Section 5. Gaussian Imagery

Size: px
Start display at page:

Download "Section 5. Gaussian Imagery"

Transcription

1 OPTI-01/0 Geoetrical ad Istruetal Optics 5-1 Sectio 5 Gaussia Iagery

2 Iagig Paraxial optics provides a sipliied etodology to deterie ray pats troug optical systes. Usig tis etod, te iage locatio or a geeral syste ca be calculated relative to te Pricipal Plaes o te syste. Tis as also allowed te ocal Legt o a geeral syste to be deied i ters o te syste s overall reractive properties. However, te pysical locatios o te Pricipal Plaes or a give optical coiguratio still eed to be derived, so te aalysis i ot yet coplete. Te details o deteriig te syste ocal legt o a cobiatio o optical eleets is also ot clear. I additio, te iage sie is as o yet ukow. As will be deostrated later, paraxial raytrace ca be used to aswer tese questios. 5- OPTI-01/0 Geoetrical ad Istruetal Optics Gaussia optics provides a alterative etod o syste aalysis tat treats iagig as a appig ro object space ito iage space.

3 Gaussia Optics Gaussia optics treats iagig as a appig ro object space ito iage space. It is a special case o a colliear trasoratio applied to rotatioally syetric systes, ad it aps poits to poits, lies to lies ad plaes to plaes. Te correspodig object ad iage eleets are called cojugate eleets. Assuptios: Axial Syetry Meridioal Plae - cotais te axis o syetry - usually te y- plae Sig Covetios ad deiitios o reerece locatios will be applied as usual. Te trasverse agiicatio or lateral agiicatio is te ratio o te iage poit eigt ro te axis to te cojugate object poit eigt: 5-3 OPTI-01/0 Geoetrical ad Istruetal Optics Te udaetal priciple o geoetrical optics olds: ay tree-diesioal object scee is represeted by a collectio o idepedetly radiatig poit sources. ac poit source is idepedetly iaged troug te syste to its cojugate iage poit (perect iagery). Te tree-diesioal iage is te superpositio o all o te poit iages.

4 Gaussia Optics Teores Teores (all derive ro rotatioal syetry o te syste): - Plaes perpedicular to te axis i oe space are apped to plaes perpedicular to te axis i te oter space. - Lies parallel to te axis i oe space ap to cojugate lies i te oter space tat eiter itersect te axis at a coo poit (ocal syste), or are also parallel to te axis (aocal syste). ocal Syste: 5-4 OPTI-01/0 Geoetrical ad Istruetal Optics Aocal Syste: I a Gaussia or colliear appig, lies ust ap to lies ad are cojugate eleets.

5 Gaussia Optics Teores - Cotiued - Te trasverse agiicatio is costat i cojugate plaes perpedicular to te axis. I te agiicatio i a plae were ot costat, te iages o te grid lies would becoe curved or distorted. 5-5 OPTI-01/0 Geoetrical ad Istruetal Optics

6 Cardial Poits ad Plaes Te cardial poits ad plaes copletely describe te ocal appig. Tey are deied by speciic agiicatios: rot ocal poit/plae = ' ear ocal poit/plae = 0 P rot pricipal plae = 1 P' ear pricipal plae = 1 Cosider cojugate lie eleets i object ad iage space: H P H OPTI-01/0 Geoetrical ad Istruetal Optics P 0 H 1 H Te rot ad rear ocal legts ( ) are deied as te directed distaces ro te ad rot ad rear pricipal plaes to te respective ocal poits.

7 Cardial Poits ad Plaes Upried variables are i object space. Pried variables are i iage space. AP rot atipricipal plae/poit = -1 AP' ear atipricipal plae/poit = -1 N rot odal poit Agular Mag = 1 N' ear odal poit Agular Mag = 1 Te two odal poits o a syste are cojugate poits. θ N N θ 5-7 OPTI-01/0 Geoetrical ad Istruetal Optics or a ocal iagig syste, a object plae locatio is related to its cojugate iage plae locatio troug te trasverse agiicatio associated wit tose plaes. Newtoia equatios easure object ad iage distaces ro te ocal plaes. Gaussia equatios easure object ad iage distaces ro te pricipal plaes.

8 Locatig a Iage wit te Cardial Poits Te optical syste ca be represeted as a set o Pricipal Plaes ad a set o ocal Poits (wit te respective ocal legts). - A ray icidet o te rot pricipal plae will eerge ro te rear pricipal plae at te sae eigt. - A ray parallel to te optical axis i object space passes troug te rear ocal poit. - A ray troug te rot ocal poit eerges parallel to te optical axis. - Te itersectio o two rays deies a object or iage poit. P P 5-8 OPTI-01/0 Geoetrical ad Istruetal Optics Te Pricipal Plaes serve as te plaes o eective reractio betwee object space ad iage space.

9 OPTI-01/0 Geoetrical ad Istruetal Optics Locatig a Iage wit te Cardial Poits xaple 1 Positive Syste eal object to te let o te rot ocal poit 5-9 P P P P

10 Locatig a Iage wit te Cardial Poits xaple Positive Syste eal object betwee te rot ocal poit ad te rot pricipal plae P P 5-10 OPTI-01/0 Geoetrical ad Istruetal Optics P P Te two iage space rays diverge ad ave a virtual crossig. A elarged, erect virtual iage is produced. Te iage is i iage space.

11 Locatig a Iage wit te Cardial Poits xaple 3 Negative Syste eal object Note te locatios o te ocal Poits. Te rot ocal Poit ad rot Pricipal Plae P are bot i te syste object space. Siilarly, te ear ocal Poit ' ad ear Pricipal Plae P' are bot i te syste iage space. Te sae iage oratio rules apply. P P 5-11 OPTI-01/0 Geoetrical ad Istruetal Optics P P Oce agai, te two iage space rays diverge ad ave a virtual crossig. A iiied, erect virtual iage is produced. Te iage is i iage space.

12 Newtoia quatios Te Newtoia equatios caracterie tis Gaussia appig we te axial locatios o te cojugate object ad iage plaes are easured relative to te respective ocal Poits. By deiitio, te rot ad rear ocal legts cotiue to be easured relative to te pricipal plaes. Use siilar triagles. ta 1 1 P ta P 0 0 Object Distace = Iage Distace = 5-1 Magiicatio: - Idepedet o te object eigt - Proportioal to te iage distace - Iversely proportioal to te object distace OPTI-01/0 Geoetrical ad Istruetal Optics

13 Newtoia quatios Applied to a Syste Te Newtoia equatios caracterie tis Gaussia appig we te axial locatios o te cojugate object ad iage plaes are easured relative to te respective ocal Poits. P P 5-13 OPTI-01/0 Geoetrical ad Istruetal Optics I air: 1

14 Gaussia quatios Te Gaussia equatios describe te ocal appig we te respective Pricipal Plaes are te reereces or easurig te locatios o te cojugate object ad iage plaes. Use te sae siilar triagles. ta P P ta Object Distace = Iage Distace = OPTI-01/0 Geoetrical ad Istruetal Optics Add: atio: Magiicatio: - Idepedet o te object eigt - Proportioal to te ratio o te iage distace to te object distace /

15 Gaussia quatios Applied to a Syste Te Gaussia equatios describe te ocal appig we te respective Pricipal Plaes are te reereces or easurig te locatios o te cojugate object ad iage plaes P P OPTI-01/0 Geoetrical ad Istruetal Optics 1 1 / 1 / I air:

16 Cojugate plaes or = 1, 0 ad Use te Gaussia equatios. = 1 = Object plae located at P Iage plae located at P P ad P' are cojugate. Object plae located at Iage plae located at ' OPTI-01/0 Geoetrical ad Istruetal Optics ad ' are cojugate. = Object plae located at Iage plae located at ad are cojugate.

