dy ds dz ds dx ds ds ds ds ds ds 4-1 DEFORMATION OF A BODY Let there be a line segment PQ in the body with coordinates as:
|
|
- Chad Carroll
- 5 years ago
- Views:
Transcription
1 5:44 (58:54 ENERGY PRNCPLES N SRUCURAL MECHANCS 4- DEFORMAON OF A BODY Q Q Let there be a ie seget PQ i the bo with cooriates as: P(,,, Q(,, Legth of the ifferetia eeet: P P Uit taget vector aog PQ: e i j k i j k After eforatio, P P' a Q Q' with cooriates as P' ( Q' (, v, w, v v, w w Legth of P'Q': ( ( v ( w SRAN Strai at the poit P is efie as (4- or ; > Sqare both sies a re-arrage ( Lectre #7
2 5:44 (58:54 ENERGY PRNCPLES N SRUCURAL MECHANCS Now, sbstitte for a a epa: ( v v w ( v v ( w v ( w w w Usig the chai re, we get v v v v v v v w w w w w w w Sbstittig i the above, we get (4- where v w ( ( ( v v v w w w Lectre #7
3 5:44 (58:54 ENERGY PRNCPLES N SRUCURAL MECHANCS v w v w v v v v v v w w w w w w (4- the ie otatio, Eq. (4- ca be writte as j i where ( i, j j,i a,i a, j he setric atri, is kow as the strai tesor, sice it obes the tesor aw of trasforatio whe the cooriates are rotate. SHEARNG SRAN Cosier two ie eeets PQ a P that are iitia perpeicar to each other. After eforatio, these ie segets becoe P'Q' a P''. Let θ be the age betwee the after the eforatio. he age [ ( π θ ] is kow as the shearig strai betwee the two ie eeets. Derivatio of Eq. (4-4 Lectre #7
4 5:44 (58:54 ENERGY PRNCPLES N SRUCURAL MECHANCS Let P, P Q, PQ, P P, P Q, PQ, P he iitia irectio cosies of PQ a P Q Q are ( a ( P P θ Sice the two ie eeets are iitia perpeicar to each other, 0 (i.e., 0 he ier proct of two vectors after eforatio isgive as cosθ Diviig both sies b a sig Eq. (4-, cos? ( ( cosθ (a where a are strais of the respective ie eeets. o erive a epressio for the vector after eforatio, cosier a eeet PQ that aps o the eeet P'Q' after eforatio. he cooriates of the poits are give as P(, Q(, P'(, Q'( Now is erive as (, [( (] ( ; [ i, j ] Lectre #7 4
5 5:44 (58:54 ENERGY PRNCPLES N SRUCURAL MECHANCS Lectre #7 5 Foregoig epressio appie to the two ie eeets, Pttig a ito Eq. (a, we get ( ( ( ( cos ( ( θ (4-4 PRNCPAL SRANS hrogh a poit i the efore ei, there are three ta perpeicar ie eeets that reai perpeicar after eforatio. he strais of these three ie eeets are cae the pricipa strais at the give poit. he are eote b,, (.
6 5:44 (58:54 ENERGY PRNCPLES N SRUCURAL MECHANCS he correspoig vaes of the qatit are eote b (,,. he qatities (,, are the three roots of the eteriat eqatio, i δ i 0 (4-5 For a otrivia sotio, δ 0 Sice is setric a rea, the roots i are awas rea. Sice i > -, i -/. he above characteristic eqatio ca be writte as 0 (4-6 where tr( et( ii tr cofactor( (4-9 he sotio i of Eq. (4-5 is the set of irectio cosies of the pricipa irectio i the efore ei correspoig to the pricipa strai i. he three irectios correspoig to the three roots (,, are ta perpeicar. Sice the pricipa strais are iepeet of cooriate sste, Eq. (4-7 shows that the qatities,, are iepeet of the Lectre #7 6
7 5:44 (58:54 ENERGY PRNCPLES N SRUCURAL MECHANCS cooriate sste. hese are cae the first, seco a thir ivariats of the strai tesor. Rotatio. B the theor of rigi-bo ispaceets, there eists for a poit of a eforabe bo, a agar ispaceet that carries the pricipa aes of strai of the efore bo ito the pricipa aes of the efore bo. We a eterie the ais of rotatio a the agar ispaceet of a partice of the ei i ters of the ispaceet vector fie i. Besies the rotatio, the partice receives the trasatio i. Aso, it eperieces the pricipa strais (,, aog the pricipa aes. his eforatio is cae a iatatio. hs the ispaceets i a eighborhoo of a poit are resove ito a trasatio, a rotatio, a a iatatio. he trasatio a the rotatio cotribte othig to the strais. VOLUMERC SRAN Let a eeet of a straie ei have the iitia voe V a the fia voe e V. he voetric strai is efie b V V (4-0 V A voe eeet V i the for of a ifiitesia rectagar paraeepipe with its eges i the pricipa irectios reais a rectagar paraeepipe after eforatio. he strais of the Lectre #7 7
