u x A j Stress in the Ocean
|
|
- Angela Harrell
- 6 years ago
- Views:
Transcription
1 Strss in t Ocan T tratmnt of strss and strain in fluids is comlicatd and somwat bond t sco of tis class. Tos rall intrstd sould look into tis rtr in Batclor Introduction to luid Dnamics givn as a rfrnc at t bginning of t class. T scaling u of turbulnt strsss to ocanic scals is vn mor sotric. Hr w will mrl summari wat is known about t form of t bod forc for ocanic flows. It involvs a tnsor wit tr comonnts of strss for ac of t tr momntum quations. [ [ [ T notation is tat t surscrit rfrs to t strss dirction of t comonnt and t subscrit rfrs to t drivativ of t comonnt. T rlationsi of strss to strain fluid rsons) can b comlicatd. or fluids lik watr and for scals tical of tos w will b discussing in tis class it can tak on t following form: i j A j u i j wr A j is a constant known as t dd viscosit. or fluids tat ar sallow rlativ to tir widts ts tr assumd constants A A A ) ar not t sam: t vrtical comonnt is muc smallr tan t two oriontal comonnts wic ar usuall not alwas) assumd to b qual. If w insrt tis dfinition into t abov quation for t vctor bod forc w can gt t following rlationsi wr A A A ) [ A u A u A u [ A v A v A v [ A w A w A w
2 Our dnamical quations av alrad incororatd on of ts bod forcs into tm: it is t tird comonnt of ac of t abov sar strsss. Considr t vrtical intgral of on of ts tr forcs sa t first on). 0 d d[ d d H 0) H ) T aroimation is bcaus w ar taking H to b a constant dt lvl and assuming t surfac to b flat or it s variation to b small comard to t dt of t ocan). Considr now t last two trms in t abov. T last on is roortional to t strss on t bottom and t rcding on is du to t strss on t to or fr surfac. If w writ t bottom strss as u H ) A H ) ru w s tat w can now associat tis trm wit t siml bottom friction w av rtaind from our ucks on ic modl. It is tis trm tat las an imortant rol in t Stomml modl for t wstrn intnsification of t wind-drivn grs. As for t wind driving tis is t just t surfac comonnt of t strss 0). or t Svrdru balanc w obtaind t following quation βv ) or β dv d 0) wr w av again ignord satial variations of H st t strss at t bottom to ro and ignord t oriontal comonnts of t strss as wll. As ou will rcall tis was t assumtion inrnt in t Svrdru balanc tat w could ignor bottom strss or t frictional trm roortional to r. T abov statmnt of t Svrdru balanc is mor gnral tat t rvious on bcaus w av licitl includd t surfac wind strss and w av NOT mad t assumtion tat t vlocit v is constant wit dt. So t Svrdru balanc can b alid to t vrtical intgral of wind-drivn currnts including t baroclinic vlocit) and subjct to som rquirmnts on t lvlnss of t ocan dt is a usl dscrition of wind-drivn frictionlss ct for t surfac lar) flow. Not tat it dos
3 includ t wind driving and trfor t dnamics of t surfac Ekman lar. Gostroic and Ekman comonnts of t Svrdru Circulation T Svrdru balanc is a vrtical intgral of t simlifid otntial vorticit quation. Lt s st back now and look at wat vlocitis mak u tis balanc. W will b using t mor gnral quations for baroclinic flow r wic ou will rcall ar t following: du dt ru dv dt 0 g u v w 0 d dt rv u v t w W could now insrt t form of t bod forc drivd abov and w would av t ll turbulnt quations usd in larg-scal sical ocanogra. But r w ar intrstd in t Svrdru balanc wic is for a stad linarid ocan wit wind forcing but otrwis no licit friction. So t abov bcom
4 0 0 w v u g Suos w writ t vlocit comonnts as t sum of a gostroic art and an Ekman art wit surscrits dnoting ts. T first two quations bcom g g W av alrad lookd at t gostroic balanc. Now w look rtr at t Ekman balanc intgrating t Ekman balanc btwn t fr surfac 0) and a dt at t bas of t Ekman lar -Z ). f Z w v u d f dv f du 0) ) 0) 0) In t first two quations abov w s tat t intgratd Ekman transort is to t rigt of t alid wind strss. In t tird w s tat tr is a
