LAPLACE S EQUATION IN SPHERICAL COORDINATES. With Applications to Electrodynamics

Size: px
Start display at page:

Download "LAPLACE S EQUATION IN SPHERICAL COORDINATES. With Applications to Electrodynamics"

Transcription

1 LALACE S EQUATION IN SHERICAL COORDINATES With Appitions to Eetodynmis We hve seen tht Lpe s eqution is one of the most signifint equtions in physis. It is the soution to pobems in wide viety of fieds inuding themodynmis nd eetodynmis. In you ees s physis students nd sientists, you wi enounte this eqution in viety of ontets. It is impotnt to know how to sove Lpe s eqution in vious oodinte systems. The oodinte systems you wi enounte most fequenty e Ctesin, yindi nd sphei po. We investigted Lpe s eqution in Ctesin oodintes in ss nd just begn investigting its soution in sphei oodintes. Let s epnd tht disussion hee. We begin with Lpe s eqution: ( We n wite the Lpin in sphei oodintes s: ( + (sinθ + ( sinθ sin θ φ ( whee θ is the po nge mesued down fom the noth poe, nd φ is the zimuth nge, nogous to ongitude in eth mesuing oodintes. (In tems of eth mesuing oodintes, the po nge is 9 minus the titude, often temed the otitude. To mke ou initi utions itte simpe, et s ssume zimuth symmety; tht mens tht ou pmete does not vy in the φ dietion. In othe wods, / φ, so we n wite the Lpin in ( bit moe simpy. Assuming zimuth symmety, eq. ( beomes: ( + (sinθ sinθ ( This is the fom of Lpe s eqution we hve to sove if we wnt to find the eeti potenti in sphei oodintes. Fist, et s ppy the method of sepbe vibes to this eqution to obtin gene soution of Lpe s eqution, nd then we wi use ou gene soution to sove few diffeent pobems. To sove Lpe s eqution in sphei oodintes, we wite:

2 Fist Step: The Ti Soution ( + (sinθ sinθ (4 The fist step in soving pti diffeenti equtions using sepbe vibes is to ssume soution of the fom: R( Θ( θ ( whee R( is funtion ony of, nd Θ(θ is funtion ony of θ. This mens tht we n set: ( ( θ R Θ ; R( ( θ Θ (6 Substituting the etionships in (6 into (4 podues: Θ( ( θ R ( R ( + (sinθ Θ ( θ sinθ (7 If we mutipy eh tem in (7 by nd then divide eh tem by R( Θ(θ, we obtin: d d ( R ( + (sinθ Θ ( θ R( d Θ( θ sinθ dθ (8 Notie tht the deivtes in (8 e no onge pti deivtives. This is beuse the method of sepbe vibes hs podued two tems; one is soey funtion of nd the othe is soey funtion of θ. Seond Step: Septing ibes Eqution (8 ows us to septe Lpe s eqution into two septe odiny diffeenti equtions; one being funtion of nd the othe funtion of θ. As we hve disussed in ss, we eize tht eh tem on the ight hnd side of (8 is equ to onstnt. This mens we n septe (8 into: d ( R( d d R ( ( + nd (sinθ Θ ( θ ( + Θ( θ sinθ dθ (9 We now hve two diffeent odiny diffeenti equtions whih we wi sove. We eize tht the podut of soutions wi ow us to use eq. ( (ong with ppopite boundy onditions to detemine the soution to Lpe s eqution. You my wonde we we hoose to wite the seption onstnt s something s non-obvious s (+.