8 5:44 (58:54 ENERGY PRNCPLES N SRUCURAL MECHANCS eges of the paraeepipe are the pricipa strais (e,e,e. Coseqet, herefore, V ( ( ( V e (4- (4- e e 4 (4- SMALL DSPLACEMEN HEORY the sa ispaceet theor, Eq. (4- is approiate b e. he qaratic ters i strai ispaceet reatios are egecte to obtai ( j,i i, j (4-4 Whe the qaratic ters are egecte,,, are the strais of ie eeets that iitia ie parae to the, a aes. Aso,,,, are the shearig strais betwee pairs of the ie eeets that iitia ie parae to the aes iicate b the sbscripts. COMPABLY EQUAONS as ch as there are si eqatios for three kow fctios i, the sste of Eq. (4-4 wi ot have a sige-vae sotio i Lectre #7 8
9 5:44 (58:54 ENERGY PRNCPLES N SRUCURAL MECHANCS geera, if the fctios e were arbitrari assige. Oe st epect that a sotio a eist o if the fctios e satisf certai coitios. t ca be obtaie b eiiatig i fro Eq. (4-4: 0 (4-5, k k, ik, j j,ik Eqs. (4.5 are kow as the copatibiit eqatios i the sa ispaceet theor. t ca be show that if give fctios satisf the copatibiit coitios (4-5, the there eist fctios i that are sotios of the strai-ispaceet reatios (4-4. Lectre #7 9
Definition 2.1 (The Derivative) (page 54) is a function. The derivative of a function f with respect to x, represented by. f ', is defined by
Chapter DACS Lok 004/05 CHAPTER DIFFERENTIATION. THE GEOMETRICAL MEANING OF DIFFERENTIATION (page 54) Defiitio. (The Derivative) (page 54) Let f () is a fctio. The erivative of a fctio f with respect to,
More informationCalculus 2 Quiz 1 Review / Fall 2011
Calcls Qiz Review / Fall 0 () The fctio is f a the iterval is [, ]. Here are two formlas yo may ee. ( ) ( ) ( ) 6 (a.) Use a left-e, right-e, a mipoit sm of "" rectagles to approimate. The withs of all
More informationAnalytic Number Theory Solutions
Aalytic Number Theory Solutios Sea Li Corell Uiversity sl6@corell.eu Ja. 03 Itrouctio This ocumet is a work-i-progress solutio maual for Tom Apostol s Itrouctio to Aalytic Number Theory. The solutios were
More information3.3 Rules for Differentiation Calculus. Drum Roll please [In a Deep Announcer Voice] And now the moment YOU VE ALL been waiting for
. Rules or Dieretiatio Calculus. RULES FOR DIFFERENTIATION Drum Roll please [I a Deep Aoucer Voice] A ow the momet YOU VE ALL bee waitig or Rule #1 Derivative o a Costat Fuctio I c is a costat value, the
More informationLecture #3. Math tools covered today
Toay s Program:. Review of previous lecture. QM free particle a particle i a bo. 3. Priciple of spectral ecompositio. 4. Fourth Postulate Math tools covere toay Lecture #3. Lear how to solve separable
More informationAP Calculus BC Review Chapter 12 (Sequences and Series), Part Two. n n th derivative of f at x = 5 is given by = x = approximates ( 6)
AP Calculus BC Review Chapter (Sequeces a Series), Part Two Thigs to Kow a Be Able to Do Uersta the meaig of a power series cetere at either or a arbitrary a Uersta raii a itervals of covergece, a kow
More informationOrthogonal Function Solution of Differential Equations
Royal Holloway Uiversity of Loo Departet of Physics Orthogoal Fuctio Solutio of Differetial Equatios trouctio A give oriary ifferetial equatio will have solutios i ters of its ow fuctios Thus, for eaple,
More information(average number of points per unit length). Note that Equation (9B1) does not depend on the
EE603 Class Notes 9/25/203 Joh Stesby Appeix 9-B: Raom Poisso Poits As iscusse i Chapter, let (t,t 2 ) eote the umber of Poisso raom poits i the iterval (t, t 2 ]. The quatity (t, t 2 ) is a o-egative-iteger-value
More information3. Calculus with distributions
6 RODICA D. COSTIN 3.1. Limits of istributios. 3. Calculus with istributios Defiitio 4. A sequece of istributios {u } coverges to the istributio u (all efie o the same space of test fuctios) if (φ, u )
More informationFAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES
LECTURE Third Editio FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES A. J. Clark School of Egieerig Departmet of Civil ad Evirometal Egieerig Chapter 7.4 b Dr. Ibrahim A. Assakkaf SPRING 3 ENES
