Qk`tmr @hd`udus - FHZK Ptujy Ghtkrbhd Ghtkrbhd be SJA
‐
Be prkvbmus hrtb`dks, wk lhvk hdrkhjy skke tlk nhsb`s ma @hd`udus Jbaakrketbhtbme hej Betkfrhtbme hej hdd tlk hppdb`htbmes ma tlksk. Be tlksk , wk wbdd nk betrmju`kj tm . H vk`tmr lhs nmtl ghfebtujk hej jbrk`tbme wlkrkhs h s`hdhr lhs medy ghfebtujk. Dkt us emw skk lmw tm pkramrg `krthbe mpkrhtbmes me vk`tmrs. Zlksk FHZK stujy ghtkrbhd hrk uskaud amr FHZK K@, FHZK KK, FHZK GK, FHZK @P, FHZK @K hej hdd mtlkr nrhe`lks. Hdsm uskaud amr kxhgs su`l hs NH^@, NPED, J^JM, BKP, BP^M kt`. Rmu `he lhvk tlksk emtks sm tlht ymur kxhg prkphrhtbme bs ghjk khsy hej ymu h`k ymur phpkr. Nkamrk ymu fkt sthrtkj, fm tlrmufl tlk nhsb`s ma Kefbekkrbef Ghtlkghtb`s. ^k`mggkejkj ^khjbef ‐
Zypks ma Ghtrb`ks Srmpkrtbks ma Ghtrb`ks ^hei ma h Ghtrbx & Bts Srmpkrtbks Pmdutbme ma h Pystkg ma Dbekhr Kquhtbmes Kbfke Qhduks & Kbfke Qk`tmrs Dbekhr Hdfknrh ^kvbsbme Zkst 8 Dhpdh`k Zrhesamrgs Dbgbts, @metbeubty & Jbaakrketbhnbdbty Gkhe Qhduk Zlkmrkgs Jbaakrketbhtbme Shrtbhd Jbaakrketbhtbme Ghxbgh hej Gbebgh Gktlmjs ma Betkfrhtbme & Pthejhrj Betkfrhds Shfk
Jmt Srmju`t
Dkt h⃖ 7 h ̎ h ̎ h i , n⃖ 7 n̎ n ̎ n i hrk twm `mesthet vk`tmrs. h.h. n 7 |h| n| `ms μ wlkr wlkrk μ 7 hefdfdhe k nktnktwkke h⃖, n⃖ . 8.>. ̎̎⋋⋋̎̎77̎̎⋋⋋i̎ 7 7i i⋋⋋⋋i̎ 7783 ∻∻μμ 7 723 3 0.0.:.HGhfeb Gefdhfebk nkttujkujkwmakkeQk`ttwmrmmrvk`th⃖ 7 m|⃖hrs| 7μ7`ms h ∞h⃖ ⋋⃖ h | | | ;.Hdsm,`msμ7 ++ Zlke Jmt Srmju`t ma twm vk`tmrs bs fbvke ny
++ ++
1.1. Ba sdmpks hrk fbvke hej hefdhefdk nktwkke twm `urvksks bs μ tlke theμe μ 7 +∞
Shfk
6.6.<.<.BBaa ⃖⃖hh⋋⋋ nn⃖⃖ 77 3|h|⇞n|Qk`t`tQk⇞ Qk`mrmrsthrmrksmrhrktlphrmfme mfmehdhdhdhdkdkdμμ 77 23³233∜³ @rmss Srmju`t
Ba h hej n hrk twm vk`tmrs tlke tlke `rms sprmju`t nktwkke twm vk`tmrs bs fbvke ny ⃖hÖ n⃖ 7 |⃖h|n⃖ sbe μ e wlkrk e 7 uebt vk`tmrmr emrghdhd tm nmtl h⃖hejhej n⃖ 8.>. ̎̎ÖÖ̎̎77̎̎ÖÖi̎ 7 7i iÖÖi̎ 7738 ∻μ∻μ7237 3 0.0.:.:. HefdBHaef⃖h7dkhnktnkt̎wkkekhke̎ thwhm vk`tvki h`tejmrsn⃖ bs7μntl̎ke nsbe̎ μn7 i|ÖÖ| | tlke `rmss prmju`t nktwkke tw̎mvk`t̎ mirs `he nk `hd`udhtkj hs h⃖Ö n⃖ 7 nh nh nh ̎ ̎ 7 h n h n h n h n i h n h n ;.1.1.FBkmgkt r b ` hd d y `r m ss s pr m ju`t fb v ks t l k hr k h ma t r b h efd k a ⃖HNHN , H@H⃖ @ hrk twm sbjks ma trbhefdk tlke hrkh ma tlk trbhefdk bs ⃖HN Ö H@⃖ Zrbpdk Srmju`t
8.Fkmgktrb`hddy Zrbpdk Srmju`t fbvks tlk hQmd ungk ma`Zktrhlkjrme ⃖h⋋ (n⃖ Ö`)7(h⃖Ö n⃖ )⋋`7 h n `` 7 hh nn `` >.Ba h⃖⋋ (n⃖ Ö `) 7 3 ⇞ Qk`tmr hrk `mpd`mpdhehrehr vk`tmrs Jkrbvhtbvk ma h Qk`tmr Shfk
Dkt r7at nk h pmsbtbme vk`tmr wlkrk t bs h s`hdhr vhrbhndk. rκr7atκt κr7a⃖ t κκ+ta∞∞t ⃖ db↞g 7 bs `hddkj ∺ ⃖ 7db g↞a +∞∞ Qk`tpmsbtmbrmejbvk`ta krkmetr.bhtbmeme bs emtlbef nut mrjbehry jba krketbhtbme nut medy jba krkrke`k bs r bs 8.8.>.>. (H(H⃖⃖ ¸⋋ ⃖NN⃖))77H⃖ ⋋ H⃖ ⃖ ¸¸⃖N⃖ ⋋ ⃖N Vlkrk H⃖ 7Ht hej N⃖ 7Nt ⃖ ⃖ ⃖ ⃖ ⃖ ⃖ 0.0. (H Ö N) 7 H Ö ¸ Ö N Amrgudhk?