3 The eson is tht we hve ed hed in the sipt, nd know tht by witing the seption onstnt in this wy we wi podue we known diffeenti eqution whose soution we edy know. Notie tht seption onstnt is positive in one eqution (the di pt nd negtive in the othe (the ngu pt; this is neessy so tht the sum of equtions is zeo s equied by Lpe s eqution. The di eqution Let s stt by soving the di eqution of eq. (9. We mutipy though by R( nd epnd the deivte to find: d R dr + ( + R ( d d This is fiy simpe empe of Fobenius (see diffeenti eqution. This is so n empe of n Eue (o Cuhy diffeenti eqution. See fo moe detis bout soving Eue s eqution. Using eithe the method of Fobenius o methods of Eue s equtions, we n find the soution to eqution (: ( + R ( A + B ( whee A nd B e onstnts whih wi be detemined one we ppy speifi boundy equtions. The ngu eqution We sove the ngu potion of eqution (9 by mutipying though by Θ(θ nd epnding the deivtive to obtin: d Θ osθ dθ + + ( + Θ dθ sinθ dθ ( This is tuy diffeenti eqution you e vey fmii with, though pehps not in this et fom. Refe bk to the soutions fo the vey fist homewok set of the semeste, nd eview gin the soutions to pobems 4 nd. You wi see tht the eqution you hve deived in ( is just the we known Legende eqution. We know tht the soutions to the Legende eqution e the Legende poynomis, (os θ. Thid Step: Constuting the ompete soution

4 Hving septed Lpe s eqution into two odiny diffeenti equtions, we n use the esuts bove to substitute into eq. ( to eize tht the gene soution to Lpe s eqution in sphei oodintes wi be onstuted of sum of soutions of the fom: ( + (, θ ( A + B ( Fom ou epeiene with Lpe s eqution in Ctesin oodintes, we know tht the fu soution wi be onstuted by tking sum of soutions of the fom of (; in othe wods, ou gene soution to Lpe s eqution in sphei oodintes is: ( + (, θ ( A + B (4 Now, we need e boundy onditions to detemine the vues of the oeffiients A nd BB. Appying Boundy Conditions Fist Empe (Bos pp Let s see how we n use (4 s the stting point to detemine soution to Lpe s eqution with speifi boundy onditions. Fo this pupose, et s use the empe in Bos pp Without ny oss of mening, we n use tk bout finding the potenti inside sphee the thn the tempetue inside sphee. So, et s ssume thee is sphee of dius, nd the potenti of the uppe hf of the sphee is kept t onstnt +, nd the potenti of the owe hf of the sphee is hed t. How n we detemine the potenti t ny point inside the sphee? Fist, et s wite the boundy onditions s:, < θ < π / o < osθ <, π / < θ < π o < osθ < Remembe tht the noth poe of the sphee oesponds to θ, nd θ π/ in the equtoi pne. Now, et s ook bit moe osey t (4. We e sked to find the potenti t ny point inside the sphee. This egime inudes, of ouse, the point, nd we n ook t (4 nd eize tht the soution diveges t uness BB. Appying the neessity fo meningfu physi soution to this pobem ows us to set oeffiients B B to zeo, so tht (4 simpifies to: (, θ A (

5 Now, we use the boundy ondition fo the sufe of the sphee. When, we know tht in the uppe hf sphee nd in the owe hf sphee. This mens we n wite ( s: (, θ A fo < osθ < (6 The epession in (6 shoud ook fmii to us: we e seeking to wite funtion (in this se the funtion equs the onstnt in tems of n infinite seies. We hve seen how to do this using both Fouie seies nd Legende oynomis. We know tht ou funtion n be epnded in seies if nd ony if we n epnd tht funtion in tems of ompete set of othogon funtions. Fouie seies e possibe beuse sin nd os epesent ompete set of othogon funtions on (-π, π; epnsion in tems of Legende poynomis is possibe sine we hve ened tht Legende poynomis e ompete set of othogon funtions on (-,. Thus, we n epnd ny funtion f( on (-, s: whee the oeffiients, e detemined by: f ( ( (7 + f ( ( d (8 We n see tht eqution (7 ppies to eq. (6 with f(, nd A, o A (9 A we hve to do now is detemine the vues of the oeffiients fom (8, substitute these vues into (9 nd then use those vues of A in ( to detemine the ompete soution to the potenti inside the sphee. We n detemine seve of the oeffiients esiy by diet integtion; in ft this is done in Bos on p. 8. Using these Legende oeffiients with f( nd substituting into (6 we obtin n epiit epnsion of ou soution fo (, θ: 7 (, θ [ ( ] ( 4 6 nd you n epnd the vious Legende poynomis epiity in tems of osθ if you wish, but thee is ey no need to go beyond the epession s it is witten in (.

