Lectures # He-like systems. October 31 November 4,6

Size: px
Start display at page:

Download "Lectures # He-like systems. October 31 November 4,6"

Transcription

1 Lectue #5-7 7 Octoe 3 oveme 4,6 Self-conitent field Htee-Foc eqution: He-lie ytem Htee-Foc eqution: cloed-hell hell ytem Chpte 3, pge 6-77, Lectue on Atomic Phyic He-lie ytem H (, h ( + h ( + h ( Z Z: He Z3: Li + hψ ( ε ψ ( Solution: Coulom (H-lie wve function ψ nlm( Pnl ( Ylm ( θ, φ One-electon wve function, π Z4: Be ++ Coulom epulion etween two electon H (, Ψ (, E Ψ(, 3/ P ( Z e Ψ (, P ( P ( {,, } 4 Z Two-electon gound tte wve function

2 Deivtion of Htee-Foc eqution fo oitl E Ψ h ( + h ( + Ψ Z Ψ h ( Ψ 4 π d P ( 4 π d P ( P ( ( 4π d P ( omliztion condition d P ( Z dp ( Z d P ( P ( d P ( d d (integting y pt dp ( Z h ( h ( d P ( d Ψ + Ψ Deivtion of Htee-Foc eqution fo oitl Ψ 4π Ψ < + + > q Y * q ( θ, φ Y ( θ, φ q The pheicl hmonic e othonoml on the unit phee: dω Y ( θ, φ Y ( θ, φ δ δ * ' q' q ' qq' * Theefoe: dω Y ( θ, φ Y δ δ nd only one tem q q q contiute fom the um ove nd q fo integl. d P d P ( ( Ψ Ψ, >

3 Deivtion of Htee-Foc eqution fo oitl d P d P ( ( Ψ Ψ d P v (, ( > >, ew deigntion Thi i potentil t of pheiclly ymmetic chge ditiution with dil denity P (. d P Ψ Ψ ( v (, Put it ll togethe E Ψ h ( + h ( + Ψ dp ( Z d P ( + v(, P ( d Ψ Ψ dp ( omliztion condition Vitionl pinciple: we equie tht enegy e ttiony with epect to vition of the dil function uject to nomliztion contnt. δ ( E λ Lgnge multiplie 3

4 HF eqution δ P ( δ P ( dp δ d d δ P d dp ( Z δ ( E λ dδ δ[ P ( ] + v(, δ[ P ( ] λδ[ P ( ] d dp ( dp ( Z + d δ d d P ( δp ( v(, P ( δp ( λδp ( δp ( d ( Z P ( + v(, P ( λ P ( δ P ( d d P (integting y pt HF eqution δ d P ( Z ( E λ d P ( (, P ( P ( P ( d λ + v δ d P ( Z P v P P ( (, ( ( d + ε HF eqution fo oitl Jut the dil Schödinge eqution fo pticle with l moving in potentil Z V ( +v(, 4

5 HF eqution: olution pocedue d P ( Z P v P P ( (, ( ( d + ε. Pic function P ( which i ou et nown ppoximtion to wve function (ceened Coulom wve function with effective chge β 3/ β 5 P ( β e, β Z 6. Ue it to clculte the potentil v (, : v (, d P ( > 3. Sutitute thi potentil to HF eqution nd olve it fo P nd ε. 4. ow ue ou new wve function to evlute to potentil v (, gin. 5. Repet until ε convege (eqution i olved itetively. Convegence pmete ε δ ε ( n ε ( n ( n ( n ε enegy fte itetion n Bc to the clcultion of totl enegy Itetion of ε : enegy convege to digit fte 8 itetion ε u. E Ψ h ( + h ( + Ψ h + v (, ε v (,.86.u ev The impovement i mll fo He ut it i the et which cn e otined within the fmewo of Independent-pticle ppoximtion. 5

6 Why do we need ppoximtion method? H (,,, h ( + i i i j ij H (,,, Ψ (,,, E Ψ(,,, Why do we need ppoximte method? Let te n ion tom. It h 6 electon: wve function depend on vile. Uing gid of only point we need 78 nume to tulte ion wve function! Ψ (,,, Thi i lge thn the etimted nume of pticle in the Sol ytem! Thi i why ppoximtion to exct olution nd the method of impoving ccucy of thee olution e of uch inteet. Independent-pticle pticle ppoximtion H (,,, h ( + H + V i i i j ij H (,,, h ( + U ( V (,,, U ( i i i i i i j ij i i h( i dd nd utct Why? To hve ette lowet ode nd mlle V. ote: we edefined ou lowet ode nd ou lowet ode wve function e olution of hψ ( ε ψ (. Wht e ou indice? Lowet ode enegy E ε + ε + + ε ( n n Full et of quntum nume which define oitl. Fo exmple: ( n, l, m, µ 6

7 Mtix element How to evlute mtix element of H nd V? H (,,, h ( + U ( V (,,, U ( i i i i i i j ij i one-ody mtix element eed to evlute: two-ody mtix element Ψ h ( Ψ Ψ U ( Ψ n i n n i n i i Ψ Ψ n n i j ij me et of indice Sytem of pticle: mny-pticle opeto F i i f ( One-pticle opeto Exmple: H i h i Let deignte ou Slte deteminte function Ψ n How to evlute the coeponding mtix element? Ψ' ' n' F Ψ n i f ii If the et of indice { n } nd { n} e the me. f f d ψ ( f ( ψ ( 3 7

8 Sytem of pticle: mny-pticle opeto G g( ij i j Two-pticle opeto Exmple: i j ij Ψ G Ψ g g ' ' n' n ( ijij ijji i, j If the et of indice { n } nd { n} e the me. g g cd d d ψ ( ψ ( g( ψ ( ψ ( 3 3 cd c d F i i Mtix element How to evlute mtix element of H nd V? f ( Ψ Ψ Ψ n F Ψ n i G g( ij Ψ n G Ψ n ( gijij gijji i, j i j h ( Ψ U ( Ψ n i n i i n n i j ij i, j ( h n i n ii i i Ψ Ψ U ii f ii ( gijij gijji Coulom mtix element 8