More informationOrthogonal transformations
Orthogoal trasformatios October 12, 2014 1 Defiig property The squared legth of a vector is give by takig the dot product of a vector with itself, v 2 v v g ij v i v j A orthogoal trasformatio is a liear
More informationSCHOOL OF MATHEMATICS AND STATISTICS. Mathematics II (Materials)
Dt proie: Form Sheet MAS5 SCHOOL OF MATHEMATICS AND STATISTICS Mthemtics II (Mteris) Atm Semester -3 hors Mrks wi e wre or swers to qestios i Sectio A or or est THREE swers to qestios i Sectio. Sectio
More informationHWA CHONG INSTITUTION JC1 PROMOTIONAL EXAMINATION Wednesday 1 October hours. List of Formula (MF15)
HWA CHONG INSTITUTION JC PROMOTIONAL EXAMINATION 4 MATHEMATICS Higher 974/ Paper Wedesda October 4 hors Additioal materials: Aswer paper List of Formla (MF5) READ THESE INSTRUCTIONS FIRST Write or ame
More informationLecture 10: Bounded Linear Operators and Orthogonality in Hilbert Spaces
Lecture : Bouded Liear Operators ad Orthogoality i Hilbert Spaces 34 Bouded Liear Operator Let ( X, ), ( Y, ) i i be ored liear vector spaces ad { } X Y The, T is said to be bouded if a real uber c such
More information1 = 2 d x. n x n (mod d) d n
HW2, Problem 3*: Use Dirichlet hyperbola metho to show that τ 2 + = 3 log + O. This ote presets the ifferet ieas suggeste by the stuets Daiel Klocker, Jürge Steiiger, Stefaia Ebli a Valerie Roiter for
More informationRepresenting Functions as Power Series. 3 n ...
Math Fall 7 Lab Represetig Fuctios as Power Series I. Itrouctio I sectio.8 we leare the series c c c c c... () is calle a power series. It is a uctio o whose omai is the set o all or which it coverges.
More informationTHE LEGENDRE POLYNOMIALS AND THEIR PROPERTIES. r If one now thinks of obtaining the potential of a distributed mass, the solution becomes-
THE LEGENDRE OLYNOMIALS AND THEIR ROERTIES The gravitatioal potetial ψ at a poit A at istace r from a poit mass locate at B ca be represete by the solutio of the Laplace equatio i spherical cooriates.
More informationFundamental Concepts: Surfaces and Curves
UNDAMENTAL CONCEPTS: SURACES AND CURVES CHAPTER udametal Cocepts: Surfaces ad Curves. INTRODUCTION This chapter describes two geometrical objects, vi., surfaces ad curves because the pla a ver importat
More informationAP Calculus BC Review Applications of Derivatives (Chapter 4) and f,
AP alculus B Review Applicatios of Derivatives (hapter ) Thigs to Kow ad Be Able to Do Defiitios of the followig i terms of derivatives, ad how to fid them: critical poit, global miima/maima, local (relative)
More information2.710 Optics Spring 09 Solutions to Problem Set #3 Due Wednesday, March 4, 2009
MASSACHUSETTS INSTITUTE OF TECHNOLOGY.70 Optics Sprig 09 Solutios to Problem Set #3 Due Weesay, March 4, 009 Problem : Waa s worl a) The geometry or this problem is show i Figure. For part (a), the object
More informationCALCULUS BASIC SUMMER REVIEW
CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=
More informationApril 1980 TR/96. Extrapolation techniques for first order hyperbolic partial differential equations. E.H. Twizell
TR/96 Apri 980 Extrapoatio techiques for first order hyperboic partia differetia equatios. E.H. Twize W96086 (0) 0. Abstract A uifor grid of step size h is superiposed o the space variabe x i the first
More information+ {JEE Advace 03} Sept 0 Name: Batch (Day) Phoe No. IT IS NOT ENOUGH TO HAVE A GOOD MIND, THE MAIN THING IS TO USE IT WELL Marks: 00. If A (α, β) = (a) A( α, β) = A( α, β) (c) Adj (A ( α, β)) = Sol : We
More informationk=1 s k (x) (3) and that the corresponding infinite series may also converge; moreover, if it converges, then it defines a function S through its sum
0. L Hôpital s rule You alreay kow from Lecture 0 that ay sequece {s k } iuces a sequece of fiite sums {S } through S = s k, a that if s k 0 as k the {S } may coverge to the it k= S = s s s 3 s 4 = s k.