6 Seond Empe Conside sphee of dius tht hs potenti on its sufe given by: (, θ os θ ( nd we e sked to find the potenti t points eteio to the sphee. We go bk to eq. (4 nd begin to ppy boundy onditions. Fist, we eize tht A must go to zeo sine n get vey ge, owing us to simpify (4 s: ( + (, θ B ( Now, we ppy the boundy ondition ( nd obtin: (, ( + θ B os θ ( This is just nothe fom of eq. (7. Hee, the funtion f( is os θ, nd the -(+ oeffiient BB stnds in the pe of. So, ou tsk now is vey fmii: ompute the oeffiients using (8, use these to detemine the vues of B B, nd substitute these vues of BB into ( to find ou ompete soution. Let s begin by finding the oeffiients. We n set os θ; sine θ vies fom to π then vies fom - to, whih is vey onvenient in uting Legende oeffiients sine the Legende poynomis e ompete, othogon set on (-,. With this substitution, we wi ute ou oeffiients fom: + + f ( ( d ( d ( + d (4 The fin integ on the ight is petty esy to do; Legende poynomis e, we, poynomis, nd mutipying them by just podues nothe poynomi whih is esy to integte between these imits. But et s think bit moe nd mke ou ives even esie. We e tht Legende poynomis e even funtions fo even vues of, nd e odd funtions fo odd vues of. This mens tht fo odd vnish sine the integnd in (4 beomes the podut of n even funtion ( nd n odd funtion ( ( fo n odd. This mens the integnd in (4 is odd wheneve is odd, nd the integ of n odd funtion between imits symmeti with espet to the oigin vnishes. Let s ompute oeffiients:

7 ( d d ( ( d ( d You wi find tht highe inde oeffiients vnish; does it mke sense tht this funtion is epessibe in tems of ony ( nd (? Thee e ony two tems whih wi ontibute to the seies epnsion of, nmey the nd tems. We emembe fom befoe tht we use ou vues of to find the vues of BB tht substitute bk into (; eq. ( tes us tht: B ; B, + so B (6 Using these vues of BB in ou gene soution ( gives us the ompete nswe to this pobem: (, θ B + B [( + ( ] (7 Thid Empe Let s sy now tht we wnt to find the potenti outside sphee of dius whose sufe is hed t potenti given by os(θ. We know tht sine we e deing with eteio points ou soution wi be of the fom of eq. (, nd tht we wi hve to find the oeffiients BB. The poess we foow is identi to the empe immeditey bove, eept now f( os(θ the thn os θ. We sw in the empe bove how we oud simpify ou utions by eizing we oud set os θ; we woud ike to epess ou uent f( in tems of os θ, but we wi hve to do itte tig nd geb mnipution to ompish this. Let s stt by witing os(θ os(θ+θ. We now epnd this s: os(θ + θ os θ osθ sin θ sinθ (os θ sin θ osθ sin osθ[ {os θ ( os θ } ( os θ ] 4os θ osθ whee os θ. θ osθ 4 (8 Now it is fiy stightfowd tsk to find the neessy oeffiients to sove ou pobem. We foow the empe of eq. (4, now with f( (4, nd sove fo :

8 ( (4 ( (4 d d 8 ( [ (4 7 ( (4 7 d d These esuts te us tht: 8 ; 4 4 B B nd we use the fom of (4 to wite the fin nswe s: ] (os 8( (os ( [ (os (os, ( 4 4 θ θ θ θ θ B B + +

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

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

More information

Orbits and Kepler s Laws

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

More information

Adaptive Control of a Production and Maintenance System with Unknown Deterioration and Obsolescence Rates

Adaptive Control of a Production and Maintenance System with Unknown Deterioration and Obsolescence Rates Int J of Mthemtic Sciences nd Appictions, Vo, No 3, Septembe Copyight Mind Rede Pubictions wwwjounshubcom Adptive Conto of Poduction nd Mintennce System with Unknown Deteiotion nd Obsoescence Rtes Fwzy

More information

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

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

More information

r (1+cos(θ)) sin(θ) C θ 2 r cos θ 2

r (1+cos(θ)) sin(θ) C θ 2 r cos θ 2 icles xmple 66: Rounding one ssume we hve cone of ngle θ, nd we ound it off with cuve of dius, how f wy fom the cone does the ound stt? nd wht is the chod length? (1+cos(θ)) sin(θ) θ 2 cos θ 2 xmple 67:

More information

2.016 Hydrodynamics Prof. A.H. Techet

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

More information

Angles and Triangles

Angles and Triangles nges nd Tringes n nge is formed when two rys hve ommon strting point or vertex. The mesure of n nge is given in degrees, with ompete revoution representing 360 degrees. Some fmiir nges inude nother fmiir

More information

32. The Tangency Problem of Apollonius.

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

More information

UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

More information

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

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

More information

Mechanics 1: Work, Power and Kinetic Energy

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

More information

A Note on Risky Bond Valuation

A Note on Risky Bond Valuation A Note on Risky Bond Valuation C. H. Hui Banking Poliy Depatment Hong Kong Monetay Authoity 0th Floo,, Gaden Road, Hong Kong Email: Cho-Hoi_Hui@hkma.gov.hk C. F. Lo Physis Depatment The Chinese Univesity

More information

Continuous Compounding and Annualization

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

More information

Mechanics 1: Motion in a Central Force Field

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

More information

Basic Principles of Homing Guidance

Basic Principles of Homing Guidance si Piniples of Homing Guidne Neil F Plumbo, Ross A luwkmp, nd Justin M Lloyd his tile povides oneptul foundtion with espet to homing guidne upon whih the next sevel tiles e nhoed To this end, bsi geometi

More information

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

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

More information

Newton s Law of Universal Gravitation and the Scale Principle

Newton s Law of Universal Gravitation and the Scale Principle Newton s Law of Univesal avitation and the ale iniple RODOLO A. RINO July 0 Eletonis Enginee Degee fo the National Univesity of Ma del lata - Agentina (odolfo_fino@yahoo.o.a) Ealie this yea I wote a pape

More information

Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapter 3 Savings, Present Value and Ricardian Equivalence Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

More information

2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES

2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES . TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an

More information

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

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

More information

Chapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.

Chapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere. Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium-39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming

More information

Skills Needed for Success in Calculus 1

Skills Needed for Success in Calculus 1 Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell

More information

Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined

More information

Week 3-4: Permutations and Combinations

Week 3-4: Permutations and Combinations Week 3-4: Pemutations and Combinations Febuay 24, 2016 1 Two Counting Pinciples Addition Pinciple Let S 1, S 2,, S m be disjoint subsets of a finite set S If S S 1 S 2 S m, then S S 1 + S 2 + + S m Multiplication

More information

12. Rolling, Torque, and Angular Momentum

12. Rolling, Torque, and Angular Momentum 12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.

More information

Intro to Circle Geometry By Raymond Cheong

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

More information

Random Variables and Distribution Functions

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

More information

Gauss Law. Physics 231 Lecture 2-1

Gauss Law. Physics 231 Lecture 2-1 Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

More information

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

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

More information

Coordinate Systems L. M. Kalnins, March 2009

Coordinate Systems L. M. Kalnins, March 2009 Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

More information

PY1052 Problem Set 8 Autumn 2004 Solutions

PY1052 Problem Set 8 Autumn 2004 Solutions PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ight-hand end. If H 6.0 m and h 2.0 m, what

More information

HUT, TUT, LUT, OU, ÅAU / Engineering departments Entrance examination in mathematics May 25, 2004

HUT, TUT, LUT, OU, ÅAU / Engineering departments Entrance examination in mathematics May 25, 2004 HUT, TUT, LUT, OU, ÅAU / Engineeing depamens Enane examinaion in mahemais May 5, 4 Insuions. Reseve a sepaae page fo eah poblem. Give you soluions in a lea fom inluding inemediae seps. Wie a lean opy of

More information

9.3 Surface Area of Pyramids

9.3 Surface Area of Pyramids Page 1 of 9 9.3 Suface Aea of Pyamids and Cones Goa Find the suface aeas of pyamids and cones. Key Wods pyamid height of a pyamid sant height of a pyamid cone height of a cone sant height of a cone The

More information

The Casino Experience. Let us entertain you

The Casino Experience. Let us entertain you The Csio Expeiee Let us eteti you The Csio Expeiee If you e lookig fo get ight out, Csio Expeiee is just fo you. 10 The Stight Flush Expeiee 25 pe peso This is get itodutio to gmig tht sves you moey Kik

More information

Lesson 7 Gauss s Law and Electric Fields

Lesson 7 Gauss s Law and Electric Fields Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual

More information

www.sakshieducation.com

www.sakshieducation.com Viscosity. The popety of viscosity in gas is due to ) Cohesive foces between the moecues ) Coisions between the moecues ) Not having a definite voume ) Not having a definite size. When tempeatue is inceased

More information

Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!

Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360! 1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the

More information

Combinatorial Testing for Tree-Structured Test Models with Constraints

Combinatorial Testing for Tree-Structured Test Models with Constraints Comintoil Testing fo Tee-Stutued Test Models with Constints Tkshi Kitmu, Akihis Ymd, Goo Htym, Cyille Atho, Eun-Hye Choi, Ngo Thi Bih Do, Yutk Oiw, Shiny Skugi Ntionl Institute of Advned Industil Siene

More information

MATH 150 HOMEWORK 4 SOLUTIONS

MATH 150 HOMEWORK 4 SOLUTIONS MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive

More information

GFI MilEssentils & GFI MilSecuity vs Symntec Bightmil 6 & Anti Vius GFI Softwe www.gfi.com GFI MilEssentils & GFI MilSecuity vs Symntec Bightmil 6 & Anti Vius GFI MilEssentils & GFI MilSecuity Bightmil

More information

Chapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6

Chapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6 Chapte 9 lectic Chages, Foces, an Fiels 6 9. One in a million (0 ) ogen molecules in a containe has lost an electon. We assume that the lost electons have been emove fom the gas altogethe. Fin the numbe

More information

4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to

4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to . Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate

More information

Screentrade Car Insurance Policy Summary

Screentrade Car Insurance Policy Summary Sceentde C Insunce Policy Summy This is summy of the policy nd does not contin the full tems nd conditions of the cove, which cn be found in the policy booklet nd schedule. It is impotnt tht you ed the

More information

GFI MilEssentils & GFI MilSecuity vs Bcud Spm Fiewll GFI Softwe www.gfi.com GFIMilEssentils & GFI MilSecuity vs Bcud Spm Fiewll GFI MilEssentils 12 & GFI MilSecuity 10 Bcud Spm Fiewll Who we e Integtes

More information

2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration

2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 25-6 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting

More information

Physics 235 Chapter 5. Chapter 5 Gravitation

Physics 235 Chapter 5. Chapter 5 Gravitation Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus

More information

GFI MilEssentils & GFI MilSecuity vs Tend Mico ScnMil Suite fo Micosoft Exchnge GFI Softwe www.gfi.com GFI MilEssentils & GFI MilSecuity vs Tend Mico ScnMil Suite fo Micosoft Exchnge Exchnge Seve 2000/2003

More information

Fluids Lecture 15 Notes

Fluids Lecture 15 Notes Fluids Lectue 15 Notes 1. Unifom flow, Souces, Sinks, Doublets Reading: Andeson 3.9 3.12 Unifom Flow Definition A unifom flow consists of a velocit field whee V = uî + vĵ is a constant. In 2-D, this velocit

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

More information

Finite-Length Analysis of Low-Density Parity-Check Codes on the Binary Erasure Channel

Finite-Length Analysis of Low-Density Parity-Check Codes on the Binary Erasure Channel 1570 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 6, JUNE 2002 Finite-Length Anaysis of Low-Density Paity-Check Codes on the Binay Easue Channe Changyan Di, Student Membe, IEEE, David Poietti,

More information

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

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

More information

A r. (Can you see that this just gives the formula we had above?)

A r. (Can you see that this just gives the formula we had above?) 24-1 (SJP, Phys 1120) lectic flux, and Gauss' law Finding the lectic field due to a bunch of chages is KY! Once you know, you know the foce on any chage you put down - you can pedict (o contol) motion

More information

Graphs on Logarithmic and Semilogarithmic Paper

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

More information

G.GMD.1 STUDENT NOTES WS #5 1 REGULAR POLYGONS

G.GMD.1 STUDENT NOTES WS #5 1 REGULAR POLYGONS G.GMD.1 STUDENT NOTES WS #5 1 REGULAR POLYGONS Regul polygon e of inteet to u becue we begin looking t the volume of hexgonl pim o Tethedl nd to do thee type of clcultion we need to be ble to olve fit

More information

The invention of line integrals is motivated by solving problems in fluid flow, forces, electricity and magnetism.

