ϕ be a scalar field. The gradient is the vector field defined by
|
|
- בַּעַל־זְבוּל Σπανού
- 7 χρόνια πριν
- Προβολές:
Transcript
1 Amn Halloc Math Ecss E-mal : amn@sthths bpa : sthths/amn MATH EXERCISES GRADIENT DIVERGENCE CURL DEL NABLA OERATOR LALACIAN OERATOR CONTINUITY AND NAVIER-STOKES EQUATIONS VECTOR RODUCTS I and thn scala o dot podct cto o s podct In som boos s also consdd ot podct dnd b GRADIENT DIVERGENCE CURL DEL NABLA OERATOR LALACIAN OERATOR GRADIENT Lt ϕ b a scala ld Th adnt s th cto ld dnd b ad ϕ ϕ ϕ ϕ DIVERGENCE Lt R Q b a cto ld contnosl dntabl th spct to and Thn th dnc o s th scala ld dnd b R Q d CURL Th cl o s th cto ld dnd b Q R Q R R Q cl o Q R Q R cl DEL NABLA OERATOR Th cto dntal opato s calld dl o nabla / 7
2 Amn Halloc Math Ecss U can dnot ad d and cl as blo: ad ϕ ϕ d cl Not that s not th sam as Q R LALACIAN OERATOR Th Laplacan opato s dnd o a scala ld U b U U U U U and o a cto ld Q R b Q R Som omlas o pola and clndcal coodnats ola coodnats dm ϑ ϑ ϑ ϑ tansomaton: aa lmnt: da d d standad bass: [Rma : Not that a dpnd on hn mo om pont to pont ths s th ason h ths bass n som boos s calld local bass ] I n Catsan cood and ϑ th sam cto n pola coodnats thn ϑ [Rma : V can d ths omlas b calclatn th componnts o n th dctons o and Ths / 7
3 Amn Halloc Math Ecss smlal ϑ ϑ ] Clndcal coodnats : tansomaton: olm lmnt: d d d dv ϑ ϑ standad bass: I n Catsan cood and ϑ th sam cto n clndcal coodnats thn ha ollon cto componnts latonshp: ϑ [Rma : o ampl can t n th ollon a: ] scala ld: adnt: ad laplacan: cto ld: dnc: d cl: cl / 7
4 Amn Halloc Math Ecss EXERCISES nd a d b ad d and c cl nd ad d cl Whch on o th ollon nctons a b ln c p satss th Laplac qaton 0? 4 nd 5 Wt th nal tanspot qaton ϕ ϕu Γ ϕ S t φ thot opatos d cl o ad H U nctons ϕ Γ S a al nctons o t and 6 Whch on an o th ollon nctons 4 a ϕ b ϕ ϕ c satss th qaton ϕu Γ ϕ S? H Γ 5 U and S 4 7 nd hch on an o th ollon nctons a ϕ b ϕ 5 5 ϕ c satss th qaton ϕ d ϕu d Γadϕ S t h Γ U 4 and S / 7
5 Amn Halloc Math Ecss 8 am 008 A Wt th nal tanspot qaton ϕ ϕu Γ ϕ S q t thot opatos d cl o ad H U nctons ϕ Γ S a al nctons o t and B Lt Γ U 4 nd S n th qaton q no that th ncton ϕ satss th qaton 9 Q6 am 008 Consd th ollon qaton ϕ ϕu Γ ϕ U 6 4 q t Lt Γ constant U nd th constant Γ n th qaton q no that th ncton ϕ t satss th qaton 0 I possbl nd o th n patal dats and a and b and c and d and 5 Hnt: Ncssa condton: I has contns dats thn th md dats o shold b qal Ths * s th ncssa condton o th stnc o a ncton that has th n dats Dtmn th al o a o hch th sstm o patal dntal qatons a and has soltons Thn nd pondn to ths al o a 5 / 7
6 Amn Halloc Math Ecss I possbl nd o th n patal dats and a and b and c and d and Hnt: Ncssa condton: I has contnos dats thn th md dats o shold b qal Ths Con : Con : Con : a th ncssa condton o th stnc o a ncton that has th n dats Dtmn th als o a and b o hch th sstm o patal dntal qatons a and b has soltons Thn nd pondn to ths als o a and b 4 W consd an ncompssbl dnst const stad stat aabls do not dpnd on tm sothmal Ntonan lo th a n loct ld V Us th ollon qatons contnt and Na Stos qatons to nd n psson o pss as a ncton o and h constant µ constant 00 0 and h 98m / s Incompssbl contnt qaton: 0 q Na Stos qatons: componnt: 6 / 7
7 Amn Halloc Math Ecss µ q t componnt: µ q t componnt: µ q4 t a V 4 0 b V 4 c V 4 5 am 009 A Consd th ollon qaton ϕ ϕu Γ ϕ U t Lt Γ constant U 4 4 nd th constant Γ n th qaton q no that th ncton ϕ t satss th qaton q B W consd an ncompssbl dnst const stad stat aabls do not dpnd on tm sothmal Ntonan lo th a n loct ld V Us th ollon qatons contnt and Na Stos qatons to nd n psson o pss as a ncton o and h constant µ constant 00 0 and h 98m / s and V am 009 W consd an ncompssbl dnst const stad stat aabls do not dpnd on tm sothmal Ntonan lo th a n loct ld V Us th ollon qatons contnt and Na Stos qatons to nd st paamt a and thn n psson o pss as a ncton o and h constant µ constant 00 0 and h 98m / s and V 5 a 7 / 7
8 Amn Halloc Math Ecss 7 Consd stad ncompssbl sothmal lamna statona Ntonan lo n a lon ond pp n th -dcton th constant ccla s-scton o ads R m Us th contnt and th Na-Stos qatons n clndcal coodnats to nd th loct ld V and th pss ld th ld lo satss th ollon condtons: c0 All patal dats th spct to tm t a 0 Stad lo c μ000 /m s and 000 /m c A Constant pss adnt / /50 a/m s appld n th hoontal as -as n o notaton: / /50 c Th lo s paalll to th as that s 0 and 0 c4 W assm that th lo s asmmtc Th loct dos not dpnd on that s 0 c5 Bonda cond No-slp bonda condton V ld V all : I thn 0 c6 Bonda condton : has mamm at 0 that s Th contnt and th Na-Stos qatons o an ncompssbl sothmal Ntonan lo dnst const st µ const th a loct ld V n Clndcal coodnats : Incompssbl contnt qaton 0 q a Na-Stos qatons n Clndcal coodnats: -componnt: t µ q b -componnt: t µ q c 8 / 7
