ECE 307: Electricity and Magnetism Fall 2012

Size: px
Start display at page:

Download "ECE 307: Electricity and Magnetism Fall 2012"

Transcription

1 ECE 7: Electct nd Mgnetsm Fll Instucto: J.D. Wllms, ssstnt Pofesso Electcl nd Compute Engneeng Unvest of lbm n Huntsvlle 46 Optcs uldng, Huntsvlle, l 5899 Pone: (56) , eml: jon.wllms@u.edu Couse mtel posted on UH ngel couse mngement webste Tetboo: M.N.O. Sdu, Elements of Electomgnetcs 5 t ed. Ofod Unvest Pess, 9. Optonl Redng: H.M. Se, Dv Gd Cul nd ll tt: n nfoml tet on vecto clculus, 4 t ed. Noton Pess, 5. ll fgues ten fom pm tetboo unless otewse cted.

2 Couse Mtel Cptes -. (Revew Mtel) ectos lgeb, Coodnte Tnsfomtons, ecto Clculus Cpte 4. Coulomb s Lw, Electc Feld Intenst, Cge Dstbuton, Electc Flu Denst, Guss Lw, Electc Potentl, Eneg Cptes 5-6. Popetes of Mtels, Cuents, Contnut Equton, Posson s Equton, Lplce s Equton, Resstnce, Cpctnce, Imge Teo (Opt.) Cptes 7-8. ot-svt Lw, mpee s Lw, Mgnetc Flu Denst, Mwell s Equtons (Sttc), Mgnetc ecto Potentls, Mgnetc Foces, Mgnetc Mtels, ound Condtons Cpte 9. (Ptl) Fd s Lw, Mwell s Equtons (Tme ng), Optonl: Tme-Hmonc Felds, Plne Wves Pontng ectos Gdng: 8 -%, 7-8%, 6-7% C. Homewo (%): Tuned n weel. Gded on ttempted effot. Ems (5%): pe Semeste. Em scoes stocll ncese ove te semeste Em : Cptes 4, 5 nd 6 Em : Cptes 7, 8 nd 9 Fnl Compeensve Em (%): Includes mtel fom Cptes 4 toug 9

3 Mwell s Tme Dependent Equtons It ws Jmes Cl Mwell tt put ll of ts togete nd educed electomgnetc feld teo to 4 smple equtons. It ws onl toug ts clfcton tt te dscove of electomgnetc wves wee dscoveed nd te teo of lgt ws developed. Te equtons Mwell s cedted wt to completel descbe n electomgnetc feld (ete sttcll o dnmcll) e wtten s: Dffeentl Fom Integl Fom Rems D E t H v J D t DdS L S S ds E dl H dl Guss s Lw Nonestence of te Mgnetc Monopole Fd s Lw mpee s Ccut Lw L S 8/7/ dv v t J S ds D ds t

4 ECE 7: Electct nd Mgnetsm Cptes -: Revew of Mtemtcl Essentls Cpte : ecto nlss Scls nd ectos Unt ecto ecto ddton nd Subtcton Poston nd Dstnce ectos ecto Multplcton Components of ecto Cpte : Coodnte Sstems nd Tnsfomtons Ctesn Coodntes Ccul Clndcl Coodntes Specl Coodntes Constnt Coodnte Sufces Cpte : ecto Clculus Dffeentl Lengt, e, nd olume Lne, Sufce, nd olume Integls Del Opeto Gdent of Scl Dvegence of ecto nd te Dvegence Teoem Cul of ecto nd Stoes s Teoem Lplcn of Scl Clssfcton of ecto Felds Homewo C. : 5c, 6, 8, 8b,, 6 C. : 7, 8,, 5, 7 C. : b, b, c, b, 6,, b,, Note: Students wll not be tested specfcll on Cptes -. Howeve, te nfomton contned wtn tem wll be used n lmost eve spect of te couse. Knowledge nd slled pctce of tese concepts wll be equed to complete ll omewo, nd emntons tougout te semeste

5 ECE 7: Electct nd Mgnetsm Cptes -: Revew of Mtemtcl Essentls Cpte : ecto nlss Scls nd ectos Unt ecto ecto ddton nd Subtcton Poston nd Dstnce ectos ecto Multplcton Components of ecto Note: Students wll not be tested specfcll on Cptes -. Howeve, te nfomton contned wtn tem wll be used n lmost eve spect of te couse. Knowledge nd slled pctce of tese concepts wll be equed to complete ll omewo, nd emntons tougout te semeste

6 ectos nd Scls Scl: quntt defned onl b ts mgntude speed: 4 m/s Cge: Coulombs Cpctnce: 5 fds ecto: quntt defned b bot ts mgntude nd decton n spce Foce: F ( 5 4 ) N Feld: Functon tt specfes ptcul quntt evewee wtn sptl domn. Electc feld (vecto feld): oltge (scl feld): q E N / C q

7 Unt ecto vecto s bot mgntude nd decton Te mgntude of s scl wtten s II. unt vecto long s defned s vecto long te decton of wt mgntude of. s suc, te vecto m be defned s nd tus Emple:

8 ecto ddton nd Subtcton Two vectos, nd, cn be dded togete to genete td vecto, C Me sue to bde b te followng bsc lws of lgeb s ppled to vectos ddton Multplcton C C l l Commuttve ssoctve Dstbutve C C

9 Dstnce ectos ectos cn lso be used to defne te dstnce between two ponts n coodnte sstem, o between lne nd plne wtn coodnte sstem If gven two ponts, P nd Q, one cn fnd te dstnce between tem s te vecto,,,,, ),, ( ),, ( Q P P Q PQ Q P

10 ecto Multplcton: Dot Poduct Two vectos, nd, cn be multpled togete to genete td vecto, C Scl Poduct ecto Poduct Scl Tple Poduct ecto Tple Poduct Me sue to bde b te followng bsc lws of lgeb s ppled to dot poducts D C l C C C C Commuttve ssoctve cos Note: otogonl vecto dot poducts multpl to cosne vlue of eo pllel vecto dot poducts multpl to cosne vlue of

11 ecto Multplcton: Coss Poduct Me sue to bde b te followng bsc lws of lgeb s ppled to coss poducts C C C C nt-commuttve Not ssoctve Dstbutve n sn Note: otogonl vecto dot poducts multpl to sne vlue of C C C C C C C C C C Scl Tple Poduct ecto Tple Poduct