17 Sig Covetios evisited Newtoia quatios Derivatio Origial Coiguratio Object to te let o ;, ad are egative. 0 P New Coiguratio Object to te rigt o : ow oly is egative. ta 5-17 OPTI-01/0 Geoetrical ad Istruetal Optics P 0 ta Sae result!

18 Sig Covetios evisited Origial igure Object to te rigt o ; ow,, ad are egative. P 0 ta Sae result! 5-18 OPTI-01/0 Geoetrical ad Istruetal Optics As te object distace goes positive to place te object to te rigt o, te object eigt becoes egative to copesate i te equatios. Te et result is always te sae. Note tat i all o tese igures, te reerece locatio used to deie te quatities as ot caged. Set te equatios up to be cosistet wit te igure, te sig covetios will allow te equatio to be valid or dieret coiguratios.

19 Distaces Betwee Pairs o Cojugate Plaes Tickess Magiicatio Te tickess agiicatio relates te distaces betwee pairs o cojugate plaes. Use Newtoia equatios: P P OPTI-01/0 Geoetrical ad Istruetal Optics / 1 / / Te tickesses ad are idepedet o te origis used.

20 Tickess Magiicatio ad Logitudial Magiicatio 1 / 1 / 1 Te tickess agiicatio equatios are valid or widely separated plaes. Sice it is a dierece i positio, te result is idepedet o te coice o origis. As te plae separatio approaces ero, te local logitudial or axial agiicatio is obtaied. 5-0 OPTI-01/0 Geoetrical ad Istruetal Optics / li 0 / Sice varies wit positio, te logitudial agiicatio ad te tickess agiicatio are a uctio o ad.

21 Tickess Magiicatio ad Logitudial Magiicatio Syste Te tickess agiicatio relates te distaces betwee pairs o cojugate plaes. 1 1 P P I air: OPTI-01/0 Geoetrical ad Istruetal Optics Te tickess agiicatio equatios are valid or widely separated plaes. As te plae separatio approaces ero, te local logitudial or axial agiicatio is obtaied: li 0 1 I air: Sice varies wit positio, te logitudial agiicatio ad te tickess agiicatio are a uctio o ad.

22 Nodal Poits Two additioal cardial poits are te rot ad rear odal poits (N ad N') tat deie te locatio o uit agular agiicatio or a ocal syste. A ray passig troug oe odal poit o a syste is apped to a ray passig troug te oter odal poit avig te sae agle wit respect to te optical axis. Te odal poits are cojugate poits. P N PN P N PN ay 1 ay ays 1 ad ust be parallel i iage space, sice teir cojugate rays cross i te rot ocal plae. Te idicated triagles are ot oly siilar, but idetical. 5- OPTI-01/0 Geoetrical ad Istruetal Optics PN PN PN PN PN Usig te tickess agiicatio (ad te locatios o te pricipal plaes ad odal poits): N P PN N P PN P N 1 P 1 N N

23 Nodal Poits o a Syste PN PN N I te sae idex occurs i object space ad iage space: 5-3 OPTI-01/0 Geoetrical ad Istruetal Optics PN PN 0 N 1 Te odal poits are located at te respective pricipal plaes i te iage space idex o reractio equals te object space idex o reractio.

24 Origis at te Nodal Poits I te object ad iage locatios are easured relative to te Nodal poits, a iterestig ad iportat result is obtaied. Use te tickess agiicatio relatiosip: N N N N N N ad are te object ad iage positios easured ro soe origis ad are te odal poit positios easured ro te sae origis N 1 N N N 5-4 OPTI-01/0 Geoetrical ad Istruetal Optics Te agular subtese o a iage as see ro te rear odal poit equals te agular subtese o te object as see ro te rot odal poit. u N N N N u=u u u N N N N

25 Cardial Poits ad Plaes o a Syste Te syste ocal legt is Te object space idex o reractio is Te iage space idex o reractio is Optical Syste V P P V d d BD 5-5 OPTI-01/0 Geoetrical ad Istruetal Optics D Te rot ocal Distace D is te distace ro te syste rot vertex to te rot ocal poit Te Back ocal Distace BD is te distace ro te syste back vertex to te back ocal poit d is te sit o te syste rot Pricipal Plae P ro te syste rot vertex V d is te sit o te syste rear Pricipal Plae P ro te syste back vertex V

26 Nodal Slide We a les syste is i air, te odal poits coicide wit te pricipal poits. Te use o a odal slide allows te pricipal plaes ad te ocal legt to be experietally deteried. We te les is rotated about its rear odal poit, te rays will coverge to te sae poit. Te iage will ot ove eve toug te ray budle orig te iage is skewed, ad ' is sited to oe side. P,N P,N V 5-6 OPTI-01/0 Geoetrical ad Istruetal Optics P,N P,N By ivertig te les, te rot cardial poits (N,, P) ca also be located.

27 Nodal Slide Procedure P,N P,N d V BD P,N P,N Mout te les syste o a traslatio stage wic is o a rotatio stage. Positio te rear vertex over te rotatio axis. We properly positioed, te vertex will ot traslate we te les is rotated. Use colliated illuiatio. Use a icroscope (wit a icroeter) to easure te distace betwee te rear vertex V' ad te ocus (rear ocal poit '). Tis is te Back ocal Distace BD te distace ro te rear vertex to te rear ocal poit. Wile observig te iage, repositio te les wit te traslatio stage so tat te iage does ot traslate we te les is rotated. Te rear odal poit (ad te rear pricipal poit) are ow over te rotatio poit. Te aout te les was oved is te separatio d' betwee te rear vertex ad te rear pricipal plae. Te syste ocal legt is oud by BDd 5-7 OPTI-01/0 Geoetrical ad Istruetal Optics

28 Gaussia Properties o a Sigle eractig Surace Pricipal Plaes: Te rot ad rear pricipal plaes are coicidet ad located at te surace vertex V. A ray icidet o te surace at soe eigt will eerge ro tat surace at te sae eigt. Uit agiicatio occurs at te surace. Nodal Poits: Bot odal poits are located at te ceter o te curvature CC o te optical surace. A ray eadig towards te ceter o curvature is oral to te reractig surace ad is ot reracted. Te ray as te sae agle i object space ad iage space (agular agiicatio = 1), ad te ray s axial itercept at te CC deies bot odal poits. V N, N 5-8 CC OPTI-01/0 Geoetrical ad Istruetal Optics Surace Power ad ocal Poits: C Te rot ad rear ocal legts are easured ro te surace vertex.