9 Amn Halloc Math Ecss -componnt: t µ q d 8 Eam Mach 0 qston A 4ponts W consd an ncompssbl dnst const stad stat aabls do not dpnd on tm sothmal Ntonan lo th a n loct ld V c 4 b a Us th ollon qatons contnt and Na Stos qatons h constant µ constant 00 0 and 98m / s to nd: paamts a b and c n psson o pss as a ncton o and Th GRADIENT VECTOR th chan o aabls and bass Th adnt cto o th ncton s dnd as ad * I chan aabls to and plac bass ctos th n lnal ndpndnt ctos thn can pss th sam adnt cto ad n tms o aabls and ctos W smpl calclat th dats and n n aabls and pss as a lna combnatons o Thn sbsttt thos als nto * S th ollon ampl 7 W consd a scala ld n n clndcal coodnats h and bass ctos a nd th psson o th adnt ad n clndcal coodnats that s n tms o and a p th sam bass 9 / 7
10 Amn Halloc Math Ecss b c am 0; Q5 B ponts d ths s otn sd as a local bass o clndcal coodnats 8 am 06; Q6 A ponts D th Cach momntm qaton DV σ Dt Solton: S ANSWERS AND SOLUTIONS: Solton: Q R a Snc d ha d 0 0 Ans a d ϕ ϕ ϕ b Snc ad ϕ ha o ϕ d ad d 00 Ans b ad d 00 d c cl Q R Ans c cl Solton: 0 / 7
11 Amn Halloc Math Ecss cl 0 Ths d cl 0 and tho ad d cl Ans: ad d ot Ans: Th ncton ln satss th Laplac qaton 4 Ans: d cl ad 6 5 Solton: ϕ d ϕu d Γadϕ Sφ t ϕ ϕ ϕ ϕ d ϕ ϕ ϕ d Γ Γ Γ Sφ t ϕ ϕ ϕ ϕ ϕ ϕ Γ Γ t ϕ Γ S φ 6 Whch on an o th ollon nctons 4 a ϕ b ϕ ϕ c satss th qaton ϕu Γ ϕ S? H Γ 5 U and S 4 Solton : Th qaton ϕu Γ ϕ S can b ttn as / 7
12 Amn Halloc d ϕu d Γadϕ S 5ϕ 5ϕ 5ϕ d ϕ ϕ ϕ d 4 ϕ ϕ ϕ 5 ϕ 5 ϕ 5 ϕ 4 Math Ecss q 4 a Lt ϕ ϕ V calclat th dats oϕ and sbsttt n th lt hand sd LHS and ht hand sd o th qaton q ϕ ϕ ϕ 4 4 ϕ 5 ϕ 5 ϕ 4 LHS: 5 RHS 60 4 Whnc LHS RHS Ths th ncton ϕ 4 s not a solton to th qaton b ϕ ϕ LHS 4 RHS 4 Whnc LHS RHS and th ncton ϕ s not a solton to th qaton ϕ ϕ c Lt Thn LHS 4 6 RHS 7 4 Ths LHS RHS and th ncton ϕ s not a solton to th qaton Ans: Non o th nctons satss th qaton 7 Ans: ncton ϕ 5 satss th qaton 8 am 98 A Wt th nal tanspot qaton ϕ ϕu Γ ϕ S q t thot opatos d cl o ad H U nctons ϕ Γ S a al nctons o t and / 7
13 Amn Halloc Math Ecss B Lt Γ U 4 nd S n th qaton q no that th ncton ϕ satss th qaton Solton: A ϕ ϕu Γ ϕ S t ϕ d ϕu d Γadϕ S t ϕ ϕ ϕ ϕ d ϕ ϕ ϕ d Γ Γ Γ S t ϕ ϕ ϕ ϕ ϕ ϕ ϕ Γ Γ Γ S q t B W sbsttt Γ U 4 and ϕ n th qaton q and t ϕ 4ϕ 8φ ϕ ϕ ϕ 0 S S Consqntl S Q6 am 008 Consd th ollon qaton ϕ ϕu Γ ϕ U 6 4 q t Lt Γ constant U nd th constant Γ n th qaton q no that th ncton ϕ t satss th qaton Solton: ϕ ϕu Γ ϕ U 6 4 t ϕ d ϕu d Γadϕ d cl U 6 4 t c cl U 0 ha d cl U 0 / 7
14 Amn Halloc Math Ecss ϕ ϕ ϕ ϕ d ϕ ϕ ϕ d Γ Γ Γ t ϕ ϕ ϕ ϕ ϕ ϕ ϕ Γ Γ Γ q t W sbsttt U and ϕ t n th qaton q and t ϕ ϕ ϕ φ ϕ ϕ ϕ Γ Γ Γ 6 4 t Not that Γ s a constant 6 0 Γ Γ Γ Γ Ans: Γ 0 I possbl nd o th n patal dats and a and b and c and d and 5 Hnt: Ncssa condton: I has contnos dats thn th md dats o shold b qal * s th ncssa condton o th stnc o a ncton that has th n dats Ans: a C b C c C d No solton c th condton * s not llld 5 Solton a Snc and th dats a contnos th condton * s llld and can nd o th n dats In od to nd ntat th spct to th st o th qatons q 4 / 7
15 Amn Halloc Math Ecss and t q d C Ths C W ha ntatd th spct to tho th constant stll dpnd on No to nd C dntat and sbsttt n q and t: C C C C C nall sbstttn C C n ha C h C s a constant Ans: om a a Thn o a ha C Ans: a C b C c C d No solton c th condton Con s not llld Solton a a and Snc th condtons Con a llld and can nd o th n dats In od to nd ntat th spct to th st o th qatons q q q and t d C Ths 5 / 7
16 Amn Halloc Math Ecss C W ha ntatd th spct to tho th constant stll dpnd on and No to nd C dntat and sbsttt n q and t: C C C C C W ha ntatd th spct to tho th constant stll dpnd on and Ths C No sbstttn n q ha C C C C C nall sbstttn C C n ha C h C s a constant Ans: om a a b b b b Ths all th condtons a llld a and b o ths als o a and b t C Calclaton o th pss ld o a non loct ld o an ncompssbl stad stat sothmal Ntonan lo 4 Ans: a 8 8 C 6 / 7
17 Amn Halloc Math Ecss 7 7 b C c C Solton a W sbsttt 4 0 n q4 and t not that al dats th spct to t a 0: Contnt qaton: 0 0 q dntcall llld Na Stos qatons: componnt: 6 q componnt: 6 q componnt: 0 q4 No q s 8 C * Sbsttton n q mpls C 6 C 8 C Hnc om * ha 8 8 C ** No sbsttt ** n q4 and t 0 0 C C C h C s a constant nall sbstttn C C n ** ha 8 8 C h C s a constant 5 Solton A: ϕ ϕ U Γ ϕ U t ϕ d ϕu d Γadϕ d cl U t c cl U 0 ha d cl U 0 ϕ ϕ ϕ ϕ d ϕ ϕ ϕ d Γ Γ Γ t 7 / 7