9 Cloed-hell hell ytem: He, Be, e, Helium He : Beyllium Be : eon e : p 6 Mgneium Mg : p 3 6 Agon Ag : p 3 3p Clcium p p C : 3 Cloed-hell hell ytem: He, Be, e, Let ue the following deigntion gin: um ove coe (cloed-hell oitl i deignted y the indice fom the eginning of the lphet:,,c,d ( ( Ψ h ( Ψ h h n i n ii i i Ψ U ( Ψ U U n i n ii i i Ψ Ψ ( gijij gijji ( g g n n i j ij i, j Sum ove men um ove the entie et of quntum nume of ll the coe electon Fo exmple, Be: nl : (, n l m µ 9

10 Cloed-hell hell ytem: He, Be, e, ( E Ψ H Ψ h + U (... n n n E Ψ V Ψ g g U ( ( ( (... n n n E h + g g... n ote: ou lowet-ode eigenvlue ε nd eigenfunction e olution of hψ ( ε ψ ( h h + U ( nd not of the hψ ( ε ψ (. ψ ( i Pn ( i ( i, i ( l Y lm θ φ χ i µ i Wht do we need to clculte to deive HF eqution? We ledy hve the expeion fo the enegy: omliztion condition: (dil function with the me vlue of l e othonoml Vitionl pincipl δ (We intoduce Lgnge multiplie λ to ccommodte nomliztion contin.. eed to clculte:. Apply the vitionl pincipl. E h + g g ( (... n ( h, g, g. n l n l dpn l Pn l δ, nn E... n λn l, n l, nl nl nnl

11 h. Evlution of ( ( h dp P P d Pn l l ( l + Z nl + nl nl d (Rememe dil Schödinge eqution fo pticle with ngul momentum l dpn l l ( l + Z d P ( nl P + nl I n l d (integting y pt. Evlution of g nd g. Let otin the genel expeion fo the Coulom mtix element g cd nd then clculte g nd. ote: we deived it in the peviou lectue. g g d d ψ ( ψ ( ψ ( ψ ( 3 3 cd c d The function ψ e given y ψ ( P ( Y ( θ, φ nlm nl lm

12 Coulom mtix element The / cn e expnded 4π < + + > q Y * q ( θ, φ Y ( θ, φ q Thi expeion my e e-witten uing C-teno defined y 4π C ( ˆ q Yq ( θ, φ ( + q ( C ( ˆ ˆ q C q ( < + > q Coulom mtix element We now utitute the expeion fo ψ nd / c into ou mtix element nd epte d nd dω integl < q ( C ( ˆ ψ ( ( ˆ nlm Pnl ( Ylm ( θ, φ q C q + > q g d d ψ ( ψ ( ψ ( ψ ( cd 3 3 c d < d d Pn l P nl P ncl c Pn dl + d > ( ( ( ( R ( cd dil integl ( q dω Y ( θ, φ C ( θ, φ Y ( θ, φ q dω Y lm q lcmc ( θ, φ C ( θ, φ Y ( θ, φ lm q ld md l m C l m q c c l m C l m q d d

13 Coulom mtix element g R ( cd ( l m C l m l m C l m q cd q c c q d d q ext, we ue Wigne-Ect theoem fo oth of the mtix element: l m l m q cd ( ( c d q q -q g R cd l C l l C l l c m c l d m d l m l m We ue -q ( q q ote: nd q e intege l d m d l d m d Coulom mtix element l m l m cd ( ( c d q q q g R cd l C l l C l l c m c l d m d l m l m ( R ( cd l C l l C l c d + l c m c l d m d l m l m ( ( cd + c d g R cd l C l l C l l c m c l d m d 3

14 Summy: Coulom mtix element (non-eltivitic ce l m l m ( ( cd + c d g R cd l C l l C l l c m c l d m d R ( cd d d P ( P ( P ( P ( < nl nl + nclc nd ld > l C l l l l ( ( + ( + l C l ( l C l l l ote : l + + l i n even intege g. Evlution of. l m l m ( + ( l m l m g R l C l l C l Let um ove m nd µ : l m l m ( + ( m µ g R l C l l C l l m l m Sum ove µ give fcto of. + ( l R ( l C l l C l l m l m l + δ l + 4

15 g. Evlution of. µ m l + g R ( l C l l C l l + l + l + l + R ( (l + R ( l + l C l l + R ( d d P ( P ( P ( P ( nl nl nl nl > g. Evlution of. l m l m ( + l + l + g ( + R l C l l C l m µ m µ l m l m l C l ( l C l ote : l + + l i n even intege l l m + l + l + l m δ µ ( R ( l C l µ + l m µ g l C l l + R ( 5

16 Summy dpn l l ( l + Z + nl nl d ( h I( n l d P P m µ m µ g (l + R ( g l C l l + R ( E h + g g ( (... n Putting it ll togethe E... n ( h + ( g g + ( h ( g g nl mµ nl mµ nl mµ Λ l l l C l I( nl + (l + R ( R ( n m ( ( l µ nl l l + + Doe not depend on m, µ o we cn um ove thee indice y multiplying y (l + E... n (l + I( nl + (l + R ( Λl ( l R nl nl 6

17 Summy E... n (l + I( nl + (l + R ( Λl ( l R nl nl dpn l l ( l + Z + nl nl d I( n l d P P δ E λ l C l l l Λ l l ( l ( l n n,, l nl nl nl nnl Thi expeion mut e ttiony with epect to vition δ P (. nl n l n l dpn l Pn l δ, nn Some deigntion E (l + I( n l + (l + R ( Λ R (... n l l nl nl R ( d P ( d P ( dp ( (, v > v(, ote deigntion fo indice nd : index lel n oitl with nn nd ll, {n,l } now. Fo exmple,, p, < ( ( ( ( ( + > R d P P d P P v (,, d P ( P ( v (,, OTE : v (,, v (, P ( P ( n l 7

18 Ou fomul with new deigntion E... n (l + I( nl + (l + R ( Λl ( l R nl nl (l + d + + P P dp l ( l Z d + (l + P ( v(, Λl ( ( (, l P P v, δ E HF eqution fo cloed-hell hell ytem λ... n n l, n l, nl nl nnl εn l, n l λ nl, nl /(4l + ε ε λ /(4l + nl, nl nl, nl d P ( l ( l + Z + P ( P ( + d + (4l + v(, P ( Λl (,, ( l v P ε P ( + ε P ( nl, nl nl n n 8