29 Nodal Poits o a Sigle eractig Surace Veriy te Nodal Poit Pricipal Plae separatio: PN PN PN PN 1 PN PN C C 5-9 OPTI-01/0 Geoetrical ad Istruetal Optics Te pricipal plaes are at te surace vertex Te odal poits are at te surace ceter o curvature. Nodal poit agiicatio or a sigle reractig surace: N

30 Gaussia Properties o a eractig Surace Suary Power: ocal Legts: 1 C Pricipal plaes/poits located at te surace vertex. Nodal poits located at te ceter o curvature o te surace. N Te rot ad rear ocal legts are easured ro te surace vertex (pricipal plaes). V CC > 0 > 0 > 5-30 OPTI-01/0 Geoetrical ad Istruetal Optics Pricipal plae - Nodal poit separatio: 1 PN PN C Iagig: 1 Magiicatio: / /

31 educed Distace quivalece Cosider te exaple o a reractig surace ad its ti les equivalet. Bot ave te sae power. 1 u = 1 = 1 = / u=u Te reduced ocal legt o te surace equals te ocal legt o te ti les. I te ray agle or slope or te reractig surace is u', te te ray agle is 'u' or te ti les. A ray agle ultiplied by te reractive idex o its optical space is called a optical agle: u. Te paraxial raytrace equatio is i ters o optical agles ad te eective (or reduced) ocal legt o te syste: y y u u y u u / 5-31 OPTI-01/0 Geoetrical ad Istruetal Optics Te use o reduced distaces ad optical agles allows ay syste to be represeted as a air-equivalet syste coprised o ti leses. I te object is ot at iiity, te iage distace or te ti les becoes te reduced iage distace o te reractig surace.

32 Gaussia Iagery Te cardial poits, alog wit te associated ocal legts ad power, copletely speciy te appig ro object space ito iage space or a ocal syste. Gaussia iagery ais to reduce ay ocal iagig syste, regardless o te uber o suraces, to a sigle, uique set o cardial poits. 5-3 OPTI-01/0 Geoetrical ad Istruetal Optics P P P P

33 Iagig xaple 1 Positive ocal Syste Object: 00 to let o = 10 Use Newtoia equatios: OPTI-01/0 Geoetrical ad Istruetal Optics P P' ' Note: Te pysical separatio betwee P ad P' is ot kow.

34 Iagig xaple 1 Gaussia quatios Use Gaussia equatios: Distaces ro Pricipal plaes OPTI-01/0 Geoetrical ad Istruetal Optics 50 to te rigt o ' P P' '

35 Iagig xaple Sae Positive ocal Syste Object: 40 to rigt o = 10 Use Newtoia equatios: OPTI-01/0 Geoetrical ad Istruetal Optics P P' '

36 OPTI-01/0 Geoetrical ad Istruetal Optics xaple Suary 5-36 P P' B A ' A B or coveiece, te Pricipal Plaes are sow as coicidet.

37 5-37 Tickess ad Logitudial Magiicatio ocal Systes / or / ad are te lateral agiicatios or te two plaes Gaussia quatios (distaces easured ro P, P') / / 1 1 Newtoia quatios (distaces easured ro, ') 1/ / / / / / 1 OPTI-01/0 Geoetrical ad Istruetal Optics / / We Δ is sall, te logitudial agiicatio is obtaied 1 1 Gaussia: Newtoia: Te iage space spacig is iversely proportioal to te Newtoia object distace squared. / / / / 1

38 Iagig xaple 3 Logitudial Magiicatio Sae Positive ocal Syste Objects: 410 to let o 400 to let o Use Newtoia equatios: OPTI-01/0 Geoetrical ad Istruetal Optics xact tickess agiicatio:

39 Iagig xaple 3A Logitudial Magiicatio Sae Positive ocal Syste Objects closer to te syste: Use Newtoia equatios: 1 50 to let o 40 to let o OPTI-01/0 Geoetrical ad Istruetal Optics 49.4 xact tickess agiicatio:

40 Object Iage elatiosips Te geeral iagery relatiosips ca be writte eiter i ters o te rot ad rear ocal legts or i ters o te eective ocal legt ad idices o reractio. 1 or clarity, te expressios will explicitly ote te ocal legt as. As te course progresses, te subscript will be dropped, ad te ore coo expressio or te ocal legt will be used. Newtoia quatios (Origis at, '): 5-40 OPTI-01/0 Geoetrical ad Istruetal Optics 1

41 Object Iage elatiosips Page Gaussia quatios (Origis at P, P'): / / OPTI-01/0 Geoetrical ad Istruetal Optics Tickess ad Logitudial Magiicatio ocal Syste: Magiicatio o te Nodal Poits: N N

42 educed Distaces ad Optical Agles We te Gaussia iagery equatios are expressed i ters o te L or power ( or ), all o te axial distaces appear as a ratio o te pysical distace to te idex o reractio i te sae optical space. Tis ratio is called a reduced distace ad is usually deoted by a Greek letter. or exaple represets te reduced distace associated wit te tickess t: t Te L is te reduced ocal legt: it equals te reduced rear ocal legt or ius te reduced rot ocal legt. A ray agle ultiplied by te reractive idex o its optical space is called a optical agle: 5-4 OPTI-01/0 Geoetrical ad Istruetal Optics u

43 Iagig quatios i Air 1 Newtoia quatios (Origis at, '): 1 Gaussia quatios (Origis at P, P'): 1 1 Tickess ad Logitudial Magiicatio ocal Syste: OPTI-01/0 Geoetrical ad Istruetal Optics 1 Magiicatio o te Nodal Poits: Aocal Systes: N 1 1 1

Section 7. Gaussian Reduction

Section 7. Gaussian Reduction 7- Sectio 7 Gaussia eductio Paraxial aytrace Equatios eractio occurs at a iterace betwee two optical spaces. The traser distace t' allows the ray height y' to be determied at ay plae withi a optical space

More information

Section 12. Afocal Systems

Section 12. Afocal Systems OPTI-0/0 Geoetrical and Instruental Optics Copyrigt 08 Jon E. Greivenkap - Section Aocal Systes Gaussian Optics Teores In te initial discussion o Gaussian optics, one o te teores deined te two dierent

More information

(s)h(s) = K( s + 8 ) = 5 and one finite zero is located at z 1

(s)h(s) = K( s + 8 ) = 5 and one finite zero is located at z 1 ROOT LOCUS TECHNIQUE 93 should be desiged differetly to eet differet specificatios depedig o its area of applicatio. We have observed i Sectio 6.4 of Chapter 6, how the variatio of a sigle paraeter like

More information

Atomic Physics 4. Name: Date: 1. The de Broglie wavelength associated with a car moving with a speed of 20 m s 1 is of the order of. A m.