More informationInhomogeneous Poisson process
Chapter 22 Ihomogeeous Poisso process We coclue our stuy of Poisso processes with the case of o-statioary rates. Let us cosier a arrival rate, λ(t), that with time, but oe that is still Markovia. That
More informationPartial Differential Equations
EE 84 Matematical Metods i Egieerig Partial Differetial Eqatios Followig are some classical partial differetial eqatios were is assmed to be a fctio of two or more variables t (time) ad y (spatial coordiates).
More informationTECHNIQUES OF INTEGRATION
7 TECHNIQUES OF INTEGRATION Simpso s Rule estimates itegrals b approimatig graphs with parabolas. Because of the Fudametal Theorem of Calculus, we ca itegrate a fuctio if we kow a atiderivative, that is,
More information1. Complex numbers. Chapter 13: Complex Numbers. Modulus of a complex number. Complex conjugate. Complex numbers are of the form
Comple umbers ad comple plae Comple cojugate Modulus of a comple umber Comple umbers Comple umbers are of the form Sectios 3 & 32 z = + i,, R, i 2 = I the above defiitio, is the real part of z ad is the
More informationas best you can in any three (3) of a f. [15 = 3 5 each] e. y = sec 2 (arctan(x)) f. y = sin (e x )
Mathematics Y Calculus I: Calculus of oe variable Tret Uiversity, Summer Solutios to the Fial Examiatio Time: 9: :, o Weesay, August,. Brought to you by Stefa. Istructios: Show all your work a justify
More informationSolution: APPM 1360 Final Spring 2013
APPM 36 Fial Sprig 3. For this proble let the regio R be the regio eclosed by the curve y l( ) ad the lies, y, ad y. (a) (6 pts) Fid the area of the regio R. (b) (6 pts) Suppose the regio R is revolved
More informationBENDING FREQUENCIES OF BEAMS, RODS, AND PIPES Revision S
BENDING FREQUENCIES OF BEAMS, RODS, AND PIPES Revisio S By Tom Irvie Email: tom@vibratioata.com November, Itrouctio The fuametal frequecies for typical beam cofiguratios are give i Table. Higher frequecies
More informationConsortium of Medical Engineering and Dental Colleges of Karnataka (COMEDK) Undergraduate Entrance Test(UGET) Maths-2012
Cosortium of Medical Egieerig ad Detal Colleges of Karataka (COMEDK) Udergraduate Etrace Test(UGET) Maths-0. If the area of the circle 7 7 7 k 0 is sq. uits, the the value of k is As: (b) b) 0 7 K 0 c)
More informationChapter 2 Transformations and Expectations
Chapter Trasformatios a Epectatios Chapter Distributios of Fuctios of a Raom Variable Problem: Let be a raom variable with cf F ( ) If we efie ay fuctio of, say g( ) g( ) is also a raom variable whose
More informationTopic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or
Topic : Sequeces ad Series A sequece is a ordered list of umbers, e.g.,,, 8, 6, or,,,.... A series is a sum of the terms of a sequece, e.g. + + + 8 + 6 + or... Sigma Notatio b The otatio f ( k) is shorthad
More informationHKDSE Exam Questions Distribution
HKDSE Eam Questios Distributio Sample Paper Practice Paper DSE 0 Topics A B A B A B. Biomial Theorem. Mathematical Iductio 0 3 3 3. More about Trigoometric Fuctios, 0, 3 0 3. Limits 6. Differetiatio 7
More informationLecture 7: Polar representation of complex numbers
Lecture 7: Polar represetatio of comple umbers See FLAP Module M3.1 Sectio.7 ad M3. Sectios 1 ad. 7.1 The Argad diagram I two dimesioal Cartesia coordiates (,), we are used to plottig the fuctio ( ) with
More informationExpectation maximization
Motivatio Expectatio maximizatio Subhrasu Maji CMSCI 689: Machie Learig 14 April 015 Suppose you are builig a aive Bayes spam classifier. After your are oe your boss tells you that there is o moey to label
More informationThe Stokes Theorem. (Sect. 16.7) The curl of a vector field in space
The tokes Theorem. (ect. 6.7) The curl of a vector field i space. The curl of coservative fields. tokes Theorem i space. Idea of the proof of tokes Theorem. The curl of a vector field i space Defiitio