19 v HF eqution fo cloed-hell hell ytem Exmple: He tom: / fo,, Λ ll Λ fo Line : (4l + v(, P ( Λl (,, ( l v P ( v(, P ( Λ l (,, ( (, ( (, ( l v P P P v v (, P ( d P ( Z P v P P ( (, ( ( d + ε (Jut we otined elie. HF eqution fo cloed-hell hell ytem Exmple: Be tom: / fo,, Λ ll Λ fo HF eqution fo oitl (,, Line : (4 l + v(, P ( Λl (,, ( l v P v(, P ( + v(, P ( v(,, P ( v(,, P ( v (, P ( + v (, P ( v (,, P ( d P ( Z + + v(, + v(, P ( v(,, P ( d ε P ( + ε P (, 9

20 HF eqution fo cloed-hell hell ytem Exmple: Be tom: HF eqution fo oitl (,, Line : (4 l + v(, P ( Λl (,, ( l v P v(, P ( + v(, P ( v(,, P ( v(,, P ( v (, P ( + v (, P ( v (,, P ( d P ( Z + + v(, + v(, P ( v(,, P ( d ε P ( + ε P (, ote: we cn choe off-digonl Lgnge multiplie to e zeo fo cloed-hell ytem, i.e. ε, ε,. HF potentil V HF HF potentil i defined y pecifying it ction on n ity oitl P * ( V P ( V P ( + V P ( HF * di * exc * V P ( V di * exc * ( P ( (4l + v (, P ( Diect potentil VHF ( g g * (4l + Λ v (,*, P ( Exchnge potentil l l* [ Follow fom the deivtion] Uing thi deigntion we cn e-wite ou HF eqution fo oitl of the cloed-hell ytem d P ( Z l ( l + + V HF P ( ( ε P d +

21 ( Clcultion of enegy ( ( VHF ( g g E h + U + g g U U V... n HF, h + ( g g ( VHF Hee ε + ( g g ( g g ε ( g g (l + ε (l + R ( Λl ( l R nl nl ε i otined fom the itetive olution of the HF eqution nlm µ

FI 2201 Electromagnetism

FI 2201 Electromagnetism FI 1 Electomgnetism Alexnde A. Isknd, Ph.D. Physics of Mgnetism nd Photonics Resech Goup Electosttics ELECTRIC PTENTIALS 1 Recll tht we e inteested to clculte the electic field of some chge distiution.

More information

U>, and is negative. Electric Potential Energy

U>, and is negative. Electric Potential Energy Electic Potentil Enegy Think of gvittionl potentil enegy. When the lock is moved veticlly up ginst gvity, the gvittionl foce does negtive wok (you do positive wok), nd the potentil enegy (U) inceses. When

More information

Lecture 4. Beyond the Hückel π-electron theory. Charge density is an important parameter that is used widely to explain properties of molecules.

Lecture 4. Beyond the Hückel π-electron theory. Charge density is an important parameter that is used widely to explain properties of molecules. Lectue 4. Beyond the Hückel π-electon theoy 4. Chge densities nd bond odes Chge density is n impotnt pmete tht is used widely to explin popeties of molecules. An electon in n obitl ψ = c φ hs density distibution

More information

Fourier-Bessel Expansions with Arbitrary Radial Boundaries

Fourier-Bessel Expansions with Arbitrary Radial Boundaries Applied Mthemtics,,, - doi:./m.. Pulished Online My (http://www.scirp.og/jounl/m) Astct Fouie-Bessel Expnsions with Aity Rdil Boundies Muhmmd A. Mushef P. O. Box, Jeddh, Sudi Ai E-mil: mmushef@yhoo.co.uk

More information

Electric Potential. and Equipotentials

Electric Potential. and Equipotentials Electic Potentil nd Euipotentils U Electicl Potentil Review: W wok done y foce in going fom to long pth. l d E dl F W dl F θ Δ l d E W U U U Δ Δ l d E W U U U U potentil enegy electic potentil Potentil

More information

( ) D x ( s) if r s (3) ( ) (6) ( r) = d dr D x

( ) D x ( s) if r s (3) ( ) (6) ( r) = d dr D x SIO 22B, Rudnick dpted fom Dvis III. Single vile sttistics The next few lectues e intended s eview of fundmentl sttistics. The gol is to hve us ll speking the sme lnguge s we move to moe dvnced topics.

More information

This immediately suggests an inverse-square law for a "piece" of current along the line.

This immediately suggests an inverse-square law for a piece of current along the line. Electomgnetic Theoy (EMT) Pof Rui, UNC Asheville, doctophys on YouTube Chpte T Notes The iot-svt Lw T nvese-sque Lw fo Mgnetism Compe the mgnitude of the electic field t distnce wy fom n infinite line

More information

Radial geodesics in Schwarzschild spacetime

Radial geodesics in Schwarzschild spacetime Rdil geodesics in Schwzschild spcetime Spheiclly symmetic solutions to the Einstein eqution tke the fom ds dt d dθ sin θdϕ whee is constnt. We lso hve the connection components, which now tke the fom using

More information

Chapter Direct Method of Interpolation More Examples Mechanical Engineering

Chapter Direct Method of Interpolation More Examples Mechanical Engineering Chpte 5 iect Method o Intepoltion Moe Exmples Mechnicl Engineeing Exmple Fo the pupose o shinking tunnion into hub, the eduction o dimete o tunnion sht by cooling it though tempetue chnge o is given by

More information

Physics 505 Fall 2005 Midterm Solutions. This midterm is a two hour open book, open notes exam. Do all three problems.