12 Components of ecto Gven two vectos, nd, one cn dectl fnd te scl component of long s cos Ts scl poduct s nown s te pojecton of long te decton. Te vecto component of long s smpl scl component multpled b te unt vecto long cos One cn lso fnd te ngle between nd usng te coss nd dot poduct of te two cos sn

13 ECE 7: Electct nd Mgnetsm Cptes -: Revew of Mtemtcl Essentls Cpte : Coodnte Sstems nd Tnsfomtons Ctesn Coodntes Ccul Clndcl Coodntes Specl Coodntes Constnt Coodnte Sufces Note: Students wll not be tested specfcll on Cptes -. Howeve, te nfomton contned wtn tem wll be used n lmost eve spect of te couse. Knowledge nd slled pctce of tese concepts wll be equed to complete ll omewo, nd emntons tougout te semeste

14 Ctesn Coodntes Coodnte sstem epesented b (,,) tt e tee otogonl vectos n stt lnes tt ntesect t sngle pont (te ogn). Te vecto n ts coodnte sstem cn be wtten s

15 Clndcl Coodntes Coodnte sstem epesented b (,,) tt e tee otogonl vectos Te vecto n ts coodnte sstem cn be wtten s Wee te followng equtons cn be used to convet between clndcl nd ctesn coodnte sstems tn cos sn

16 Mt Tnsfomtons: Ct. nd Cl. cos sn sn cos cos sn sn cos

17 Specl Coodntes Coodnte sstem epesented b (,,) tt e tee otogonl vectos emntng fom o evolvng ound te ogn Te vecto n ts coodnte sstem cn be wtten s Wee te followng equtons cn be used to convet between specl nd Ctesn coodnte sstems tn tn sn cos sn sn cos

18 Mt Tnsfomtons: Ct. nd Sp. sn cos cos sn cos sn sn sn cos cos cos sn cos sn sn sn cos cos cos cos sn sn cos sn

19 ECE 7: Electct nd Mgnetsm Cptes -: Revew of Mtemtcl Essentls Cpte : ecto Clculus Dffeentl Lengt, e, nd olume Lne, Sufce, nd olume Integls Del Opeto Gdent of Scl Dvegence of ecto nd te Dvegence Teoem Cul of ecto nd Stoes s Teoem Lplcn of Scl Clssfcton of ecto Felds Note: Students wll not be tested specfcll on Cptes -. Howeve, te nfomton contned wtn tem wll be used n lmost eve spect of te couse. Knowledge nd slled pctce of tese concepts wll be equed to complete ll omewo, nd emntons tougout te semeste

20 Dffeentl Lengt, e, nd olume Ctesn Coodntes Dffeentl Dsplcement (dl) dl d Dffeentl Sufce e (ds) d ds dd ds dd ds dd d olume Dffeentl (dv) dv ddd Note: Dffeentl lengt nd sufce es e vectos. Dffeentl volume s scl

21 Dffeentl Lengt, e, nd olume Clndcl Coodntes Dffeentl Dsplcement (dl) dl d d d Dffeentl Sufce e (ds) ds dd ds dd ds dd olume Dffeentl (dv) dv ddd Note: tem s used s multple to complete unts ssocted wt c dns of d

22 Dffeentl Lengt, e, nd olume Specl Coodntes Dffeentl Dsplcement (dl) dl d d snd Dffeentl Sufce e (ds) ds sn d d ds sn dd ds dd olume Dffeentl (dv) dv sn dd d Note: Dffeentl lengt nd sufce es e vectos. Dffeentl volume s scl

23 Lne Integls Te lne ntegl s te ntegl of te tngentl component of long te cuve L Reques L be smoot, contnuous cuve. Te vecto, m be vecto feld component Lne ntegls e sd to be pt ndependent f te soluton of te tngentl component of s ndependent to te pt L ten wtn te feld. Te most common emple of pt ndependent ntegls used e wo (eneg) solutons ntegtng te foce ove te pt lengt, L. Pt ndependence of occus f = (te cul of s equl to eo) L dl dl b cosdl Fo contou of lengt, L Gves te cculton of ound te contou, L

24 Emple of Lne Integl Gven te followng equton fo F = [,- ] Fnd te lne ntegl of F long te followng pts between (,) nd (,) Pt : stt lne Pt : pbol d d d d d W d d / d d d d W d d d d d F W d d d F d d d,, Pt Dependent!!! Sow (on ou own) tt te cul does not equl

25 Note: Te sufce ntegl wll become te bss fo flu of te electc feld toug Gussn sufce n Cpte 4 Sufce Integls Te sufce ntegl s te ntegl of te vecto feld,, ove te closed contou, S povdes te net outwd flu of toug te sufce, S ds n S ds ds ds b cosds noml _ to _ te _ sufce S n ds

26 Sufce Integls: Fndng te e of te Sufce Sufce ntegls e double ntegls tt clculte te e of closed sufce Emples Ccle Clnde ( two ccles wt tubul sufce between tem L d L d d d d ds ds ds ds ds L S S S S S d d d d d ds S S S L

27 Sufce Integls (cont.) Unt convesons llow one to smplf poblems to ese clcultons Emple: Fnd te e cut fom te uppe lf of spee b te clnde Ts s te sum of te e of te spee wc pojects onto te ds n te (,) plne. Tus, we wnt to ntegte te e of te ds. Te geomet pesented sows te e tt wll be ntegted. Integton e Te desed e cn ten be clculted s: S ds / wee, d sn e dd d / cos dd dd / sn dd

28 e nd olume dv v v ddd S ds S S dd

29 olume Integls Used to clculte te volume, mss, centod, cge, etc of sold vdv v v dv v v s volumetc denst functon olume of spee v dv v snddd sndd d 4 olume of clnde v dv v olume ddd ( e L dd L _ of _ te _ ccle )( egt )

30 Del Opeto Del s vecto dffeentl opeto. Te del opeto wll be used n 4 dffeentl opetons tougout n couse on feld teo. Te followng equton s te del opeto fo dffeent coodnte sstems Gdent of Scl, s wtten s vecto Te dvegence of vecto,, s wtten s scl Te cul of vecto,, s wtten s vecto Te Lplcn of scl,, s wtten s scl sn,,