5 oriontal divrgnc of t nt Ekman transort wic will roduc a vrtical vlocit at t bottom of t Ekman lar t vrtical vlocit at t surfac bing ro). Ts rssions ar not valid at t quator wr t Corilios aramtr vaniss! T vrtical vlocit is suc tat tr is sinking wit subtroical grs and uwlling witin subolar grs irrsctiv of misr. Now if w know t intgratd Ekman transort and w also know t vrticall intgratd Svrdru transort tn t vrticall intgratd gostroic transort must just b t diffrnc btwn t Svrdru and Ekman transorts sinc b dfinition v s v g v. Som invstigators av usd tis constraint to l dtrmin t unknown rfrnc lvl vlocit for gostroic motion but tis is onl as good as t assumtions bind it. In fact ou will b amining tis in on of our nt omwork roblms. T fact tat tr ar larg diffrnc btwn t actual and Svrdru volum transorts in t Gulf Stram 50 vs. 30 Sv. rsctivl) sould mak on aroac tis mtod wit caution! W will b sing tis diffrnc latr. Munk s gr vs. Stomml s gr Munk 950) roducd anotr modl of t N. Atlantic wind-drivn circulation and laind its wstward intnsification using a diffrnt friction modl tan Stomml. Instad of vrtical friction as containd in t frictional aramtr r oosing t motion Munk usd a oriontal dd viscosit. W will not go into tis modl as it also rquird a ositiv β to av t Gulf Stram strongr on t wstrn boundar. W will just writ down t linar vorticit balanc wit bot frictional trms. A ψ ψ ) r ψ ψ ) βψ [ 0) 0) Hr w av rlacd t bod forc wit t wind strss at t surfac and t stramnction is now t vrtical intgral of t rvious stramnction: dv ψ & du ψ
6 Munk solvd t abov nglcting t bottom friction trm roortional to r) but rtaining oriontal friction roortional to A ) Bcaus tr ar now four -drivitivs to t stramnction t wstrn boundar currnt is vn sarr and mor intns tan Stomml s wic ad 2 -drivativs. As w indicatd arlir t intrior Svrdru Balanc olds in t intrior of t basin but in t boundar nar t wstrn boundar a diffrnt dnamical balanc must b rsnt. W can dtrmin t widt of t wstrn boundar currnt in ac of t two modls vr asil witout actuall solving t roblm. In ac modl w av t following aroimat balanc nar t wstrn boundar wr tings cang raidl in t -dirction: rψ A ψ βψ βψ 0 s 0 r Stomml' s wbc widt β m A β / 3 Munk' s wbc widt Tim-dndnt & non-linar numrical modls of t ocan circulation using itr bottom or latral friction must b abl to rsolv t structur of t wbc. T abov satial scals for t two ts of boundar currnts bcom imortant minimum scals rquird for numrical ocan modls. Tis comlts t tra nots on t wind-drivn circulation. urtr discussion will cntr around t ttbook until w mov on to otr arts of t ocan.
Physics 43 HW #9 Chapter 40 Key
Pysics 43 HW #9 Captr 4 Ky Captr 4 1 Aftr many ours of dilignt rsarc, you obtain t following data on t potolctric ffct for a crtain matrial: Wavlngt of Ligt (nm) Stopping Potntial (V) 36 3 4 14 31 a) Plot
More informationTrigonometric functions
Robrto s Nots on Diffrntial Calculus Captr 5: Drivativs of transcndntal functions Sction 5 Drivativs of Trigonomtric functions Wat you nd to know alrady: Basic trigonomtric limits, t dfinition of drivativ,
More informationAP Calculus BC AP Exam Problems Chapters 1 3
AP Eam Problms Captrs Prcalculus Rviw. If f is a continuous function dfind for all ral numbrs and if t maimum valu of f() is 5 and t minimum valu of f() is 7, tn wic of t following must b tru? I. T maimum
More informationDivision of Mechanics Lund University MULTIBODY DYNAMICS. Examination Name (write in block letters):.
Division of Mchanics Lund Univrsity MULTIBODY DYNMICS Examination 7033 Nam (writ in block lttrs):. Id.-numbr: Writtn xamination with fiv tasks. Plas chck that all tasks ar includd. clan copy of th solutions
More information3-2-1 ANN Architecture
ARTIFICIAL NEURAL NETWORKS (ANNs) Profssor Tom Fomby Dpartmnt of Economics Soutrn Mtodist Univrsity Marc 008 Artificial Nural Ntworks (raftr ANNs) can b usd for itr prdiction or classification problms.