More informationRAYLEIGH'S METHOD Revision D
RAYGH'S METHOD Revisio D B To Irvie Eail: toirvie@aol.co Noveber 5, Itroductio Daic sstes ca be characterized i ters of oe or ore atural frequecies. The atural frequec is the frequec at which the sste
More informationSolutions for May. 3 x + 7 = 4 x x +
Solutios for May 493. Prove that there is a atural umber with the followig characteristics: a) it is a multiple of 007; b) the first four digits i its decimal represetatio are 009; c) the last four digits
More informationChapter 13: Complex Numbers
Sectios 13.1 & 13.2 Comple umbers ad comple plae Comple cojugate Modulus of a comple umber 1. Comple umbers Comple umbers are of the form z = + iy,, y R, i 2 = 1. I the above defiitio, is the real part
More informationDiagonalization of Quadratic Forms. Recall in days past when you were given an equation which looked like
Diagoalizatio of Qadratic Forms Recall i das past whe o were gie a eqatio which looked like ad o were asked to sketch the set of poits which satisf this eqatio. It was ecessar to complete the sqare so
More informationMATHEMATICS. 61. The differential equation representing the family of curves where c is a positive parameter, is of
MATHEMATICS 6 The differetial equatio represetig the family of curves where c is a positive parameter, is of Order Order Degree (d) Degree (a,c) Give curve is y c ( c) Differetiate wrt, y c c y Hece differetial
More informationd dx where k is a spring constant
Vorlesug IX Harmoic Oscillator 1 Basic efiitios a properties a classical mechaics Oscillator is efie as a particle subject to a liear force fiel The force F ca be epresse i terms of potetial fuctio V F
More informationAP Calculus BC Summer Math Packet
AP Calculus BC Summer Math Packet This is the summer review a preparatio packet for stuets eterig AP Calculus BC. Dear Bear Creek Calculus Stuet, The first page is the aswer sheet for the attache problems.
More informationClassical Electrodynamics
A First Look at Quatum Physics Classical Electroyamics Chapter Itrouctio a Survey Classical Electroyamics Prof. Y. F. Che Cotets A First Look at Quatum Physics. Coulomb s law a electric fiel. Electric
More informationMATH CALCULUS II Objectives and Notes for Test 4
MATH 44 - CALCULUS II Objectives ad Notes for Test 4 To do well o this test, ou should be able to work the followig tpes of problems. Fid a power series represetatio for a fuctio ad determie the radius
More informationChapter 2 The Solution of Numerical Algebraic and Transcendental Equations
Chapter The Solutio of Numerical Algebraic ad Trascedetal Equatios Itroductio I this chapter we shall discuss some umerical methods for solvig algebraic ad trascedetal equatios. The equatio f( is said
More informationCAMI Education linked to CAPS: Mathematics. Grade The main topics in the FET Mathematics Curriculum NUMBER
- 1 - CAMI Eucatio like to CAPS: Grae 1 The mai topics i the FET Curriculum NUMBER TOPIC 1 Fuctios Number patters, sequeces a series 3 Fiace, growth a ecay 4 Algebra 5 Differetial Calculus 6 Probability
More informationStrauss PDEs 2e: Section Exercise 4 Page 1 of 5. u tt = c 2 u xx ru t for 0 < x < l u = 0 at both ends u(x, 0) = φ(x) u t (x, 0) = ψ(x),
Strauss PDEs e: Sectio 4.1 - Exercise 4 Page 1 of 5 Exercise 4 Cosider waves i a resistat medium that satisfy the probem u tt = c u xx ru t for < x < u = at both eds ux, ) = φx) u t x, ) = ψx), where r
More informationTransmissibility Properties of MDOF Systems
rasmissibiity Proerties of MDOF Systems Weie Liu a D J Ewis Deartmet of Mechaica Egieerig Imeria Coege Loo SW7 X U K bstract theorem eaig with the reatioshi betwee FRFs is roose for chai-ie mass-srig systems,
More informationA Note on the form of Jacobi Polynomial used in Harish-Chandra s Paper Motion of an Electron in the Field of a Magnetic Pole.