The invention of line integrals is motivated by solving problems in fluid flow, forces, electricity and magnetism. Instrutor: Longfei Li Mth 43 Leture Notes 16. Line Integrls The invention of line integrls is motivted by solving problems in fluid flow, fores, eletriity nd mgnetism. Line Integrls of Funtion We n integrte

More information

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

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

More information

Symmetric polynomials and partitions Eugene Mukhin

Symmetric polynomials and partitions Eugene Mukhin Symmetic polynomials and patitions Eugene Mukhin. Symmetic polynomials.. Definition. We will conside polynomials in n vaiables x,..., x n and use the shotcut p(x) instead of p(x,..., x n ). A pemutation

More information

Solar wind speed theory and the nonextensivity of solar corona

Solar wind speed theory and the nonextensivity of solar corona AXiv:080.170 Sola wind speed theoy and the nonextensivity of sola oona Du Jiulin *, Song Yeli Depatment of Physis, Shool of Siene, Tianjin Univesity, Tianjin 30007, China Abstat. The sola oona is a omplex

More information

Lesson 8 Ampère s Law and Differential Operators

Lesson 8 Ampère s Law and Differential Operators Lesson 8 Ampèe s Law and Diffeential Opeatos Lawence Rees 7 You ma make a single cop of this document fo pesonal use without witten pemission 8 Intoduction Thee ae significant diffeences between the electic

More information

Forces & Magnetic Dipoles. r r τ = μ B r

Forces & Magnetic Dipoles. r r τ = μ B r Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent

More information

Valuation of Floating Rate Bonds 1

Valuation of Floating Rate Bonds 1 Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned

More information

payments Excess demand (Expenditure>output) r > r Excess demand (Expenditure>output) r > r Excess supply (Expenditure<output) Excess supply

payments Excess demand (Expenditure>output) r > r Excess demand (Expenditure>output) r > r Excess supply (Expenditure<output) Excess supply Chpte Review Questions. The IS-- model defines six egions, eh oesponding to disequiliium in the money mket, goods mket nd/o the lne of pyments. Identify nd desie eh of these when pitl is pefetly moile,

More information

Things to Remember. r Complete all of the sections on the Retirement Benefit Options form that apply to your request.

Things to Remember. r Complete all of the sections on the Retirement Benefit Options form that apply to your request. Retiement Benefit 1 Things to Remembe Complete all of the sections on the Retiement Benefit fom that apply to you equest. If this is an initial equest, and not a change in a cuent distibution, emembe to

More information

Vindforsk report Project 30988-1/V-238

Vindforsk report Project 30988-1/V-238 Repot Vindfosk 3988-/V-38 Vindfosk epot Pojet 3988-/V-38 Ny teknik fö isning indkftsing New Tehnologies fo de-iing Wind Tuines Ls Bååth nd Hns Löfgen Hlmstd Uniesity ls.th@hh.se phone: 46 ()75-65735 Vindfosk:

More information

XML Data Integration using Fragment Join

XML Data Integration using Fragment Join XML Dt Integtion using Fgment Join Jin Gong, Dvi W. Cheung, Nikos Mmoulis, n Ben Ko Deptment of Compute Siene, The Univesity of Hong Kong Pokfulm, Hong Kong, Chin {jgong,heung,nikos,ko}@s.hku.hk Astt.

More information

Voltage ( = Electric Potential )

Voltage ( = Electric Potential ) V-1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is

More information

1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2

1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2 Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the

More information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers. 2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

More information

4.1 - Trigonometric Functions of Acute Angles

4.1 - Trigonometric Functions of Acute Angles 4.1 - Tigonometic Functions of cute ngles a is a half-line that begins at a point and etends indefinitel in some diection. Two as that shae a common endpoint (o vete) fom an angle. If we designate one

More information

The force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges

The force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee

More information

Functions of a Random Variable: Density. Math 425 Intro to Probability Lecture 30. Definition Nice Transformations. Problem