More informationFinite Element Models for Steady Flows of Viscous Incompressible Fluids
Finit Elmnt Modls for Stad Flows of Viscous Incomprssibl Fluids Rad: Chaptr 10 JN Rdd CONTENTS Govrning Equations of Flows of Incomprssibl Fluids Mid (Vlocit-Prssur) Finit Elmnt Modl Pnalt Function Mthod
More informationExponential Functions
Eponntial Functions Dinition: An Eponntial Function is an unction tat as t orm a, wr a > 0. T numbr a is calld t bas. Eampl: Lt i.. at intgrs. It is clar wat t unction mans or som valus o. 0 0,,, 8,,.,.
More informationSection 11.6: Directional Derivatives and the Gradient Vector
Sction.6: Dirctional Drivativs and th Gradint Vctor Practic HW rom Stwart Ttbook not to hand in p. 778 # -4 p. 799 # 4-5 7 9 9 35 37 odd Th Dirctional Drivativ Rcall that a b Slop o th tangnt lin to th
More informationdy 1. If fx ( ) is continuous at x = 3, then 13. If y x ) for x 0, then f (g(x)) = g (f (x)) when x = a. ½ b. ½ c. 1 b. 4x a. 3 b. 3 c.
AP CALCULUS BC SUMMER ASSIGNMENT DO NOT SHOW YOUR WORK ON THIS! Complt ts problms during t last two wks of August. SHOW ALL WORK. Know ow to do ALL of ts problms, so do tm wll. Itms markd wit a * dnot
More informationCompton Scattering. There are three related processes. Thomson scattering (classical) Rayleigh scattering (coherent)
Comton Scattring Tr ar tr rlatd rocsss Tomson scattring (classical) Poton-lctron Comton scattring (QED) Poton-lctron Raylig scattring (cornt) Poton-atom Tomson and Raylig scattring ar lasticonly t dirction
More informationLagrangian Analysis of a Class of Quadratic Liénard-Type Oscillator Equations with Exponential-Type Restoring Force function
agrangian Analysis of a Class of Quadratic iénard-ty Oscillator Equations wit Eonntial-Ty Rstoring Forc function J. Akand, D. K. K. Adjaï,.. Koudaoun,Y. J. F. Komaou,. D. onsia. Dartmnt of Pysics, Univrsity
More informationCase Study 1 PHA 5127 Fall 2006 Revised 9/19/06
Cas Study Qustion. A 3 yar old, 5 kg patint was brougt in for surgry and was givn a /kg iv bolus injction of a muscl rlaxant. T plasma concntrations wr masurd post injction and notd in t tabl blow: Tim
More informationSundials and Linear Algebra
Sundials and Linar Algbra M. Scot Swan July 2, 25 Most txts on crating sundials ar dirctd towards thos who ar solly intrstd in making and using sundials and usually assums minimal mathmatical background.
More informationEinstein Equations for Tetrad Fields
Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for
More informationMassachusetts Institute of Technology Department of Mechanical Engineering
Massachustts Institut of Tchnolog Dpartmnt of Mchanical Enginring. Introduction to Robotics Mid-Trm Eamination Novmbr, 005 :0 pm 4:0 pm Clos-Book. Two shts of nots ar allowd. Show how ou arrivd at our
More informationAS 5850 Finite Element Analysis
AS 5850 Finit Elmnt Analysis Two-Dimnsional Linar Elasticity Instructor Prof. IIT Madras Equations of Plan Elasticity - 1 displacmnt fild strain- displacmnt rlations (infinitsimal strain) in matrix form
More informationPARTICLE MOTION IN UNIFORM GRAVITATIONAL and ELECTRIC FIELDS
VISUAL PHYSICS ONLINE MODULE 6 ELECTROMAGNETISM PARTICLE MOTION IN UNIFORM GRAVITATIONAL and ELECTRIC FIELDS A fram of rfrnc Obsrvr Origin O(,, ) Cartsian coordinat as (X, Y, Z) Unit vctors iˆˆj k ˆ Scif
More informationUnit 6: Solving Exponential Equations and More
Habrman MTH 111 Sction II: Eonntial and Logarithmic Functions Unit 6: Solving Eonntial Equations and Mor EXAMPLE: Solv th quation 10 100 for. Obtain an act solution. This quation is so asy to solv that
More informationDifferential Equations
Prfac Hr ar m onlin nots for m diffrntial quations cours that I tach hr at Lamar Univrsit. Dspit th fact that ths ar m class nots, th should b accssibl to anon wanting to larn how to solv diffrntial quations
More informationAP Calculus Multiple-Choice Question Collection
AP Calculus Multipl-Coic Qustion Collction 985 998 . f is a continuous function dfind for all ral numbrs and if t maimum valu of f () is 5 and t minimum valu of f () is 7, tn wic of t following must b