e -Joral of Sciece & Techology (e-jst) A Note o the form of Jacobi Polyomial se i Harish-Chara s Paper Motio of a Electro i the Fiel of a Magetic Pole Vio Kmar Yaav Jior Research Fellow (CSIR) Departmet
More informationSeptember 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1
September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright
More informationCalculus I Practice Test Problems for Chapter 5 Page 1 of 9
Calculus I Practice Test Problems for Chapter 5 Page of 9 This is a set of practice test problems for Chapter 5. This is i o way a iclusive set of problems there ca be other types of problems o the actual
More informationComplex Numbers. Brief Notes. z = a + bi
Defiitios Complex Numbers Brief Notes A complex umber z is a expressio of the form: z = a + bi where a ad b are real umbers ad i is thought of as 1. We call a the real part of z, writte Re(z), ad b the
More informationHauptman and Karle Joint and Conditional Probability Distributions. Robert H. Blessing, HWI/UB Structural Biology Department, January 2003 ( )
Hauptma ad Karle Joit ad Coditioal Probability Distributios Robert H Blessig HWI/UB Structural Biology Departmet Jauary 00 ormalized crystal structure factors are defied by E h = F h F h = f a hexp ihi
More informationModule 3 : Analysis of Strain
Mod/Lsso Mod : Aasis of trai.. INTROUCTION T o dfi ora strai rfr to th fooi Fir. hr i AB of a aia oadd br has sffrd dforatio to bco A B. Fir. Aia oadd bar Th th of AB is. As sho i Fir.(b) poits A ad B
More informationFor use only in Badminton School November 2011 C2 Note. C2 Notes (Edexcel)
For use oly i Badmito School November 0 C Note C Notes (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets For use oly i Badmito School November 0 C Note Copyright www.pgmaths.co.uk
More informationMathematical Description of Discrete-Time Signals. 9/10/16 M. J. Roberts - All Rights Reserved 1
Mathematical Descriptio of Discrete-Time Sigals 9/10/16 M. J. Roberts - All Rights Reserved 1 Samplig ad Discrete Time Samplig is the acquisitio of the values of a cotiuous-time sigal at discrete poits
More informationU8L1: Sec Equations of Lines in R 2
MCVU U8L: Sec. 8.9. Equatios of Lies i R Review of Equatios of a Straight Lie (-D) Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio of the lie
More informationTutorial 4: FUNDAMENTAL SOLUTIONS: I-SIMPLE AND COMPOUND OPERATORS
Boary Elemet Commicatios 00 Ttorial 4: FNDAMENTAL SOLTIONS: I-SIMPLE AND COMPOND OPERATORS YOSSEF F. RASHED Dept. o Strctral Egieerig Cairo iversity iza Egypt yosse@eg.c.e.eg Smmary a objectives I the
More informationWeek 10 Spring Lecture 19. Estimation of Large Covariance Matrices: Upper bound Observe. is contained in the following parameter space,
Week 0 Sprig 009 Lecture 9. stiatio of Large Covariace Matrices: Upper boud Observe ; ; : : : ; i.i.d. fro a p-variate Gaussia distributio, N (; pp ). We assue that the covariace atrix pp = ( ij ) i;jp
More informationElementary Linear Algebra
Elemetary Liear Algebra Ato & Rorres th Editio Lectre Set Chapter : Eclidea Vector Spaces Chapter Cotet Vectors i -Space -Space ad -Space Norm Distace i R ad Dot Prodct Orthogoality Geometry of Liear Systems
More informationThe structure of Fourier series
The structure of Fourier series Valery P Dmitriyev Lomoosov Uiversity, Russia Date: February 3, 2011) Fourier series is costructe basig o the iea to moel the elemetary oscillatio 1, +1) by the expoetial
More informationMATH 10550, EXAM 3 SOLUTIONS
MATH 155, EXAM 3 SOLUTIONS 1. I fidig a approximate solutio to the equatio x 3 +x 4 = usig Newto s method with iitial approximatio x 1 = 1, what is x? Solutio. Recall that x +1 = x f(x ) f (x ). Hece,
More informationSparsification using Regular and Weighted. Graphs
Sparsificatio usig Regular a Weighte 1 Graphs Aly El Gamal ECE Departmet a Cooriate Sciece Laboratory Uiversity of Illiois at Urbaa-Champaig Abstract We review the state of the art results o spectral approximatio
More informationLommel Polynomials. Dr. Klaus Braun taken from [GWa] source. , defined by ( [GWa] 9-6)
Loel Polyoials Dr Klaus Brau tae fro [GWa] source The Loel polyoials g (, efie by ( [GWa] 9-6 fulfill Puttig g / ( -! ( - g ( (,!( -!! ( ( g ( g (, g : g ( : h ( : g ( ( a relatio betwee the oifie Loel
More informationVibratory Motion. Prof. Zheng-yi Feng NCHU SWC. National CHung Hsing University, Department of Soil and Water Conservation
Vibratory Motio Prof. Zheg-yi Feg NCHU SWC 1 Types of vibratory motio Periodic motio Noperiodic motio See Fig. A1, p.58 Harmoic motio Periodic motio Trasiet motio impact Trasiet motio earthquake A powerful
More informationEDEXCEL STUDENT CONFERENCE 2006 A2 MATHEMATICS STUDENT NOTES
EDEXCEL STUDENT CONFERENCE 006 A MATHEMATICS STUDENT NOTES South: Thursday 3rd March 006, Lodo EXAMINATION HINTS Before the eamiatio Obtai a copy of the formulae book ad use it! Write a list of ad LEARN
More informationMath 216A Notes, Week 3
Math 26A Notes Week 3 Scrie: Parker Williams Disclaimer: These otes are ot early as polishe (a quite possily ot early as correct as a pulishe paper. Please use them at your ow risk.. Posets a Möius iversio
More informationJEE ADVANCED 2013 PAPER 1 MATHEMATICS
Oly Oe Optio Correct Type JEE ADVANCED 0 PAPER MATHEMATICS This sectio cotais TEN questios. Each has FOUR optios (A), (B), (C) ad (D) out of which ONLY ONE is correct.. The value of (A) 5 (C) 4 cot cot
More informationAP Calculus Formulas Matawan Regional High School Calculus BC only material has a box around it.