18 Amn Halloc Math Ecss ϕ ϕ ϕ ϕ ϕ ϕ ϕ Γ Γ Γ t q W sbsttt U 4 4 and ϕ t n th qaton q and t ϕ 8ϕ 8ϕ φ ϕ ϕ Γ Γ t Not that Γ s a constant Γ Γ Γ 5 Ans A: Γ 5 Solton B: W sbsttt n q4 and t not that al dats th spct to t a 0: Contnt qaton: 0 0 q dntcall llld Na Stos qatons: componnt: 6 4 q componnt: 4 8 q componnt: 4 4 q4 No q s 8 4 C * Sbsttton n q mpls C 4 8 C 8 C Hnc om * ha C ** No sbsttt ** n q4 and t C C 4 C h C s a constant nall sbstttn C C n ** ha C Ans B: Γ ϕ / 7
19 Amn Halloc Math Ecss C h C s a constant 6 Solton 5 a V st sbsttt a 5 n q and t not that al dats th spct to t a 0: Contnt qaton: 0 a a No ha 5 V U th Na Stos qatons t: componnt: 6 9 q componnt: 5 q componnt: 4 q4 No q s 6 9 C * Sbsttton n q mpls 5 5 C C C Hnc om * ha C ** W sbsttt ** n q4 and t C 4 4 C C h C s a constant nall sbstttn C n ** ha C Ans : C h C s a constant Q7 Consd stad ncompssbl sothmal lamna statona Ntonan lo n a lon ond pp n th -dcton th constant ccla s-scton o ads R m Us 9 / 7
20 Amn Halloc Math Ecss th contnt and th Na-Stos qatons n clndcal coodnats to nd th loct ld V and th pss ld th ld lo satss th ollon condtons: c0 All patal dats th spct to tm t a 0 Stad lo c μ000 /m s and 000 /m c A Constant pss adnt / /50 a/m s appld n th hoontal as -as n o notaton: / /50 c Th lo s paalll to th as that s 0 and 0 c4 W assm that th lo s asmmtc Th loct dos not dpnd on that s 0 c5 Bonda cond No-slp bonda condton V ld V all : I thn 0 c6 Bonda condton : has mamm at 0 that s 0 0 Th contnt and th Na-Stos qatons o an ncompssbl sothmal Ntonan lo dnst const st µ const th a loct ld V n Clndcal coodnats : SOLUTION Incompssbl contnt qaton 0 q a Na-Stos qatons n Clndcal coodnats: -componnt: t µ q b -componnt: t µ q c -componnt: 0 / 7
21 Amn Halloc Math Ecss t µ q d W choos as a tcal as an a n a hoontal plan and th lo s paalll th th -as W dnot loct cto V h and a -componnt - componnt and -componnt n clndcal coodnats Accodn to th assmptons ha 0 0 and dos not dpnd on Snc s th tcal as ha that cto - 00 h 98 m/s hch n clndcal coodnats s and 0 No sbsttt / /50 a/m μ000 /ms n th contnt and Na- Stos qatons: Snc 0 and 0 accodn to c contnt qaton n clndcal coodnats 0 s 0 / 7
22 Amn Halloc Math Ecss Ths tlls s that s not a ncton o thmo c loct dos not dpnd on assmpton c4 concld that dpnds onl on To smpl notaton dnot * No sbsttt and 0 / /50 a/m μ000 /ms n th Na-Stos qatons: Th -componnt o th Na-Stos qaton s: 0 q -c Th -componnt o th Na-Stos qaton: 0 q -c Th Z-componnt o th Na-Stos qaton h and 0 q -c s: Stp W nd th pss In od to nd th pss sol q -c q -c and th qaton s 50 that 50 om ths qatons t C 50 / 7
23 Amn Halloc Math Ecss Stp W nd th loct componnt W sol q -c th bondas c5 and c6: 0 q -c c5 0 0 c6 d Rma: Tchncall can t nstad d aabl om q -c ha c s no a ncton o onl on C sbsttton 0 and c6 C 0 C sbsttton and c5 C 4 4 Ths 4 and V Ans : C 50 V / 7
24 Amn Halloc Math Ecss 8 W consd an ncompssbl dnst const stad stat aabls do not dpnd on tm sothmal Ntonan lo th a n loct ld 4 a b c V Us th ollon qatons contnt and Na Stos qatons h constant µ constant 00 0 and / 98 s m to nd: paamts a b and c n psson o pss as a ncton o and Incompssbl contnt qaton: 0 q Na Stos qatons: componnt: t µ q componnt: t µ q componnt: t µ q W sbsttt 4 a b c n q4 and t not that al dats th spct to t a 0: Contnt qaton: 0 a a q Ths 4 b c Na Stos qatons: componnt: q componnt: q componnt: 4 / 7
25 Amn Halloc Math Ecss q4 Th sstm q q q4 s solabl onl md dats a qal: 4 : c c Con 0 0 : b bc Con 0 0 : b b Con Ths c and b0 W sol smpld qatons and t C Ans C W consd a scala ld n n clndcal coodnats h and bass ctos a nd th psson o th adnt ad n clndcal coodnats that s n tms o and a p th sam bass b c am 0; Q5 B ponts d ths s otn sd as a local bass o clndcal coodnats 5 / 7
26 Amn Halloc Math Ecss Solton: In aabls ha ad q o clndcal coodnats ha st t th dats and n coodnats th aabl s n both cood sstms Soln th ollon sstm o and t ** W sbsttt th dats ** n q and t *** ad To sol poblms a b c and d mst pss as a lna combnatons o and sbsttt thm nto *** a om *** c ha mmdatl ad b om nd q b Thn pt om q b nto *** and t ad 6 / 7
27 Amn Halloc Math Ecss and at collctn componnts o ad c om ha q c ttn om q c nto *** s ad d W can sol d n th sam mann as n abc bt ths tm can st collct tms and t th slt: ad Ans: a ad b ad c ad d ad 7 / 7
ϕ be a scalar field. The gradient is the vector field defined by
Amn Halloc Math Ecss E-mal : amn@sthths bpa : sthths/amn MATH EXERCISES GRADIENT DIVERGENCE CURL DEL NABLA OERATOR LALACIAN OERATOR CONTINUITY AND NAVIER-STOKES EQUATIONS VECTOR RODUCTS I an thn scala
webpage :
Amn Halloc Mah Ecss / 7 E-mal : amn@shhs bpag : shhs/amn MATH EXERCISES GRADIENT DIVERGENCE CURL DEL NABLA OERATOR LALACIAN OERATOR CONTINUITY AND NAVIER-STOKES EQUATIONS VECTOR RODUCTS I and hn scala