More informationMAT 270 Test 3 Review (Spring 2012) Test on April 11 in PSA 21 Section 3.7 Implicit Derivative
MAT 7 Tst Rviw (Spring ) Tst on April in PSA Sction.7 Implicit Drivativ Rmmbr: Equation of t tangnt lin troug t point ( ab, ) aving slop m is y b m( a ). dy Find t drivativ y d. y y. y y y. y 4. y sin(
More informationNEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA
NE APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA Mirca I CÎRNU Ph Dp o Mathmatics III Faculty o Applid Scincs Univrsity Polithnica o Bucharst Cirnumirca @yahoocom Abstract In a rcnt papr [] 5 th indinit intgrals
More informationDerivation of Electron-Electron Interaction Terms in the Multi-Electron Hamiltonian
Drivation of Elctron-Elctron Intraction Trms in th Multi-Elctron Hamiltonian Erica Smith Octobr 1, 010 1 Introduction Th Hamiltonian for a multi-lctron atom with n lctrons is drivd by Itoh (1965) by accounting
More informationDual Nature of Matter and Radiation
Higr Ordr Tinking Skill Qustions Dual Natur of Mattr and Radiation 1. Two bas on of rd ligt and otr of blu ligt of t sa intnsity ar incidnt on a tallic surfac to it otolctrons wic on of t two bas its lctrons
More informationDIFFERENTIAL EQUATION
MD DIFFERENTIAL EQUATION Sllabus : Ordinar diffrntial quations, thir ordr and dgr. Formation of diffrntial quations. Solution of diffrntial quations b th mthod of sparation of variabls, solution of homognous
More information6. The Interaction of Light and Matter
6. Th Intraction of Light and Mattr - Th intraction of light and mattr is what maks lif intrsting. - Light causs mattr to vibrat. Mattr in turn mits light, which intrfrs with th original light. - Excitd
More informationLecture Outline. Skin Depth Power Flow 8/7/2018. EE 4347 Applied Electromagnetics. Topic 3e
8/7/018 Cours Instructor Dr. Raymond C. Rumpf Offic: A 337 Phon: (915) 747 6958 E Mail: rcrumpf@utp.du EE 4347 Applid Elctromagntics Topic 3 Skin Dpth & Powr Flow Skin Dpth Ths & Powr nots Flow may contain
More informationThinking outside the (Edgeworth) Box
Tinking outsid t (dgwort) ox by Jon G. Rily Dartmnt of conomics UCL 0 Novmbr 008 To dvlo an undrstanding of Walrasian quilibrium allocations, conomists tyically start wit t two rson, two-commodity xcang
More informationPipe flow friction, small vs. big pipes
Friction actor (t/0 t o pip) Friction small vs larg pips J. Chaurtt May 016 It is an intrsting act that riction is highr in small pips than largr pips or th sam vlocity o low and th sam lngth. Friction
More informationHydrogen Atom and One Electron Ions
Hydrogn Atom and On Elctron Ions Th Schrödingr quation for this two-body problm starts out th sam as th gnral two-body Schrödingr quation. First w sparat out th motion of th cntr of mass. Th intrnal potntial
More informationMinimum Spanning Trees
Yufi Tao ITEE Univrsity of Qunslan In tis lctur, w will stuy anotr classic prolm: finin a minimum spannin tr of an unirct wit rap. Intrstinly, vn tou t prolm appars ratr iffrnt from SSSP (sinl sourc sortst
More informationHigher order derivatives
Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of
More informationLecture 37 (Schrödinger Equation) Physics Spring 2018 Douglas Fields
Lctur 37 (Schrödingr Equation) Physics 6-01 Spring 018 Douglas Filds Rducd Mass OK, so th Bohr modl of th atom givs nrgy lvls: E n 1 k m n 4 But, this has on problm it was dvlopd assuming th acclration
More informationCOHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.
MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function
More informationBrief Introduction to Statistical Mechanics
Brif Introduction to Statistical Mchanics. Purpos: Ths nots ar intndd to provid a vry quick introduction to Statistical Mchanics. Th fild is of cours far mor vast than could b containd in ths fw pags.
More informationu 3 = u 3 (x 1, x 2, x 3 )
Lctur 23: Curvilinar Coordinats (RHB 8.0 It is oftn convnint to work with variabls othr than th Cartsian coordinats x i ( = x, y, z. For xampl in Lctur 5 w mt sphrical polar and cylindrical polar coordinats.
