Phys 2310 Fri. Oct. 23, 2017 Today s Topics. Begin Chapter 6: More on Geometric Optics Reading for Next Time

Size: px
Start display at page:

Download "Phys 2310 Fri. Oct. 23, 2017 Today s Topics. Begin Chapter 6: More on Geometric Optics Reading for Next Time"

Transcription

1 Py F. Oct., 7 Today Topc Beg Capte 6: Moe o Geometc Optc eadg fo Next Tme

2 Homewok t Week HW # Homewok t week due Mo., Oct. : Capte 4: #47, 57, 59, 6, 6, 6, 6, 67, 7

3 Supplemetal: Tck ee ad e Sytem Pcple Plae of a Tck e Defto & Geomety Fo tck le code magay plae fom wc ay dvege/covege to/fom fty Te Nodal pot N, N ae te pot wee tee plae co optcal ax Fot ad Back focal legt ae te dtace fom te vetce V, V of te le. Offax ay pag toug Nodal pot coepod to te Cef ay of a t le Fg. 6. V H ad V H

4 Supplemetal: Tck ee ad e Sytem Note tat te Nodal pot ca le outde te le f t togly bet. e Make Equato fo a Tck e f Devato mla to tat fo compoud t le but we ad text wll kp: é ù l dl ú û l ê ë Te locato of te pcple plae, : f l d f l d l l l ad 4

5 Supplemetal: Multple Tck ee Code te two tck lee ow at gt. Fo two tck lee ee text: f f f d f f H H fd f ad: ad H H fd f Note tee a eo te text fo te above equato. 5

6 6 Supplemetal: Optcal Abeato ee uffe fom abeato due to ue of pecal uface o ax lgt, boke ymmety offax lgt ad dpeo of gla deped o l. ecall ftode teoy we we aume q ~ q Td ode teoy eult we we aume q ~ q q /! Devato extemely tedou wt exteve algeba o we ad text wll kp t: Specal Abeato If we deve te fomula fo efacto at a gle pecal uface wt tdode teoy we get: ú ú û ù ê ê ë é o o o Wee te tem multpled by epeet a coecto to ou ftode appoxmato. Te lage te lage te coecto. Tu we ca top dow te le to educe S.A.

7 Supplemetal: Optcal Abeato Afte lot of algeba oe ca deve a equato fo te devato of a wavefot emegg fom a pecal uface elatve to a pefect pee ceteed o te mage, te Gaua efeece pee ee Bo & Wolf, Pcple of Optc I pola cood. te devato fom te Gaua efeecepee f : 4 f B Cy co q Dy 4 Te coepodg ay Abeato ae : Dx B q Fy q coq Dy q! ad, Dy B coq Fy co q C D y coq Ey! So fo ay adu, tcke, object dtace, ad ay egt you ca compute te coeffcet. Ey coq Fy coq!

8 Supplemetal: Optcal Abeato Sedel woked out ow te coeffcet ca be calculated fo eac uface: Sum. te Sedel Tee coeffcetca te be computed ad ae kow a ad uface: So fo eac addtoal teobject ad etace pupl. te ft ufacefom beg te dtace of ad wt ad uface ad tem allowg fo apecal a Wee : Coma, : Dtoto, Feld : Cuvatue of, Atgmatm :, Abeato : Specal 4 d t t d t t b b H F t t b H E t t b H D b H C b B þ ý ü î í ì þ ý ü î í ì þ ý ü î í ì þ ý ü î í ì þ ý ü î í ì å å å å å

9 Supplemetal: Optcal Abeato Cotollg Specal Abeato SA va ape of te le SA ca be cotolled omewat by bedg a le Focal legt ema almot te ame SA ca be caged Code te ape facto: q Note tat by cagg a le ape q we ca educe but ot elmate SA. Negatve lee ave egatve SA o combato of potve ad egatve lee ae ued to educe SA. 9

10 Supplemetal: Optcal Abeato Coma eult we mage ae fomed fom offax ay ay efacted toug dffeet aul mag of te le Ca be tougt of a a cage magfcato wt ay dtace fom te le cete. Tdode devato complex but effect foud to deped o, offax agle, ad te le ape q. Alo deped o te object dtace lke SA.

11 Supplemetal: Optcal Abeato Bot Specal Abeato ad Coma deped o te object dtace a well. T expeed tem of te poto facto. p o o Mmzg Specal Abeato ad Coma va le ape. We ca mmze SA ad Coma of a le fo a gve object dtace. Fo fty ee fgue at gt. Te geeal oluto ca be foud ome text but complex.

12 eadg t Week By Wededay: F C.4 Y&F Itumet

Minimizing spherical aberrations Exploiting the existence of conjugate points in spherical lenses

Minimizing spherical aberrations Exploiting the existence of conjugate points in spherical lenses Mmzg sphecal abeatos Explotg the exstece of cojugate pots sphecal leses Let s ecall that whe usg asphecal leses, abeato fee magg occus oly fo a couple of, so called, cojugate pots ( ad the fgue below)

More information

Professor Wei Zhu. 1. Sampling from the Normal Population

Professor Wei Zhu. 1. Sampling from the Normal Population AMS570 Pofesso We Zhu. Samplg fom the Nomal Populato *Example: We wsh to estmate the dstbuto of heghts of adult US male. It s beleved that the heght of adult US male follows a omal dstbuto N(, ) Def. Smple

More information

SYSTEMS OF NON-LINEAR EQUATIONS. Introduction Graphical Methods Close Methods Open Methods Polynomial Roots System of Multivariable Equations

SYSTEMS OF NON-LINEAR EQUATIONS. Introduction Graphical Methods Close Methods Open Methods Polynomial Roots System of Multivariable Equations SYSTEMS OF NON-LINEAR EQUATIONS Itoduto Gaphal Method Cloe Method Ope Method Polomal Root Stem o Multvaale Equato Chapte Stem o No-Lea Equato /. Itoduto Polem volvg o-lea equato egeeg lude optmato olvg