Atomic Physics 4. Name: Date: 1. The de Broglie wavelength associated with a car moving with a speed of 20 m s 1 is of the order of. A m. Name: Date: Atomic Pysics 4 1. Te de Broglie wavelegt associated wit a car movig wit a speed of 0 m s 1 is of te order of A. 10 38 m. B. 10 4 m. C. 10 4 m. D. 10 38 m.. Te diagram below sows tree eergy

More information

Optimal Estimator for a Sample Set with Response Error. Ed Stanek

Optimal Estimator for a Sample Set with Response Error. Ed Stanek Optial Estiator for a Saple Set wit Respose Error Ed Staek Itroductio We develop a optial estiator siilar to te FP estiator wit respose error tat was cosidered i c08ed63doc Te first 6 pages of tis docuet

More information

) is a square matrix with the property that for any m n matrix A, the product AI equals A. The identity matrix has a ii

) is a square matrix with the property that for any m n matrix A, the product AI equals A. The identity matrix has a ii square atrix is oe that has the sae uber of rows as colus; that is, a atrix. he idetity atrix (deoted by I, I, or [] I ) is a square atrix with the property that for ay atrix, the product I equals. he

More information

( ) D) E) NOTA

( ) D) E) NOTA 016 MAΘ Natioal Covetio 1. Which Greek mathematicia do most historias credit with the discovery of coic sectios as a solutio to solvig the Delia problem, also kow as doublig the cube? Eratosthees Meaechmus

More information

U8L1: Sec Equations of Lines in R 2

U8L1: Sec Equations of Lines in R 2 MCVU U8L: Sec. 8.9. Equatios of Lies i R Review of Equatios of a Straight Lie (-D) Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio of the lie

More information

CALCULUS BASIC SUMMER REVIEW

CALCULUS BASIC SUMMER REVIEW CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=

More information

distinct distinct n k n k n! n n k k n 1 if k n, identical identical p j (k) p 0 if k > n n (k)

distinct distinct n k n k n! n n k k n 1 if k n, identical identical p j (k) p 0 if k > n n (k) THE TWELVEFOLD WAY FOLLOWING GIAN-CARLO ROTA How ay ways ca we distribute objects to recipiets? Equivaletly, we wat to euerate equivalece classes of fuctios f : X Y where X = ad Y = The fuctios are subject

More information

Balancing. Rotating Components Examples of rotating components in a mechanism or a machine. (a)

Balancing. Rotating Components Examples of rotating components in a mechanism or a machine. (a) alacig NOT COMPLETE Rotatig Compoets Examples of rotatig compoets i a mechaism or a machie. Figure 1: Examples of rotatig compoets: camshaft; crakshaft Sigle-Plae (Static) alace Cosider a rotatig shaft

More information

Name Period ALGEBRA II Chapter 1B and 2A Notes Solving Inequalities and Absolute Value / Numbers and Functions

Name Period ALGEBRA II Chapter 1B and 2A Notes Solving Inequalities and Absolute Value / Numbers and Functions Nae Period ALGEBRA II Chapter B ad A Notes Solvig Iequalities ad Absolute Value / Nubers ad Fuctios SECTION.6 Itroductio to Solvig Equatios Objectives: Write ad solve a liear equatio i oe variable. Solve

More information

Partial Differential Equations

Partial Differential Equations EE 84 Matematical Metods i Egieerig Partial Differetial Eqatios Followig are some classical partial differetial eqatios were is assmed to be a fctio of two or more variables t (time) ad y (spatial coordiates).

More information

3. Newton s Rings. Background. Aim of the experiment. Apparatus required. Theory. Date : Newton s Rings

3. Newton s Rings. Background. Aim of the experiment. Apparatus required. Theory. Date : Newton s Rings ate : Newto s igs 3. Newto s igs Backgroud Coheret light Phase relatioship Path differece Iterferece i thi fil Newto s rig apparatus Ai of the experiet To study the foratio of Newto s rigs i the air-fil

More information

U8L1: Sec Equations of Lines in R 2

U8L1: Sec Equations of Lines in R 2 MCVU Thursda Ma, Review of Equatios of a Straight Lie (-D) U8L Sec. 8.9. Equatios of Lies i R Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio

More information

Mechanical Vibrations

Mechanical Vibrations Mechaical Vibratios Cotets Itroductio Free Vibratios o Particles. Siple Haroic Motio Siple Pedulu (Approxiate Solutio) Siple Pedulu (Exact Solutio) Saple Proble 9. Free Vibratios o Rigid Bodies Saple Proble

More information

An Algorithmist s Toolkit October 20, Lecture 11

An Algorithmist s Toolkit October 20, Lecture 11 18.409 A Algorithist s Toolkit October 20, 2009 Lecture 11 Lecturer: Joatha Keler Scribe: Chaithaya Badi 1 Outlie Today we ll itroduce ad discuss Polar of a covex body. Correspodece betwee or fuctios ad

More information

Problem Cosider the curve give parametrically as x = si t ad y = + cos t for» t» ß: (a) Describe the path this traverses: Where does it start (whe t =

Problem Cosider the curve give parametrically as x = si t ad y = + cos t for» t» ß: (a) Describe the path this traverses: Where does it start (whe t = Mathematics Summer Wilso Fial Exam August 8, ANSWERS Problem 1 (a) Fid the solutio to y +x y = e x x that satisfies y() = 5 : This is already i the form we used for a first order liear differetial equatio,

More information

R is a scalar defined as follows:

R is a scalar defined as follows: Math 8. Notes o Dot Product, Cross Product, Plaes, Area, ad Volumes This lecture focuses primarily o the dot product ad its may applicatios, especially i the measuremet of agles ad scalar projectio ad

More information

Principle Of Superposition

Principle Of Superposition ecture 5: PREIMINRY CONCEP O RUCUR NYI Priciple Of uperpositio Mathematically, the priciple of superpositio is stated as ( a ) G( a ) G( ) G a a or for a liear structural system, the respose at a give

More information

2.710 Optics Spring 09 Solutions to Problem Set #2 Due Wednesday, Feb. 25, 2009

2.710 Optics Spring 09 Solutions to Problem Set #2 Due Wednesday, Feb. 25, 2009 MASSACHUSETTS INSTITUTE OF TECHNOLOGY.70 Optics Sprig 09 Solutios to Prolem Set # Due Weesay, Fe. 5, 009 Prolem : Wiper spee cotrol Figure shows a example o a optical system esige to etect the amout o

More information

Complex Numbers Solutions

Complex Numbers Solutions Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i

More information

Two or more points can be used to describe a rigid body. This will eliminate the need to define rotational coordinates for the body!