webpage :
Amin Haliloic Mah Eciss E-mail : amin@shkhs wbpa : wwwshkhs/amin MATH EXERISES GRADIENT DIVERGENE URL DEL NABLA OERATOR LALAIAN OERATOR ONTINUITY AND NAVIER-STOKES EQUATIONS VETOR RODUTS I and hn scala
ELE 3310 Tutorial 11. Reflection of plane waves Wave impedance of the total field
L 0 Tuto Rfcton of pn wvs Wv mpdnc of th tot fd Rfcton of M wvs Rfcton tks pc whn n M wv hts on bound. Pt of th wv gts fctd, nd pt of t gts tnsmttd. Popgton dctons nd mptuds of th fctd nd tnsmttd wvs dpnd
General theorems of Optical Imaging systems
Gnral thorms of Optcal Imagng sstms Tratonal Optcal Imagng Topcs Imagng qualt harp: mags a pont sourc to a pont Dstorton fr: mags a shap to a smlar shap tgmatc Imagng Imags a pont sourc to a nfntl sharp
Matrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Homework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
Answers - Worksheet A ALGEBRA PMT. 1 a = 7 b = 11 c = 1 3. e = 0.1 f = 0.3 g = 2 h = 10 i = 3 j = d = k = 3 1. = 1 or 0.5 l =
C ALGEBRA Answers - Worksheet A a 7 b c d e 0. f 0. g h 0 i j k 6 8 or 0. l or 8 a 7 b 0 c 7 d 6 e f g 6 h 8 8 i 6 j k 6 l a 9 b c d 9 7 e 00 0 f 8 9 a b 7 7 c 6 d 9 e 6 6 f 6 8 g 9 h 0 0 i j 6 7 7 k 9
Multi-dimensional Central Limit Theorem
Mult-dmensonal Central Lmt heorem Outlne () () () t as () + () + + () () () Consder a sequence of ndependent random proceses t, t, dentcal to some ( t). Assume t 0. Defne the sum process t t t t () t tme
Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
ECE 222b Applied Electromagnetics Notes Set 3b
C b Appl lcomancs Nos S 3b Insuco: Pof. Val Loman Dpamn of lccal an Compu nnn Unvs of Calfona San Do Rflcon an Tansmsson. Nomal ncnc T R T R Fs fn h manc fls: 3 Rflcon an Tansmsson T R T R T R T R R T
Multi-dimensional Central Limit Theorem
Mult-dmensonal Central Lmt heorem Outlne () () () t as () + () + + () () () Consder a sequence of ndependent random proceses t, t, dentcal to some ( t). Assume t 0. Defne the sum process t t t t () t ();
Convection Derivatives February 17, E+01 1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 1.E-08 1.E-09 1.E-10. Error
onvcton rvtvs brry 7, nt Volm Mtho or onvcton rvtvs Lrry rtto Mchncl ngnrng 69 omttonl l ynmcs brry 7, Otln Rv nmrcl nlyss bscs oncl rslts or son th sorc nlyss Introc nt-volm mtho or convcton Not n or
Mock Exam 7. 1 Hong Kong Educational Publishing Company. Section A 1. Reference: HKDSE Math M Q2 (a) (1 + kx) n 1M + 1A = (1) =
Mock Eam 7 Mock Eam 7 Section A. Reference: HKDSE Math M 0 Q (a) ( + k) n nn ( )( k) + nk ( ) + + nn ( ) k + nk + + + A nk... () nn ( ) k... () From (), k...() n Substituting () into (), nn ( ) n 76n 76n
( y) Partial Differential Equations
Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate
ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
α & β spatial orbitals in
The atrx Hartree-Fock equatons The most common method of solvng the Hartree-Fock equatons f the spatal btals s to expand them n terms of known functons, { χ µ } µ= consder the spn-unrestrcted case. We
Phasor Diagram of an RC Circuit V R
ESE Lecture 3 Phasor Dagram of an rcut VtV m snt V t V o t urrent s a reference n seres crcut KVL: V m V + V V ϕ I m V V m ESE Lecture 3 Phasor Dagram of an L rcut VtV m snt V t V t L V o t KVL: V m V
ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ ΤΜΗΜΑ ΕΠΙΣΤΗΜΗΣ ΚΑΙ ΤΕΧΝΟΛΟΓΙΑΣ ΥΛΙΚΩΝ ΦΑΙΝΟΜΕΝΑ ΜΕΤΑΦΟΡΑΣ
ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ ΤΜΗΜΑ ΕΠΙΣΤΗΜΗΣ ΚΑΙ ΤΕΧΝΟΛΟΓΙΑΣ ΥΛΙΚΩΝ ΦΑΙΝΟΜΕΝΑ ΜΕΤΑΦΟΡΑΣ ΠΕΡΙΛΗΨΗ ΣΥΜΒΟΛΙΣΜΩΝ NOTATION ΓΙΑ ΙΑΝΥΣΜΑΤΑ ΚΑΙ ΤΑΝΥΣΤΕΣ -Bd, Steat and Lghtfoot "Tanpot Phenomena" -Bd, Amtong and Haage
Chapter 4 : Linear Wire Antenna
Chapt 4 : Lina Wi Antnna nfinitsima Dipo Sma Dipo Finit Lngth Dipo Haf-Wavngth Dipo Lina mnts na o on nfinit Pfct Conductos nfinitsima Dipo Lngth
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr (T t N n) Pr (max (X 1,..., X N ) t N n) Pr (max
26 28 Find an equation of the tangent line to the curve at the given point Discuss the curve under the guidelines of Section
SECTION 5. THE NATURAL LOGARITHMIC FUNCTION 5. THE NATURAL LOGARITHMIC FUNCTION A Click here for answers. S Click here for solutions. 4 Use the Laws of Logarithms to epand the quantit.. ln ab. ln c. ln
19. ATOMS, MOLECULES AND NUCLEI HOMEWORK SOLUTIONS
. ATOMS, MOLECULES AND NUCLEI HOMEWORK SOLUTIONS. Givn :.53 Å 3?? n n ε πm n n Radius of n t Bo obit, n n ε πm n n 3 n 3 n 3 (3) () (.53).77Å n n ( ) () (.53) 53 Å. Givn : 3 7.7 x m? n n ε πm Radius of
Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών. ΗΥ-570: Στατιστική Επεξεργασία Σήµατος. ιδάσκων : Α. Μουχτάρης. εύτερη Σειρά Ασκήσεων.
Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 2015 ιδάσκων : Α. Μουχτάρης εύτερη Σειρά Ασκήσεων Λύσεις Ασκηση 1. 1. Consder the gven expresson for R 1/2 : R 1/2
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr T t N n) Pr max X 1,..., X N ) t N n) Pr max
One and two particle density matrices for single determinant HF wavefunctions. (1) = φ 2. )β(1) ( ) ) + β(1)β * β. (1)ρ RHF
One and two partcle densty matrces for sngle determnant HF wavefunctons One partcle densty matrx Gven the Hartree-Fock wavefuncton ψ (,,3,!, = Âϕ (ϕ (ϕ (3!ϕ ( 3 The electronc energy s ψ H ψ = ϕ ( f ( ϕ
ECE 222b Applied Electromagnetics Notes Set 3a
C b lid lcomagnics Nos S 3a Insuco: Pof. Viali Lomakin Damn of lcical and Comu ngining Univsi of Califonia San Digo Unifom Plan Wavs Consid Mawll s quaions: In a losslss mdium ε and µ a al and σ : Sinc
Na/K (mole) A/CNK
Li, W.-C., Chen, R.-X., Zheng, Y.-F., Tang, H., and Hu, Z., 206, Two episodes of partial melting in ultrahigh-pressure migmatites from deeply subducted continental crust in the Sulu orogen, China: GSA
Chapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Το άτομο του Υδρογόνου
Το άτομο του Υδρογόνου Δυναμικό Coulomb Εξίσωση Schrödinger h e (, r, ) (, r, ) E (, r, ) m ψ θφ r ψ θφ = ψ θφ Συνθήκες ψ(, r θφ, ) = πεπερασμένη ψ( r ) = 0 ψ(, r θφ, ) =ψ(, r θφ+, ) π Επιτρεπτές ενέργειες
Reminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Trigonometry 1.TRIGONOMETRIC RATIOS
Trigonometry.TRIGONOMETRIC RATIOS. If a ray OP makes an angle with the positive direction of X-axis then y x i) Sin ii) cos r r iii) tan x y (x 0) iv) cot y x (y 0) y P v) sec x r (x 0) vi) cosec y r (y
Finite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Spherical Coordinates
Spherical Coordinates MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Spherical Coordinates Another means of locating points in three-dimensional space is known as the spherical
Pairs of Random Variables
Pairs of Random Variabls Rading: Chaptr 4. 4. Homwork: (do at last 5 out of th following problms 4..4, 4..6, 4.., 4.3.4, 4.3.5, 4.4., 4.4.4, 4.5.3, 4.6.3, 4.6.7, 4.6., 4.7.9, 4.7., 4.8.3, 4.8.7, 4.9.,
Partial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
Example Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.
Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given
SOLUTIONS & ANSWERS FOR KERALA ENGINEERING ENTRANCE EXAMINATION-2018 PAPER II VERSION B1
SOLUTIONS & ANSWERS FOR KERALA ENGINEERING ENTRANCE EXAMINATION-8 PAPER II VERSION B [MATHEMATICS]. Ans: ( i) It is (cs5 isin5 ) ( i). Ans: i z. Ans: i i i The epressin ( i) ( ). Ans: cs i sin cs i sin
b. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!
MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.
1.19 Curvilinear Coordinates: Curved Geometries
Scton.9.9 Curlnr Coordnts: Curd Gomtrs In ths scton s mnd th spcl cs of two-dmnsonl curd surfc..9. Monoclnc Coordnt Systms Bs Vctors curd surfc cn b dfnd usn two cornt bs ctors wth th thrd bs ctor rywhr
Section 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.
EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.
Chapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Constant Elasticity of Substitution in Applied General Equilibrium
Constant Elastct of Substtuton n Appled General Equlbru The choce of nput levels that nze the cost of producton for an set of nput prces and a fed level of producton can be epressed as n sty.. f Ltng for
Reflection & Transmission
Rflc & Tasmss 4 D. Ray Kw Rflc & Tasmss - D. Ray Kw Gmc Opcs (M wavs flc fac - asmss cdc.. Sll s Law: s s 3. Ccal agl: s c / 4. Tal flc wh > c ly f > Rflc & Tasmss - D. Ray Kw Pla Wav λ wavfs λ λ. < ;
EE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Κύµατα παρουσία βαρύτητας
Κύµατα παουσία βαύτητας 8. Grait as in th ocan Sarantis Sofianos Dpt. of hsics, Unirsit of thns Was in th ocan Srfac grait as Short and long limit in grait as Wa charactristics Intrnal as Charactristic
SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
2. Chemical Thermodynamics and Energetics - I
. Chemical Thermodynamics and Energetics - I 1. Given : Initial Volume ( = 5L dm 3 Final Volume (V = 10L dm 3 ext = 304 cm of Hg Work done W = ext V ext = 304 cm of Hg = 304 atm [... 76cm of Hg = 1 atm]
Appendix A. Curvilinear coordinates. A.1 Lamé coefficients. Consider set of equations. ξ i = ξ i (x 1,x 2,x 3 ), i = 1,2,3
Appendix A Curvilinear coordinates A. Lamé coefficients Consider set of equations ξ i = ξ i x,x 2,x 3, i =,2,3 where ξ,ξ 2,ξ 3 independent, single-valued and continuous x,x 2,x 3 : coordinates of point
Appendix to On the stability of a compressible axisymmetric rotating flow in a pipe. By Z. Rusak & J. H. Lee
Appendi to On the stability of a compressible aisymmetric rotating flow in a pipe By Z. Rusak & J. H. Lee Journal of Fluid Mechanics, vol. 5 4, pp. 5 4 This material has not been copy-edited or typeset
Mean-Variance Analysis
Mean-Variance Analysis Jan Schneider McCombs School of Business University of Texas at Austin Jan Schneider Mean-Variance Analysis Beta Representation of the Risk Premium risk premium E t [Rt t+τ ] R1
ON THE MEASUREMENT OF
ON THE MEASUREMENT OF INVESTMENT TYPES: HETEROGENEITY IN CORPORATE TAX ELASTICITIES HENDRIK JUNGMANN, SIMON LORETZ WORKING PAPER NO. 2016-01 t s r t st t t2 s t r t2 r r t t 1 st t s r r t3 str t s r ts
8.324 Relativistic Quantum Field Theory II
Lecture 8.3 Relatvstc Quantum Feld Theory II Fall 00 8.3 Relatvstc Quantum Feld Theory II MIT OpenCourseWare Lecture Notes Hon Lu, Fall 00 Lecture 5.: RENORMALIZATION GROUP FLOW Consder the bare acton
C 1 D 1. AB = a, AD = b, AA1 = c. a, b, c : (1) AC 1 ; : (1) AB + BC + CC1, AC 1 = BC = AD, CC1 = AA 1, AC 1 = a + b + c. (2) BD 1 = BD + DD 1,
1 1., BD 1 B 1 1 D 1, E F B 1 D 1. B = a, D = b, 1 = c. a, b, c : (1) 1 ; () BD 1 ; () F; D 1 F 1 (4) EF. : (1) B = D, D c b 1 E a B 1 1 = 1, B1 1 = B + B + 1, 1 = a + b + c. () BD 1 = BD + DD 1, BD =
Differential equations
Differential equations Differential equations: An equation inoling one dependent ariable and its deriaties w. r. t one or more independent ariables is called a differential equation. Order of differential
Written Examination. Antennas and Propagation (AA ) April 26, 2017.
Written Examination Antennas and Propagation (AA. 6-7) April 6, 7. Problem ( points) Let us consider a wire antenna as in Fig. characterized by a z-oriented linear filamentary current I(z) = I cos(kz)ẑ
Molekulare Ebene (biochemische Messungen) Zelluläre Ebene (Elektrophysiologie, Imaging-Verfahren) Netzwerk Ebene (Multielektrodensysteme) Areale (MRT, EEG...) Gene Neuronen Synaptische Kopplung kleine
Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
α A G C T 國立交通大學生物資訊及系統生物研究所林勇欣老師
A G C T Juks and Cantor s (969) on-aramtr modl A T C G A G 0 0 0-3 C T A() A( t ) ( 3 ) ( ) A() A() ( 3 ) ( ) A( A( A( A( t ) A( 3 A( t ) ( ) A( A( Juks and Cantor s (969) on-aramtr modl A( A( t ) A( d
#%" )*& ##+," $ -,!./" %#/%0! %,!
-!"#$% -&!'"$ & #("$$, #%" )*& ##+," $ -,!./" %#/%0! %,! %!$"#" %!#0&!/" /+#0& 0.00.04. - 3 3,43 5 -, 4 $ $.. 04 ... 3. 6... 6.. #3 7 8... 6.. %9: 3 3 7....3. % 44 8... 6.4. 37; 3,, 443 8... 8.5. $; 3
Νόµοςπεριοδικότητας του Moseley:Η χηµική συµπεριφορά (οι ιδιότητες) των στοιχείων είναι περιοδική συνάρτηση του ατοµικού τους αριθµού.
Νόµοςπεριοδικότητας του Moseley:Η χηµική συµπεριφορά (οι ιδιότητες) των στοιχείων είναι περιοδική συνάρτηση του ατοµικού τους αριθµού. Περιοδικός πίνακας: α. Είναι µια ταξινόµηση των στοιχείων κατά αύξοντα
ibemo Kazakhstan Republic of Kazakhstan, West Kazakhstan Oblast, Aksai, Pramzone, BKKS office complex Phone: ; Fax:
ΓΗ ΚΑΙ ΣΥΜΠΑΝ. Εικόνα 1. Φωτογραφία του γαλαξία μας (από αρχείο της NASA)
ΓΗ ΚΑΙ ΣΥΜΠΑΝ Φύση του σύμπαντος Η γη είναι μία μονάδα μέσα στο ηλιακό μας σύστημα, το οποίο αποτελείται από τον ήλιο, τους πλανήτες μαζί με τους δορυφόρους τους, τους κομήτες, τα αστεροειδή και τους μετεωρίτες.
! " #! $ %&! '( #)!' * +#, " -! %&! "!! ! " #$ % # " &' &'... ()* ( +, # ' -. + &', - + &' / # ' -. + &' (, % # , 2**.
! " #! $ %&! '( #)!' * +#, " -! %&! "!!! " #$ % # " &' &'... ()* ( +, # ' -. + &', - + &' / 0123 4 # ' -. + &' (, % #. -5 0126, 2**., 2, + &' %., 0, $!, 3,. 7 8 ', $$, 9, # / 3:*,*2;
1 B0 C00. nly Difo. r II. on III t o. ly II II. Di XR. Di un 5.8. Di Dinly. Di F/ / Dint. mou. on.3 3 D. 3.5 ird Thi. oun F/2. s m F/3 /3.