More information

Multiple Choice Test. Chapter Adequacy of Models for Regression

Multiple Choice Test. Chapter Adequacy of Models for Regression Multple Choce Test Chapter 06.0 Adequac of Models for Regresso. For a lear regresso model to be cosdered adequate, the percetage of scaled resduals that eed to be the rage [-,] s greater tha or equal to

More information

Lecture 10: Condensed matter systems

Lecture 10: Condensed matter systems Lectue 0: Codesed matte systems Itoducg matte ts codesed state.! Ams: " Idstgushable patcles ad the quatum atue of matte: # Cosequeces # Revew of deal gas etopy # Femos ad Bosos " Quatum statstcs. # Occupato

More information

( m is the length of columns of A ) spanned by the columns of A : . Select those columns of B that contain a pivot; say those are Bi

( m is the length of columns of A ) spanned by the columns of A : . Select those columns of B that contain a pivot; say those are Bi Assgmet /MATH 47/Wte Due: Thusday Jauay The poblems to solve ae umbeed [] to [] below Fst some explaatoy otes Fdg a bass of the colum-space of a max ad povg that the colum ak (dmeso of the colum space)

More information

Chapter Linear Regression

Chapter Linear Regression Chpte 6.3 Le Regesso Afte edg ths chpte, ou should be ble to. defe egesso,. use sevel mmzg of esdul cte to choose the ght cteo, 3. deve the costts of le egesso model bsed o lest sques method cteo,. use

More information

= y and Normed Linear Spaces

= y and Normed Linear Spaces 304-50 LINER SYSTEMS Lectue 8: Solutos to = ad Nomed Lea Spaces 73 Fdg N To fd N, we eed to chaacteze all solutos to = 0 Recall that ow opeatos peseve N, so that = 0 = 0 We ca solve = 0 ecusvel backwads

More information

Triangles Technique for Time and Location Finding of the Lightning Discharge in Spherical Model of the Earth

Triangles Technique for Time and Location Finding of the Lightning Discharge in Spherical Model of the Earth Joual of Geocece ad Evomet Potecto 06 4 5-5 Publhed Ole Apl 06 ScRe http://wwwcpog/oual/gep http://dxdoog/046/gep064406 Tagle Techque fo Tme ad Locato Fdg of the Lghtg Dchage Sphecal Model of the Eath

More information

Tutorial on Strehl ratio, wavefront power series expansion, Zernike polynomials expansion in small aberrated optical systems By Sheng Yuan

Tutorial on Strehl ratio, wavefront power series expansion, Zernike polynomials expansion in small aberrated optical systems By Sheng Yuan Tutoial on Stel atio, wavefont powe seies expansion, Zenike polynomials expansion in small abeated optical systems By Seng Yuan. Stel Ratio Te wave abeation function, (x,y, is defined as te distance, in

More information

Question 1. Typical Cellular System. Some geometry TELE4353. About cellular system. About cellular system (2)

Question 1. Typical Cellular System. Some geometry TELE4353. About cellular system. About cellular system (2) TELE4353 Moble a atellte Commucato ystems Tutoal 1 (week 3-4 4 Questo 1 ove that fo a hexagoal geomety, the co-chael euse ato s gve by: Q (3 N Whee N + j + j 1/ 1 Typcal Cellula ystem j cells up cells

More information

Lecture 7 Diffusion. Our fluid equations that we developed before are: v t v mn t

Lecture 7 Diffusion. Our fluid equations that we developed before are: v t v mn t Cla ot fo EE6318/Phy 6383 Spg 001 Th doumt fo tutoal u oly ad may ot b opd o dtbutd outd of EE6318/Phy 6383 tu 7 Dffuo Ou flud quato that w dvlopd bfo a: f ( )+ v v m + v v M m v f P+ q E+ v B 13 1 4 34

More information

Sensorless A.C. Drive with Vector Controlled Synchronous Motor

Sensorless A.C. Drive with Vector Controlled Synchronous Motor Seole A.C. Dve wth Vecto Cotolle Sychoo Moto Ořej Fše VŠB-echcal Uvety of Otava, Faclty of Electcal Egeeg a Ifomatc, Deatmet of Powe Electoc a Electcal Dve, 17.ltoa 15, 78 33 Otava-Poba, Czech eblc oej.fe@vb.cz

More information

Skin Effect and Formation Damage

Skin Effect and Formation Damage Well Stimulation and Sand Poduction Management (PGE 489 ) Sin Effect and Fomation Damage By D. Moammed A. Kami 02-02-2016 Sin Facto A fomation damage model i a dynamic elationip expeing te fluid tanpot

More information

Negative Exponent a n = 1 a n, where a 0. Power of a Power Property ( a m ) n = a mn. Rational Exponents =

Negative Exponent a n = 1 a n, where a 0. Power of a Power Property ( a m ) n = a mn. Rational Exponents = Refeece Popetie Popetie of Expoet Let a ad b be eal umbe ad let m ad be atioal umbe. Zeo Expoet a 0 = 1, wee a 0 Quotiet of Powe Popety a m a = am, wee a 0 Powe of a Quotiet Popety ( a b m, wee b 0 b)

More information

Chapter 2: Descriptive Statistics

Chapter 2: Descriptive Statistics Chapte : Decptve Stattc Peequte: Chapte. Revew of Uvaate Stattc The cetal teecy of a oe o le yetc tbuto of a et of teval, o hghe, cale coe, ofte uaze by the athetc ea, whch efe a We ca ue the ea to ceate

More information

University of Pavia, Pavia, Italy. North Andover MA 01845, USA

University of Pavia, Pavia, Italy. North Andover MA 01845, USA Iteatoal Joual of Optmzato: heoy, Method ad Applcato 27-5565(Pt) 27-6839(Ole) wwwgph/otma 29 Global Ifomato Publhe (HK) Co, Ltd 29, Vol, No 2, 55-59 η -Peudoleaty ad Effcecy Gogo Gog, Noma G Rueda 2 *