Two or more points can be used to describe a rigid body. This will eliminate the need to define rotational coordinates for the body! OINTCOORDINATE FORMULATION Two or more poits ca be used to describe a rigid body. This will elimiate the eed to defie rotatioal coordiates for the body i z r i i, j r j j rimary oits: The coordiates of

More information

SHEAR LAG MODELLING OF THERMAL STRESSES IN UNIDIRECTIONAL COMPOSITES

SHEAR LAG MODELLING OF THERMAL STRESSES IN UNIDIRECTIONAL COMPOSITES ORA/POSER REFERENCE: ICF00374OR SHEAR AG MODEING OF HERMA SRESSES IN UNIDIRECIONA COMPOSIES Chad M. adis Departet o Mechaical Egieerig ad Materials Sciece MS 3 Rice Uiversity P.O. Box 89 Housto X 7705

More information

DEGENERACY AND ALL THAT

DEGENERACY AND ALL THAT DEGENERACY AND ALL THAT Te Nature of Termodyamics, Statistical Mecaics ad Classical Mecaics Termodyamics Te study of te equilibrium bulk properties of matter witi te cotext of four laws or facts of experiece

More information

Complex Analysis Spring 2001 Homework I Solution

Complex Analysis Spring 2001 Homework I Solution Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle

More information

42 Dependence and Bases

42 Dependence and Bases 42 Depedece ad Bases The spa s(a) of a subset A i vector space V is a subspace of V. This spa ay be the whole vector space V (we say the A spas V). I this paragraph we study subsets A of V which spa V

More information

Lenses and Imaging (Part II)

Lenses and Imaging (Part II) Lee ad Imagig (Part II) emider rom Part I Surace o poitive/egative power eal ad virtual image Imagig coditio Thick lee Pricipal plae 09/20/04 wk3-a- The power o urace Poitive power : eitig ray coverge

More information

Answer Key, Problem Set 1, Written

Answer Key, Problem Set 1, Written Cheistry 1 Mies, Sprig, 018 Aswer Key, Proble Set 1, Writte 1. 14.3;. 14.34 (add part (e): Estiate / calculate the iitial rate of the reactio); 3. NT1; 4. NT; 5. 14.37; 6. 14.39; 7. 14.41; 8. NT3; 9. 14.46;

More information

Chapter 4. Fourier Series

Chapter 4. Fourier Series Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,

More information

Lecture 11. Solution of Nonlinear Equations - III

Lecture 11. Solution of Nonlinear Equations - III Eiciecy o a ethod Lecture Solutio o Noliear Equatios - III The eiciecy ide o a iterative ethod is deied by / E r r: rate o covergece o the ethod : total uber o uctios ad derivative evaluatios at each step

More information

Geometry Unit 3 Notes Parallel and Perpendicular Lines

Geometry Unit 3 Notes Parallel and Perpendicular Lines Review Cocepts: Equatios of Lies Geoetry Uit Notes Parallel ad Perpedicular Lies Syllabus Objective:. - The studet will differetiate aog parallel, perpedicular, ad skew lies. Lies that DO NOT itersect:

More information

Formula List for College Algebra Sullivan 10 th ed. DO NOT WRITE ON THIS COPY.

Formula List for College Algebra Sullivan 10 th ed. DO NOT WRITE ON THIS COPY. Forula List for College Algera Sulliva 10 th ed. DO NOT WRITE ON THIS COPY. Itercepts: Lear how to fid the x ad y itercepts. Syetry: Lear how test for syetry with respect to the x-axis, y-axis ad origi.

More information

Integrals of Functions of Several Variables

Integrals of Functions of Several Variables Itegrals of Fuctios of Several Variables We ofte resort to itegratios i order to deterie the exact value I of soe quatity which we are uable to evaluate by perforig a fiite uber of additio or ultiplicatio

More information

Lecture Outline. 2 Separating Hyperplanes. 3 Banach Mazur Distance An Algorithmist s Toolkit October 22, 2009

Lecture Outline. 2 Separating Hyperplanes. 3 Banach Mazur Distance An Algorithmist s Toolkit October 22, 2009 18.409 A Algorithist s Toolkit October, 009 Lecture 1 Lecturer: Joatha Keler Scribes: Alex Levi (009) 1 Outlie Today we ll go over soe of the details fro last class ad ake precise ay details that were

More information

and then substitute this into the second equation to get 5(11 4 y) 3y

and then substitute this into the second equation to get 5(11 4 y) 3y Math E-b Lecture # Notes The priary focus of this week s lecture is a systeatic way of solvig ad uderstadig systes of liear equatios algebraically, geoetrically, ad logically. Eaple #: Solve the syste

More information

APPENDIX F Complex Numbers

APPENDIX F Complex Numbers APPENDIX F Complex Numbers Operatios with Complex Numbers Complex Solutios of Quadratic Equatios Polar Form of a Complex Number Powers ad Roots of Complex Numbers Operatios with Complex Numbers Some equatios

More information

DYNAMICS VECTOR MECHANICS FOR ENGINEERS: Mechanical Vibrations. Seventh Edition CHAPTER. Ferdinand P. Beer E. Russell Johnston, Jr.

DYNAMICS VECTOR MECHANICS FOR ENGINEERS: Mechanical Vibrations. Seventh Edition CHAPTER. Ferdinand P. Beer E. Russell Johnston, Jr. Seveth Editio CHAPTER 9 VECTOR MECHANICS FOR ENGINEERS: DYNAMICS Ferdiad P. Beer E. Russell Johsto, Jr. Mechaical Vibratios Lecture Notes: J. Walt Oler Texas Tech Uiversity 003 The McGraw-Hill Copaies,

More information

ECE Spring Prof. David R. Jackson ECE Dept. Notes 20

ECE Spring Prof. David R. Jackson ECE Dept. Notes 20 ECE 6341 Sprig 016 Prof. David R. Jackso ECE Dept. Notes 0 1 Spherical Wave Fuctios Cosider solvig ψ + k ψ = 0 i spherical coordiates z φ θ r y x Spherical Wave Fuctios (cot.) I spherical coordiates we

More information

Fundamental Concepts: Surfaces and Curves

Fundamental Concepts: Surfaces and Curves UNDAMENTAL CONCEPTS: SURACES AND CURVES CHAPTER udametal Cocepts: Surfaces ad Curves. INTRODUCTION This chapter describes two geometrical objects, vi., surfaces ad curves because the pla a ver importat

More information

Chapter 4 Postulates & General Principles of Quantum Mechanics

Chapter 4 Postulates & General Principles of Quantum Mechanics Chapter 4 Postulates & Geeral Priciples of Quatu Mechaics Backgroud: We have bee usig quite a few of these postulates already without realizig it. Now it is tie to forally itroduce the. State of a Syste

More information

PHYS-3301 Lecture 9. CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I. 5.3: Electron Scattering. Bohr s Quantization Condition

PHYS-3301 Lecture 9. CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I. 5.3: Electron Scattering. Bohr s Quantization Condition CHAPTER 5 Wave Properties of Matter ad Quatum Mecaics I PHYS-3301 Lecture 9 Sep. 5, 018 5.1 X-Ray Scatterig 5. De Broglie Waves 5.3 Electro Scatterig 5.4 Wave Motio 5.5 Waves or Particles? 5.6 Ucertaity

More information

CHAPTER 11 Limits and an Introduction to Calculus

CHAPTER 11 Limits and an Introduction to Calculus CHAPTER Limits ad a Itroductio to Calculus Sectio. Itroductio to Limits................... 50 Sectio. Teciques for Evaluatig Limits............. 5 Sectio. Te Taget Lie Problem................. 50 Sectio.