AP Clcls Formls Mtw Regiol High School Clcls BC oly mteril hs bo ro it.. floor fctio (ef) Gretest iteger tht is less th or eql to.. (grph) 3. ceilig fctio (ef) Lest iteger tht is greter th or eql to. 4.
More informationGaps between Consecutive Perfect Powers
Iteratioal Mathematical Forum, Vol. 11, 016, o. 9, 49-437 HIKARI Lt, www.m-hikari.com http://x.oi.org/10.1988/imf.016.63 Gaps betwee Cosecutive Perfect Powers Rafael Jakimczuk Divisió Matemática, Uiversia
More information19 Fourier Series and Practical Harmonic Analysis
9 Fourier Series ad Practica Harmoic Aaysis Eampe : Obtai the Fourier series of f ( ) e a i. a Soutio: Let f ( ) acos bsi sih a a a a a a e a a where a f ( ) d e d e e a a e a f ( ) cos d e cos d ( a cos
More informationCoordinate Systems. Things to think about:
Coordiate Sstems There are 3 coordiate sstems that a compter graphics programmer is most cocered with: the Object Coordiate Sstem (OCS), the World Coordiate Sstem (WCS), ad the Camera Coordiate Sstem (CCS).
More informationComplete Solutions to Supplementary Exercises on Infinite Series
Coplete Solutios to Suppleetary Eercises o Ifiite Series. (a) We eed to fid the su ito partial fractios gives By the cover up rule we have Therefore Let S S A / ad A B B. Covertig the suad / the by usig
More informationLecture 11: A Fourier Transform Primer
PHYS 34 Fall 1 ecture 11: A Fourier Trasform Primer Ro Reifeberger Birck aotechology Ceter Purdue Uiversity ecture 11 1 f() I may edeavors, we ecouter sigals that eriodically reeat f(t) T t Such reeatig
More informationMaximum and Minimum Values
Sec 4.1 Maimum ad Miimum Values A. Absolute Maimum or Miimum / Etreme Values A fuctio Similarly, f has a Absolute Maimum at c if c f f has a Absolute Miimum at c if c f f for every poit i the domai. f
More informationDiscrete Fourier Transform
Discrete Fourier Trasform ) Purpose The purpose is to represet a determiistic or stochastic siga u( t ) as a fiite Fourier sum, whe observatios of u() t ( ) are give o a reguar grid, each affected by a
More informationDefinition 2 (Eigenvalue Expansion). We say a d-regular graph is a λ eigenvalue expander if
Expaer Graphs Graph Theory (Fall 011) Rutgers Uiversity Swastik Kopparty Throughout these otes G is a -regular graph 1 The Spectrum Let A G be the ajacecy matrix of G Let λ 1 λ λ be the eigevalues of A
More informationx !1! + 1!2!
4 Euler-Maclauri Suatio Forula 4. Beroulli Nuber & Beroulli Polyoial 4.. Defiitio of Beroulli Nuber Beroulli ubers B (,,3,) are defied as coefficiets of the followig equatio. x e x - B x! 4.. Expreesio
More informationPre-Calculus 12 Practice Exam 2 MULTIPLE-CHOICE (Calculator permitted )
Pre-alculus Practice Eam MULTIPLE-HOIE (alculator permitted ). Solve cos = si, 0 0.9 0.40,.5 c. 0.79 d. 0.79,.8. Determie the equatio of a circle with cetre ( 0,0) passig through the poit P (,5) + = c.