. F/ /3 3. I F/ 7 7 0 0 Mo ode del 0 00 0 00 A 6 A C00 00 0 S 0 C 0 008 06 007 07 09 A 0 00 0 00 0 009 09 A 7 I 7 7 0 0 F/.. 6 6 8 8 0 00 0 F/3 /3. fo I t o nt un D ou s ds 3. ird F/ /3 Thi ur T ou 0 Fo
Equations. BSU Math 275 sec 002,003 Fall 2018 (Ultman) Final Exam Notes 1. du dv. FTLI : f (B) f (A) = f dr. F dr = Green s Theorem : y da
BSU Math 275 sec 002,003 Fall 2018 (Ultman) Final Exam Notes 1 Equations r(t) = x(t) î + y(t) ĵ + z(t) k r = r (t) t s = r = r (t) t r(u, v) = x(u, v) î + y(u, v) ĵ + z(u, v) k S = ( ( ) r r u r v = u
Some Geometric Properties of a Class of Univalent. Functions with Negative Coefficients Defined by. Hadamard Product with Fractional Calculus I
Itrtol Mthtcl Foru Vol 6 0 o 64 379-388 So otrc Proprts o Clss o Uvlt Fuctos wth Ntv Cocts Dd y Hdrd Product wth Frctol Clculus I Huss Jr Adul Huss Dprtt o Mthtcs d Coputr pplctos Coll o Sccs Uvrsty o
Solutions for Mathematical Physics 1 (Dated: April 19, 2015)
Solutons for Mathematcal Physcs 1 Dated: Aprl 19, 215 3.2.3 Usng the vectors P ê x cos θ + ê y sn θ, Q ê x cos ϕ ê y sn ϕ, R ê x cos ϕ ê y sn ϕ, 1 prove the famlar trgonometrc denttes snθ + ϕ sn θ cos
Sarò signor io sol. α α. œ œ. œ œ œ œ µ œ œ. > Bass 2. Domenico Micheli. Canzon, ottava stanza. Soprano 1. Soprano 2. Alto 1
Sarò signor io sol Canzon, ottava stanza Domenico Micheli Soprano Soprano 2 Alto Alto 2 Α Α Sa rò si gnor io sol del mio pen sie io sol Sa rò si gnor io sol del mio pen sie io µ Tenor Α Tenor 2 Α Sa rò
d dx x 2 = 2x d dx x 3 = 3x 2 d dx x n = nx n 1
d dx x 2 = 2x d dx x 3 = 3x 2 d dx x n = nx n1 x dx = 1 2 b2 1 2 a2 a b b x 2 dx = 1 a 3 b3 1 3 a3 b x n dx = 1 a n +1 bn +1 1 n +1 an +1 d dx d dx f (x) = 0 f (ax) = a f (ax) lim d dx f (ax) = lim 0 =
Linearized Lifting Surface Theory Thin-Wing Theory
13.021 Marine Hdrodnamics Lecture 23 Copright c 2001 MIT - Department of Ocean Engineering, All rights reserved. 13.021 - Marine Hdrodnamics Lecture 23 Linearized Lifting Surface Theor Thin-Wing Theor
The Spiral of Theodorus, Numerical Analysis, and Special Functions
Theo p. / The Spiral of Theodorus, Numerical Analysis, and Special Functions Walter Gautschi wxg@cs.purdue.edu Purdue University Theo p. 2/ Theodorus of ca. 46 399 B.C. Theo p. 3/ spiral of Theodorus 6
Lifting Entry (continued)
ifting Entry (continued) Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion Planar state equations MARYAN 1 01 avid. Akin - All rights reserved http://spacecraft.ssl.umd.edu
,, #,#, %&'(($#(#)&*"& 3,,#!4!4! +&'(#,-$#,./$012 5 # # %, )
!! "#$%&'%( (%)###**#+!"#$ ',##-.#,,, #,#, /01('/01/'#!2#! %&'(($#(#)&*"& 3,,#!4!4! +&'(#,-$#,./$012 5 # # %, ) 6###+! 4! 4! 4,*!47! 4! (! 8!9%,,#!41! 4! (! 4!5),!(8! 4! (! :!;!(7! (! 4! 4!!8! (! 8! 4!!8(!44!
StringMassCylinder2. Clear["Global`*"] r = R * {Cos[theta[t]], Sin[theta[t]]} + u * {Sin[theta[t]], -Cos[theta[t]]}
StringMassCylinder2 In[2224]:= In[297]:= Out[297]= Clear["Global`*"] r = R * {Cos[theta[t]], Sin[theta[t]]} + u * {Sin[theta[t]], -Cos[theta[t]]} {R Cos[theta[t]] + u Sin[theta[t]], - u Cos[theta[t]] +
D Alembert s Solution to the Wave Equation
D Alembert s Solution to the Wave Equation MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Objectives In this lesson we will learn: a change of variable technique
LAPLACE TRANSFORM TABLE
LAPLACE TRANSFORM TABLE Th Laplac afom of am mpl fuco a gv h Tabl. Fuco U mpul U Sp U Ramp Expoal Rpad Roo S Co Polyomal Dampd Dampd co f δ u -a -a co,,... -a -a co F / / /a /a / /!/ /a a/a Thom : Shf
ΘΕΩΡΙΑ ΤΩΝ ΗΜΙΑΓΩΓΩΝ ΔΕΥΤΕΡΗ ΕΝΟΤΗΤΑ ΟΜΟΓΕΝΕΙΣ ΗΜΙΑΓΩΓΟΙ ΦΑΙΝΟΜΕΝΑ ΜΕΤΑΦΟΡΑΣ
ΘΕΩΡΙΑ ΤΩΝ ΗΜΙΑΓΩΓΩΝ ΔΕΥΤΕΡΗ ΕΝΟΤΗΤΑ ΟΜΟΓΕΝΕΙΣ ΗΜΙΑΓΩΓΟΙ ΦΑΙΝΟΜΕΝΑ ΜΕΤΑΦΟΡΑΣ 1. Μηχανισμοί σκέδασης των φορέων (ηλεκτρόνια οπές) 2. Ηλεκτρική Αγωγιμότητα 3. Ολίσθηση φορέων (ρεύμα ολίσθησης) 4. Διάχυση
ECE 308 SIGNALS AND SYSTEMS FALL 2017 Answers to selected problems on prior years examinations
ECE 308 SIGNALS AND SYSTEMS FALL 07 Answers to selected problems on prior years examinations Answers to problems on Midterm Examination #, Spring 009. x(t) = r(t + ) r(t ) u(t ) r(t ) + r(t 3) + u(t +
Byeong-Joo Lee
yeg-j ee OTECH - ME alphad@psteh.a.k yeg-j ee www.psteh.a.k/~alphad ufae Tast ad Allyg Effet N.M. Hwag et al., 000. ue W W 0.4wt% N Vau Aealg yeg-j ee www.psteh.a.k/~alphad Abal a wth f N.M. Hwag yeg-j
!"#$ % &# &%#'()(! $ * +
,!"#$ % &# &%#'()(! $ * + ,!"#$ % &# &%#'()(! $ * + 6 7 57 : - - / :!", # $ % & :'!(), 5 ( -, * + :! ",, # $ %, ) #, '(#,!# $$,',#-, 4 "- /,#-," -$ '# &",,#- "-&)'#45)')6 5! 6 5 4 "- /,#-7 ",',8##! -#9,!"))