More informationBasic Polyhedral theory
Basic Polyhdral thory Th st P = { A b} is calld a polyhdron. Lmma 1. Eithr th systm A = b, b 0, 0 has a solution or thr is a vctorπ such that π A 0, πb < 0 Thr cass, if solution in top row dos not ist
More information1997 AP Calculus AB: Section I, Part A
997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. (4 6 ) d= 4 6 6
More informationLarge Systems (Section 2.4)
Larg Sstms (Sction.4) Rad Schrodr sction.4 about small numbrs, larg numbrs (i.. 10 3 ), and 3 10 VERY larg numbrs (i.. 10 ) (Mak sur ou can do Probs..1,.13) o Eonntial of a vr larg numbr is a larg numbr
More informationDifferential Equations
UNIT I Diffrntial Equations.0 INTRODUCTION W li in a world of intrrlatd changing ntitis. Th locit of a falling bod changs with distanc, th position of th arth changs with tim, th ara of a circl changs
More informationu x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula
7. Intgration by Parts Each drivativ formula givs ris to a corrsponding intgral formula, as w v sn many tims. Th drivativ product rul yilds a vry usful intgration tchniqu calld intgration by parts. Starting
More informationINTEGRATION BY PARTS
Mathmatics Rvision Guids Intgration by Parts Pag of 7 MK HOME TUITION Mathmatics Rvision Guids Lvl: AS / A Lvl AQA : C Edcl: C OCR: C OCR MEI: C INTEGRATION BY PARTS Vrsion : Dat: --5 Eampls - 6 ar copyrightd
More informationMath 34A. Final Review
Math A Final Rviw 1) Us th graph of y10 to find approimat valus: a) 50 0. b) y (0.65) solution for part a) first writ an quation: 50 0. now tak th logarithm of both sids: log() log(50 0. ) pand th right
More informationGeneral Notes About 2007 AP Physics Scoring Guidelines
AP PHYSICS C: ELECTRICITY AND MAGNETISM 2007 SCORING GUIDELINES Gnral Nots About 2007 AP Physics Scoring Guidlins 1. Th solutions contain th most common mthod of solving th fr-rspons qustions and th allocation
More informationλ = 2L n Electronic structure of metals = 3 = 2a Free electron model Many metals have an unpaired s-electron that is largely free
5.6 4 Lctur #4-6 pag Elctronic structur of mtals r lctron modl Many mtals av an unpaird s-lctron tat is largly fr Simplst modl: Particl in a box! or a cubic box of lngt L, ψ ( xyz) 8 xπ ny L L L n x π
More informationPrelab Lecture Chmy 374 Thur., March 22, 2018 Edited 22mar18, 21mar18
Prlab Lctur Cmy 374 Tur., Marc, 08 Editd mar8, mar8 LA REPORT:From t ClassicalTrmoISub-7.pdf andout: Was not a dry lab A partially complt spradst was postd on wb Not ruird 3 If solid is pur X Partial
More informationCollisions between electrons and ions
DRAFT 1 Collisions btwn lctrons and ions Flix I. Parra Rudolf Pirls Cntr for Thortical Physics, Unirsity of Oxford, Oxford OX1 NP, UK This rsion is of 8 May 217 1. Introduction Th Fokkr-Planck collision
More informationCharacteristics of a Terrain-Following Sigma Coordinate
ATMOSPERIC AND OCEANIC SCIENCE LETTERS, 0, VOL. 4, NO., 576 Caractristics of a Trrain-Following Sigma Coordinat LI Yi-Yuan, WANG Bin,, WANG Dong-ai Stat K Laborator of Numrical Modling for Atmospric Scincs
More informationThat is, we start with a general matrix: And end with a simpler matrix:
DIAGON ALIZATION OF THE STR ESS TEN SOR INTRO DUCTIO N By th us of Cauchy s thorm w ar abl to rduc th numbr of strss componnts in th strss tnsor to only nin valus. An additional simplification of th strss
More informationMCE503: Modeling and Simulation of Mechatronic Systems Discussion on Bond Graph Sign Conventions for Electrical Systems
MCE503: Modling and Simulation o Mchatronic Systms Discussion on Bond Graph Sign Convntions or Elctrical Systms Hanz ichtr, PhD Clvland Stat Univrsity, Dpt o Mchanical Enginring 1 Basic Assumption In a
More informationExam 1. It is important that you clearly show your work and mark the final answer clearly, closed book, closed notes, no calculator.
Exam N a m : _ S O L U T I O N P U I D : I n s t r u c t i o n s : It is important that you clarly show your work and mark th final answr clarly, closd book, closd nots, no calculator. T i m : h o u r
More information6.1 Integration by Parts and Present Value. Copyright Cengage Learning. All rights reserved.