More information

Chapter 1 Functions and Graphs

Chapter 1 Functions and Graphs Capte Functions and Gaps Section.... 6 7. 6 8 8 6. 6 6 8 8.... 6.. 6. n n n n n n n 6 n 6 n n 7. 8 7 7..8..8 8.. 8. a b ± ± 6 c ± 6 ± 8 8 o 8 6. 8y 8y 7 8y y 8y y 8 o y y. 7 7 o 7 7 Capte : Functions and

More information

Harmonic Curvatures in Lorentzian Space

Harmonic Curvatures in Lorentzian Space BULLETIN of the Bull Malaya Math Sc Soc Secod See 7-79 MALAYSIAN MATEMATICAL SCIENCES SOCIETY amoc Cuvatue Loetza Space NEJAT EKMEKÇI ILMI ACISALIOĞLU AND KĀZIM İLARSLAN Aaa Uvety Faculty of Scece Depatmet

More information

Centroids & Moments of Inertia of Beam Sections

Centroids & Moments of Inertia of Beam Sections RCH 614 Note Set 8 S017ab Cetrods & Momets of erta of Beam Sectos Notato: b C d d d Fz h c Jo L O Q Q = ame for area = ame for a (base) wdth = desgato for chael secto = ame for cetrod = calculus smbol

More information

ˆ SSE SSE q SST R SST R q R R q R R q

ˆ SSE SSE q SST R SST R q R R q R R q Bll Evas Spg 06 Sggested Aswes, Poblem Set 5 ECON 3033. a) The R meases the facto of the vaato Y eplaed by the model. I ths case, R =SSM/SST. Yo ae gve that SSM = 3.059 bt ot SST. Howeve, ote that SST=SSM+SSE

More information

Theory study about quarter-wave-stack dielectric mirrors

Theory study about quarter-wave-stack dielectric mirrors Theor tud about quarter-wave-tack delectrc rror Stratfed edu tratted reflected reflected Stratfed edu tratted cdet cdet T T Frt, coder a wave roagato a tratfed edu. A we kow, a arbtrarl olared lae wave

More information

Lesson 5. Chapter 7. Wiener Filters. Bengt Mandersson. r k s r x LTH. September Prediction Error Filter PEF (second order) from chapter 4

Lesson 5. Chapter 7. Wiener Filters. Bengt Mandersson. r k s r x LTH. September Prediction Error Filter PEF (second order) from chapter 4 Optimal Sigal oceig Leo 5 Capte 7 Wiee Filte I ti capte we will ue te model ow below. Te igal ito te eceie i ( ( iga. Nomally, ti igal i ditubed by additie wite oie (. Te ifomatio i i (. Alo, we ofte ued

More information

COORDINATE SYSTEMS, COORDINATE TRANSFORMS, AND APPLICATIONS

COORDINATE SYSTEMS, COORDINATE TRANSFORMS, AND APPLICATIONS Dola Bagaoo 0 COORDINTE SYSTEMS COORDINTE TRNSFORMS ND PPLICTIONS I. INTRODUCTION Smmet coce of coodnate sstem. In solvng Pscs poblems one cooses a coodnate sstem tat fts te poblem at and.e. a coodnate

More information

Lecturer: Ivan Kassamakov, Docent Assistant: Kalle Hanhijärvi. Course webpage:

Lecturer: Ivan Kassamakov, Docent Assistant: Kalle Hanhijärvi. Course webpage: Lecturer: Iva Kaamakov, Docet Atat: Kalle Hahjärv Coure webpage: http://electroc.phyc.helk.f/coure/optc/dex.html Peroal formato Iva Kaamakov e-mal: va.kaamakov@helk.f offce: PHYSICUM - PHY C 38 (9:00-9:00)

More information

General Method for Calculating Chemical Equilibrium Composition

General Method for Calculating Chemical Equilibrium Composition AE 6766/Setzma Sprg 004 Geeral Metod for Calculatg Cemcal Equlbrum Composto For gve tal codtos (e.g., for gve reactats, coose te speces to be cluded te products. As a example, for combusto of ydroge wt

More information

ROOT-LOCUS ANALYSIS. Lecture 11: Root Locus Plot. Consider a general feedback control system with a variable gain K. Y ( s ) ( ) K

ROOT-LOCUS ANALYSIS. Lecture 11: Root Locus Plot. Consider a general feedback control system with a variable gain K. Y ( s ) ( ) K ROOT-LOCUS ANALYSIS Coder a geeral feedback cotrol yte wth a varable ga. R( Y( G( + H( Root-Locu a plot of the loc of the pole of the cloed-loop trafer fucto whe oe of the yte paraeter ( vared. Root locu

More information

APPLICATIONS OF FD APPROXIMATIONS FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS

APPLICATIONS OF FD APPROXIMATIONS FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS LECTURE 10 APPLICATIONS OF FD APPROXIMATIONS FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS Ordinary Differential Equations Initial Value Problems For Initial Value problems (IVP s), conditions are specified

More information

1 Solution to Problem 6.40

1 Solution to Problem 6.40 1 Soluto to Problem 6.40 (a We wll wrte T τ (X 1,...,X where the X s are..d. wth PDF f(x µ, σ 1 ( x µ σ g, σ where the locato parameter µ s ay real umber ad the scale parameter σ s > 0. Lettg Z X µ σ we

More information

Fairing of Parametric Quintic Splines

Fairing of Parametric Quintic Splines ISSN 46-69 Eglad UK Joual of Ifomato ad omputg Scece Vol No 6 pp -8 Fag of Paametc Qutc Sples Yuau Wag Shagha Isttute of Spots Shagha 48 ha School of Mathematcal Scece Fuda Uvesty Shagha 4 ha { P t )}

More information

VIII Dynamics of Systems of Particles

VIII Dynamics of Systems of Particles VIII Dyacs of Systes of Patcles Cete of ass: Cete of ass Lea oetu of a Syste Agula oetu of a syste Ketc & Potetal Eegy of a Syste oto of Two Iteactg Bodes: The Reduced ass Collsos: o Elastc Collsos R whee:

More information

Idea is to sample from a different distribution that picks points in important regions of the sample space. Want ( ) ( ) ( ) E f X = f x g x dx

Idea is to sample from a different distribution that picks points in important regions of the sample space. Want ( ) ( ) ( ) E f X = f x g x dx Importace Samplg Used for a umber of purposes: Varace reducto Allows for dffcult dstrbutos to be sampled from. Sestvty aalyss Reusg samples to reduce computatoal burde. Idea s to sample from a dfferet

More information

Layered structures: transfer matrix formalism

Layered structures: transfer matrix formalism Layered tructure: trafer matrx formalm Iterface betwee LI meda Trafer matrx formalm Petr Kužel Practcally oly oe formula to be kow order to calculate ay tructure Applcato: Atreflectve coatg Delectrc mrror,

More information

Multiple Regression. More than 2 variables! Grade on Final. Multiple Regression 11/21/2012. Exam 2 Grades. Exam 2 Re-grades

Multiple Regression. More than 2 variables! Grade on Final. Multiple Regression 11/21/2012. Exam 2 Grades. Exam 2 Re-grades STAT 101 Dr. Kar Lock Morga 11/20/12 Exam 2 Grades Multple Regresso SECTIONS 9.2, 10.1, 10.2 Multple explaatory varables (10.1) Parttog varablty R 2, ANOVA (9.2) Codtos resdual plot (10.2) Trasformatos

More information

Phys 2310 Wed. Sept. 7, 2017 Today s Topics

Phys 2310 Wed. Sept. 7, 2017 Today s Topics Pys 30 Wed. Set. 7, 07 Today s Toics Proerties of Ligt Fiis Detectio of Ligt Cater 4: Proagatio of Ligt Homework Assiged Readig for Net Time Homework #4 Due Oct. 4 SZ C. 33: #33.3, 33.7, 33.9, 33., 33.,

More information

Investigation of Partially Conditional RP Model with Response Error. Ed Stanek

Investigation of Partially Conditional RP Model with Response Error. Ed Stanek Partally Codtoal Radom Permutato Model 7- vestgato of Partally Codtoal RP Model wth Respose Error TRODUCTO Ed Staek We explore the predctor that wll result a smple radom sample wth respose error whe a

More information

Detection and Estimation Theory

Detection and Estimation Theory ESE 54 Detecto ad Etmato Theoy Joeph A. O Sullva Samuel C. Sach Pofeo Electoc Sytem ad Sgal Reeach Laboatoy Electcal ad Sytem Egeeg Wahgto Uvety Ubaue Hall 34-935-473 (Lyda awe) jao@wutl.edu J. A. O'S.

More information

( ) ( ) Last Time. 3-D particle in box: summary. Modified Bohr model. 3-dimensional Hydrogen atom. Orbital magnetic dipole moment

( ) ( ) Last Time. 3-D particle in box: summary. Modified Bohr model. 3-dimensional Hydrogen atom. Orbital magnetic dipole moment Last Time 3-dimensional quantum states and wave functions Couse evaluations Tuesday, Dec. 9 in class Deceasing paticle size Quantum dots paticle in box) Optional exta class: eview of mateial since Exam

More information

cos kd kd 2 cosθ = π 2 ± nπ d λ cosθ = 1 2 ± n N db

cos kd kd 2 cosθ = π 2 ± nπ d λ cosθ = 1 2 ± n N db . (Balanis 6.43) You can confim tat AF = e j kd cosθ + e j kd cosθ N = cos kd cosθ gives te same esult as (6-59) and (6-6), fo a binomial aay wit te coefficients cosen as in section 6.8.. Tis single expession

More information

DERIVATION OF THE BASIC LAWS OF GEOMETRIC OPTICS

DERIVATION OF THE BASIC LAWS OF GEOMETRIC OPTICS DERIVATION OF THE BASIC LAWS OF GEOMETRIC OPTICS It s well kow that a lght ay eflectg off of a suface has ts agle of eflecto equal to ts agle of cdece ad that f ths ay passes fom oe medum to aothe that

More information

Chapter 6 Area and Volume

Chapter 6 Area and Volume Capte 6 Aea and Volume Execise 6. Q. (i) Aea of paallelogam ( ax)( x) Aea of ectangle ax ( x + ax)( x) x x ( + a) a x a Faction x ( + a) + a (ii) Aea of paallelogam Aea of ectangle 5 ax (( + 5 x)( x ax)

More information

Multivariate Transformation of Variables and Maximum Likelihood Estimation

Multivariate Transformation of Variables and Maximum Likelihood Estimation Marquette Uversty Multvarate Trasformato of Varables ad Maxmum Lkelhood Estmato Dael B. Rowe, Ph.D. Assocate Professor Departmet of Mathematcs, Statstcs, ad Computer Scece Copyrght 03 by Marquette Uversty

More information

Non-axial symmetric loading on axial symmetric. Final Report of AFEM

Non-axial symmetric loading on axial symmetric. Final Report of AFEM No-axal symmetc loadg o axal symmetc body Fal Repot of AFEM Ths poject does hamoc aalyss of o-axal symmetc loadg o axal symmetc body. Shuagxg Da, Musket Kamtokat 5//009 No-axal symmetc loadg o axal symmetc

More information

Maximum likelihood estimate of phylogeny. BIOL 495S/ CS 490B/ MATH 490B/ STAT 490B Introduction to Bioinformatics April 24, 2002

Maximum likelihood estimate of phylogeny. BIOL 495S/ CS 490B/ MATH 490B/ STAT 490B Introduction to Bioinformatics April 24, 2002 Mmm lkelhood eme of phylogey BIO 9S/ S 90B/ MH 90B/ S 90B Iodco o Bofomc pl 00 Ovevew of he pobblc ppoch o phylogey o k ee ccodg o he lkelhood d ee whee d e e of eqece d ee by ee wh leve fo he eqece. he

More information

Lecture 07: Poles and Zeros

Lecture 07: Poles and Zeros Lecture 07: Poles ad Zeros Defto of poles ad zeros The trasfer fucto provdes a bass for determg mportat system respose characterstcs wthout solvg the complete dfferetal equato. As defed, the trasfer fucto

More information

ENGI 4421 Propagation of Error Page 8-01

ENGI 4421 Propagation of Error Page 8-01 ENGI 441 Propagato of Error Page 8-01 Propagato of Error [Navd Chapter 3; ot Devore] Ay realstc measuremet procedure cotas error. Ay calculatos based o that measuremet wll therefore also cota a error.

