd, fid (cu F ). Soutio: F = gd i j k i j k i j (cu F) i j k k i - - j k i - - j - k - i j k 5.4 PROPERTIES OF GRADIENT, DIVERGENCE AND CURL (i) f g = f g (ii). A B =. A. B (iii) A B = A. B
Poof: (i) f g i j k f g i f g j f g k f g i f j f k i g j g g f g (ii) Lt A = Ai A j Ak B = B i B j B k. A B. A B i A B j A B k A B A B A B A A A B B B. A. B (ii) Lt A B i j k A B A B A B i A B A B i A B i A B A i i A B B
Dictio divtiv Th divtiv of poit fuctio (sc o vcto) i pticu dictio is cd its dictio divtiv og th dictio. Th dictio divtiv of sc poit fuctio φ i giv dictio is th t of chg of φ i th dictio. It is giv b th compot of gdφ i tht dictio. Th dictio divtiv of sc poit fuctio φ (,,) i th dictio of is giv b φ.. Dictio divtiv of φ is mimum i th dictio of Hc th mimum dictio divtiv is φogdφ φ. Uit om vcto to th sufc If φ (,, ) b sc fuctio, th φ (,, ) = c psts A sufc d th uit om vcto to th sufc φ is giv b φ φ Equtio of th tgt p d om to th sufc Suppos is th positio vcto of th poit,, ) O th sufc φ (,, ) = c. If ( = i j k is th positio vcto of poit (,,) o th tgt p to th sufc t, th th qutio of th tgt p to th sufc φ t giv poit o it is giv b. gdφ = If is th positio vcto of poit o th om to th sufc t th poit o it. Th vcto qutio of th om t giv poit o th sufc φ is gdφ = Th Ctsi fom of th om t,, ) o th sufc ( φ (,,) = c is φ = φ o = φ Divgc of vcto If (,, ) is cotiuous difftib vcto poit fuctio i F giv gio of spc, th th divgcs of F is dfid b F. F = divf = i F F j k
= F i If = k F j F i F F,th ).( = k F j F i F F div i.., F F F divf = Sooid Vcto A vcto F is sid to b sooid if = F div (i). = F Cu of vcto fuctio If ),, ( F is difftib vcto poit fuctio dfid t ch poit (,, ), th th cu of F is dfid b = F F cu = F k F j F i = F i If = k F j F i F F,th ) ( = k F j F i F F cu F F F k j i F cu = = F F k F F j F F i Cu F is so sid to b ottio F Iottio Vcto A vcto F is cd iottio if Cu = F (i) if = F Sc Potti If F is iottio vcto, th th ists sc fuctio φ Such tht φ = F. Such sc fuctio is cd sc potti of F Poptis of Gdit. If f d g two sc poit fuctio tht ( ) g f g f ± = ± (o) ( ) gdg gdf g f gd ± = ± Soutio: ( ) ( ) g f k j i g f ± = ±
= ( ) ( ) ( ) ± ± ± g f k g f j g f i = g k f k g j f j g i f i ± ± ± = ± g k g j g i f k f j f i = g f ±. If f d g two sc poit fuctios th ( ) f g g f fg = (o) ggdf fgdg fg gd = ) ( Soutio: ( ) = fg ( ) fg k j i = ( ) ( ) ( ) fg k fg j fg i = f g g f k f g g f j f g g f i = f k f j f i g g k g j g i f = f g g f. If f d g two sc poit fuctio th g g f f g g f = wh g Soutio: = g f g f k j i = g f i = g g f f g i = g i f f i g g = [ ] g f f g g 4. If = k j i such tht =,pov tht = Soutio: k j i = = k j i