31 Gdent of Scl Te gdent of scl feld,, s vecto tt epesents bot te mgntude nd te decton of te mmum spce te of ncese of. To elp vsule ts concept, te fo emple topogpcl mp. Lnes on te mp epesent equl mgntudes of te scl feld. Te gdent vecto cosses mp t te locton wee te lnes pced nto te most dense spce nd pependcul (o noml) to tem. Te oentton (up o down) of te gdent vecto s suc tt te feld s ncesed n mgntude long tt decton. Emple,,,4, 4

32 Gdent of Scl () Fundmentl popetes of te gdent of scl feld Te mgntude of gdent equls te mmum te of cnge n pe unt dstnce Gdent ponts n te decton of te mmum te of cnge n Gdent t n pont s pependcul to te constnt sufce tt psses toug tt pont Te pojecton of te gdent n te decton of te unt vecto, s nd s clled te dectonl devtve of long. Ts s te te of cnge of n te decton of. If s te gdent of, ten s sd to be te scl potentl of Esl Poven Mtemtcl Reltons U U U U U U U U U n n

33 Dvegence of ecto Te dvegence of vecto,, t n gven pont P s te outwd flu pe unt volume s volume sns bout P. Te followng eltons cn be esl deved fo te dvegence of feld v ds dv s v sn sn sn lm Imge fom

34 Dvegence Teoem Te dvegence teoem sttes tt te totl outwd flu of vecto feld,, toug te closed sufce, S, s te sme s te volume ntegl of te dvegence of. Ts teoem s esl sown fom te equton fo te dvegence of vecto feld. ds s dv lm v v dv ds v poof s : ds s s ds s ds v v Dvegence of vecto (ed) out of specl sufce (geen) ttp:// dv

35 Dvegence Teoem Emples: d d d d d d d d d d ds S S s Suppose we wnted to fnd te flu of toug clnde of dus nd egt

36 Cul of ecto Te cul of vecto, s n l vecto wose mgntude s te mmum cculton of pe unt e s te e tends to eo nd wose decton s te noml decton of te e wen te e s oented to me te cculton mmum. cul lm S L dl S m Te followng eltons cn be esl deved fo te dvegence of feld n Inset: Feld lnes of (n blc) defned toug te closed loop dl (geen)

37 Cul of ecto () Cul of vecto n ec of te tee pm coodnte sstems dscussed n ts couse sn sn sn sn sn sn Ctesn Clndcl Specl

38 Cul of ecto (Emple) cot 6 cos 6 sn sn sn sn

39 Stoes Teoem Stoes teoem sttes tt te cculton of vecto feld, ound closed pt, L s equl to te sufce ntegl of te cul of ove te open sufce S bounded b L. Ts teoem s been poven to old s long s nd te cul of e contnuous long te closed sufce S of closed pt L Ts teoem s esl sown fom te equton fo te cul of vecto feld. cul lm S dl ds L poof L : dl s L dl L dl S L m n dl S S ds n ds Fo emple, see lne ntegl

40 Lplcn of Scl Te Lplcn dffeentl opeto s combnton of gdent nd dvegence opetos. Te Lplcn of scl feld s te dvegence of te gdent of n monc feld Lplce s Eqn. Feld wt ntenl dvegence Poson s Eqn. Te followng vecto dentt olds tue fo ll Lplcn felds govened b vecto, sn sn sn Lplcn

41 Clssfcton of ecto Felds ecto felds e unquel defned b te dvegence nd cul. Howeve nete te dvegence o te cul lone of vecto feld s suffcent to completel descbe te feld. ot flu nd otton must be consdeed concuentl. Tus we use te fou followng equtons to mtemtcll descbe nd pedct te effects of vecto feld () (b) (c) (d),,,,

42 Clssfcton of ecto Felds Te vecto feld,, s sd to be dvegenceless ( o solenodl) f Suc felds ve no souce o sn of flu, tus ll te vecto feld lnes enteng n enclosed sufce, S, must lso leve t. Emples nclude mgnetc felds, conducton cuent denst unde sted stte, nd mcompessble fluds Te followng equtons e commonl utled to solve dvegenceless feld poblems Te vecto feld,, s sd to be potentl (o ottonl) f Suc felds e sd to be consevtve. Emples nclude gvt, nd electosttc felds. Te followng equtons e commonl used to solve potentl feld poblems ds dv S v F dl L S ds

43 Clssfcton of ecto Felds () Retun gn to ou ntl sttement. vecto feld s defned b bot ts dvegence nd ts cul. Tus, we cn descbe n feld b evlutng te followng two equtons. v wee s denst component of te feld, v epesents volume, nd S epesents sufce e. We efe to te volume denst component s te souce denst, nd te e component s te cculton denst Hemolt s Teoem sttes tt n feld stsfng te condtons bove wose denst components vns t nfnt cn be completel descbed s te sum of two vectos. One of te vectos s ottonl, nd te ote s solenodl. In ou cse, te ottonl component s te electc feld, nd te solenodl component s te mgnetc feld. s suc one m completel descbe n electomgnetc feld usng te followng eltons F q E v v S S

44 (Optonl Mtel) ltentve ppoc to ecto Devtves Usng Scle Fctos (Optonl Mtel)

45 Devtves n Spce Scl Feld: scl functon, S, of poston coodntes n fnte dmensonl vecto spce. Fo most of ou puposes, te fnte vecto dmensonl vecto spce wll be lmted to dmensons (o pscll el spce). Defnng coodnte sstem {,, }wtn, ten llows fo dffeentton to be wtten s: ds d Nme of te coodnte sstem d d o ds Ctesn Clndcl Specl ds d d d sn 45

46 Devtves n Spce: Emple Deve te scle fctos fo te specl coodnte sstem ds d d d sn cos sn sn cos d d d d d d d d d d d d d d d d d d sn Hee we te onl te postve oots 46

47 Dffeentl Lengt, e, nd olume Ctesn Coodntes Dffeentl Dsplcement (dl) dl d d d d d d d Dffeentl Sufce e (ds) ds j ddê d d j ds ds ds dd dd dd olume Dffeentl (dv) dv j j d ddd d d d j d d Note: Dffeentl lengt nd sufce es e vectos. Dffeentl volume s scl 47

48 Dffeentl Lengt, e, nd olume Clndcl Coodntes Dffeentl Dsplcement (dl) dl d d Dffeentl Sufce e (ds) ds ds ds olume Dffeentl (dv) d d d d d d d d d e e e d dv ddd ddd dd dd dd Note: tem s used s multple to complete unts ssocted wt c dns of d d 48