More information

ENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections

ENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections ENGI 441 Jot Probablty Dstrbutos Page 7-01 Jot Probablty Dstrbutos [Navd sectos.5 ad.6; Devore sectos 5.1-5.] The jot probablty mass fucto of two dscrete radom quattes, s, P ad p x y x y The margal probablty

More information

Chapter 3. Differentiation 3.2 Differentiation Rules for Polynomials, Exponentials, Products and Quotients

Chapter 3. Differentiation 3.2 Differentiation Rules for Polynomials, Exponentials, Products and Quotients 3.2 Dfferetato Rules 1 Capter 3. Dfferetato 3.2 Dfferetato Rules for Polyomals, Expoetals, Proucts a Quotets Rule 1. Dervatve of a Costat Fucto. If f as te costat value f(x) = c, te f x = [c] = 0. x Proof.

More information

Chapter 3. Many-Electron Atoms

Chapter 3. Many-Electron Atoms Capte. May-ecto Atom Readg: Bade & Joaca Capte 8 Ceta ed Appoxmato Ampto: eac of te atomc eecto move a effectve pecay ymmetc poteta ceated by te ce ad a te ote eecto. amtoa of te -eecto atom m 4πε e >

More information

Consumer theory. A. The preference ordering B. The feasible set C. The consumption decision. A. The preference ordering. Consumption bundle

Consumer theory. A. The preference ordering B. The feasible set C. The consumption decision. A. The preference ordering. Consumption bundle Thomas Soesso Mcoecoomcs Lecte Cosme theoy A. The efeece odeg B. The feasble set C. The cosmto decso A. The efeece odeg Cosmto bdle x ( 2 x, x,... x ) x Assmtos: Comleteess 2 Tastvty 3 Reflexvty 4 No-satato

More information

d dt d d dt dt Also recall that by Taylor series, / 2 (enables use of sin instead of cos-see p.27 of A&F) dsin

d dt d d dt dt Also recall that by Taylor series, / 2 (enables use of sin instead of cos-see p.27 of A&F) dsin Learzato of the Swg Equato We wll cover sectos.5.-.6 ad begg of Secto 3.3 these otes. 1. Sgle mache-fte bus case Cosder a sgle mache coected to a fte bus, as show Fg. 1 below. E y1 V=1./_ Fg. 1 The admttace

More information

This may involve sweep, revolution, deformation, expansion and forming joints with other curves.

This may involve sweep, revolution, deformation, expansion and forming joints with other curves. 5--8 Shapes ae ceated by cves that a sface sch as ooftop of a ca o fselage of a acaft ca be ceated by the moto of cves space a specfed mae. Ths may volve sweep, evolto, defomato, expaso ad fomg jots wth

More information

Infinitesimal Automorphisms in the Tangent Bundle of a Riemannian Manifold with Horizontal Lift of Affine Connection

Infinitesimal Automorphisms in the Tangent Bundle of a Riemannian Manifold with Horizontal Lift of Affine Connection Cag a J Sc 2007; 34(2) 5 Cag a J Sc 2007; 34(2) : 5-59 wwwscececmuact/joural-scece/josctml Cotrbuted Paper Iftesmal Automorpsms te Taget Budle of a Remaa afold wt orzotal Lft of Affe Coecto Ayd Gezer *,

More information

T T V e g em D e j ) a S D } a o "m ek j g ed b m "d mq m [ d, )

T T V e g em D e j ) a S D } a o m ek j g ed b m d mq m [ d, ) . ) 6 3 ; 6 ;, G E E W T S W X D ^ L J R Y [ _ ` E ) '" " " -, 7 4-4 4-4 ; ; 7 4 4 4 4 4 ;= : " B C CA BA " ) 3D H E V U T T V e g em D e j ) a S D } a o "m ek j g ed b m "d mq m [ d, ) W X 6 G.. 6 [ X

More information

REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION

REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION I lear regreo, we coder the frequecy dtrbuto of oe varable (Y) at each of everal level of a ecod varable (X). Y kow a the depedet varable. The

More information

Chapter 19 Webassign Help Problems

Chapter 19 Webassign Help Problems Chapte 9 Webaign Help Poblem 4 5 6 7 8 9 0 Poblem 4: The pictue fo thi poblem i a bit mileading. They eally jut give you the pictue fo Pat b. So let fix that. Hee i the pictue fo Pat (a): Pat (a) imply

More information

BENDING OF BEAM. Compressed layer. Elongated. layer. Un-strained. layer. NA= Neutral Axis. Compression. Unchanged. Elongation. Two Dimensional View

BENDING OF BEAM. Compressed layer. Elongated. layer. Un-strained. layer. NA= Neutral Axis. Compression. Unchanged. Elongation. Two Dimensional View BNDING OF BA Compessed laye N Compession longation Un-stained laye Unchanged longated laye NA Neutal Axis Two Dimensional View A When a beam is loaded unde pue moment, it can be shown that the beam will

More information

Chapter 4 Multiple Random Variables

Chapter 4 Multiple Random Variables Revew for the prevous lecture: Theorems ad Examples: How to obta the pmf (pdf) of U = g (, Y) ad V = g (, Y) Chapter 4 Multple Radom Varables Chapter 44 Herarchcal Models ad Mxture Dstrbutos Examples:

More information

VECTOR MECHANICS FOR ENGINEERS: Vector Mechanics for Engineers: Dynamics. In the current chapter, you will study the motion of systems of particles.