Physics 505 Fall 2005 Midterm Solutions. This midterm is a two hour open book, open notes exam. Do all three problems. Physics 55 Fll 5 Midtem Solutions This midtem is two hou open ook, open notes exm. Do ll thee polems. [35 pts] 1. A ectngul ox hs sides of lengths, nd c z x c [1] ) Fo the Diichlet polem in the inteio

More information

10 m, so the distance from the Sun to the Moon during a solar eclipse is. The mass of the Sun, Earth, and Moon are = =

10 m, so the distance from the Sun to the Moon during a solar eclipse is. The mass of the Sun, Earth, and Moon are = = Chpte 1 nivesl Gvittion 11 *P1. () The un-th distnce is 1.4 nd the th-moon 8 distnce is.84, so the distnce fom the un to the Moon duing sol eclipse is 11 8 11 1.4.84 = 1.4 The mss of the un, th, nd Moon

More information

Physics 1502: Lecture 2 Today s Agenda

Physics 1502: Lecture 2 Today s Agenda 1 Lectue 1 Phsics 1502: Lectue 2 Tod s Agend Announcements: Lectues posted on: www.phs.uconn.edu/~cote/ HW ssignments, solutions etc. Homewok #1: On Mstephsics this Fid Homewoks posted on Msteingphsics

More information

General Physics II. number of field lines/area. for whole surface: for continuous surface is a whole surface

General Physics II. number of field lines/area. for whole surface: for continuous surface is a whole surface Genel Physics II Chpte 3: Guss w We now wnt to quickly discuss one of the moe useful tools fo clculting the electic field, nmely Guss lw. In ode to undestnd Guss s lw, it seems we need to know the concept

More information

Lecture 10. Solution of Nonlinear Equations - II

Lecture 10. Solution of Nonlinear Equations - II Fied point Poblems Lectue Solution o Nonline Equtions - II Given unction g : R R, vlue such tht gis clled ied point o the unction g, since is unchnged when g is pplied to it. Whees with nonline eqution

More information

10 Statistical Distributions Solutions

10 Statistical Distributions Solutions Communictions Engineeing MSc - Peliminy Reding 1 Sttisticl Distiutions Solutions 1) Pove tht the vince of unifom distiution with minimum vlue nd mximum vlue ( is ) 1. The vince is the men of the sques

More information

Physics 11b Lecture #11

Physics 11b Lecture #11 Physics 11b Lectue #11 Mgnetic Fields Souces of the Mgnetic Field S&J Chpte 9, 3 Wht We Did Lst Time Mgnetic fields e simil to electic fields Only diffeence: no single mgnetic pole Loentz foce Moving chge

More information

Ch 26 - Capacitance! What s Next! Review! Lab this week!

Ch 26 - Capacitance! What s Next! Review! Lab this week! Ch 26 - Cpcitnce! Wht s Next! Cpcitnce" One week unit tht hs oth theoeticl n pcticl pplictions! Cuent & Resistnce" Moving chges, finlly!! Diect Cuent Cicuits! Pcticl pplictions of ll the stuff tht we ve

More information

Week 8. Topic 2 Properties of Logarithms

Week 8. Topic 2 Properties of Logarithms Week 8 Topic 2 Popeties of Logithms 1 Week 8 Topic 2 Popeties of Logithms Intoduction Since the esult of ithm is n eponent, we hve mny popeties of ithms tht e elted to the popeties of eponents. They e

More information

Lecture 11: Potential Gradient and Capacitor Review:

Lecture 11: Potential Gradient and Capacitor Review: Lectue 11: Potentil Gdient nd Cpcito Review: Two wys to find t ny point in spce: Sum o Integte ove chges: q 1 1 q 2 2 3 P i 1 q i i dq q 3 P 1 dq xmple of integting ove distiution: line of chge ing of

More information

1.3 Using Formulas to Solve Problems

1.3 Using Formulas to Solve Problems Section 1.3 Uing Fomul to Solve Polem 73 1.3 Uing Fomul to Solve Polem OBJECTIVES 1 Solve fo Vile in Fomul 2 Ue Fomul to Solve Polem Peping fo Fomul Befoe getting tted, tke ti edine quiz. If you get polem

More information

defined on a domain can be expanded into the Taylor series around a point a except a singular point. Also, f( z)

defined on a domain can be expanded into the Taylor series around a point a except a singular point. Also, f( z) 08 Tylo eie nd Mcluin eie A holomophic function f( z) defined on domin cn be expnded into the Tylo eie ound point except ingul point. Alo, f( z) cn be expnded into the Mcluin eie in the open dik with diu

More information

EECE 260 Electrical Circuits Prof. Mark Fowler

EECE 260 Electrical Circuits Prof. Mark Fowler EECE 60 Electicl Cicuits Pof. Mk Fowle Complex Numbe Review /6 Complex Numbes Complex numbes ise s oots of polynomils. Definition of imginy # nd some esulting popeties: ( ( )( ) )( ) Recll tht the solution

More information

Physics 604 Problem Set 1 Due Sept 16, 2010

Physics 604 Problem Set 1 Due Sept 16, 2010 Physics 64 Polem et 1 Due ept 16 1 1) ) Inside good conducto the electic field is eo (electons in the conducto ecuse they e fee to move move in wy to cncel ny electic field impessed on the conducto inside

More information

Energy Dissipation Gravitational Potential Energy Power

Energy Dissipation Gravitational Potential Energy Power Lectue 4 Chpte 8 Physics I 0.8.03 negy Dissiption Gvittionl Potentil negy Powe Couse wesite: http://fculty.uml.edu/andiy_dnylov/teching/physicsi Lectue Cptue: http://echo360.uml.edu/dnylov03/physicsfll.html

More information

Discrete Model Parametrization

Discrete Model Parametrization Poceedings of Intentionl cientific Confeence of FME ession 4: Automtion Contol nd Applied Infomtics Ppe 9 Discete Model Pmetition NOKIEVIČ, Pet Doc,Ing,Cc Deptment of Contol ystems nd Instumenttion, Fculty

More information

Math 4318 : Real Analysis II Mid-Term Exam 1 14 February 2013

Math 4318 : Real Analysis II Mid-Term Exam 1 14 February 2013 Mth 4318 : Rel Anlysis II Mid-Tem Exm 1 14 Febuy 2013 Nme: Definitions: Tue/Flse: Poofs: 1. 2. 3. 4. 5. 6. Totl: Definitions nd Sttements of Theoems 1. (2 points) Fo function f(x) defined on (, b) nd fo

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

9.4 The response of equilibrium to temperature (continued)

9.4 The response of equilibrium to temperature (continued) 9.4 The esponse of equilibium to tempetue (continued) In the lst lectue, we studied how the chemicl equilibium esponds to the vition of pessue nd tempetue. At the end, we deived the vn t off eqution: d

More information

Quantum Mechanics Qualifying Exam - August 2016 Notes and Instructions

Quantum Mechanics Qualifying Exam - August 2016 Notes and Instructions Quntum Mechnics Qulifying Exm - August 016 Notes nd Instructions There re 6 problems. Attempt them ll s prtil credit will be given. Write on only one side of the pper for your solutions. Write your lis