49 Dffeentl Lengt, e, nd olume Specl Coodntes Dffeentl Dsplcement (dl) Dffeentl Sufce e (ds) olume Dffeentl (dv) d dd d dd dv sn d d d d d d d dl sn dd dd ds dd dd ds d d d d ds sn sn Note: Dffeentl lengt nd sufce es e vectos. Dffeentl volume s scl 49

50 Del Opeto Del s vecto dffeentl opeto. Te del opeto wll be used n 4 dffeentl opetons tougout n couse on feld teo. Te followng equton s te del opeto fo dffeent coodnte sstems Gdent of Scl, s wtten s vecto Te dvegence of vecto,, s wtten s scl Te cul of vecto,, s wtten s vecto Te Lplcn of scl,, s wtten s scl sn,, 5 e e

51 Devtves n Spce: Te Gdent Defnton: Te gdent s vecto feld geneted b te poduct of te dffeentl opeto,, on scl feld,. gd ds ds d s suc, scl s smpl te poduct of two vectos. d ds cos ds Te gdent s te mmum te of cnge of te scl feld on te pont t wc te devtve s ten. It follows tt s te mmum te of cnge of d ds Fnll, s lws pependcul to equpotentl lnes of te sufce Te genel fom of te t component of te gdent vecto follows fom te defnton of te gdent lm ds d ds Yeldng te genel equton gd e 5

52 Devtves n Spce: Dvegence olume of te sufce: ddd Te dvegence of te scl feld ove te sufce s wtten s: dv j fo j j 5

53 Genel Equtons fo te cul n n otogonl cuvlne coodnte sstem cn be etended fom ou clcultons of te vecto coss poduct: Devtves n Spce: Te Cul 5 e e e e cul ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

54 Devtves n Spce: Lplcn Defnton: Te Lplcn of scl feld s defned s te dvegence of te gdent tt scl feld,. Geneled fom of te Lplcn of vecto feld,, n otogonl cuvlne coodntes 54 Lplcn

Summary: Vectors. This theorem is used to find any points (or position vectors) on a given line (direction vector). Two ways RT can be applied:

Summary: Vectors. This theorem is used to find any points (or position vectors) on a given line (direction vector). Two ways RT can be applied: Summ: Vectos ) Rtio Theoem (RT) This theoem is used to find n points (o position vectos) on given line (diection vecto). Two ws RT cn e pplied: Cse : If the point lies BETWEEN two known position vectos

More information

Bending Stresses for Simple Shapes

Bending Stresses for Simple Shapes -6 Bendng Stesses fo Smple Sapes In bendng, te maxmum stess and amount of deflecton can be calculated n eac of te followng stuatons. Addtonal examples ae avalable n an engneeng andbook. Secton Modulus

More information

Orbits and Kepler s Laws

Orbits and Kepler s Laws Obits nd Keple s Lws This web pge intoduces some of the bsic ides of obitl dynmics. It stts by descibing the bsic foce due to gvity, then consides the ntue nd shpe of obits. The next section consides how

More information

16. Mean Square Estimation

16. Mean Square Estimation 6 Me Sque stmto Gve some fomto tht s elted to uow qutty of teest the poblem s to obt good estmte fo the uow tems of the obseved dt Suppose epeset sequece of dom vbles bout whom oe set of obsevtos e vlble

More information

Simulation of Spacecraft Attitude and Orbit Dynamics

Simulation of Spacecraft Attitude and Orbit Dynamics Smulton o Spcect Atttude nd Obt Dynmcs Ps Rhmäk, Jen-Pete Ylén Contol Engneeng Lbotoy Helsnk Unvesty o Technology PL-, TKK E-ml: ps.hmk@tkk., pete.ylen@tkk. KEYWORDS Smulton Model, Stellte, FDIR, Qutenon

More information

Orbit dynamics and kinematics with full quaternions

Orbit dynamics and kinematics with full quaternions bt dynamcs and knematcs wth full quatenons Davde Andes and Enco S. Canuto, Membe, IEEE Abstact Full quatenons consttute a compact notaton fo descbng the genec moton of a body n the space. ne of the most

More information

THE GEOMETRY OF PYRAMIDS

THE GEOMETRY OF PYRAMIDS TE GEOMETRY OF PYRAMIDS One of te more interesting solid structures wic s fscinted individuls for tousnds of yers going ll te wy bck to te ncient Egyptins is te pyrmid. It is structure in wic one tkes

More information

Gravitation. Definition of Weight Revisited. Newton s Law of Universal Gravitation. Newton s Law of Universal Gravitation. Gravitational Field

Gravitation. Definition of Weight Revisited. Newton s Law of Universal Gravitation. Newton s Law of Universal Gravitation. Gravitational Field Defnton of Weght evsted Gavtaton The weght of an object on o above the eath s the gavtatonal foce that the eath exets on the object. The weght always ponts towad the cente of mass of the eath. On o above

More information

(Ch. 22.5) 2. What is the magnitude (in pc) of a point charge whose electric field 50 cm away has a magnitude of 2V/m?

(Ch. 22.5) 2. What is the magnitude (in pc) of a point charge whose electric field 50 cm away has a magnitude of 2V/m? Em I Solutions PHY049 Summe 0 (Ch..5). Two smll, positively chged sphees hve combined chge of 50 μc. If ech sphee is epelled fom the othe by n electosttic foce of N when the sphees e.0 m pt, wht is the

More information

Vector Geometry for Computer Graphics

Vector Geometry for Computer Graphics Vector Geometry for Computer Grphcs Bo Getz Jnury, 7 Contents Prt I: Bsc Defntons Coordnte Systems... Ponts nd Vectors Mtrces nd Determnnts.. 4 Prt II: Opertons Vector ddton nd sclr multplcton... 5 The

More information

Electric Potential. otherwise to move the object from initial point i to final point f

Electric Potential. otherwise to move the object from initial point i to final point f PHY2061 Enched Physcs 2 Lectue Notes Electc Potental Electc Potental Dsclame: These lectue notes ae not meant to eplace the couse textbook. The content may be ncomplete. Some topcs may be unclea. These

More information

Scalar and Vector Quantities. A scalar is a quantity having only magnitude (and possibly phase). LECTURE 2a: VECTOR ANALYSIS Vector Algebra

Scalar and Vector Quantities. A scalar is a quantity having only magnitude (and possibly phase). LECTURE 2a: VECTOR ANALYSIS Vector Algebra Sclr nd Vector Quntities : VECTO NLYSIS Vector lgebr sclr is quntit hving onl mgnitude (nd possibl phse). Emples: voltge, current, chrge, energ, temperture vector is quntit hving direction in ddition to

More information

Curvature. (Com S 477/577 Notes) Yan-Bin Jia. Oct 8, 2015

Curvature. (Com S 477/577 Notes) Yan-Bin Jia. Oct 8, 2015 Cuvtue Com S 477/577 Notes Yn-Bin Ji Oct 8, 205 We wnt to find mesue of how cuved cuve is. Since this cuvtue should depend only on the shpe of the cuve, it should not be chnged when the cuve is epmetized.

