VEKTORANALYS FLUX INTEGRAL LINE INTEGRAL. and. Kursvecka 2. Kapitel 4 5. Sidor 29 50
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1 VEKTORANAYS Ksecka INE INTEGRA and UX INTEGRA Kaptel 4 5 Sdo 9 5
2 A wnd TARGET PROBEM We want to psh a mne cat along a path fom A to B. Bt the wnd s blowng. How mch enegy s needed? (.e. how mch s the wok? B We wll ae to the fnal answe n thee steps: - Path: staght lne Decton: the path has the same decton of the wnd - Path: staght lne Decton: the path has not the same decton of the wnd 3- Path: ced lne Wnd not constant B path A wnd path B B wnd A α A path wnd
3 Step : B path A wnd W Step : α // W cos α // Step 3: z B ( W A ( y W W W ( W ( o ey small segments: ( ( W lm d s the lne ntegal of along the path d We need to: - ntodce a VECTOR IED, ( - Defne the nfntesmal dsplacement d along the path 3
4 VECTOR IED A ecto feld assocates a ecto A(,y,z to each pont (,y,z of the space. Eamples: elocty dstbton n a fld magnetc feld aond a magnet electc feld aond an electc chage Two typcal ways to epesent a ecto feld: Aow feld The aow length s popotonal to the feld ampltde The aow decton shows the feld decton ne feld The tangent to the ces s paallel to the ecto feld n all ponts. The densty of the lnes s popotonal to the stength of the feld. 4
5 VECTOR IED The aplane wng eample (elocty feld of a aond a wng ow elocty (hozontal Hgh elocty (hozontal ow elocty (hozontal Hgh elocty (hozontal ( (, EXERCISE: - plot the ecto feld - wte on the ce defned by: ( ( (, 5
6 z B W d ( AND THE INE INTEGRA z + A y y s descbed by ( + + ( ( + + d lm lm [ ( ( ] + lm [ ( ( ] + So, the lne ntegal can be calclated as: b d ( d ( ( d a d EXERCISE: Calclate ( d wth ( (, and defned by ( (, : d d d 6
7 A INE INTEGRA (some sefl popetes THEOREM (4. n the tetbook B - A B ( d ( d PROO If all lne elements change sgn then also the ntegal wll change sgn. DEINITION The lne of ntegal of A along a closed ce C s called cclaton of a A along C: A ( d C (3 (4 7
8 INE INTEGRA DEINITION: A ecto feld A s called conseate f: A ( C d THEOREM (4.3 n the tetbook The cclaton of A along a close ce s zeo f and only f fo all ponts P and Q the lne ntegal of A fom P to Q s ndependent fom the ntegaton path between P and Q. PROO Assme that and ae two ces fom P to Q. P Then - s a closed ce. ( The cclaton s zeo the lne ntegal fom P to Q s ndependent fom the path A ( d A ( d A ( d A ( d A ( d Q A ( d A( d The lne ntegal s ndependent fom the ntegaton path! ( The lne ntegal fom P to Q s ndependent fom the path the cclaton s zeo. A ( d A( d A ( d A ( d A ( d A ( d A ( d The cclaton s zeo 8
9 INE INTEGRA THEOREM 3 (4.4 n the tetbook If Agadφ : Q A ( d φ ( Q φ ( P P So the lne ntegal s ndependent fom the ntegaton path and depends only on the statng pont and on the endng pont (5 PROO If ( s a ce fom P to Q then, sng the chan le fo the patal deate: q d q φ φ φ d dy dz A ( d gad φ d,,,, d d y z d d d ( p p q φ d φ dy φ dz q d d φ( ( d φ( Q φ( P p + + d yd zd p d O, ease: A ( d gad φ d d φ φ( B φ( A 9