More information

COMP 2804 Solutions Assignment 1

COMP 2804 Solutions Assignment 1 COMP 2804 Solutios Assiget 1 Questio 1: O the first page of your assiget, write your ae ad studet uber Solutio: Nae: Jaes Bod Studet uber: 007 Questio 2: I Tic-Tac-Toe, we are give a 3 3 grid, cosistig

More information

Section 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations

Section 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations Differece Equatios to Differetial Equatios Sectio. Calculus: Areas Ad Tagets The study of calculus begis with questios about chage. What happes to the velocity of a swigig pedulum as its positio chages?

More information

ALLOCATING SAMPLE TO STRATA PROPORTIONAL TO AGGREGATE MEASURE OF SIZE WITH BOTH UPPER AND LOWER BOUNDS ON THE NUMBER OF UNITS IN EACH STRATUM

ALLOCATING SAMPLE TO STRATA PROPORTIONAL TO AGGREGATE MEASURE OF SIZE WITH BOTH UPPER AND LOWER BOUNDS ON THE NUMBER OF UNITS IN EACH STRATUM ALLOCATING SAPLE TO STRATA PROPORTIONAL TO AGGREGATE EASURE OF SIZE WIT BOT UPPER AND LOWER BOUNDS ON TE NUBER OF UNITS IN EAC STRATU Lawrece R. Erst ad Cristoper J. Guciardo Erst_L@bls.gov, Guciardo_C@bls.gov

More information

Math 21C Brian Osserman Practice Exam 2

Math 21C Brian Osserman Practice Exam 2 Math 1C Bria Osserma Practice Exam 1 (15 pts.) Determie the radius ad iterval of covergece of the power series (x ) +1. First we use the root test to determie for which values of x the series coverges

More information

Orthogonal Functions

Orthogonal Functions Royal Holloway Uiversity of odo Departet of Physics Orthogoal Fuctios Motivatio Aalogy with vectors You are probably failiar with the cocept of orthogoality fro vectors; two vectors are orthogoal whe they

More information

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2015

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2015 Uiversity of Wasigto Departmet of Cemistry Cemistry 453 Witer Quarter 15 Lecture 14. /11/15 Recommeded Text Readig: Atkis DePaula: 9.1, 9., 9.3 A. Te Equipartitio Priciple & Eergy Quatizatio Te Equipartio

More information

9.5 Young s Double-Slit Experiment

9.5 Young s Double-Slit Experiment 9.5 Youg s Double-Slit Experiet Physics Tool box Early attepts to deostrate the iterferece of light were usuccessful because the two sources were too far apart ad out of phase, ad the wavelegth of light

More information

The Hypergeometric Coupon Collection Problem and its Dual

The Hypergeometric Coupon Collection Problem and its Dual Joural of Idustrial ad Systes Egieerig Vol., o., pp -7 Sprig 7 The Hypergeoetric Coupo Collectio Proble ad its Dual Sheldo M. Ross Epstei Departet of Idustrial ad Systes Egieerig, Uiversity of Souther

More information

d y f f dy Numerical Solution of Ordinary Differential Equations Consider the 1 st order ordinary differential equation (ODE) . dx

d y f f dy Numerical Solution of Ordinary Differential Equations Consider the 1 st order ordinary differential equation (ODE) . dx umerical Solutio o Ordiar Dieretial Equatios Cosider te st order ordiar dieretial equatio ODE d. d Te iitial coditio ca be tae as. Te we could use a Talor series about ad obtai te complete solutio or......!!!

More information

Solutions. tan 2 θ(tan 2 θ + 1) = cot6 θ,

Solutions. tan 2 θ(tan 2 θ + 1) = cot6 θ, Solutios 99. Let A ad B be two poits o a parabola with vertex V such that V A is perpedicular to V B ad θ is the agle betwee the chord V A ad the axis of the parabola. Prove that V A V B cot3 θ. Commet.

More information

Lecture 10: Bounded Linear Operators and Orthogonality in Hilbert Spaces

Lecture 10: Bounded Linear Operators and Orthogonality in Hilbert Spaces Lecture : Bouded Liear Operators ad Orthogoality i Hilbert Spaces 34 Bouded Liear Operator Let ( X, ), ( Y, ) i i be ored liear vector spaces ad { } X Y The, T is said to be bouded if a real uber c such

More information

STRAIGHT LINES & PLANES

STRAIGHT LINES & PLANES STRAIGHT LINES & PLANES PARAMETRIC EQUATIONS OF LINES The lie "L" is parallel to the directio vector "v". A fixed poit: "( a, b, c) " o the lie is give. Positio vectors are draw from the origi to the fixed

More information

2.710 Optics Spring 09 Solutions to Problem Set #3 Due Wednesday, March 4, 2009

2.710 Optics Spring 09 Solutions to Problem Set #3 Due Wednesday, March 4, 2009 MASSACHUSETTS INSTITUTE OF TECHNOLOGY.70 Optics Sprig 09 Solutios to Problem Set #3 Due Weesay, March 4, 009 Problem : Waa s worl a) The geometry or this problem is show i Figure. For part (a), the object

More information

MATH spring 2008 lecture 3 Answers to selected problems. 0 sin14 xdx = x dx. ; (iv) x +

MATH spring 2008 lecture 3 Answers to selected problems. 0 sin14 xdx = x dx. ; (iv) x + MATH - sprig 008 lecture Aswers to selected problems INTEGRALS. f =? For atiderivatives i geeral see the itegrals website at http://itegrals.wolfram.com. (5-vi (0 i ( ( i ( π ; (v π a. This is example

More information

Poornima University, For any query, contact us at: ,18

Poornima University, For any query, contact us at: ,18 AIEEE/1/MAHS 1 S. No Questios Solutios Q.1 he circle passig through (1, ) ad touchig the axis of x at (, ) also passes through the poit (a) (, ) (b) (, ) (c) (, ) (d) (, ) Q. ABCD is a trapezium such that

More information

X. Perturbation Theory

X. Perturbation Theory X. Perturbatio Theory I perturbatio theory, oe deals with a ailtoia that is coposed Ĥ that is typically exactly solvable of two pieces: a referece part ad a perturbatio ( Ĥ ) that is assued to be sall.

