Vector Integration. Line integral: Let F ( x y,
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1 Vetor Integrtion Thi hpter tret integrtion in vetor field. It i the mthemti tht engineer nd phiit ue to deribe fluid flow, deign underwter trnmiion ble, eplin the flow of het in tr, nd put tellite in orbit. In prtiulr, we define line integrl, whih re ued to find the wor done b fore field in moving n obet long pth through the field. We lo define urfe integrl o we n find the rte tht fluid flow ro urfe. Along the w we develop e onept nd reult, uh onervtive fore field nd Green theorem, to implif our lultion of thee new integrl b onneting them to the ingle, double, nd triple integrl we hve lred tudied. Obetive : At the end of thi unit he will be ble to undertnd : Line integrl of vetor funtion re helpful in determining totl wor done b fore F & find the irultion of F. Green theorem etblihe onnetion between double integrl nd line integrl & Green theorem in plne pplie to impl onneted region bounded b loed urve Green theorem n lo be etended to line integrl in pe Stoe theorem relte the line integrl of vetor funtion to the urfe integrl of the url of the vetor funtion Stoe theorem i ueful in trnforming line integrl in to urfe integrl nd vie ver Green theorem in plne i peil e of toe theorem. The ignifine of the Divergene theorem lie in the ft tht urfe integrl m be epreed Volume integrl nd vie ver Line integrl: Let F (,, be vetor funtion nd urve AB. Line integrl of vetor funtion F long urve AB i defined integrl of the omponent of F long the tngent to the urve AB. omponent of F long tngent PT t P. Line Integrl Therefore Line integrl Dot produt of F nd unit vetor long PT dr dr F i unit vetor long PT d d dr F from A to B long the urve d dr F d d F dr
2 Note: ( Wor. If F repreent the vrible fore ting on prtile long r AB, then the totl wor done F dr ( irultion: If V repreent the veloit of liquid then V dr i lled the irultion of V round the loed urve. If the irultion of V round ever loed urve i ero then V i id to be irrottionl there. ( When the pth of integrtion i loed urve then the nottion i in ple of. Emple:. If fore F i diple prtile in the -plne from (, to (, long urve Solution: wor done F dr r i putting the vlue nd d 8d,. Find the wor done. dr di d d ( i ( di d ( d we get [ ( d ( 8d]
3 d. Evlute F dr where F i nd i the boundr of the qure in the plne nd bounded b the line,, nd. Solution: From the figure,we hve F dr F dr F dr F dr F OA AB B O dr Here r i dr di d, F i F dr d d On OA,, F dr d F dr d OA On AB,, d F dr d F dr d On B,, d F dr d B F dr d
4 On O,, F dr O F dr F dr.. A vetor field i given F ( i ( (. Evlute F dr Along the pth i t, t, t from t to t. Solution: we hve b definition [ ( d ( d ( d ] F dr [( t ( dt ( t ( t dt ( t t ( t dt ] 7 t t t t t t t t t t. If F i 7. 87, evlute dr o t, in t, ot from t to Solution: We hve r i F long the urve π t. dr di d d i F dr d d d
5 ( d d i ( d d ( d d [ ot( in t dt ot( ot dt] i [ in t( in t dt ot( in t ] in t( ot dt ot( in t dt [ ] ( ot in t o t i ( in t ot in t F dr π π [ ( ot in t o t i ( in t ot in t ]dt o t in t i dt π ( o t in t dt π π o t in t t i t in t o t π oπ inπ o in ( i π inπ oπ in o π i π π i π. The elertion of prtile t time t i given b 8 ot i 8in t t. If the veloit v nd diplement r be ero t t, find v nd r t n point t. d r Solution: Here 8o t i 8in t dt On integrting, we hve t dr v i8o tdt 8in t dt tdt dt v in t i o t t (
6 At t, v putting t nd v, we get in t i dr v dt Agin integrting, we get ( o t t r i t dt ( o t dt in t dt r ot i ( At, t, r ( in t t t putting t nd r in, ( we get Hene i i r.. If A ( i ( ot i ( in t t t, Evlute the line integrl A dr from (,, to (,, long the urve Solution: We hve t, t, t. [ i ] [ i d d d] A dr [( d d] If t, t, t, then point (,, nd (,, orrepond to t nd t repetivel. [ ] Now, A dr ( t t dt t t d( t t( t d( t 7 A dr [( 9 t dt t t dt ( t t dt] 7 t t t A dr 9 8 7
7 7. A vetor field i given b F ( in i ( o irulr pth,. Solution: We hve Wor done F dr. Evlute the line integrl over d [( in i ( o ] [ di d ] Q & ( in d ( o d ( in d o d d F dr ( d in d The prmetri eqution of the given pth re oθ, inθ. θ vrie from to π π [ oθ in( inθ ] F dr d π π [ oθ in( inθ ] F dr d 8. Evlute oθ oθ dθ π π o θdθ o θ dθ π [ oθ in( inθ ] in θ θ π π π. A n ds, where A 8 i nd S i the prt of the plne inluded in the firt otnt. Solution: Here A 8 i (,, Given f f i i Unit norml vetor t n point (,, of i i 9 7 Norml vetor ( n ( (
8 d d ds n Now, 7 d d ( i d d 7 7 d d A n ds 7 ( 8 i ( i d d 7 d d Putting the vlue of --, we get ( 8 ( dd ( ( ( ( ( d dd ( d ( [ ] ( d d ( ( ( 7 d dd
9 [ 9 ] [ ] SURFAE INTEGRAL A urfe r f(u,v i lled mooth if f(u,v poe ontinuou firt order prtil derivtive. Let F be vetor funtion nd S be the given urfe. Surfe integrl of vetor funtion F over the urfe S i defined the integrl of the omponent of F long the norml to the urfe. omponent of F long the norml F n, where n i the unit norml vetor to n element d nd grd f d d n d grd f ( n Surfe integrl of F over S F n ( F nd Note: ( Flu ( F nd where, F repreent the veloit of liquid. d If ( F n, then F i id to be olenoidl vetor point funtion.