More information

Exam in physics, El-grunder (Electromagnetism), 2014-03-26, kl 9.00-15.00

Exam in physics, El-grunder (Electromagnetism), 2014-03-26, kl 9.00-15.00 Umeå Univesitet, Fysik 1 Vitly Bychkov Em in physics, El-gunde (Electomgnetism, 14--6, kl 9.-15. Hjälpmedel: Students my use ny book(s. Mino notes in the books e lso llowed. Students my not use thei lectue

More information

Volumes as integrals of cross-sections (Sect. 6.1) Volumes as integrals of cross-sections (Sect. 6.1)

Volumes as integrals of cross-sections (Sect. 6.1) Volumes as integrals of cross-sections (Sect. 6.1) Volumes s integrls of cross-sections (ect. 6.1) Te volume of simple regions in spce Volumes integrting cross-sections: Te generl cse. Certin regions wit oles. Volumes s integrls of cross-sections (ect.

More information

Positive Integral Operators With Analytic Kernels

Positive Integral Operators With Analytic Kernels Çnky Ünverte Fen-Edeyt Fkülte, Journl of Art nd Scence Sy : 6 / Arl k 006 Potve ntegrl Opertor Wth Anlytc Kernel Cn Murt D KMEN Atrct n th pper we contruct exmple of potve defnte ntegrl kernel whch re

More information

N V V L. R a L I. Transformer Equation Notes

N V V L. R a L I. Transformer Equation Notes Tnsfome Eqution otes This file conts moe etile eivtion of the tnsfome equtions thn the notes o the expeiment 3 wite-up. t will help you to unestn wht ssumptions wee neee while eivg the iel tnsfome equtions

More information

EN3: Introduction to Engineering. Teach Yourself Vectors. 1. Definition. Problems

EN3: Introduction to Engineering. Teach Yourself Vectors. 1. Definition. Problems EN3: Introducton to Engneerng Tech Yourself Vectors Dvson of Engneerng Brown Unversty. Defnton vector s mthemtcl obect tht hs mgntude nd drecton, nd stsfes the lws of vector ddton. Vectors re used to represent

More information

Newton-Raphson Method of Solving a Nonlinear Equation Autar Kaw

Newton-Raphson Method of Solving a Nonlinear Equation Autar Kaw Newton-Rphson Method o Solvng Nonlner Equton Autr Kw Ater redng ths chpter, you should be ble to:. derve the Newton-Rphson method ormul,. develop the lgorthm o the Newton-Rphson method,. use the Newton-Rphson

More information

Perturbation Theory and Celestial Mechanics

Perturbation Theory and Celestial Mechanics Copyght 004 9 Petubaton Theoy and Celestal Mechancs In ths last chapte we shall sketch some aspects of petubaton theoy and descbe a few of ts applcatons to celestal mechancs. Petubaton theoy s a vey boad

More information

Vector Algebra. Lecture programme. Engineering Maths 1.2

Vector Algebra. Lecture programme. Engineering Maths 1.2 Leue pogmme Engneeng Mh. Veo lge Conen of leue. Genel noduon. Sl nd veo. Cen omponen. Deon one. Geome epeenon. Modulu of veo. Un veo. Pllel veo.. ddon of veo: pllelogm ule; ngle lw; polgon lw; veo lw fo

More information

(Semi)Parametric Models vs Nonparametric Models

(Semi)Parametric Models vs Nonparametric Models buay, 2003 Pobablty Models (Sem)Paametc Models vs Nonpaametc Models I defne paametc, sempaametc, and nonpaametc models n the two sample settng My defnton of sempaametc models s a lttle stonge than some

More information

Derivatives and Rates of Change

Derivatives and Rates of Change Section 2.1 Derivtives nd Rtes of Cnge 2010 Kiryl Tsiscnk Derivtives nd Rtes of Cnge Te Tngent Problem EXAMPLE: Grp te prbol y = x 2 nd te tngent line t te point P(1,1). Solution: We ve: DEFINITION: Te

More information

CLASS XI CHAPTER 3. Theorem 1 (sine formula) In any triangle, sides are proportional to the sines of the opposite angles. That is, in a triangle ABC

CLASS XI CHAPTER 3. Theorem 1 (sine formula) In any triangle, sides are proportional to the sines of the opposite angles. That is, in a triangle ABC CLASS XI Anneue I CHAPTER.6. Poofs and Simple Applications of sine and cosine fomulae Let ABC be a tiangle. By angle A we mean te angle between te sides AB and AC wic lies between 0 and 80. Te angles B

More information

Table of Information and Equation Tables for AP Physics Exams

Table of Information and Equation Tables for AP Physics Exams le of Infomton n Eton le fo P Py Em e ompnyng le of Infomton n Eton le wll e pove to tent wen tey tke te P Py Em. eefoe, tent my NO ng te own ope of tee tle to te em oom, ltog tey my e tem togot te ye

More information

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100 hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

More information

LINES ON BRIESKORN-PHAM SURFACES

LINES ON BRIESKORN-PHAM SURFACES LIN ON BRIKORN-PHAM URFAC GUANGFNG JIANG, MUTUO OKA, DUC TAI PHO, AND DIRK IRMA Abstact By usng toc modfcatons and a esult of Gonzalez-pnbeg and Lejeune- Jalabet, we answe the followng questons completely

More information

Formulas and Units. Transmission technical calculations Main Formulas. Size designations and units according to the SI-units.