More information

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007 School of Electicl nd Compute Engineeing, Conell Univesity ECE 303: Electomgnetic Fields nd Wves Fll 007 Homewok 4 Due on Sep. 1, 007 by 5:00 PM Reding Assignments: i) Review the lectue notes. ii) Relevnt

More information

The Area of a Triangle

The Area of a Triangle The e of Tingle tkhlid June 1, 015 1 Intodution In this tile we will e disussing the vious methods used fo detemining the e of tingle. Let [X] denote the e of X. Using se nd Height To stt off, the simplest

More information

7.5-Determinants in Two Variables

7.5-Determinants in Two Variables 7.-eteminnts in Two Vibles efinition of eteminnt The deteminnt of sque mti is el numbe ssocited with the mti. Eve sque mti hs deteminnt. The deteminnt of mti is the single ent of the mti. The deteminnt

More information

Example 2: ( ) 2. $ s ' 9.11" 10 *31 kg ( )( 1" 10 *10 m) ( e)

Example 2: ( ) 2. $ s ' 9.11 10 *31 kg ( )( 1 10 *10 m) ( e) Emple 1: Two point chge e locted on the i, q 1 = e t = 0 nd q 2 = e t =.. Find the wok tht mut be done b n etenl foce to bing thid point chge q 3 = e fom infinit to = 2. b. Find the totl potentil eneg

More information

Chapter 2: Electric Field

Chapter 2: Electric Field P 6 Genel Phsics II Lectue Outline. The Definition of lectic ield. lectic ield Lines 3. The lectic ield Due to Point Chges 4. The lectic ield Due to Continuous Chge Distibutions 5. The oce on Chges in

More information

Language Processors F29LP2, Lecture 5

Language Processors F29LP2, Lecture 5 Lnguge Pocessos F29LP2, Lectue 5 Jmie Gy Feuy 2, 2014 1 / 1 Nondeteministic Finite Automt (NFA) NFA genelise deteministic finite utomt (DFA). They llow sevel (0, 1, o moe thn 1) outgoing tnsitions with

More information

3.1 Magnetic Fields. Oersted and Ampere

3.1 Magnetic Fields. Oersted and Ampere 3.1 Mgnetic Fields Oested nd Ampee The definition of mgnetic induction, B Fields of smll loop (dipole) Mgnetic fields in mtte: ) feomgnetism ) mgnetiztion, (M ) c) mgnetic susceptiility, m d) mgnetic field,

More information

Solutions to Midterm Physics 201

Solutions to Midterm Physics 201 Solutions to Midtem Physics. We cn conside this sitution s supeposition of unifomly chged sphee of chge density ρ nd dius R, nd second unifomly chged sphee of chge density ρ nd dius R t the position of

More information

Answers to test yourself questions

Answers to test yourself questions Answes to test youself questions opic Descibing fields Gm Gm Gm Gm he net field t is: g ( d / ) ( 4d / ) d d Gm Gm Gm Gm Gm Gm b he net potentil t is: V d / 4d / d 4d d d V e 4 7 9 49 J kg 7 7 Gm d b E

More information

u(r, θ) = 1 + 3a r n=1

u(r, θ) = 1 + 3a r n=1 Mth 45 / AMCS 55. etuck Assignment 8 ue Tuesdy, Apil, 6 Topics fo this week Convegence of Fouie seies; Lplce s eqution nd hmonic functions: bsic popeties, computions on ectngles nd cubes Fouie!, Poisson

More information

SPA7010U/SPA7010P: THE GALAXY. Solutions for Coursework 1. Questions distributed on: 25 January 2018.

SPA7010U/SPA7010P: THE GALAXY. Solutions for Coursework 1. Questions distributed on: 25 January 2018. SPA7U/SPA7P: THE GALAXY Solutions fo Cousewok Questions distibuted on: 25 Jnuy 28. Solution. Assessed question] We e told tht this is fint glxy, so essentilly we hve to ty to clssify it bsed on its spectl

More information

Friedmannien equations

Friedmannien equations ..6 Fiedmnnien equtions FLRW metic is : ds c The metic intevl is: dt ( t) d ( ) hee f ( ) is function which detemines globl geometic l popety of D spce. f d sin d One cn put it in the Einstein equtions

More information

dx was area under f ( x ) if ( ) 0

dx was area under f ( x ) if ( ) 0 13. Line Integls Line integls e simil to single integl, f ( x) dx ws e unde f ( x ) if ( ) 0 Insted of integting ove n intevl [, ] (, ) f xy ds f x., we integte ove cuve, (in the xy-plne). **Figue - get

More information

Chapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s:

Chapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s: Chpte 7 Kleene s Theoem 7.1 Kleene s Theoem The following theoem is the most impotnt nd fundmentl esult in the theoy of FA s: Theoem 6 Any lnguge tht cn e defined y eithe egul expession, o finite utomt,

More information

EE Control Systems LECTURE 8

EE Control Systems LECTURE 8 Coyright F.L. Lewi 999 All right reerved Udted: Sundy, Ferury, 999 EE 44 - Control Sytem LECTURE 8 REALIZATION AND CANONICAL FORMS A liner time-invrint (LTI) ytem cn e rereented in mny wy, including: differentil

More information

Homework 3 MAE 118C Problems 2, 5, 7, 10, 14, 15, 18, 23, 30, 31 from Chapter 5, Lamarsh & Baratta. The flux for a point source is:

Homework 3 MAE 118C Problems 2, 5, 7, 10, 14, 15, 18, 23, 30, 31 from Chapter 5, Lamarsh & Baratta. The flux for a point source is: . Homewok 3 MAE 8C Poblems, 5, 7, 0, 4, 5, 8, 3, 30, 3 fom Chpte 5, msh & Btt Point souces emit nuetons/sec t points,,, n 3 fin the flux cuent hlf wy between one sie of the tingle (blck ot). The flux fo

More information

Electricity & Magnetism Lecture 6: Electric Potential

Electricity & Magnetism Lecture 6: Electric Potential Electicity & Mgnetism Lectue 6: Electic Potentil Tody s Concept: Electic Potenl (Defined in tems of Pth Integl of Electic Field) Electicity & Mgnesm Lectue 6, Slide Stuff you sked bout:! Explin moe why

More information

Physics 2A Chapter 10 - Moment of Inertia Fall 2018

Physics 2A Chapter 10 - Moment of Inertia Fall 2018 Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.