VECTOR MECHANICS FOR ENGINEERS: Vector Mechanics for Engineers: Dynamics. In the current chapter, you will study the motion of systems of particles. Seeth Edto CHPTER 4 VECTOR MECHNICS FOR ENINEERS: DYNMICS Fedad P. ee E. Russell Johsto, J. Systems of Patcles Lectue Notes: J. Walt Ole Texas Tech Uesty 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed. Seeth

More information

Born-Oppenheimer Approximation. Kaito Takahashi

Born-Oppenheimer Approximation. Kaito Takahashi o-oppehee ppoato Kato Takahah toc Ut Fo quatu yte uch a ecto ad olecule t eae to ue ut that ft the=tomc UNT Ue a of ecto (ot kg) Ue chage of ecto (ot coulob) Ue hba fo agula oetu (ot kg - ) Ue 4pe 0 fo

More information

The Geometric Proof of the Hecke Conjecture

The Geometric Proof of the Hecke Conjecture The Geometc Poof of the Hecke Cojectue Kada Sh Depatmet of Mathematc Zhejag Ocea Uvety Zhouha Cty 6 Zhejag Povce Cha Atact Begg fom the eoluto of Dchlet fucto ug the e poduct fomula of two fte-dmeoal vecto

More information

RECAPITULATION & CONDITIONAL PROBABILITY. Number of favourable events n E Total number of elementary events n S

RECAPITULATION & CONDITIONAL PROBABILITY. Number of favourable events n E Total number of elementary events n S Fomulae Fo u Pobablty By OP Gupta [Ida Awad We, +91-9650 350 480] Impotat Tems, Deftos & Fomulae 01 Bascs Of Pobablty: Let S ad E be the sample space ad a evet a expemet espectvely Numbe of favouable evets

More information

CHAPTER 5 : SERIES. 5.2 The Sum of a Series Sum of Power of n Positive Integers Sum of Series of Partial Fraction Difference Method

CHAPTER 5 : SERIES. 5.2 The Sum of a Series Sum of Power of n Positive Integers Sum of Series of Partial Fraction Difference Method CHAPTER 5 : SERIES 5.1 Seies 5. The Sum of a Seies 5..1 Sum of Powe of Positive Iteges 5.. Sum of Seies of Patial Factio 5..3 Diffeece Method 5.3 Test of covegece 5.3.1 Divegece Test 5.3. Itegal Test 5.3.3

More information

Objectives. Learning Outcome. 7.1 Centre of Gravity (C.G.) 7. Statics. Determine the C.G of a lamina (Experimental method)

Objectives. Learning Outcome. 7.1 Centre of Gravity (C.G.) 7. Statics. Determine the C.G of a lamina (Experimental method) Ojectves 7 Statcs 7. Cete of Gavty 7. Equlum of patcles 7.3 Equlum of g oes y Lew Sau oh Leag Outcome (a) efe cete of gavty () state the coto whch the cete of mass s the cete of gavty (c) state the coto

More information

ESS Line Fitting

ESS Line Fitting ESS 5 014 17. Le Fttg A very commo problem data aalyss s lookg for relatoshpetwee dfferet parameters ad fttg les or surfaces to data. The smplest example s fttg a straght le ad we wll dscuss that here

More information

Lecture 11: Introduction to nonlinear optics I.

Lecture 11: Introduction to nonlinear optics I. Lectue : Itoducto to olea optcs I. Pet Kužel Fomulato of the olea optcs: olea polazato Classfcato of the olea pheomea Popagato of wea optc sgals stog quas-statc felds (descpto usg eomalzed lea paametes)!

More information

Consider two masses m 1 at x = x 1 and m 2 at x 2.

Consider two masses m 1 at x = x 1 and m 2 at x 2. Chapte 09 Syste of Patcles Cete of ass: The cete of ass of a body o a syste of bodes s the pot that oes as f all of the ass ae cocetated thee ad all exteal foces ae appled thee. Note that HRW uses co but

More information

8.6 Order-Recursive LS s[n]

8.6 Order-Recursive LS s[n] 8.6 Order-Recurive LS [] Motivate ti idea wit Curve Fittig Give data: 0,,,..., - [0], [],..., [-] Wat to fit a polyomial to data.., but wic oe i te rigt model?! Cotat! Quadratic! Liear! Cubic, Etc. ry

More information

Midterm Exam 1, section 1 (Solution) Thursday, February hour, 15 minutes

Midterm Exam 1, section 1 (Solution) Thursday, February hour, 15 minutes coometrcs, CON Sa Fracsco State Uversty Mchael Bar Sprg 5 Mdterm am, secto Soluto Thursday, February 6 hour, 5 mutes Name: Istructos. Ths s closed book, closed otes eam.. No calculators of ay kd are allowed..

More information

L. Yaroslavsky. Selected Topics in Image Processing Part 1. Imaging transforms in digital computers

L. Yaroslavsky. Selected Topics in Image Processing Part 1. Imaging transforms in digital computers L. Yolv. Selecte Topc Imge Poceg Pt. Imgg tom gtl compte Lecte. Covolto tegl gtl lte.. Imgg tom pcple o te cete epeetto Te cotec pcple te mtl coepoece pcple etwee coto gtl tomto. ( M mpto: gl cete epeetto

More information

Auchmuty High School Mathematics Department Sequences & Series Notes Teacher Version

Auchmuty High School Mathematics Department Sequences & Series Notes Teacher Version equeces ad eies Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio A sequece takes the fom,,7,0,, while 7 0 is a seies. Thee ae two types of sequece/seies aithmetic ad geometic.

More information

Reaction Time VS. Drug Percentage Subject Amount of Drug Times % Reaction Time in Seconds 1 Mary John Carl Sara William 5 4

Reaction Time VS. Drug Percentage Subject Amount of Drug Times % Reaction Time in Seconds 1 Mary John Carl Sara William 5 4 CHAPTER Smple Lear Regreo EXAMPLE A expermet volvg fve ubject coducted to determe the relatohp betwee the percetage of a certa drug the bloodtream ad the legth of tme t take the ubject to react to a tmulu.