Formulas and Units. Transmission technical calculations Main Formulas. Size designations and units according to the SI-units. Fomuls nd Units Tnsmission technicl clcultions Min Fomuls Size designtions nd units ccoding to the SI-units Line movement: s v = m/s t s = v t m s = t m v = m/s t P = F v W F = m N Rottion ω = π f d/s

More information

Resistive Network Analysis. The Node Voltage Method - 1

Resistive Network Analysis. The Node Voltage Method - 1 esste Network Anlyss he nlyss of n electrcl network conssts of determnng ech of the unknown rnch currents nd node oltges. A numer of methods for network nlyss he een deeloped, sed on Ohm s Lw nd Krchoff

More information

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a. Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles

More information

An Algorithm For Factoring Integers

An Algorithm For Factoring Integers An Algothm Fo Factong Integes Yngpu Deng and Yanbn Pan Key Laboatoy of Mathematcs Mechanzaton, Academy of Mathematcs and Systems Scence, Chnese Academy of Scences, Bejng 100190, People s Republc of Chna

More information

Mechanics 1: Work, Power and Kinetic Energy

Mechanics 1: Work, Power and Kinetic Energy Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).

More information

Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chapter 40 Key Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

More information

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one. 5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

More information

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus Section 5.4 Te Funmentl Teorem of Clculus Kiryl Tsiscnk Te Funmentl Teorem of Clculus EXAMPLE: If f is function wose grp is sown below n g() = f(t)t, fin te vlues of g(), g(), g(), g(3), g(4), n g(5).

More information

Cypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period:

Cypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period: Nme: SOLUTIONS Dte: Period: Directions: Solve ny 5 problems. You my ttempt dditionl problems for extr credit. 1. Two blocks re sliding to the right cross horizontl surfce, s the drwing shows. In Cse A

More information

Additional File 1 - A model-based circular binary segmentation algorithm for the analysis of array CGH data

Additional File 1 - A model-based circular binary segmentation algorithm for the analysis of array CGH data 1 Addtonal Fle 1 - A model-based ccula bnay segmentaton algothm fo the analyss of aay CGH data Fang-Han Hsu 1, Hung-I H Chen, Mong-Hsun Tsa, Lang-Chuan La 5, Ch-Cheng Huang 1,6, Shh-Hsn Tu 6, Ec Y Chuang*

More information

WHAT HAPPENS WHEN YOU MIX COMPLEX NUMBERS WITH PRIME NUMBERS?

WHAT HAPPENS WHEN YOU MIX COMPLEX NUMBERS WITH PRIME NUMBERS? WHAT HAPPES WHE YOU MIX COMPLEX UMBERS WITH PRIME UMBERS? There s n ol syng, you n t pples n ornges. Mthemtns hte n t; they love to throw pples n ornges nto foo proessor n see wht hppens. Sometmes they

More information

AMPERE S LAW. by Kirby Morgan MISN-0-138

AMPERE S LAW. by Kirby Morgan MISN-0-138 MISN-0-138 AMPERE S LAW by Kiby Mogn 1. Usefullness................................................ 1 AMPERE S LAW 2. The Lw................................................... 1. The Integl Reltionship...............................

More information

Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Luby s Alg. for Maximal Independent Sets using Pairwise Independence Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent

More information

Or more simply put, when adding or subtracting quantities, their uncertainties add.

Or more simply put, when adding or subtracting quantities, their uncertainties add. Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re

More information

PCA vs. Varimax rotation

PCA vs. Varimax rotation PCA vs. Vamax otaton The goal of the otaton/tansfomaton n PCA s to maxmze the vaance of the new SNP (egensnp), whle mnmzng the vaance aound the egensnp. Theefoe the dffeence between the vaances captued

More information

(1) continuity equation: 0. momentum equation: u v g (2) u x. 1 a

(1) continuity equation: 0. momentum equation: u v g (2) u x. 1 a Comment on The effect of vible viscosity on mied convection het tnsfe long veticl moving sufce by M. Ali [Intentionl Jounl of Theml Sciences, 006, Vol. 45, pp. 60-69] Asteios Pntoktos Associte Pofesso

More information

Moment and couple. In 3-D, because the determination of the distance can be tedious, a vector approach becomes advantageous. r r

Moment and couple. In 3-D, because the determination of the distance can be tedious, a vector approach becomes advantageous. r r Moment and couple In 3-D, because the detemination of the distance can be tedious, a vecto appoach becomes advantageous. o k j i M k j i M o ) ( ) ( ) ( + + M o M + + + + M M + O A Moment about an abita

More information

AREA OF A SURFACE OF REVOLUTION

AREA OF A SURFACE OF REVOLUTION AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.

More information

AREA COVERAGE SIMULATIONS FOR MILLIMETER POINT-TO-MULTIPOINT SYSTEMS USING STATISTICAL MODEL OF BUILDING BLOCKAGE

AREA COVERAGE SIMULATIONS FOR MILLIMETER POINT-TO-MULTIPOINT SYSTEMS USING STATISTICAL MODEL OF BUILDING BLOCKAGE Radoengneeng Aea Coveage Smulatons fo Mllmete Pont-to-Multpont Systems Usng Buldng Blockage 43 Vol. 11, No. 4, Decembe AREA COVERAGE SIMULATIONS FOR MILLIMETER POINT-TO-MULTIPOINT SYSTEMS USING STATISTICAL

More information

ALABAMA ASSOCIATION of EMERGENCY MANAGERS

ALABAMA ASSOCIATION of EMERGENCY MANAGERS LBM SSOCTON of EMERGENCY MNGERS ON O PCE C BELLO MER E T R O CD NCY M N G L R PROFESSONL CERTFCTON PROGRM .. E. M. CERTFCTON PROGRM 2014 RULES ND REGULTONS 1. THERE WLL BE FOUR LEVELS OF CERTFCTON. BSC,

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

Data Mining for extraction of fuzzy IF-THEN rules using Mamdani and Takagi-Sugeno-Kang FIS

Data Mining for extraction of fuzzy IF-THEN rules using Mamdani and Takagi-Sugeno-Kang FIS Engneeng Lettes, 5:, EL_5 3 Dt Mnng fo extcton of fuzzy IF-THEN ules usng Mmdn nd Tkg-Sugeno-Kng FIS Jun E. Moeno, Osc Cstllo, Jun R. Csto, Lus G. Mtínez, Ptc Meln Abstct Ths ppe pesents clusteng technques

More information

C o a t i a n P u b l i c D e b tm a n a g e m e n t a n d C h a l l e n g e s o f M a k e t D e v e l o p m e n t Z a g e bo 8 t h A p i l 2 0 1 1 h t t pdd w w wp i j fp h D p u b l i c2 d e b td S t

More information

Intro to Circle Geometry By Raymond Cheong

Intro to Circle Geometry By Raymond Cheong Into to Cicle Geomety By Rymond Cheong Mny poblems involving cicles cn be solved by constucting ight tingles then using the Pythgoen Theoem. The min chllenge is identifying whee to constuct the ight tingle.