6.1 Intgration by Parts and Prsnt Valu Copyright Cngag Larning. All rights rsrvd. Warm-Up: Find f () 1. F() = ln(+1). F() = 3 3. F() =. F() = ln ( 1) 5. F() = 6. F() = - Objctivs, Day #1 Studnts will b
More information2008 AP Calculus BC Multiple Choice Exam
008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl
More information1997 AP Calculus AB: Section I, Part A
997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs x for which f (x) is a ral numbr.. (4x 6 x) dx=
More informationUniversity of Washington Department of Chemistry Chemistry 453 Winter Quarter 2014 Lecture 20: Transition State Theory. ERD: 25.14
Univrsity of Wasinton Dpartmnt of Cmistry Cmistry 453 Wintr Quartr 04 Lctur 0: Transition Stat Tory. ERD: 5.4. Transition Stat Tory Transition Stat Tory (TST) or ctivatd Complx Tory (CT) is a raction mcanism
More informationThomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Unit C Algbra Trigonomtr Gomtr Calculus Vctor gomtr Pag Algbra Molus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv
More informationAtomic Physics. Final Mon. May 12, 12:25-2:25, Ingraham B10 Get prepared for the Final!
# SCORES 50 40 30 0 10 MTE 3 Rsults P08 Exam 3 0 30 40 50 60 70 80 90 100 SCORE Avrag 79.75/100 std 1.30/100 A 19.9% AB 0.8% B 6.3% BC 17.4% C 13.1% D.1% F 0.4% Final Mon. Ma 1, 1:5-:5, Ingraam B10 Gt
More information1.2 Faraday s law A changing magnetic field induces an electric field. Their relation is given by:
Elctromagntic Induction. Lorntz forc on moving charg Point charg moving at vlocity v, F qv B () For a sction of lctric currnt I in a thin wir dl is Idl, th forc is df Idl B () Elctromotiv forc f s any
More informationu r du = ur+1 r + 1 du = ln u + C u sin u du = cos u + C cos u du = sin u + C sec u tan u du = sec u + C e u du = e u + C
Tchniqus of Intgration c Donald Kridr and Dwight Lahr In this sction w ar going to introduc th first approachs to valuating an indfinit intgral whos intgrand dos not hav an immdiat antidrivativ. W bgin
More informationNote If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark.
. (a) Eithr y = or ( 0, ) (b) Whn =, y = ( 0 + ) = 0 = 0 ( + ) = 0 ( )( ) = 0 Eithr = (for possibly abov) or = A 3. Not If th candidat blivs that = 0 solvs to = 0 or givs an tra solution of = 0, thn withhold
More informationA Propagating Wave Packet Group Velocity Dispersion
Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to
More informationThe van der Waals interaction 1 D. E. Soper 2 University of Oregon 20 April 2012
Th van dr Waals intraction D. E. Sopr 2 Univrsity of Orgon 20 pril 202 Th van dr Waals intraction is discussd in Chaptr 5 of J. J. Sakurai, Modrn Quantum Mchanics. Hr I tak a look at it in a littl mor
More information22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches.
Subjct Chmistry Papr No and Titl Modul No and Titl Modul Tag 8/ Physical Spctroscopy / Brakdown of th Born-Oppnhimr approximation. Slction ruls for rotational-vibrational transitions. P, R branchs. CHE_P8_M
More informationThe Transmission Line Wave Equation
1//5 Th Transmission Lin Wav Equation.doc 1/6 Th Transmission Lin Wav Equation Q: So, what functions I (z) and V (z) do satisfy both tlgraphr s quations?? A: To mak this asir, w will combin th tlgraphr
More informationThe Quantum Efficiency and Thermal Emittance of Metal Cathodes
T Quantum fficincy and Trmal mittanc of Mtal Catods David H. Dowll Tory Sminar Jun, 6 I. Introduction II. Q and Trmal mittanc Tory III. Ral World Issus Surfac Rougnss Masurmnts Diamond Turning vs. Polising
More informationFinite element discretization of Laplace and Poisson equations
Finit lmnt discrtization of Laplac and Poisson quations Yashwanth Tummala Tutor: Prof S.Mittal 1 Outlin Finit Elmnt Mthod for 1D Introduction to Poisson s and Laplac s Equations Finit Elmnt Mthod for 2D-Discrtization
More informationEngineering 323 Beautiful HW #13 Page 1 of 6 Brown Problem 5-12
Enginring Bautiful HW #1 Pag 1 of 6 5.1 Two componnts of a minicomputr hav th following joint pdf for thir usful liftims X and Y: = x(1+ x and y othrwis a. What is th probability that th liftim X of th