More information

Previously. Extensions to backstepping controller designs. Tracking using backstepping Suppose we consider the general system

Previously. Extensions to backstepping controller designs. Tracking using backstepping Suppose we consider the general system 436-459 Advnced contol nd utomtion Extensions to bckstepping contolle designs Tcking Obseves (nonline dmping) Peviously Lst lectue we looked t designing nonline contolles using the bckstepping technique

More information

DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING FLUID MECHANICS III Solutions to Problem Sheet 3

DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING FLUID MECHANICS III Solutions to Problem Sheet 3 DEPATMENT OF CIVIL AND ENVIONMENTAL ENGINEEING FLID MECHANICS III Solutions to Poblem Sheet 3 1. An tmospheic vote is moelle s combintion of viscous coe otting s soli boy with ngul velocity Ω n n iottionl

More information

10.3 The Quadratic Formula

10.3 The Quadratic Formula . Te Qudti Fomul We mentioned in te lst setion tt ompleting te sque n e used to solve ny qudti eqution. So we n use it to solve 0. We poeed s follows 0 0 Te lst line of tis we ll te qudti fomul. Te Qudti

More information

(a) Counter-Clockwise (b) Clockwise ()N (c) No rotation (d) Not enough information

(a) Counter-Clockwise (b) Clockwise ()N (c) No rotation (d) Not enough information m m m00 kg dult, m0 kg bby. he seesw stts fom est. Which diection will it ottes? ( Counte-Clockwise (b Clockwise ( (c o ottion ti (d ot enough infomtion Effect of Constnt et oque.3 A constnt non-zeo toque

More information

E-Companion: Mathematical Proofs

E-Companion: Mathematical Proofs E-omnon: Mthemtcl Poo Poo o emm : Pt DS Sytem y denton o t ey to vey tht t ncee n wth d ncee n We dene } ] : [ { M whee / We let the ttegy et o ech etle n DS e ]} [ ] [ : { M w whee M lge otve nume oth

More information

Course Updates. Reminders: 1) Assignment #8 available. 2) Chapter 28 this week.

Course Updates. Reminders: 1) Assignment #8 available. 2) Chapter 28 this week. Couse Updtes http://www.phys.hwii.edu/~vne/phys7-sp1/physics7.html Remindes: 1) Assignment #8 vilble ) Chpte 8 this week Lectue 3 iot-svt s Lw (Continued) θ d θ P R R θ R d θ d Mgnetic Fields fom long

More information

Problem Set 10 Solutions

Problem Set 10 Solutions Chemisty 6 D. Jean M. Standad Poblem Set 0 Solutions. Give the explicit fom of the Hamiltonian opeato (in atomic units) fo the lithium atom. You expession should not include any summations (expand them

More information

Physics 505 Homework No. 9 Solutions S9-1

Physics 505 Homework No. 9 Solutions S9-1 Physics 505 Homewok No 9 s S9-1 1 As pomised, hee is the tick fo summing the matix elements fo the Stak effect fo the gound state of the hydogen atom Recall, we need to calculate the coection to the gound

More information

PHYS 2421 Fields and Waves

PHYS 2421 Fields and Waves PHYS 242 Felds nd Wves Instucto: Joge A. López Offce: PSCI 29 A, Phone: 747-7528 Textook: Unvesty Physcs e, Young nd Feedmn 23. Electc potentl enegy 23.2 Electc potentl 23.3 Clcultng electc potentl 23.4

More information

General Physics (PHY 2140)

General Physics (PHY 2140) Genel Physics (PHY 40) Lightning Review Lectue 3 Electosttics Lst lectue:. Flux. Guss s s lw. simplifies computtion of electic fields Q Φ net Ecosθ ε o Electicl enegy potentil diffeence nd electic potentil

More information

Section 35 SHM and Circular Motion

Section 35 SHM and Circular Motion Section 35 SHM nd Cicul Motion Phsics 204A Clss Notes Wht do objects do? nd Wh do the do it? Objects sometimes oscillte in simple hmonic motion. In the lst section we looed t mss ibting t the end of sping.

More information

New Approach to Many-Body-QED Calculations: Merging Quantum-Electro- Dynamics with Many-Body Perturbation

New Approach to Many-Body-QED Calculations: Merging Quantum-Electro- Dynamics with Many-Body Perturbation 1 New Appoch to Mny-Body-QED Clcultions: Meging Quntum-Electo- Dynmics with Mny-Body Petubtion Ingv Lindgen, Sten Slomonson, nd Dniel Hedendhl 1. Intoduction Abstct: A new method fo bound-stte QED clcultions

More information

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10 University of Wshington Deprtment of Chemistry Chemistry 45 Winter Qurter Homework Assignment 4; Due t 5p.m. on // We lerned tht the Hmiltonin for the quntized hrmonic oscilltor is ˆ d κ H. You cn obtin

More information

Lecture 7: Angular Momentum, Hydrogen Atom

Lecture 7: Angular Momentum, Hydrogen Atom Lectue 7: Angula Momentum, Hydogen Atom Vecto Quantization of Angula Momentum and Nomalization of 3D Rigid Roto wavefunctions Conside l, so L 2 2 2. Thus, we have L 2. Thee ae thee possibilities fo L z

More information

1.2 Differential cross section

1.2 Differential cross section .2. DIFFERENTIAL CROSS SECTION Febuay 9, 205 Lectue VIII.2 Diffeential coss section We found that the solution to the Schodinge equation has the fom e ik x ψ 2π 3/2 fk, k + e ik x and that fk, k = 2 m

More information

2.Decision Theory of Dependence

2.Decision Theory of Dependence .Deciio Theoy of Depedece Theoy :I et of vecto if thee i uet which i liely depedet the whole et i liely depedet too. Coolly :If the et i liely idepedet y oepty uet of it i liely idepedet. Theoy : Give

More information

ELECTROSTATICS. 4πε0. E dr. The electric field is along the direction where the potential decreases at the maximum rate. 5. Electric Potential Energy:

ELECTROSTATICS. 4πε0. E dr. The electric field is along the direction where the potential decreases at the maximum rate. 5. Electric Potential Energy: LCTROSTATICS. Quntiztion of Chge: Any chged body, big o smll, hs totl chge which is n integl multile of e, i.e. = ± ne, whee n is n intege hving vlues,, etc, e is the chge of electon which is eul to.6