More information

A New replenishment Policy in a Two-echelon Inventory System with Stochastic Demand

A New replenishment Policy in a Two-echelon Inventory System with Stochastic Demand A ew eplenshment Polcy n a wo-echelon Inventoy System wth Stochastc Demand Rasoul Haj, Mohammadal Payesh eghab 2, Amand Babol 3,2 Industal Engneeng Dept, Shaf Unvesty of echnology, ehan, Ian (haj@shaf.edu,

More information

Green's function integral equation methods for plasmonic nanostructures

Green's function integral equation methods for plasmonic nanostructures Geens functon ntegal equaton methods fo plasmonc nanostuctues (Ph Couse: Optcal at the Nanoscale) Thomas Søndegaad epatment of Phscs and Nanotechnolog, Aalbog Unvest, Senve 4A, K-9 Aalbog Øst, enma. Intoducton

More information

Lecture 3: Force of Interest, Real Interest Rate, Annuity

Lecture 3: Force of Interest, Real Interest Rate, Annuity Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annuty-mmedate, and ts present value Study annuty-due, and

More information

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

More information

Electronic Structure Methods

Electronic Structure Methods Electonc Stuctue Methods One-electon models e.g., Huckel theoy Semempcal methods e.g., AM, PM3, MNDO Sngle-efeence based ab nto methods Hatee-Fock Petubaton theoy MP, MP3, MP4 Coupled cluste theoy e.g.,

More information

Recurrence. 1 Definitions and main statements

Recurrence. 1 Definitions and main statements Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

More information

Keyframe Control of Smoke Simulations SIGGRAPH 2003. 2005 Roland Angst

Keyframe Control of Smoke Simulations SIGGRAPH 2003. 2005 Roland Angst Keyfame Contol of Smoke Smulatons SIGGRAPH 2003 Keyfame Contol of Smoke Smulatons SIGGRAPH 2003 Authos: Aden eulle (Unesty of Washngton) Antone McNamaa (Unesty of Washngton) Zoan Popoc (Unesty of Washngton)

More information

1 Example 1: Axis-aligned rectangles

1 Example 1: Axis-aligned rectangles COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton

More information

SOLUTIONS TO CONCEPTS CHAPTER 5

SOLUTIONS TO CONCEPTS CHAPTER 5 1. m k S 10m Let, ccelertion, Initil velocity u 0. S ut + 1/ t 10 ½ ( ) 10 5 m/s orce: m 5 10N (ns) 40000. u 40 km/hr 11.11 m/s. 3600 m 000 k ; v 0 ; s 4m v u ccelertion s SOLUIONS O CONCEPS CHPE 5 0 11.11

More information

Continuous Compounding and Annualization

Continuous Compounding and Annualization Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem

More information

SPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:

SPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background: SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and

More information

Goals Rotational quantities as vectors. Math: Cross Product. Angular momentum

Goals Rotational quantities as vectors. Math: Cross Product. Angular momentum Physcs 106 Week 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap 11.2 to 3 Rotatonal quanttes as vectors Cross product Torque expressed as a vector Angular momentum defned Angular momentum as a

More information

32. The Tangency Problem of Apollonius.

32. The Tangency Problem of Apollonius. . The Tngeny olem of Apollonius. Constut ll iles tngent to thee given iles. This eleted polem ws posed y Apollinius of eg (. 60-70 BC), the getest mthemtiin of ntiquity fte Eulid nd Ahimedes. His mjo wok

More information

Victims Compensation Claim Status of All Pending Claims and Claims Decided Within the Last Three Years

Victims Compensation Claim Status of All Pending Claims and Claims Decided Within the Last Three Years Claim#:021914-174 Initials: J.T. Last4SSN: 6996 DOB: 5/3/1970 Crime Date: 4/30/2013 Status: Claim is currently under review. Decision expected within 7 days Claim#:041715-334 Initials: M.S. Last4SSN: 2957

More information

Geometry 7-1 Geometric Mean and the Pythagorean Theorem

Geometry 7-1 Geometric Mean and the Pythagorean Theorem Geometry 7-1 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the

More information

The Cox-Ross-Rubinstein Option Pricing Model

The Cox-Ross-Rubinstein Option Pricing Model Fnance 400 A. Penat - G. Pennacc Te Cox-Ross-Rubnsten Opton Prcng Model Te prevous notes sowed tat te absence o arbtrage restrcts te prce o an opton n terms o ts underlyng asset. However, te no-arbtrage

More information

We are now ready to answer the question: What are the possible cardinalities for finite fields?

We are now ready to answer the question: What are the possible cardinalities for finite fields? Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the

More information

CANKAYA UNIVERSITY FACULTY OF ENGINEERING MECHANICAL ENGINEERING DEPARTMENT ME 212 THERMODYNAMICS II HW# 11 SOLUTIONS

CANKAYA UNIVERSITY FACULTY OF ENGINEERING MECHANICAL ENGINEERING DEPARTMENT ME 212 THERMODYNAMICS II HW# 11 SOLUTIONS CNKY UNIVESIY FCULY OF ENGINEEING MECHNICL ENGINEEING DEMEN ME HEMODYNMICS II HW# SOLUIONS Deterne te ulton reure of wter or t -60 0 C ung dt lle n te te tle. Soluton Ste tle do not ge turton reure for

More information

Lesson 28 Psychrometric Processes

Lesson 28 Psychrometric Processes 1 Lesson 28 Psychrometrc Processes Verson 1 ME, IIT Khrgpur 1 2 The specfc objectves of ths lecture re to: 1. Introducton to psychrometrc processes nd ther representton (Secton 28.1) 2. Importnt psychrometrc

More information

2.016 Hydrodynamics Prof. A.H. Techet

2.016 Hydrodynamics Prof. A.H. Techet .016 Hydodynmics Reding #5.016 Hydodynmics Po. A.H. Techet Fluid Foces on Bodies 1. Stedy Flow In ode to design oshoe stuctues, suce vessels nd undewte vehicles, n undestnding o the bsic luid oces cting

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

More information

Paper Technics Orientation Course in Papermaking 2009:

Paper Technics Orientation Course in Papermaking 2009: P P Otto Cou Pmkg 2009: g to mk u tt you ol o tgt P Wo ould ttd? Otto Cou Pmkg wll b of vlu to t followg gou of ol:- 1. P mll mloy, wo dl dtly wt t o of mkg d w to mov t udtdg of t o d t mll oto t bod