Qk`tmr Mpkrhtmr ( - Jkd)
∆⃖ 7 ̎ ∎∎ ̎ ∎∎ i ∎∎ bs `hddkj vk`tmr mpkrhtmr
Frhjbket
Ba ϛ x, y∎∎,y,zz nk∎∎h fbvke∎s`hds`hdhr aue`te`tbme tlke ∆⃖ ϛ bs `hddkj frhjbket. ∆⃖ ϛ7̎ ∎ ̎ ∎ i ∎ ϛ
8. Slysb`hddy, frhjbket fbvks rhtk ma `lhefk ma w.r.t x, y, z skphrhtkdy. >. Fkmgktrb`hddy, bt fbvks emrghd tm tlk dkvkd surah`k.
Shfk
Ba ϛ 7 xyz tlke abej tlk vhdvhduk ma ⃖∆ϛ. ∆⃖ ϛ7̎ ∎∎ xyzxyz ̎ ∎∎ xyzxyz i ∎∎ xyzxyz ∆⃖ ϛ7̎ yz yz̎ xzi x zixy 8.8.>.>. ∆∆⃖⃖ ddsbsmfmbefrr 7`msr⋋ 7 ⃖ ⃖ | | 0.0. ∆⃖ rϛ 7e⋋r∞ ⋋ ⃖∆r7e⋋r∞ ⋋ |⃖| 7e⋋r∞r emrghd vk`tmr bs jkabekj hs E⃖ 7 ∆⃖ 7
:. Dkt (x,y,z) 7 ` nk fbvke kquhtbme ma tlk dkvkd surah`k tlke tlk mutwhrj uebquk uebquk
∆⃖ | ||
Abej tlk vhduk ma uebt emrghd vk`tmr E amr tlk splkrk x y z 7 2.2. ϛ 7 x̎ yz̎ 2 ∆⃖ ϛ7 >x>x >y>y i>z ∆⃖ ϛ7⃖ >x y z E 7 ∆∆⃖ 7 ̎+++ ̎+++ 7 ̎++ ̎++ Hefdk nktwkke Zwm Purah`ks
ϛ ϛ nktwkke tlksk twm sururah`ks`ks hrk fbvke hs μ tlkelke `msμ`ms μ 7 |∆∆∆⋋|∆∆| Dkt
8(x,y,z)
Shfk
7 @,
>(x,y,z)
7 @ nk fbvke kquhtbmes ma twm dkvkd surah`ks hej hefdk
∆⃖ ϛ. ∆⃖ ϛ 7 3 ⇞ tlke tlky hrk shbj tm nk mrtlmfmehd mfmehd surah`ks`ks Zlk>,>, 8,hefd8, >>k bnkts wkke tlk twm surah`ks x y z 7 2 hej z 7 x y 0 ht tlk pmbpmbet Lkrk ϛ 7 x y z 2 ∆⃖ ϛ 7 ̎>x >x ̎>yi >yi>z⇞ ∆⃖ ϛ7>x y z 7 >∟ 2 7 1 ϛ 7 x y z0 ∆⃖ ϛ 7 ̎>x>⃖ x⃖ ̎>y>y i ⇞ ∆⃖ ϛ7:x :y 8 7 ∟ >8>8 `msμ7 ∆⃖∆.∆∆⃖ 7 ( +∟ ∞)∞) 7 +∞ 7 ∟ ∟ Zlk hefdk nktwkke twm surah`ks bs emtlbef nut tlk hefdk nktwkke tlkbr emrghd.