= k j i = k j i = k j i = 5. Fid uit om to th sufc 4 = t (,-, ) Soutio: Giv tht = φ ) ( k j i = φ = ( ) ( ) ( ) k j i At (,-, ) ( ) ) (4 (4) 6 8 = k j i φ = k j i 4 4 6 6 6 6 4 = = = φ Uit om to th giv sufc t (,-,) φ φ = 6 4 4 k j i = k j i 6. Fid th dictio divtiv of = 4 φ t (,,) i th dictio of k j i Soutio: Giv = 4 φ ) 4 ( k j i = φ = ( ) ( ) ( ) k j i 8 4 At (,, ) = k j i 8 6 54 φ Giv: = k j i 6 4 = =
= D D.. φ = 6. 8 6 54 k j i k j i = [ ] [ ] 86 6 8 6 8 6 = 7. Fid th g btw th sufc 5 = d 5 = t (,,) Soutio: Lt = φ d = φ,, = = = φ φ φ,, = = = φ φ φ = k j i φ = k j i ) ( φ At (o,,) = k j 4 φ = k j i 4 φ Cos 6 4 4 4 6 4. 4. = = k j i k j φ φ φ φ θ 4 4 6 4 cos = = θ = 4 cos θ = 4 cos 8. Fid th g btw th sufcs og = d = t th poit (,,) Soutio: t og = φ d = φ = = =,, og φ φ φ,, = = = φ φ φ = k k j i ) og ( φ = k j φ
Cos 6 5 4 4. = = = φ φ φ φ θ = 6 5 cos θ 9. Fid ( ) Soutio: ( ) = ( ). = ( ) ( ) ( ) k j i = k j i = k j i = = = = = = = = = ( )= k j i = k j i = Sic = u div u u φ φ φ. ( ) = = ( ).. = k j i k j i. = = ( ) ( ) =. = ( )( ) 4. = ( )( ) ( ) [ ] ( ) = =
. If = k j i d =.Pov tht is sooid if = d is iottio fo vctos of. Soutio: = k j i div ( ) ( ) ( ) = () Now = Difftitig pti w..to, = = Simi, = = = = Now ( ) ( ) =. =. ( ) = ( ) = Fom () w hv ( ) div = = = ( ) Th vcto is sooid if div = ( ) = = = is sooid o if = - Now k j i cu = = ( ) ( ) i = i
= i = ( ) i = Cu ( ) = k j i = Cu ( ) = fo vus of Hc is iottio fo vus of.. Pov tht ( ) ( ) = k j i F 4 si cos is iottio d fid its sc potti Soutio: 4 si cos k j i F cu = = [ ] [ ] [ ] cos cos = k j i F is iottio. To Fid φ such tht φ gd F = ( ) ( ) k j i k j i = φ φ φ 4 si cos Itgtig th qutio pti w..to,, spctiv ), si ( f = φ ), ( 4 si f = φ ), ( f = φ, 4 si C = φ is sc potti. Pov tht ).( ).( = B cu A A cu B B A div Poof : ).( = B A B A div = B A i = B A i B A i = B A i A B i
= B A i. A i. B = cu B. A cua. B.Pov tht Soutio: cu cuf = F F cu cuf = F B usig b c =. c b. b c. F. F = ( ) =. F F Pobms bsd o i Itg Emp : If F ( 6) (,,) og th cuv VECTOR INTEGRATION Li, sufc d Voum Itgs = i 4j k Evut F. d fom (,,) to = t =, = t, t Soutio: Th d poits (,, ) d (,, ) Ths poits cospod to t = d t = d = dt, d = t, d = t F. d = ( 6) d 4d d C C 5 7 = ( t 6t ) dt 4t ( tdt) t ( t ) dt 6 9 = ( t 8t 6t ) 9 dt t 4t 6t 7 = ( ) = [( 4 6) ] = 5 C Emp : Show tht F = i j k is cosvtiv vcto fid.