More information

Chapter #3 EEE Subsea Control and Communication Systems

Chapter #3 EEE Subsea Control and Communication Systems EEE 87 Chter #3 EEE 87 Sube Cotrol d Commuictio Sytem Cloed loo ytem Stedy tte error PID cotrol Other cotroller Chter 3 /3 EEE 87 Itroductio The geerl form for CL ytem: C R ', where ' c ' H or Oe Loo (OL)

More information

Chapter #5 EEE Control Systems

Chapter #5 EEE Control Systems Sprig EEE Chpter #5 EEE Cotrol Sytem Deig Bed o Root Locu Chpter / Sprig EEE Deig Bed Root Locu Led Cotrol (equivlet to PD cotrol) Ued whe the tedy tte propertie of the ytem re ok but there i poor performce,

More information

THEORY OF EQUATIONS OBJECTIVE PROBLEMS. If the eqution x 6x 0 0 ) - ) 4) -. If the sum of two oots of the eqution k is -48 ) 6 ) 48 4) 4. If the poduct of two oots of 4 ) -4 ) 4) - 4. If one oot of is

More information

Prerna Tower, Road No 2, Contractors Area, Bistupur, Jamshedpur , Tel (0657) ,

Prerna Tower, Road No 2, Contractors Area, Bistupur, Jamshedpur , Tel (0657) , R Pen Towe Rod No Conttos Ae Bistupu Jmshedpu 8 Tel (67)89 www.penlsses.om IIT JEE themtis Ppe II PART III ATHEATICS SECTION I (Totl ks : ) (Single Coet Answe Type) This setion ontins 8 multiple hoie questions.

More information

Quality control. Final exam: 2012/1/12 (Thur), 9:00-12:00 Q1 Q2 Q3 Q4 Q5 YOUR NAME

Quality control. Final exam: 2012/1/12 (Thur), 9:00-12:00 Q1 Q2 Q3 Q4 Q5 YOUR NAME Qulity contol Finl exm: // (Thu), 9:-: Q Q Q3 Q4 Q5 YOUR NAME NOTE: Plese wite down the deivtion of you nswe vey clely fo ll questions. The scoe will be educed when you only wite nswe. Also, the scoe will

More information

Chapter 9 Many Electron Atoms

Chapter 9 Many Electron Atoms Chem 356: Introductory Quntum Mechnics Chpter 9 Mny Electron Atoms... 11 MnyElectron Atoms... 11 A: HrtreeFock: Minimize the Energy of Single Slter Determinnt.... 16 HrtreeFock Itertion Scheme... 17 Chpter

More information

π,π is the angle FROM a! TO b

π,π is the angle FROM a! TO b Mth 151: 1.2 The Dot Poduct We hve scled vectos (o, multiplied vectos y el nume clled scl) nd dded vectos (in ectngul component fom). Cn we multiply vectos togethe? The nswe is YES! In fct, thee e two

More information

9.4. The Vector Product. Introduction. Prerequisites. Learning Outcomes

9.4. The Vector Product. Introduction. Prerequisites. Learning Outcomes The Vector Product 9.4 Introduction In this section we descrie how to find the vector product of two vectors. Like the sclr product its definition my seem strnge when first met ut the definition is chosen

More information

STD: XI MATHEMATICS Total Marks: 90. I Choose the correct answer: ( 20 x 1 = 20 ) a) x = 1 b) x =2 c) x = 3 d) x = 0

STD: XI MATHEMATICS Total Marks: 90. I Choose the correct answer: ( 20 x 1 = 20 ) a) x = 1 b) x =2 c) x = 3 d) x = 0 STD: XI MATHEMATICS Totl Mks: 90 Time: ½ Hs I Choose the coect nswe: ( 0 = 0 ). The solution of is ) = b) = c) = d) = 0. Given tht the vlue of thid ode deteminnt is then the vlue of the deteminnt fomed

More information

Math 259 Winter Solutions to Homework #9

Math 259 Winter Solutions to Homework #9 Mth 59 Winter 9 Solutions to Homework #9 Prolems from Pges 658-659 (Section.8). Given f(, y, z) = + y + z nd the constrint g(, y, z) = + y + z =, the three equtions tht we get y setting up the Lgrnge multiplier

More information

Topics for Review for Final Exam in Calculus 16A

Topics for Review for Final Exam in Calculus 16A Topics fo Review fo Finl Em in Clculus 16A Instucto: Zvezdelin Stnkov Contents 1. Definitions 1. Theoems nd Poblem Solving Techniques 1 3. Eecises to Review 5 4. Chet Sheet 5 1. Definitions Undestnd the

More information

Chapter 25 Electric Potential

Chapter 25 Electric Potential Chpte 5 lectic Potentil consevtive foces -> potentil enegy - Wht is consevtive foce? lectic potentil = U / : the potentil enegy U pe unit chge is function of the position in spce Gol:. estblish the eltionship

More information

= ΔW a b. U 1 r m 1 + K 2

= ΔW a b. U 1 r m 1 + K 2 Chpite 3 Potentiel électiue [18 u 3 mi] DEVOIR : 31, 316, 354, 361, 35 Le potentiel électiue est le tvil p unité de chge (en J/C, ou volt) Ce concept est donc utile dns les polèmes de consevtion d énegie

More information

Important design issues and engineering applications of SDOF system Frequency response Functions

Important design issues and engineering applications of SDOF system Frequency response Functions Impotnt design issues nd engineeing pplictions of SDOF system Fequency esponse Functions The following desciptions show typicl questions elted to the design nd dynmic pefomnce of second-ode mechnicl system

More information

3.23 Electrical, Optical, and Magnetic Properties of Materials

3.23 Electrical, Optical, and Magnetic Properties of Materials MIT OpenCouseWae http://ocw.mit.edu 3.23 Electical, Optical, and Magnetic Popeties of Mateials Fall 27 Fo infomation about citing these mateials o ou Tems of Use, visit: http://ocw.mit.edu/tems. 3.23 Fall

More information

Elastic scattering of 4 He atoms at the surface of liquid helium

Elastic scattering of 4 He atoms at the surface of liquid helium Indin Jounl of Pue & Applied Physics Vol. 48, Octobe, pp. 743-748 Elstic sctteing of 4 He toms t the sufce of liquid helium P K Toongey, K M Khnn, Y K Ayodo, W T Skw, F G Knyeki, R T Eki, R N Kimengichi