More informationTEMASEK JUNIOR COLLEGE, SINGAPORE JC One Promotion Examination 2014 Higher 2
TEMASEK JUNIOR COLLEGE, SINGAPORE JC Oe Promotio Eamiatio 04 Higher MATHEMATICS 9740 9 Septemer 04 Additioal Materials: Aswer paper 3 hours List of Formulae (MF5) READ THESE INSTRUCTIONS FIRST Write your
More informationComplex Numbers Solutions
Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i
More informationAreas and Distances. We can easily find areas of certain geometric figures using well-known formulas:
Areas ad Distaces We ca easily fid areas of certai geometric figures usig well-kow formulas: However, it is t easy to fid the area of a regio with curved sides: METHOD: To evaluate the area of the regio
More informationMathematics 1 Outcome 1a. Pascall s Triangle and the Binomial Theorem (8 pers) Cumulative total = 8 periods. Lesson, Outline, Approach etc.
prouce for by Tom Strag Pascall s Triagle a the Biomial Theorem (8 pers) Mathematics 1 Outcome 1a Lesso, Outlie, Approach etc. Nelso MIA - AH M1 1 Itrouctio to Pascal s Triagle via routes alog a set of
More informationMATHEMATICAL METHODS
8 Practice Exam A Letter STUDENT NUMBER MATHEMATICAL METHODS Writte examiatio Sectio Readig time: 5 miutes Writig time: hours WORKED SOLUTIONS Number of questios Structure of book Number of questios to
More informationR is a scalar defined as follows:
Math 8. Notes o Dot Product, Cross Product, Plaes, Area, ad Volumes This lecture focuses primarily o the dot product ad its may applicatios, especially i the measuremet of agles ad scalar projectio ad
More informationApply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j.
Eigevalue-Eigevector Istructor: Nam Su Wag eigemcd Ay vector i real Euclidea space of dimesio ca be uiquely epressed as a liear combiatio of liearly idepedet vectors (ie, basis) g j, j,,, α g α g α g α
More informationThe Probabilities of Large Deviations for the Chi-square and Log-likelihood Ratio Statistics Sherzod Mirakhmedov
The Probabiities of Large Deiatios for the Chi-square a Log-ieihoo Ratio Statistics Sherzo Miraheo Istitute of Matheatics. atioa Uiersity of Uzbeista 005 Tashet Duro yui st. 9 e-ai: shiraheo@yahoo.co Abstract.
More informationA Recurrence Formula for Packing Hyper-Spheres
A Recurrece Formula for Packig Hyper-Spheres DokeyFt. Itroductio We cosider packig of -D hyper-spheres of uit diameter aroud a similar sphere. The kissig spheres ad the kerel sphere form cells of equilateral
More informationDe Moivre s Theorem - ALL
De Moivre s Theorem - ALL. Let x ad y be real umbers, ad be oe of the complex solutios of the equatio =. Evaluate: (a) + + ; (b) ( x + y)( x + y). [6]. (a) Sice is a complex umber which satisfies = 0,.
More informationThe z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j
The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.
More informationPRACTICE FINAL/STUDY GUIDE SOLUTIONS
Last edited December 9, 03 at 4:33pm) Feel free to sed me ay feedback, icludig commets, typos, ad mathematical errors Problem Give the precise meaig of the followig statemets i) a f) L ii) a + f) L iii)
More informationMathematics Extension 2
009 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etesio Geeral Istructios Readig time 5 miutes Workig time hours Write usig black or blue pe Board-approved calculators may be used A table of stadard
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE 1 MATHEMATICS P NOVEMBER 01 MARKS: 150 TIME: 3 hours This questio paper cosists of 13 pages, 1 diagram sheet ad 1 iformatio sheet. Please tur over Mathematics/P DBE/November
More informationREVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains.
The mai ideas are: Idetities REVISION SHEET FP (MEI) ALGEBRA Before the exam you should kow: If a expressio is a idetity the it is true for all values of the variable it cotais The relatioships betwee
More informationECE Spring Prof. David R. Jackson ECE Dept. Notes 20
ECE 6341 Sprig 016 Prof. David R. Jackso ECE Dept. Notes 0 1 Spherical Wave Fuctios Cosider solvig ψ + k ψ = 0 i spherical coordiates z φ θ r y x Spherical Wave Fuctios (cot.) I spherical coordiates we
More informationMath 21B-B - Homework Set 2
Math B-B - Homework Set Sectio 5.:. a) lim P k= c k c k ) x k, where P is a partitio of [, 5. x x ) dx b) lim P k= 4 ck x k, where P is a partitio of [,. 4 x dx c) lim P k= ta c k ) x k, where P is a partitio
More information1 Review and Overview
CS229T/STATS231: Statistical Learig Theory Lecturer: Tegyu Ma Lecture #12 Scribe: Garrett Thomas, Pega Liu October 31, 2018 1 Review a Overview Recall the GAN setup: we have iepeet samples x 1,..., x raw
More information