More information

Halley s Comet Project. Calculus III

Halley s Comet Project. Calculus III Hlle s Come Projec Clculus III Come Hlle from Moun Wlson, 1986 "The dvers of he phenomen of nure s so gre, nd he resures hdden n he hevens so rch, precsel n order h he humn mnd shll never be lcng n fresh

More information

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by 6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng

More information

v a 1 b 1 i, a 2 b 2 i,..., a n b n i.

v a 1 b 1 i, a 2 b 2 i,..., a n b n i. SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are

More information

The Mathematical Derivation of Least Squares

The Mathematical Derivation of Least Squares Pscholog 885 Prof. Federco The Mathematcal Dervaton of Least Squares Back when the powers that e forced ou to learn matr algera and calculus, I et ou all asked ourself the age-old queston: When the hell

More information

c. Values in statements are broken down by fiscal years; many projects are

c. Values in statements are broken down by fiscal years; many projects are Lecture 18: Finncil Mngement (Continued)/Csh Flow CEE 498 Construction Project Mngement L Schedules A. Schedule.of Contrcts Completed See Attchment # 1 ll. 1. Revenues Erned 2. Cost of Revenues 3. Gross

More information

Dispersion Relation and Wave Loads on a Vertical Cylinder in Water due to First Order Diffraction

Dispersion Relation and Wave Loads on a Vertical Cylinder in Water due to First Order Diffraction Dspeson Relon nd Wve Lods on Vecl Clnde n We due o s Ode Dffcon Dbu B Absc Hee we consde e fs ode wve dffcon b clndcl sucue. Te clndcl sucue s ccul vecl sufce pecn n we of fne dep. s we se e bound vlue

More information

The Can-Order Policy for One-Warehouse N-Retailer Inventory System: A Heuristic Approach

The Can-Order Policy for One-Warehouse N-Retailer Inventory System: A Heuristic Approach Atcle Te Can-Ode Polcy fo One-Waeouse N-Retale Inventoy ystem: A Heustc Appoac Vaapon Pukcanon, Paveena Caovaltongse, and Naagan Pumcus Depatment of Industal Engneeng, Faculty of Engneeng, Culalongkon

More information

Lecture 2: Single Layer Perceptrons Kevin Swingler

Lecture 2: Single Layer Perceptrons Kevin Swingler Lecture 2: Sngle Layer Perceptrons Kevn Sngler kms@cs.str.ac.uk Recap: McCulloch-Ptts Neuron Ths vastly smplfed model of real neurons s also knon as a Threshold Logc Unt: W 2 A Y 3 n W n. A set of synapses

More information

Random Variables and Distribution Functions

Random Variables and Distribution Functions Topic 7 Rndom Vibles nd Distibution Functions 7.1 Intoduction Fom the univese of possible infomtion, we sk question. To ddess this question, we might collect quntittive dt nd ognize it, fo emple, using

More information

Lecture 5. Inner Product

Lecture 5. Inner Product Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right

More information

MATHCAD'S PROGRAM FUNCTION and APPLICATION IN TEACHING OF MATH

MATHCAD'S PROGRAM FUNCTION and APPLICATION IN TEACHING OF MATH . About Matcad MATHCAD'S PROGRAM FUNCTION and APPLICATION IN TEACHING OF MATH DE TING WU Depart of Mat Moreouse College Atlanta, GA.33, USA dtwu@moreouse.edu. Introducton Matcad s one of popular computer

More information

Vectors 2. 1. Recap of vectors

Vectors 2. 1. Recap of vectors Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

More information

The LCOE is defined as the energy price ($ per unit of energy output) for which the Net Present Value of the investment is zero.

The LCOE is defined as the energy price ($ per unit of energy output) for which the Net Present Value of the investment is zero. Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the

More information

www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)

www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values) www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input

More information

21 Vectors: The Cross Product & Torque

21 Vectors: The Cross Product & Torque 21 Vectors: The Cross Product & Torque Do not use our left hand when applng ether the rght-hand rule for the cross product of two vectors dscussed n ths chapter or the rght-hand rule for somethng curl

More information

Mechanics 1: Motion in a Central Force Field

Mechanics 1: Motion in a Central Force Field Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.

More information

Drag force acting on a bubble in a cloud of compressible spherical bubbles at large Reynolds numbers

Drag force acting on a bubble in a cloud of compressible spherical bubbles at large Reynolds numbers Euopean Jounal of Mechancs B/Fluds 24 2005 468 477 Dag foce actng on a bubble n a cloud of compessble sphecal bubbles at lage Reynolds numbes S.L. Gavlyuk a,b,,v.m.teshukov c a Laboatoe de Modélsaton en

More information

SCO TT G LEA SO N D EM O Z G EB R E-

SCO TT G LEA SO N D EM O Z G EB R E- SCO TT G LEA SO N D EM O Z G EB R E- EG Z IA B H ER e d it o r s N ) LICA TIO N S A N D M ETH O D S t DVD N CLUDED C o n t e n Ls Pr e fa c e x v G l o b a l N a v i g a t i o n Sa t e llit e S y s t e

More information

+ + + - - This circuit than can be reduced to a planar circuit

+ + + - - This circuit than can be reduced to a planar circuit MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to

More information

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.

More information

Math 135 Circles and Completing the Square Examples

Math 135 Circles and Completing the Square Examples Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

More information

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt. Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces

More information

Institut für Halle Institute for Economic Research Wirtschaftsforschung Halle

Institut für Halle Institute for Economic Research Wirtschaftsforschung Halle Insttut fü Halle Insttute fo Economc Reseach Wtschaftsfoschung Halle A Smple Repesentaton of the Bea-Jaque-Lee Test fo Pobt Models Joachm Wlde Dezembe 2007 No. 13 IWH-Dskussonspapee IWH-Dscusson Papes

More information

GRAVITATION 1. BASIC FORCES IN NATURE

GRAVITATION 1. BASIC FORCES IN NATURE GRAVITATION. BASIC ORCES IN NATURE POINTS TO REMEMBER. Bsing on the ntue nd eltive stength the bsic foces in ntue e clssified into fou ctegoies. They e ) Gvittionl foce ) Electomgnetic foce 3) Stong Nucle

More information

Simple Interest Loans (Section 5.1) :

Simple Interest Loans (Section 5.1) : Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information