More informationCHAPTER 1. Introductory Concepts Elements of Vector Analysis Newton s Laws Units The basis of Newtonian Mechanics D Alembert s Principle
CHPTER 1 Introductory Concpts Elmnts of Vctor nalysis Nwton s Laws Units Th basis of Nwtonian Mchanics D lmbrt s Principl 1 Scinc of Mchanics: It is concrnd with th motion of matrial bodis. odis hav diffrnt
More informationMor Tutorial at www.dumblittldoctor.com Work th problms without a calculator, but us a calculator to chck rsults. And try diffrntiating your answrs in part III as a usful chck. I. Applications of Intgration
More informationsurface of a dielectric-metal interface. It is commonly used today for discovering the ways in
Surfac plasmon rsonanc is snsitiv mchanism for obsrving slight changs nar th surfac of a dilctric-mtal intrfac. It is commonl usd toda for discovring th was in which protins intract with thir nvironmnt,
More informationQuasi-Classical States of the Simple Harmonic Oscillator
Quasi-Classical Stats of th Simpl Harmonic Oscillator (Draft Vrsion) Introduction: Why Look for Eignstats of th Annihilation Oprator? Excpt for th ground stat, th corrspondnc btwn th quantum nrgy ignstats
More informationBifurcation Theory. , a stationary point, depends on the value of α. At certain values
Dnamic Macroconomic Thor Prof. Thomas Lux Bifurcation Thor Bifurcation: qualitativ chang in th natur of th solution occurs if a paramtr passs through a critical point bifurcation or branch valu. Local
More informationChapter Taylor Theorem Revisited
Captr 0.07 Taylor Torm Rvisitd Atr radig tis captr, you sould b abl to. udrstad t basics o Taylor s torm,. writ trascdtal ad trigoomtric uctios as Taylor s polyomial,. us Taylor s torm to id t valus o
More informationis an appropriate single phase forced convection heat transfer coefficient (e.g. Weisman), and h
For t BWR oprating paramtrs givn blow, comput and plot: a) T clad surfac tmpratur assuming t Jns-Lotts Corrlation b) T clad surfac tmpratur assuming t Tom Corrlation c) T clad surfac tmpratur assuming
More informationAnswer Homework 5 PHA5127 Fall 1999 Jeff Stark
Answr omwork 5 PA527 Fall 999 Jff Stark A patint is bing tratd with Drug X in a clinical stting. Upon admiion, an IV bolus dos of 000mg was givn which yildd an initial concntration of 5.56 µg/ml. A fw
More informationThomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Cor rvision gui Pag Algbra Moulus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv blow th ais in th ais. f () f () f
More informationDifferentiation of Exponential Functions
Calculus Modul C Diffrntiation of Eponntial Functions Copyright This publication Th Northrn Albrta Institut of Tchnology 007. All Rights Rsrvd. LAST REVISED March, 009 Introduction to Diffrntiation of
More informationCharacteristics of Gliding Arc Discharge Plasma
Caractristics of Gliding Arc Discarg Plasma Lin Li( ), Wu Bin(, Yang Ci(, Wu Cngkang ( Institut of Mcanics, Cins Acadmy of Scincs, Bijing 8, Cina E-mail: linli@imc.ac.cn Abstract A gliding arc discarg
More information10. The Discrete-Time Fourier Transform (DTFT)
Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w
More informationThe Frequency Response of a Quarter-Wave Matching Network
4/1/29 Th Frquncy Rsons o a Quartr 1/9 Th Frquncy Rsons o a Quartr-Wav Matchg Ntwork Q: You hav onc aga rovidd us with conusg and rhas uslss ormation. Th quartr-wav matchg ntwork has an xact SFG o: a Τ
More informationEquations of motion - summary
MATH 2240 Wk 8 Summary Equations of motion - summary Starting from Nwton s Laws w av sown ΣF a m du 1 dp ru + fv + dt ρo d ρ dv 1 dp y ru fu + dt ρo dy ρ dw 1 dp + g + coriolis + friction dt ρo dz du dv
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
More informationRESPONSE OF DUFFING OSCILLATOR UNDER NARROW-BAND RANDOM EXCITATION
Th rd Intrnational Confrnc on Comutational Mchanics and Virtual Enginring COMEC 9 9 OCTOBER 9, Brasov, Romania RESPONSE O DUING OSCILLATOR UNDER NARROW-BAND RANDOM EXCITATION Ptr STAN, Mtallurgical High
More informationPartial Derivatives: Suppose that z = f(x, y) is a function of two variables.