More information

JEE ADVANCED 2013 PAPER 1 MATHEMATICS

JEE ADVANCED 2013 PAPER 1 MATHEMATICS Oly Oe Optio Correct Type JEE ADVANCED 0 PAPER MATHEMATICS This sectio cotais TEN questios. Each has FOUR optios (A), (B), (C) ad (D) out of which ONLY ONE is correct.. The value of (A) 5 (C) 4 cot cot

More information

Cooking with Complementarity: A Recipe Guide for Complementarity Based Rigid-Multi-Body Dynamics Simulation

Cooking with Complementarity: A Recipe Guide for Complementarity Based Rigid-Multi-Body Dynamics Simulation Cookig wit Complemetarity: A Recipe Guide or Complemetarity Based Rigid-Multi-Body Dyamics Simulatio Stepe Berard August 25, 26 1 Itroductio Te ocus o tis report is to clariy te complemetarity based ormulatio

More information

Statistics for Applications Fall Problem Set 7

Statistics for Applications Fall Problem Set 7 18.650. Statistics for Applicatios Fall 016. Proble Set 7 Due Friday, Oct. 8 at 1 oo Proble 1 QQ-plots Recall that the Laplace distributio with paraeter λ > 0 is the cotiuous probaλ bility easure with

More information

05 - PERMUTATIONS AND COMBINATIONS Page 1 ( Answers at the end of all questions )

05 - PERMUTATIONS AND COMBINATIONS Page 1 ( Answers at the end of all questions ) 05 - PERMUTATIONS AND COMBINATIONS Page 1 ( Aswers at the ed of all questios ) ( 1 ) If the letters of the word SACHIN are arraged i all possible ways ad these words are writte out as i dictioary, the

More information

6.4 Binomial Coefficients

6.4 Binomial Coefficients 64 Bioial Coefficiets Pascal s Forula Pascal s forula, aed after the seveteeth-cetury Frech atheaticia ad philosopher Blaise Pascal, is oe of the ost faous ad useful i cobiatorics (which is the foral ter

More information

Differentiation Techniques 1: Power, Constant Multiple, Sum and Difference Rules

Differentiation Techniques 1: Power, Constant Multiple, Sum and Difference Rules Differetiatio Teciques : Power, Costat Multiple, Sum ad Differece Rules 97 Differetiatio Teciques : Power, Costat Multiple, Sum ad Differece Rules Model : Fidig te Equatio of f '() from a Grap of f ()

More information

An Insight into Differentiation and Integration

An Insight into Differentiation and Integration Differetiatio A Isigt ito Differetiatio a Itegratio Differetiatio is basically a task to fi out ow oe variable is cagig i relatio to aoter variable, te latter is usually take as a cause of te cage. For

More information

Contents Two Sample t Tests Two Sample t Tests

Contents Two Sample t Tests Two Sample t Tests Cotets 3.5.3 Two Saple t Tests................................... 3.5.3 Two Saple t Tests Setup: Two Saples We ow focus o a sceario where we have two idepedet saples fro possibly differet populatios. Our

More information

AVERAGE MARKS SCALING

AVERAGE MARKS SCALING TERTIARY INSTITUTIONS SERVICE CENTRE Level 1, 100 Royal Street East Perth, Wester Australia 6004 Telephoe (08) 9318 8000 Facsiile (08) 95 7050 http://wwwtisceduau/ 1 Itroductio AVERAGE MARKS SCALING I

More information

Mathematics Extension 2

Mathematics Extension 2 009 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etesio Geeral Istructios Readig time 5 miutes Workig time hours Write usig black or blue pe Board-approved calculators may be used A table of stadard

More information

Summer MA Lesson 13 Section 1.6, Section 1.7 (part 1)

Summer MA Lesson 13 Section 1.6, Section 1.7 (part 1) Suer MA 1500 Lesso 1 Sectio 1.6, Sectio 1.7 (part 1) I Solvig Polyoial Equatios Liear equatio ad quadratic equatios of 1 variable are specific types of polyoial equatios. Soe polyoial equatios of a higher

More information

We will conclude the chapter with the study a few methods and techniques which are useful

We will conclude the chapter with the study a few methods and techniques which are useful Chapter : Coordiate geometry: I this chapter we will lear about the mai priciples of graphig i a dimesioal (D) Cartesia system of coordiates. We will focus o drawig lies ad the characteristics of the graphs

More information

x x x 2x x N ( ) p NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS By Newton-Raphson formula

x x x 2x x N ( ) p NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS By Newton-Raphson formula NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS. If g( is cotiuous i [a,b], te uder wat coditio te iterative (or iteratio metod = g( as a uique solutio i [a,b]? '( i [a,b].. Wat

More information

Objective Mathematics

Objective Mathematics 6. If si () + cos () =, the is equal to :. If <

More information

NUMERICAL DIFFERENTIAL 1

NUMERICAL DIFFERENTIAL 1 NUMERICAL DIFFERENTIAL Ruge-Kutta Metods Ruge-Kutta metods are ver popular ecause o teir good eiciec; ad are used i most computer programs or dieretial equatios. Te are sigle-step metods as te Euler metods.

More information

Discrete Mathematics: Lectures 8 and 9 Principle of Inclusion and Exclusion Instructor: Arijit Bishnu Date: August 11 and 13, 2009

Discrete Mathematics: Lectures 8 and 9 Principle of Inclusion and Exclusion Instructor: Arijit Bishnu Date: August 11 and 13, 2009 Discrete Matheatics: Lectures 8 ad 9 Priciple of Iclusio ad Exclusio Istructor: Arijit Bishu Date: August ad 3, 009 As you ca observe by ow, we ca cout i various ways. Oe such ethod is the age-old priciple

More information

Fuzzy n-normed Space and Fuzzy n-inner Product Space

Fuzzy n-normed Space and Fuzzy n-inner Product Space Global Joural o Pure ad Applied Matheatics. ISSN 0973-768 Volue 3, Nuber 9 (07), pp. 4795-48 Research Idia Publicatios http://www.ripublicatio.co Fuzzy -Nored Space ad Fuzzy -Ier Product Space Mashadi

More information

Appendix F: Complex Numbers

Appendix F: Complex Numbers Appedix F Complex Numbers F1 Appedix F: Complex Numbers Use the imagiary uit i to write complex umbers, ad to add, subtract, ad multiply complex umbers. Fid complex solutios of quadratic equatios. Write

More information

AH Checklist (Unit 3) AH Checklist (Unit 3) Matrices

AH Checklist (Unit 3) AH Checklist (Unit 3) Matrices AH Checklist (Uit 3) AH Checklist (Uit 3) Matrices Skill Achieved? Kow that a matrix is a rectagular array of umbers (aka etries or elemets) i paretheses, each etry beig i a particular row ad colum Kow

More information

FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES

FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES LECTURE Third Editio FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES A. J. Clark School of Egieerig Departmet of Civil ad Evirometal Egieerig Chapter 7.4 b Dr. Ibrahim A. Assakkaf SPRING 3 ENES

More information

SAFE HANDS & IIT-ian's PACE EDT-10 (JEE) SOLUTIONS

SAFE HANDS & IIT-ian's PACE EDT-10 (JEE) SOLUTIONS . If their mea positios coicide with each other, maimum separatio will be A. Now from phasor diagram, we ca clearly see the phase differece. SAFE HANDS & IIT-ia's PACE ad Aswer : Optio (4) 5. Aswer : Optio