Functions of a Random Variable: Density. Math 425 Intro to Probability Lecture 30. Definition Nice Transformations. Problem Intoduction One Function of Random Vaiables Functions of a Random Vaiable: Density Math 45 Into to Pobability Lectue 30 Let gx) = y be a one-to-one function whose deiatie is nonzeo on some egion A of the

More information

Radius of the Earth - Radii Used in Geodesy James R. Clynch Naval Postgraduate School, 2002

Radius of the Earth - Radii Used in Geodesy James R. Clynch Naval Postgraduate School, 2002 dius of the Erth - dii Used in Geodesy Jmes. Clynh vl Postgrdute Shool, 00 I. Three dii of Erth nd Their Use There re three rdii tht ome into use in geodesy. These re funtion of ltitude in the ellipsoidl

More information

ABSORPTION-FREE SUPERLUMINAL LIGHT PROPAGATION IN A V-TYPE SYSTEM

ABSORPTION-FREE SUPERLUMINAL LIGHT PROPAGATION IN A V-TYPE SYSTEM Amenin Jounl of Physis 0 vol. 4 issue. 38-48 ABSORPTION-FREE SUPERLUMINAL LIGHT PROPAGATION IN A V-TYPE SYSTEM S. W. Rbiei* Kh. Sidi B. Ruzbhni M. Mhmoudi 3 Islmi Azd Univesity - Snndj Bnh Snndj In Physis

More information

Graphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities.

Graphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities. Gaphs of Equations CHAT Pe-Calculus A coodinate sstem is a wa to gaphicall show the elationship between quantities. Definition: A solution of an equation in two vaiables and is an odeed pai (a, b) such

More information

Financing Terms in the EOQ Model

Financing Terms in the EOQ Model Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad

More information

Questions for Review. By buying bonds This period you save s, next period you get s(1+r)

Questions for Review. By buying bonds This period you save s, next period you get s(1+r) MACROECONOMICS 2006 Week 5 Semina Questions Questions fo Review 1. How do consumes save in the two-peiod model? By buying bonds This peiod you save s, next peiod you get s() 2. What is the slope of a consume

More information

Chapter 22 The Electric Field II: Continuous Charge Distributions

Chapter 22 The Electric Field II: Continuous Charge Distributions Chpte The lectic Field II: Continuous Chge Distibutions Conceptul Poblems [SSM] Figue -7 shows n L-shped object tht hs sides which e equl in length. Positive chge is distibuted unifomly long the length

More information

Words Symbols Diagram. abcde. a + b + c + d + e

Words Symbols Diagram. abcde. a + b + c + d + e Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To

More information

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding 1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

More information

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

More information

Lab M4: The Torsional Pendulum and Moment of Inertia

Lab M4: The Torsional Pendulum and Moment of Inertia M4.1 Lab M4: The Tosional Pendulum and Moment of netia ntoduction A tosional pendulum, o tosional oscillato, consists of a disk-like mass suspended fom a thin od o wie. When the mass is twisted about the

More information

Derivation of Annuity and Perpetuity Formulae. A. Present Value of an Annuity (Deferred Payment or Ordinary Annuity)

Derivation of Annuity and Perpetuity Formulae. A. Present Value of an Annuity (Deferred Payment or Ordinary Annuity) Aity Deivatios 4/4/ Deivatio of Aity ad Pepetity Fomlae A. Peset Vale of a Aity (Defeed Paymet o Odiay Aity 3 4 We have i the show i the lecte otes ad i ompodi ad Discoti that the peset vale of a set of

More information

CURVES ANDRÉ NEVES. that is, the curve α has finite length. v = p q p q. a i.e., the curve of smallest length connecting p to q is a straight line.

CURVES ANDRÉ NEVES. that is, the curve α has finite length. v = p q p q. a i.e., the curve of smallest length connecting p to q is a straight line. CURVES ANDRÉ NEVES 1. Problems (1) (Ex 1 of 1.3 of Do Crmo) Show tht the tngent line to the curve α(t) (3t, 3t 2, 2t 3 ) mkes constnt ngle with the line z x, y. (2) (Ex 6 of 1.3 of Do Crmo) Let α(t) (e

More information

Chapter 23 Electrical Potential

Chapter 23 Electrical Potential hpte Electicl Potentil onceptul Polems [SSM] A poton is moved to the left in unifom electic field tht points to the ight. Is the poton moving in the diection of incesing o decesing electic potentil? Is