More information

Java Applets / Flash Java Applet vs. Flash

Java Applets / Flash Java Applet vs. Flash ava Alet / Flah ava Alet v. Flah oltal oblem wth Mooft hhl otable moe dfflt develomet ot a oblem le o exellet val develomet tool Alet / Flah ood fo tato volv omlex e t whee XTML t elemet ae ot adeqate

More information

are positive, and the pair A, B is controllable. The uncertainty in is introduced to model control failures.

are positive, and the pair A, B is controllable. The uncertainty in is introduced to model control failures. Lectue 4 8. MRAC Desg fo Affe--Cotol MIMO Systes I ths secto, we cosde MRAC desg fo a class of ult-ut-ult-outut (MIMO) olea systes, whose lat dyacs ae lealy aaetezed, the ucetates satsfy the so-called

More information

Exam 1. Sept. 22, 8:00-9:30 PM EE 129. Material: Chapters 1-8 Labs 1-3

Exam 1. Sept. 22, 8:00-9:30 PM EE 129. Material: Chapters 1-8 Labs 1-3 Eam ept., 8:00-9:30 PM EE 9 Mateal: Chapte -8 Lab -3 tandadzaton and Calbaton: Ttaton: ue of tandadzed oluton to detemne the concentaton of an unknown. Rele on a eacton of known tochomet, a oluton wth

More information

ETIKA V PROFESII PSYCHOLÓGA

ETIKA V PROFESII PSYCHOLÓGA P r a ž s k á v y s o k á š k o l a p s y c h o s o c i á l n í c h s t u d i í ETIKA V PROFESII PSYCHOLÓGA N a t á l i a S l o b o d n í k o v á v e d ú c i p r á c e : P h D r. M a r t i n S t r o u

More information

Physics Courseware Electromagnetism

Physics Courseware Electromagnetism Pysics Cousewae lectomagnetism lectic field Poblem.- a) Find te electic field at point P poduced by te wie sown in te figue. Conside tat te wie as a unifom linea cage distibution of λ.5µ C / m b) Find

More information

FORMULAE. 8. a 2 + b 2 + c 2 ab bc ca = 1 2 [(a b)2 + (b c) 2 + (c a) 2 ] 10. (a b) 3 = a 3 b 3 3ab (a b) = a 3 3a 2 b + 3ab 2 b 3

FORMULAE. 8. a 2 + b 2 + c 2 ab bc ca = 1 2 [(a b)2 + (b c) 2 + (c a) 2 ] 10. (a b) 3 = a 3 b 3 3ab (a b) = a 3 3a 2 b + 3ab 2 b 3 FORMULAE Algeba 1. (a + b) = a + b + ab = (a b) + 4ab. (a b) = a + b ab = (a + b) 4ab 3. a b = (a b) (a + b) 4. a + b = (a + b) ab = (a b) + ab 5. (a + b) + (a b) = (a + b ) 6. (a + b) (a b) = 4ab 7. (a

More information

Fall 2004/05 Solutions to Assignment 5: The Stationary Phase Method Provided by Mustafa Sabri Kilic. I(x) = e ixt e it5 /5 dt (1) Z J(λ) =

Fall 2004/05 Solutions to Assignment 5: The Stationary Phase Method Provided by Mustafa Sabri Kilic. I(x) = e ixt e it5 /5 dt (1) Z J(λ) = 8.35 Fall 24/5 Solution to Aignment 5: The Stationay Phae Method Povided by Mutafa Sabi Kilic. Find the leading tem fo each of the integal below fo λ >>. (a) R eiλt3 dt (b) R e iλt2 dt (c) R eiλ co t dt

More information

Centers of Gravity - Centroids

Centers of Gravity - Centroids RCH Note Set 9. S205ab Ceters of Gravt - Cetrods Notato: C Fz L O Q Q t tw = ame for area = desgato for chael secto = ame for cetrod = force compoet the z drecto = ame for legth = ame for referece org

More information

THIS PAGE DECLASSIFIED IAW E

THIS PAGE DECLASSIFIED IAW E THS PAGE DECLASSFED AW E0 2958 BL K THS PAGE DECLASSFED AW E0 2958 THS PAGE DECLASSFED AW E0 2958 B L K THS PAGE DECLASSFED AW E0 2958 THS PAGE DECLASSFED AW EO 2958 THS PAGE DECLASSFED AW EO 2958 THS

More information

Exponential Generating Functions - J. T. Butler

Exponential Generating Functions - J. T. Butler Epoetal Geeatg Fuctos - J. T. Butle Epoetal Geeatg Fuctos Geeatg fuctos fo pemutatos. Defto: a +a +a 2 2 + a + s the oday geeatg fucto fo the sequece of teges (a, a, a 2, a, ). Ep. Ge. Fuc.- J. T. Butle

More information

Optical Remote Sensing with DIfferential Absorption Lidar (DIAL)

Optical Remote Sensing with DIfferential Absorption Lidar (DIAL) Optcal emote esg wth DIffeetal Absopt Lda DIAL Pat : Theoy hstoph eff IE Uvesty of oloado & OAA/EL/D/Atmosphec emote esg Goup http://www.esl.oaa.gov/csd/goups/csd3/ Guest lectue fo AE-659 Lda emote esg

More information

MEI Structured Mathematics. Module Summary Sheets. Numerical Methods (Version B reference to new book)

MEI Structured Mathematics. Module Summary Sheets. Numerical Methods (Version B reference to new book) MEI Matematics in Education and Industy MEI Stuctued Matematics Module Summay Seets (Vesion B efeence to new book) Topic : Appoximations Topic : Te solution of equations Topic : Numeical integation Topic

More information

Determining the Best Linear Unbiased Predictor of PSU Means with the Data. included with the Random Variables. Ed Stanek

Determining the Best Linear Unbiased Predictor of PSU Means with the Data. included with the Random Variables. Ed Stanek Detemining te Bet Linea Unbiaed Pedicto of PSU ean wit te Data included wit te andom Vaiable Ed Stanek Intoduction We develop te equation fo te bet linea unbiaed pedicto of PSU mean in a two tage andom

More information

FI 2201 Electromagnetism

FI 2201 Electromagnetism FI Electomagnetim Aleande A. Ikanda, Ph.D. Phyic of Magnetim and Photonic Reeach Goup ecto Analyi CURILINEAR COORDINAES, DIRAC DELA FUNCION AND HEORY OF ECOR FIELDS Cuvilinea Coodinate Sytem Cateian coodinate:

More information

A stopping criterion for Richardson s extrapolation scheme. under finite digit arithmetic.