Chaptr Functions o Two Variabls Applid Calculus 61 Sction : Calculus o Functions o Two Variabls Now that ou hav som amiliarit with unctions o two variabls it s tim to start appling calculus to hlp us solv
More informationFourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.
Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform
More informationOn the Hamiltonian of a Multi-Electron Atom
On th Hamiltonian of a Multi-Elctron Atom Austn Gronr Drxl Univrsity Philadlphia, PA Octobr 29, 2010 1 Introduction In this papr, w will xhibit th procss of achiving th Hamiltonian for an lctron gas. Making
More informationare given in the table below. t (hours)
CALCULUS WORKSHEET ON INTEGRATION WITH DATA Work th following on notbook papr. Giv dcimal answrs corrct to thr dcimal placs.. A tank contains gallons of oil at tim t = hours. Oil is bing pumpd into th
More information1973 AP Calculus AB: Section I
97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=
More information4037 ADDITIONAL MATHEMATICS
CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Ordinary Lvl MARK SCHEME for th Octobr/Novmbr 0 sris 40 ADDITIONAL MATHEMATICS 40/ Papr, maimum raw mark 80 This mark schm is publishd as an aid to tachrs and candidats,
More informationCalculus Revision A2 Level
alculus Rvision A Lvl Tabl of drivativs a n sin cos tan d an sc n cos sin Fro AS * NB sc cos sc cos hain rul othrwis known as th function of a function or coposit rul. d d Eapl (i) (ii) Obtain th drivativ
More informationPHA 5127 Answers Homework 2 Fall 2001
PH 5127 nswrs Homwork 2 Fall 2001 OK, bfor you rad th answrs, many of you spnt a lot of tim on this homwork. Plas, nxt tim if you hav qustions plas com talk/ask us. Thr is no nd to suffr (wll a littl suffring
More informationComplex Powers and Logs (5A) Young Won Lim 10/17/13
Complx Powrs and Logs (5A) Copyright (c) 202, 203 Young W. Lim. Prmission is grantd to copy, distribut and/or modify this documnt undr th trms of th GNU Fr Documntation Licns, Vrsion.2 or any latr vrsion
More informationExercise 1. Sketch the graph of the following function. (x 2
Writtn tst: Fbruary 9th, 06 Exrcis. Sktch th graph of th following function fx = x + x, spcifying: domain, possibl asymptots, monotonicity, continuity, local and global maxima or minima, and non-drivability
More informationSec 2.3 Modeling with First Order Equations
Sc.3 Modling with First Ordr Equations Mathmatical modls charactriz physical systms, oftn using diffrntial quations. Modl Construction: Translating physical situation into mathmatical trms. Clarly stat
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More informationReview of Exponentials and Logarithms - Classwork
Rviw of Eponntials and Logarithms - Classwork In our stud of calculus, w hav amind drivativs and intgrals of polnomial prssions, rational prssions, and trignomtric prssions. What w hav not amind ar ponntial
More informationMathematics. Complex Number rectangular form. Quadratic equation. Quadratic equation. Complex number Functions: sinusoids. Differentiation Integration
Mathmatics Compl numbr Functions: sinusoids Sin function, cosin function Diffrntiation Intgration Quadratic quation Quadratic quations: a b c 0 Solution: b b 4ac a Eampl: 1 0 a= b=- c=1 4 1 1or 1 1 Quadratic
More informationElements of Statistical Thermodynamics
24 Elmnts of Statistical Thrmodynamics Statistical thrmodynamics is a branch of knowldg that has its own postulats and tchniqus. W do not attmpt to giv hr vn an introduction to th fild. In this chaptr,
More informationMid Year Examination F.4 Mathematics Module 1 (Calculus & Statistics) Suggested Solutions
Mid Ya Eamination 3 F. Matmatics Modul (Calculus & Statistics) Suggstd Solutions Ma pp-: 3 maks - Ma pp- fo ac qustion: mak. - Sam typ of pp- would not b countd twic fom wol pap. - In any cas, no pp maks
More informationPHYSICS 489/1489 LECTURE 7: QUANTUM ELECTRODYNAMICS
PHYSICS 489/489 LECTURE 7: QUANTUM ELECTRODYNAMICS REMINDER Problm st du today 700 in Box F TODAY: W invstigatd th Dirac quation it dscribs a rlativistic spin /2 particl implis th xistnc of antiparticl
More information