More information

Engineering Mechanics Dynamics & Vibrations. Engineering Mechanics Dynamics & Vibrations Plane Motion of a Rigid Body: Equations of Motion

Engineering Mechanics Dynamics & Vibrations. Engineering Mechanics Dynamics & Vibrations Plane Motion of a Rigid Body: Equations of Motion 1/5/013 Egieerig Mechaics Dyaics ad Vibratios Egieerig Mechaics Dyaics & Vibratios Egieerig Mechaics Dyaics & Vibratios Plae Motio of a Rigid Body: Equatios of Motio Motio of a rigid body i plae otio is

More information

Chapter 1. Complex Numbers. Dr. Pulak Sahoo

Chapter 1. Complex Numbers. Dr. Pulak Sahoo Chapter 1 Complex Numbers BY Dr. Pulak Sahoo Assistat Professor Departmet of Mathematics Uiversity Of Kalyai West Begal, Idia E-mail : sahoopulak1@gmail.com 1 Module-2: Stereographic Projectio 1 Euler

More information

Orthogonal transformations

Orthogonal transformations Orthogoal trasformatios October 12, 2014 1 Defiig property The squared legth of a vector is give by takig the dot product of a vector with itself, v 2 v v g ij v i v j A orthogoal trasformatio is a liear

More information

Mathematical Foundations -1- Sets and Sequences. Sets and Sequences

Mathematical Foundations -1- Sets and Sequences. Sets and Sequences Mathematical Foudatios -1- Sets ad Sequeces Sets ad Sequeces Methods of proof 2 Sets ad vectors 13 Plaes ad hyperplaes 18 Liearly idepedet vectors, vector spaces 2 Covex combiatios of vectors 21 eighborhoods,

More information

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense, 3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [

More information

SEQUENCES AND SERIES

SEQUENCES AND SERIES Sequeces ad 6 Sequeces Ad SEQUENCES AND SERIES Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives rise to what is called a sequece. Sequeces

More information

TEACHER CERTIFICATION STUDY GUIDE

TEACHER CERTIFICATION STUDY GUIDE COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra

More information

2D DSP Basics: Systems Stability, 2D Sampling

2D DSP Basics: Systems Stability, 2D Sampling - Digital Iage Processig ad Copressio D DSP Basics: Systes Stability D Saplig Stability ty Syste is stable if a bouded iput always results i a bouded output BIBO For LSI syste a sufficiet coditio for stability:

More information

Bernoulli Polynomials Talks given at LSBU, October and November 2015 Tony Forbes

Bernoulli Polynomials Talks given at LSBU, October and November 2015 Tony Forbes Beroulli Polyoials Tals give at LSBU, October ad Noveber 5 Toy Forbes Beroulli Polyoials The Beroulli polyoials B (x) are defied by B (x), Thus B (x) B (x) ad B (x) x, B (x) x x + 6, B (x) dx,. () B 3

More information

Binomial transform of products

Binomial transform of products Jauary 02 207 Bioial trasfor of products Khristo N Boyadzhiev Departet of Matheatics ad Statistics Ohio Norther Uiversity Ada OH 4580 USA -boyadzhiev@ouedu Abstract Give the bioial trasfors { b } ad {

More information

Lecture 7: Polar representation of complex numbers

Lecture 7: Polar representation of complex numbers Lecture 7: Polar represetatio of comple umbers See FLAP Module M3.1 Sectio.7 ad M3. Sectios 1 ad. 7.1 The Argad diagram I two dimesioal Cartesia coordiates (,), we are used to plottig the fuctio ( ) with

More information

ANSWER KEY WITH SOLUTION PAPER - 2 MATHEMATICS SECTION A 1. B 2. B 3. D 4. C 5. B 6. C 7. C 8. B 9. B 10. D 11. C 12. C 13. A 14. B 15.

ANSWER KEY WITH SOLUTION PAPER - 2 MATHEMATICS SECTION A 1. B 2. B 3. D 4. C 5. B 6. C 7. C 8. B 9. B 10. D 11. C 12. C 13. A 14. B 15. TARGET IIT-JEE t [ACCELERATION] V0 to V BATCH ADVANCED TEST DATE : - 09-06 ANSWER KEY WITH SOLUTION PAPER - MATHEMATICS SECTION A. B. B. D. C 5. B 6. C 7. C 8. B 9. B 0. D. C. C. A. B 5. C 6. D 7. A 8.

More information

MODEL TEST PAPER II Time : hours Maximum Marks : 00 Geeral Istructios : (i) (iii) (iv) All questios are compulsory. The questio paper cosists of 9 questios divided ito three Sectios A, B ad C. Sectio A

More information

Chapter 2 The Solution of Numerical Algebraic and Transcendental Equations

Chapter 2 The Solution of Numerical Algebraic and Transcendental Equations Chapter The Solutio of Numerical Algebraic ad Trascedetal Equatios Itroductio I this chapter we shall discuss some umerical methods for solvig algebraic ad trascedetal equatios. The equatio f( is said

More information

PROPERTIES OF AN EULER SQUARE

PROPERTIES OF AN EULER SQUARE PROPERTIES OF N EULER SQURE bout 0 the mathematicia Leoard Euler discussed the properties a x array of letters or itegers ow kow as a Euler or Graeco-Lati Square Such squares have the property that every

More information

The Random Walk For Dummies

The Random Walk For Dummies The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli

More information

RADICAL EXPRESSION. If a and x are real numbers and n is a positive integer, then x is an. n th root theorems: Example 1 Simplify

RADICAL EXPRESSION. If a and x are real numbers and n is a positive integer, then x is an. n th root theorems: Example 1 Simplify Example 1 Simplify 1.2A Radical Operatios a) 4 2 b) 16 1 2 c) 16 d) 2 e) 8 1 f) 8 What is the relatioship betwee a, b, c? What is the relatioship betwee d, e, f? If x = a, the x = = th root theorems: RADICAL

More information

Math 155 (Lecture 3)

Math 155 (Lecture 3) Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,

More information

ON THE PROPORTION OF DIGITS IN REDUNDANT NUMERATION SYSTEMS*

ON THE PROPORTION OF DIGITS IN REDUNDANT NUMERATION SYSTEMS* ON TE PROPORTION OF DIITS IN REDUNDANT NUMERATION SYSTEMS* by o T Butler Dept of Electrical ad Cop Eg Naval Postgraduate Scool, Code EC/Bu Moterey, CA 93943-5 USA ad Tsutou Sasao Dept of Coputer Sci ad

More information

Linear Complementarity Problem Simulation of The Meter Stick Trick

Linear Complementarity Problem Simulation of The Meter Stick Trick Liear Complemetarity Problem Simulatio o Te Meter Stick Trick Yoke Peg Leog, Ze Lv Abstract I tis paper, we ocus o aturally icorporatig cotact modes estimatio i a dyamical simulatio o a object wit multiple

More information