10 Emple 9. Evlute ( d i where S i the urfe of the phere in the firt otnt. Solution: Let φ Vetor norml to the urfe i φ φ φ φ ( i i i i i n φ φ Given ( i F Therefore ( i i n F Now ( ( d d n d d n F n d F d d d ( d 8 Emple : Show tht ( d n F, where i F nd S i the urfe of the ube bounded b the plne,,,,,,
11 Solution: But ( F n d ( F n d ( F n d ( F n d ( F n d ( F n d ( F nd OAB DEFG OAB DEFG OAGF BED ABDG ( F n d ( i ( d d ( dd Q OAB ( F n d ( i ( d d ( dd DEFG d d [ ] ( i ( d d ( dd Q OAGF ( i ( d d ( dd Q BED d d [ ] [ ] ( i ( i d d dd dd Q ABDG [ ] OEF
12 ( i ( i d d ( dd Q OEF On putting l thee vlue, we get ( F n d VOLUME INTEGRAL Let F be vetor point funtion nd volume V enloed b loed urfe. The volume integrl Emple : If F v F dv region bounded b the urfe, Solution: Fdv ( i ddd v i, Evlute F dv where, v i the,,,,,. d d ( i d d d d d [ i ] [ i i ] d i i [ i 8 i 8 ]d v
13 8 i i 8 i i i [ i ] Sttement: If (, ϕ(, GREEN S THEOREM φ ϕ φ,, nd be ontinuou funtion over region R bounded b imple loed urve in -plne, then ϕ φ ( φ d ϕ d d d Emple : A vetor field F i given b F in i ( o R. Evlute the line integrl F dr where i the irulr pth given b Solution:.
14 Given F in i ( o F dr in i ( o i [ ] ( d d [ in d ( o d ] On ppling Green Theorem, we hve ϕ φ ( φ d ϕ d d d S [( o ] d d o Where i the irulr plne urfe of rdiu. d d Are of the irle π. S Emple : Uing Green theorem, evlute ( d d, where i the boundr deribed ounter lowie of the tringle with vertie (,,(,,(,. Solution: B Green theorem, we hve ϕ φ ( φ d ϕ d d d ( d d ( d d ( d d ( d[ ] R ( d[ ] ( d
15 Emple : Uing Green theorem, evlute ( 8 d ( d i the boundr of the region defined b nd. Solution: Given tht ( 8 d ( d B Green theorem, we hve ϕ φ ( φ d ϕ d d d ( ( 8 d d [( ] d d [( ] d d d [ ]d Emple : Uing Green theorem, evlute [( d ( d], where, where i the boundr of the re enloed b X-i nd the upper hlf of the irle.. Solution: Given tht [( d ( d]
16 B Green theorem, we hve ( d d d d φ ϕ ϕ φ ( ( d d ( [ ] d d ( [ ] d d d ( d ( d d ( STOKE S THEOREM ( Reltionhip between line integrl nd urfe integrl
17 Sttement: Surfe integrl of the omponent of url F long the norml to the urfe S, ten over the urfe S bounded b urve i equl to the line integrl of the vetor point funtion F ten long the loed urve. Therefore F dr S url F n d Where n o α i o β oγ i unit eternl norml to n urfe d. Emple : Evlute b Stoe theorem ( d d d urve,. Solution : Given ( d d d Here F ( i, where i the ( i ( i d d d F d r url F dr url F n d S S n d F Emple 7: Evlute b Stoe theorem ( i url F ( i ( ( [ d d d], where i the urve, orreponding to the urfe of phere of unit rdiu. Solution : Given ( [ d d d] [( i ] ( i d d d
18 B Stoe theorem url F i F dr S url F n d ( i ( ( Therefore d d n d n d d Are of the irle n π Q d d d n Gu Divergene Theorem (Reltion between urfe integrl nd volume integrl Sttement: The urfe integrl of the norml omponent of vetor funtion F ten round loed urfe S i equl to the integrl of the divergene of F ten over the volume V enloed b the urfe S. Mthemtill nd S F V div F dv Emple. Evlute F nd uing Gu divergene theorem where S i the urfe S of the phere nd F i. Solution: Given F i nd rdiu of the phere r Therefore F i ( i F Then b theorem, we hve F nd div F dv dv v S Beue v i the volume of the phere 7π π ( V V
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