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

set is not closed under matrix [ multiplication, ] and does not form a group.

set is not closed under matrix [ multiplication, ] and does not form a group. Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed

More information

Satellite Orbits. Orbital Mechanics. Circular Satellite Orbits

Satellite Orbits. Orbital Mechanics. Circular Satellite Orbits Obitl Mechnic tellite Obit Let u tt by king the quetion, Wht keep tellite in n obit ound eth?. Why doen t tellite go diectly towd th, nd why doen t it ecpe th? The nwe i tht thee e two min foce tht ct

More information

Introductions to ArithmeticGeometricMean

Introductions to ArithmeticGeometricMean Intoductions to AitheticGeoeticMen Intoduction to the Aithetic-Geoetic Men Genel The ithetic-geoetic en eed in the woks of J Lnden (77, 775) nd J-L Lgnge (784-785) who defined it though the following quite-ntul

More information

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007 School of Electicl nd Compute Engineeing, Conell Univesity ECE 303: Electomgnetic Fields nd Wves Fll 007 Homewok 3 Due on Sep. 14, 007 by 5:00 PM Reding Assignments: i) Review the lectue notes. ii) Relevnt

More information

Physics 215 Quantum Mechanics 1 Assignment 2

Physics 215 Quantum Mechanics 1 Assignment 2 Physics 15 Quntum Mechnics 1 Assignment Logn A. Morrison Jnury, 16 Problem 1 Clculte p nd p on the Gussin wve pcket α whose wve function is x α = 1 ikx x 1/4 d 1 Solution Recll tht where ψx = x ψ. Additionlly,

More information

CHAPTER 25 ELECTRIC POTENTIAL

CHAPTER 25 ELECTRIC POTENTIAL CHPTE 5 ELECTIC POTENTIL Potential Diffeence and Electic Potential Conside a chaged paticle of chage in a egion of an electic field E. This filed exets an electic foce on the paticle given by F=E. When

More information

Mark Scheme (Results) January 2008

Mark Scheme (Results) January 2008 Mk Scheme (Results) Jnuy 00 GCE GCE Mthemtics (6679/0) Edecel Limited. Registeed in Englnd nd Wles No. 4496750 Registeed Office: One90 High Holbon, London WCV 7BH Jnuy 00 6679 Mechnics M Mk Scheme Question

More information

Collection of Formulas

Collection of Formulas Collection of Fomuls Electomgnetic Fields EITF8 Deptment of Electicl nd Infomtion Technology Lund Univesity, Sweden August 8 / ELECTOSTATICS field point '' ' Oigin ' Souce point Coulomb s Lw The foce F

More information

Continuous Charge Distributions

Continuous Charge Distributions Continuous Chge Distibutions Review Wht if we hve distibution of chge? ˆ Q chge of distibution. Q dq element of chge. d contibution to due to dq. Cn wite dq = ρ dv; ρ is the chge density. = 1 4πε 0 qi

More information

Math 2142 Homework 2 Solutions. Problem 1. Prove the following formulas for Laplace transforms for s > 0. a s 2 + a 2 L{cos at} = e st.

Math 2142 Homework 2 Solutions. Problem 1. Prove the following formulas for Laplace transforms for s > 0. a s 2 + a 2 L{cos at} = e st. Mth 2142 Homework 2 Solution Problem 1. Prove the following formul for Lplce trnform for >. L{1} = 1 L{t} = 1 2 L{in t} = 2 + 2 L{co t} = 2 + 2 Solution. For the firt Lplce trnform, we need to clculte:

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

22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 20

22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 20 .615, MHD Theoy of Fusion Systes Pof. Feideg Lectue Resistive Wll Mode 1. We hve seen tht pefectly conducting wll, plced in close poxiity to the pls cn hve stong stilizing effect on extenl kink odes..

More information

Optimization. x = 22 corresponds to local maximum by second derivative test

Optimization. x = 22 corresponds to local maximum by second derivative test Optimiztion Lectue 17 discussed the exteme vlues of functions. This lectue will pply the lesson fom Lectue 17 to wod poblems. In this section, it is impotnt to emembe we e in Clculus I nd e deling one-vible

More information

2. Elementary Linear Algebra Problems

2. Elementary Linear Algebra Problems . Eleety e lge Pole. BS: B e lge Suoute (Pog pge wth PCK) Su of veto opoet:. Coputto y f- poe: () () () (3) N 3 4 5 3 6 4 7 8 Full y tee Depth te tep log()n Veto updte the f- poe wth N : ) ( ) ( ) ( )

More information

PHYSICS 211 MIDTERM I 22 October 2003

PHYSICS 211 MIDTERM I 22 October 2003 PHYSICS MIDTERM I October 3 Exm i cloed book, cloed note. Ue onl our formul heet. Write ll work nd nwer in exm booklet. The bck of pge will not be grded unle ou o requet on the front of the pge. Show ll

More information

Class Summary. be functions and f( D) , we define the composition of f with g, denoted g f by

Class Summary. be functions and f( D) , we define the composition of f with g, denoted g f by Clss Summy.5 Eponentil Functions.6 Invese Functions nd Logithms A function f is ule tht ssigns to ech element D ectly one element, clled f( ), in. Fo emple : function not function Given functions f, g:

More information

CHAPTER 18: ELECTRIC CHARGE AND ELECTRIC FIELD

CHAPTER 18: ELECTRIC CHARGE AND ELECTRIC FIELD ollege Physics Student s Mnul hpte 8 HAPTR 8: LTRI HARG AD LTRI ILD 8. STATI LTRIITY AD HARG: OSRVATIO O HARG. ommon sttic electicity involves chges nging fom nnocoulombs to micocoulombs. () How mny electons

More information

Artificial Intelligence Markov Decision Problems

Artificial Intelligence Markov Decision Problems rtificil Intelligence Mrkov eciion Problem ilon - briefly mentioned in hpter Ruell nd orvig - hpter 7 Mrkov eciion Problem; pge of Mrkov eciion Problem; pge of exmple: probbilitic blockworld ction outcome

More information

A Hartree-Fock Example Using Helium

A Hartree-Fock Example Using Helium Univesity of Connecticut DigitalCommons@UConn Chemisty Education Mateials Depatment of Chemisty June 6 A Hatee-Fock Example Using Helium Cal W. David Univesity of Connecticut, Cal.David@uconn.edu Follow

More information