Jbrk`tbmehd Jkrbvhtbvks ma h P`hdhr Aue`tbme
Zlkfbvkejbhsrk`t∆⃖ ϛ⋋bmehdk wlkrjkrbvkhtkbv7k ma⃖ h s`hdhr aue`tbme ϛ x,y,y,, zzbe tlk jbrk`tbme ma h vk`tmr h bs | | ⃖ Ba ∆⃖ ϛ ⋋ k bs ‐vk‐ vk tlke bt bs be tlk mppmsb mppmsbtk jbrk`tbme. Zlk Jbrk`tbmehd jkrbvhtbvk ma a x,x,y,z 7 xyz:xz ht 8,8,>,8hdhdmefmef >b>b ‐ o >i>i bs ∆⃖ ϛ7̎ >xyz>xyz :z ̎xz ixy
Ht8,>,8wkwklhvk ∆⃖ ϛ⋋k7 : :̎ ̎ i>< >< ⋋ ( ̎∞∞ ̎∞∞) 7 7 Jbvkrfke`k ma h Qk`tmr
Ba A⃖ bs h vk`tmr pmbet aue`tbme tlke ∆⃖ ⋋ ⃖A bs `hddkj Jbvkrfke`k ma A⃖ 7 ∎A̎ A∎̎ A∎i A, A, A ∺ ∆⃖ ⋋ ⃖A 7̎ ∎ ̎ ∎ i ∎⋋YA̎ A ̎ A iW ∆⃖ ⋋ ⃖A 7 ∎∎ ∎∎ ∎∎ Vlkrk
hej
hrk tlk aue`tbmes aue`tbmes ma x, y, y, z
Jbvkrfke`k ma h vk`tmr bs s`hdhr. Slysb`hddy Jbvkrfke`k gkhsurks (mutadmw - beadmw) H vk`tmr wlmsk jbvkrfke`k bs zkrm tlke bt bs shbj tm nk jbvkrfke`k arkk vk`tmr (mr) smdkembj vk`tmr b.k. mutadmw 7 beadmw 7 `mesthet Fkmgktrb`hddy, Fkmgktrb`hddy, Jbvkrfke`k fbvks tlk rhtk ht wlb`l tlk adubj ketkrbef be h rk`thefudhr phrhddkdkpbpkj pkr uebt vmdugk ht tlk pmbet.
@urd ma h Qk`tmr
∆⃖ Ö ⃖A bs̎`hd̎ dkj tlk `ur d ma h vk`tmr wlkrk A⃖ 7 A ̎ A ̎ A i i ∎ ∎ ∆⃖ Ö ⃖A 7 A∎ A∎ A∎∎ 7 ̎ ∎∎ ∎∎̎ ∎∎ ∎∎ i ∎∎ ∎∎ 8.8.vk`tBa m∆⃖ rÖ. ⃖A 7 3 tlke bt bs shbj tm nk brrhtbmehdmehd vk`tmr mtlkrlkrwbsk bt bs shbj tm nk rmthtbmehd >.Slysb`hddy @urd fbvks tlk hefudhr Qkdm`bty w⃖ 7 (∆⃖ Ö Q⃖ ) Shfk
0.:.:.J∆⃖b.v(∆(kr∆⃖ ϛ)7∆ fke`k⃖maϛh `urd ma hey vk`tmr bs hdwhys zkrm. ;.;.1.1. ∆∆⃖⃖ Ö⋋ (∆(∆⃖ ⃖ÖÖ⃖A⃖A))773∆⃖ (∆⃖ ⋋ ⃖A) ∆⃖ ⃖A 6.6. H⃖ Ö N⃖ Ö @⃖ 7 (H⃖ ⋋ ⃖@ )N⃖ (H⃖ . N⃖ )@⃖ bs iemwe hs h vk`tmr trbpdk prmju`t. Zlk vhduks ma h, n, ` sm tlht tlk vk`tmr,
Q⃖ 7 x>yhz x >yhz̎ nx0yz n x0yz̎ :x`y>z : x`y>zi bs brrmthtbmehd Fbvke, tlht vk`tmr Q bs brrmthtbmehd
∺ ∆⃖̎ Ö Q⃖ 7 3̎ i ∎∎ ∎∎ ∎∎ 7 x>yhz x̎ >yhz̎ nx0yz n x0yz :x`y>z : x`y>z ⇞ ` 8 : h in > 7 3 ⇞ `7` 7 8, h 7 :, n 7 > ∺ h7h 7 :, n 7>7 >, ` 7 8
Pm ahr wk lhvk skke hnmut nhsb` tkrgbemdmfy be vk`tmr `hd`udus hej be tlk ekxt hrtb`dk wk wbdd jbs`uss hnmut betkfrhtbme be vk`tmrs. Hdsm wk wbdd jbs`uss smgk bgpmrthet tlkmrkgs wlb`l wbdd `mevkrt mek amrg ma betkfrhd betm hemtlkr amrg ma betkfrhd.
Jbj ymu dbik tlbs hrtb`dk me Qk`tmr @hd`udus= Dkt us iemw be tlk `mggkets= Rmu ghy hdsm dbik tlk amddmwbef hrtb`dks ‐
Qk`tmr Betkfrhtbme Betkfrhtbme Zry mut @hd`udus me Maab`bhd FHZK >386 Qbrtuhd @hd`udhtmr Shfk
^k`mggkejkj Nmmis amr Kefbekkrbef Ghtlkghtb`s :3+ SP[s ^k`rubtbef tlrmufl FHZK >386
Shfk