MathCity.org Merging man and maths
MathCity.org Merging man and maths Exercise 10. (s) Page Textbook of Algebra and Trigonometry for Class XI Available online @, Version:.0 Question # 1 Find the values of sin, and tan when: 1 π (i) (ii)
Parts Manual. Trio Mobile Surgery Platform. Model 1033
Trio Mobile Surgery Platform Model 1033 Parts Manual For parts or technical assistance: Pour pièces de service ou assistance technique : Für Teile oder technische Unterstützung Anruf: Voor delen of technische
Laplace s Equation in Spherical Polar Coördinates
Laplace s Equation in Spheical Pola Coödinates C. W. David Dated: Januay 3, 001 We stat with the pimitive definitions I. x = sin θ cos φ y = sin θ sin φ z = cos θ thei inveses = x y z θ = cos 1 z = z cos1
derivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Commutative Monoids in Intuitionistic Fuzzy Sets
Commutative Monoids in Intuitionistic Fuzzy Sets S K Mala #1, Dr. MM Shanmugapriya *2 1 PhD Scholar in Mathematics, Karpagam University, Coimbatore, Tamilnadu- 641021 Assistant Professor of Mathematics,
1. (a) (5 points) Find the unit tangent and unit normal vectors T and N to the curve. r(t) = 3cost, 4t, 3sint
1. a) 5 points) Find the unit tangent and unit normal vectors T and N to the curve at the point P, π, rt) cost, t, sint ). b) 5 points) Find curvature of the curve at the point P. Solution: a) r t) sint,,
m i N 1 F i = j i F ij + F x
N m i i = 1,..., N m i Fi x N 1 F ij, j = 1, 2,... i 1, i + 1,..., N m i F i = j i F ij + F x i mi Fi j Fj i mj O P i = F i = j i F ij + F x i, i = 1,..., N P = i F i = N F ij + i j i N i F x i, i = 1,...,
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
m 1, m 2 F 12, F 21 F12 = F 21
m 1, m 2 F 12, F 21 F12 = F 21 r 1, r 2 r = r 1 r 2 = r 1 r 2 ê r = rê r F 12 = f(r)ê r F 21 = f(r)ê r f(r) f(r) < 0 f(r) > 0 m 1 r1 = f(r)ê r m 2 r2 = f(r)ê r r = r 1 r 2 r 1 = 1 m 1 f(r)ê r r 2 = 1 m
HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
Fundamental Equations of Fluid Mechanics
Fundamental Equations of Fluid Mechanics 1 Calculus 1.1 Gadient of a scala s The gadient of a scala is a vecto quantit. The foms of the diffeential gadient opeato depend on the paticula geomet of inteest.
SUPPLEMENTAL INFORMATION. Fully Automated Total Metals and Chromium Speciation Single Platform Introduction System for ICP-MS
Electronic Supplementary Material (ESI) for Journal of Analytical Atomic Spectrometry. This journal is The Royal Society of Chemistry 2018 SUPPLEMENTAL INFORMATION Fully Automated Total Metals and Chromium
ΕΠΙΤΡΟΠΗ ΔΙΑΓΩΝΙΣΜΩΝ ΕΜΕ 28 η Ελληνική Μαθηματική Ολυμπιάδα "Ο Αρχιμήδης" ΣΑΒΒΑΤΟ, 26 ΦΕΒΡΟΥΑΡΙΟΥ 2011 ( )
ΕΠΙΤΡΟΠΗ ΔΙΑΓΩΝΙΣΜΩΝ ΕΜΕ 8 η Ελληνική Μαθηματική Ολυμπιάδα "Ο Αρχιμήδης" ΣΑΒΒΑΤΟ 6 ΦΕΒΡΟΥΑΡΙΟΥ 0 Ενδεικτικές Λύσεις θεμάτων μεγάλων τάξεων ΠΡΟΒΛΗΜΑ Να λύσετε στους ακέραιους την εξίσωση 4 xy y x = xy 6.
Forced Pendulum Numerical approach
Numerical approach UiO April 8, 2014 Physical problem and equation We have a pendulum of length l, with mass m. The pendulum is subject to gravitation as well as both a forcing and linear resistance force.
DISPLAY SUPPLY: FILTER STANDBY
ircuit iagrams and PW Layouts. ircuit iagrams and PW Layouts J.0 P. 0 isplay Supply P: ilter Standby MNS NPUT -Vac 00 P-V- V_OT 0 0 0 0 0 0 0 0 SPLY SUPPLY: LT STNY 0 M0 V 0 T,/0V MSU -VOLTS NOML... STNY
TeSys contactors a.c. coils for 3-pole contactors LC1-D
References a.c. coils for 3-pole contactors LC1-D Control circuit voltage Average resistance Inductance of Reference (1) Weight Uc at 0 C ± 10 % closed circuit For 3-pole " contactors LC1-D09...D38 and