10 OTHER KINDS O INE INTEGRAS It s possble to combne scala and ecto lne elements n many dffeent ways along a ce and ths get dffeent knds of lne ntegals φ Some eamples: ( ds φ ( d whee dêds To calclate the ntegals: A ( d ( [, ] : a b whee a b d d d d o ds d d d (6 (7 (8
11 EXAMPE z The foce s: (-yz,z,- Q The path s : (cos, sn, wth : 4π Calclate the wok fom P(,, tll Q(,,4π y fom defnton ( ( b 4 π (( d W d d a d P ( ( ( (sn,(cos, d d ( sn,cos, d ( ( ( sn + cos d + (sn cos 4 π 4 π ( ( 8 4 W d d π π
12 WHICH STATEMENT IS WRONG? WHICH STATEMENT IS WRONG? - The mage aea of a ecto feld A s composed of ectos (yellow - The lne ntegal d s a scala (ed 3- The sgn of the lne ntegal ddepends on the ntegaton path (geen gad A 4- The gadent of a ecto feld can be wtten as: (ble
13 TARGET PROBEM We ae makng canbey jce. Afte canbees ae sqeezed, It s bette to flte the jce! How mch jce flows togh the cloth each second? We need : ( to ndestand how to calclate the fl of a VECTOR IED (, y, z ( a method to ntegate the fl oe the whole sface. 3
14 THE UX In the jce eample, the fl s the olme of the fld V that flows thogh the sface n the tme t. t V STEP : - the fld elocty s pependcla to the sface - the sface s not ced V V t STEP : - the fld elocty s NOT pependcla to the sface - the sface s not ced S S n ˆ S // S t S S S S t STEP 3: - the sface s ced nˆ S d S S S lm ds s the fl ntegal of on the sface S S S S nˆ 4
15 ds AND UX INTEGRA Assme that the sface S s paametezed by (, z S et s consde two dsplacements, - de a change n : ( +, - de a change n : (, + y α snα S The aea S s S sn α nˆ s pependcla to S. Bt also s pependcla to S S n ˆ S 5
16 dd ds d d d d S ds + +, (, (, (, ( lm, (, ( lm lm lm lm AND UX INTEGRA ds ( S dd ds, ( So, the fl ntegal of the ecto feld on the sface S can be calclated as: n the same way: 6
17 EXAMPE Calclate the fl of the ecto feld A(y,,z thogh the sface S: z +y +y n ê z < SOUTION: - fge - Paametezaton of S 3- l calclaton sng eqaton z +y z +y z y y Ths defnes n whch decton we wll calclate the fl. n s chosen so that z-component s negate. z z +y +y n y 7
18 EXAMPE Paametezaton of S z +y +y ϕ (ρ,ϕ y ρ cos ϕ ρsn ϕ z + y ( ρ sn ϕ + ( ρsn ϕ ρ ρ ϕ π ρ and ϕ ae pola coodnates z π (ρ,ϕ y ρ Paametezaton of the ecto feld: A(y,,z (ρ snϕ cosϕ,, ρ 4 8
19 EXAMPE l calclaton sng eqaton A ds A (( ρ, ϕ d ρd ϕ ρ ϕ S S ρ ϕ ( cos ϕ,sn ϕ, ρ ( ρsn ϕ, ρcos ϕ, eˆ ˆ ˆ ey ez cos ϕ sn ϕ ρ ρ ϕ ρsn ϕ ρcos ϕ ( ρ cos ϕ, ρ sn ϕ, ρcos ϕ+ ρsn ϕ ρ ϕ ρ ϕ ρ ( cos, sn, Note that ρ ϕ has a poste z-component, whle the fl was n the othe decton. So we odnay sole the ntegal bt then we change the sgn n the answe! 9
20 EXAMPE AdS A ( ( ρϕ, d ρdϕ ρ ϕ S S π π ( ( 4 sn cos + 5 sn cos,, ( cos, sn, d d 4 ρ ϕ ϕ ρ ρ ϕ ρ ϕ ρ ρ ϕ ρ ϕ ϕ+ ρ dρd ϕ π π 5 6 ρ sn ϕ cos ϕ+ ρ dϕ sn ϕ cos ϕ+ dϕ cos ϕ π + ϕ π Bt we mst change sgn! The answe s 3 π
21 WHICH STATEMENT IS WRONG? WHICH STATEMENT IS WRONG? - The fl ntegal s a scala (yellow - l ntegals can be calclated also on a closed sface. (ed 3- The pependcla to the ntegaton sface ponts ot fom an abtay sde. (geen 4- The fl thogh a membane can be calclated wth fl ntegals. (ble
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