A stopping criterion for Richardson s extrapolation scheme. under finite digit arithmetic. A stoppg crtero for cardso s extrapoato sceme uder fte dgt artmetc MAKOO MUOFUSHI ad HIEKO NAGASAKA epartmet of Lbera Arts ad Sceces Poytecc Uversty 4-1-1 Hasmotoda,Sagamara,Kaagawa 229-1196 JAPAN Abstract:

More information

BINOMIAL THEOREM An expression consisting of two terms, connected by + or sign is called a

BINOMIAL THEOREM An expression consisting of two terms, connected by + or sign is called a BINOMIAL THEOREM hapte 8 8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4 5y, etc., ae all biomial epessios. 8.. Biomial theoem If

More information

Kinematics. Redundancy. Task Redundancy. Operational Coordinates. Generalized Coordinates. m task. Manipulator. Operational point

Kinematics. Redundancy. Task Redundancy. Operational Coordinates. Generalized Coordinates. m task. Manipulator. Operational point Mapulato smatc Jot Revolute Jot Kematcs Base Lks: movg lk fed lk Ed-Effecto Jots: Revolute ( DOF) smatc ( DOF) Geealzed Coodates Opeatoal Coodates O : Opeatoal pot 5 costats 6 paametes { postos oetatos

More information

Module Title: Business Mathematics and Statistics 2

Module Title: Business Mathematics and Statistics 2 CORK INSTITUTE OF TECHNOLOGY INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ Semeste Eamatos 009/00 Module Ttle: Busess Mathematcs ad Statstcs Module Code: STAT 6003 School: School of Busess ogamme Ttle: Bachelo of

More information

BINOMIAL THEOREM NCERT An expression consisting of two terms, connected by + or sign is called a

BINOMIAL THEOREM NCERT An expression consisting of two terms, connected by + or sign is called a 8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4, etc., ae all biomial 5y epessios. 8.. Biomial theoem BINOMIAL THEOREM If a ad b ae

More information

Rotational Kinetic Energy

Rotational Kinetic Energy Add Impotant Rotational Kinetic Enegy Page: 353 NGSS Standad: N/A Rotational Kinetic Enegy MA Cuiculum Famewok (006):.1,.,.3 AP Phyic 1 Leaning Objective: N/A, but olling poblem have appeaed on peviou

More information

Inference for A One Way Factorial Experiment. By Ed Stanek and Elaine Puleo

Inference for A One Way Factorial Experiment. By Ed Stanek and Elaine Puleo Infeence fo A One Way Factoial Expeiment By Ed Stanek and Elaine Puleo. Intoduction We develop etimating equation fo Facto Level mean in a completely andomized one way factoial expeiment. Thi development

More information

Photographing a time interval

Photographing a time interval Potogaping a time inteval Benad Rotenstein and Ioan Damian Politennia Univesity of imisoaa Depatment of Pysis imisoaa Romania benad_otenstein@yaoo.om ijdamian@yaoo.om Abstat A metod of measuing time intevals

More information

THE ROYAL STATISTICAL SOCIETY 2016 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5

THE ROYAL STATISTICAL SOCIETY 2016 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5 THE ROYAL STATISTICAL SOCIETY 06 EAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5 The Socety s provdg these solutos to assst cadtes preparg for the examatos 07. The solutos are teded as learg ads ad should

More information

Physics 111 Lecture 5 (Walker: 3.3-6) Vectors & Vector Math Motion Vectors Sept. 11, 2009

Physics 111 Lecture 5 (Walker: 3.3-6) Vectors & Vector Math Motion Vectors Sept. 11, 2009 Physics 111 Lectue 5 (Walke: 3.3-6) Vectos & Vecto Math Motion Vectos Sept. 11, 2009 Quiz Monday - Chap. 2 1 Resolving a vecto into x-component & y- component: Pola Coodinates Catesian Coodinates x y =

More information

Suppose we have observed values t 1, t 2, t n of a random variable T.

Suppose we have observed values t 1, t 2, t n of a random variable T. Sppose we have obseved vales, 2, of a adom vaable T. The dsbo of T s ow o belog o a cea ype (e.g., expoeal, omal, ec.) b he veco θ ( θ, θ2, θp ) of ow paamees assocaed wh s ow (whee p s he mbe of ow paamees).

More information

CH E 374 Computational Methods in Engineering Fall 2007

CH E 374 Computational Methods in Engineering Fall 2007 CH E 7 Computatoal Methods Egeerg Fall 007 Sample Soluto 5. The data o the varato of the rato of stagato pressure to statc pressure (r ) wth Mach umber ( M ) for the flow through a duct are as follows:

More information

Chapter 3. Differentiation 3.3 Differentiation Rules

Chapter 3. Differentiation 3.3 Differentiation Rules 3.3 Dfferetato Rules 1 Capter 3. Dfferetato 3.3 Dfferetato Rules Dervatve of a Costat Fucto. If f as te costat value f(x) = c, te f x = [c] = 0. x Proof. From te efto: f (x) f(x + ) f(x) o c c 0 = 0. QED

More information

Ideal multigrades with trigonometric coefficients

Ideal multigrades with trigonometric coefficients Ideal multgrades wth trgoometrc coeffcets Zarathustra Brady December 13, 010 1 The problem A (, k) multgrade s defed as a par of dstct sets of tegers such that (a 1,..., a ; b 1,..., b ) a j = =1 for all

More information