Soutio: If F is cosvtiv th = F is cosvtiv vcto fid. i j k Now F = = i j k = F Sufc Itgs Dfiitio: Cosid sufc S. Lt dot th uit outwd om to th sufc S. Lt R b th pojctio of th sufc o th XY p. Lt f b vcto vud dfid i som gio cotiig th sufc S. Th th sufc itg of f is dfid to b Emp ; Evut S S f. f. ds = d. d R. k f. ds wh F = i j k d S is th sufc of th cid = icudd i th fist octt btw th ps = d =. Soutio: Giv φ F = i j k = φ = i j φ = = = 4 4 Th uit om to th sufc = φ φ i j = = i j F. = i j k. i j =
(i) (ii) (iii) INTEGRAL THEOREMS Guss s divgc thom Stok s thom G s thom i th p G s Thom Sttmt: If M(,) d N(,) cotiuous fuctios with cotiuous pti divtivs i gio R of th p boudd b simp cosd cuv C, th N M Md d = dd, wh C is th cuv dscibd i th c R positiv dictio.
Vif G s thom i p fo th itg ( ) d d tk oud th cic = 4 Soutio: G s thom givs N M Md Nd = dd c R Cosid ( ) d d c M = N = M N =, = N M dd R dd dd ( ) = R R = [A of th cic] = π =. π. 4 = π () Now Md Nd W kow tht th pmtic qutio of th cic = 4 = cosθ = siθ d = siθdθ, d = cosθdθ Md Nd = ( ) d d cosθ 4siθ siθdθ cosθ cosθ d = ( )( ) ( ) θ = cosθ siθ 8si θ 4cos θdθ Wh θ vious fom to π π ( cosθ siθ 4si θ 4) Md Nd = dθ C π cos θ = si θ 4 4 d θ π = ( si θ 6 cos θ ) d θ cos θ si θ = 6θ = π = π.() Fom () d () π c
N M Md Nd = dd c R Hc G s Thom is vifid. Emp Usig G s thoms fid th of cic of dius. Soutio: B G s thom w kow tht A cosd b C = d d Th pmtic qutio of cic of dius is = Wh θ π π C A of th cic = cosθ ( cosθ ) siθ ( siθ ) dθ Emp : π = ( cos θ si θ ) dθ π = dθ θ π = π = [ ] Evut [( ) d cosd] c cosθ, = siθ si wh c is th tig with π vtics (,),(,) d ( π,) Soutio: Equtio of OB is = π = π
N M B G s thom Md Nd = dd c R M H M = si, = N N = cos, = si [( si ) d cosd] = ( si )dd C R π π I th gio R, vis fom = to d vis fom = to = C π ( si ) d cosd = ( si ) π = [ cos ] dd π π d π π π = cos d π π π = si π 4 π π π = = π 4 π Emp 4 Vif G s thom i th p fo 8 d 4 6d wh C is th boud of th gio dfid ( ) ( ) C b X =, =, = Soutio: W hv to pov tht
c N M Md Nd = dd R M = 8, N = 4 6 M N = 6, = 6 B G s thom i th p N M Md Nd = dd c R = ( ) dd = = ( ) 5 d ( ) = 5 Cosid Md Nd = c OA Aog OA, =, vis fom to OA AB Md Nd = 5 = BO d = [ ] = Aog AB, = - d = d d vis fom to. AB STOKE S THEOREM Md Nd = [ 8( ) 4( ) 6( ) ]d ( ) 4( ) 8 = 8 = = 8
If S is op sufc boudd b simp cosd cuv C d if vcto fuctio F is cotiuous d hs cotiuous pti divtivs i S d o C, th cu F. ds = F. d wh is th uit vcto om to th c sufc (i) Th sufc itg of th om compot of to th itg of th tgti compot of F tk oud C. Emp Vif Stok s thom fo F ( ) cu F is qu = i j k wh S is th upp hf of th sph = d C is th cicu boud o = p. Soutio: B Stok s thom H =k ( ) c F = i j k cuf = F. d = cuf. ds i j k = i [ ] j( ) k( ) s = k sic C is th cicu boud o = p S = of th cic cu F. ds = S dd = π ( ) = π.() ON =, c O C, = cosθ, = siθ d = si θdθ, d = cosθdθ θ vis fom to π F. d = cuf. ds s