More information

9 CONTINUOUS DISTRIBUTIONS

9 CONTINUOUS DISTRIBUTIONS 9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete

More information

GFI EventsMnge vs Netikus.net EventSenty GFI Softwe www.gfi.com GFI EventsMnge vs Netikus.net EventSenty GFI EventsMnge EventSenty Who we e Suppot fo MS SQL Seve Suppot fo MSDE / MS SQL Expess Suppot fo

More information

Density Curve. Continuous Distributions. Continuous Distribution. Density Curve. Meaning of Area Under Curve. Meaning of Area Under Curve

Density Curve. Continuous Distributions. Continuous Distribution. Density Curve. Meaning of Area Under Curve. Meaning of Area Under Curve Continuous Distributions Rndom Vribles of the Continuous Tye Density Curve Perent Density funtion f () f() A smooth urve tht fit the distribution 6 7 9 Test sores Density Curve Perent Probbility Density

More information

tools for Web data extraction

tools for Web data extraction HTML-we tools fo Web dt extction Thesis pesenttion 1 Student: Xvie Azg Supeviso: Andes Tho Tble of contents Intoduction Dt Extction Pocess Dt Extction Tools Relized tests Futue Wok 2 Intoduction We e going

More information

Binary Representation of Numbers Autar Kaw

Binary Representation of Numbers Autar Kaw Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

More information

Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27

Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27 Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew - electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field

More information

SR-Phlx-2016-26 Page 39 of 43 NASDAQ OMX PHLX LLC 1 PRICING SCHEDULE THE EXCHANGE CALCULATES FEES ON A TRADE DATE BASIS.

SR-Phlx-2016-26 Page 39 of 43 NASDAQ OMX PHLX LLC 1 PRICING SCHEDULE THE EXCHANGE CALCULATES FEES ON A TRADE DATE BASIS. SR-Phlx-216-26 Page 39 of 43 Deleted text is [baketed]. New text is undelined. NASDAQ OMX PHLX LLC 1 PRICING SCHEDULE THE EXCHANGE CALCULATES FEES ON A TRADE DATE BASIS. EXHIBIT POLICY FOR AMENDING BILLING

More information

Voltage ( = Electric Potential )

Voltage ( = Electric Potential ) V-1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage

More information

CHAPTER 5 GRAVITATIONAL FIELD AND POTENTIAL

CHAPTER 5 GRAVITATIONAL FIELD AND POTENTIAL CHATER 5 GRAVITATIONAL FIELD AND OTENTIAL 5. Intoduction. This chapte deals with the calculation of gavitational fields and potentials in the vicinity of vaious shapes and sizes of massive bodies. The

More information

12.1. FÖRSTER RESONANCE ENERGY TRANSFER

12.1. FÖRSTER RESONANCE ENERGY TRANSFER ndei Tokmakoff, MIT epatment of Chemisty, 3/5/8 1-1 1.1. FÖRSTER RESONNCE ENERGY TRNSFER Föste esonance enegy tansfe (FR) efes to the nonadiative tansfe of an electonic excitation fom a dono molecule to

More information

It is required to solve the heat-condition equation for the excess-temperature function:

It is required to solve the heat-condition equation for the excess-temperature function: Jounal of Engineeing Physics and Themophysics. Vol. 73. No. 5. 2 METHOD OF PAIED INTEGAL EQUATIONS WITH L-PAAMETE IN POBLEMS OF NONSTATIONAY HEAT CONDUCTION WITH MIXED BOUNDAY CONDITIONS FO AN INFINITE

More information

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

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

More information

Turbulence simulation in wavelet domain based on Log- Poisson model: univariate and multivariate wind processes

Turbulence simulation in wavelet domain based on Log- Poisson model: univariate and multivariate wind processes he Seventh Interntion Cooquium on Buff Body Aerodynmis nd Appitions (BBAA7) Shnghi, Chin; September -6, 0 urbuene simution in wveet domin bsed on Log- Poisson mode: univrite nd mutivrite wind proesses

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

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

More information

Carter-Penrose diagrams and black holes

Carter-Penrose diagrams and black holes Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example

More information

4.11 Inner Product Spaces

4.11 Inner Product Spaces 314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

More information