On the Pairs of Orthogonal Ruled Surfaces
|
|
- Juliet McDowell
- 6 years ago
- Views:
Transcription
1 EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5 No ISSN On the Pirs of Orthogonl Ruled Surfces Filiz KANBAY Deprtment of Mthemtics Fculty of Arts nd Science Yıldız Technicl Uniersity Dutpṣ Istnbul Turkey Abstrct. In this work in three dimensionl Eucliden spce E 3 by using the ruled Bonnet surfces which he been known up to now the problem of finding some pirs of orthogonl ruled surfces is exmined nd only one pir of orthogonl ruled surfces cn be obtined Mthemtics Subject Clssifictions: 53A05 Key Words nd Phrses: Orthogonl Surfce Bonnet Surfce Ruled surfce Weingrten surfce Tngentil surfce 1. Introduction The orthogonl surfces re relted to the problem of isometric representtion. These surfces lso ply n importnt role in ll questions connected with the study of the infinitesiml bending problem. Becse the infinitesiml bending problem is reduced the problem of finding the orthogonl surfces[5 7 8]. The orthogonl ruled surfces re significnt to inestigte the isometric representtion of the ruled surfces or the infinitesiml bending of ruled surfces. For this im we cn use the Bonnet ruled surfces to exmine pirs of orthogonl ruled surfces. Generlly Bonnet surfces he been clssified into three ctegories: The surfces of constnt men curture other thn the plnes nd spheres. The isometric Weingrten surfces of non-constnt men curture which re isometric to surfce of reolution. The surfces of non-constnt men curture tht dmit single non-triil isometry. In[2] by using the method gien in[4] some specil Bonnet ruled surfces of non-constnt men curture re gien. It is shown tht the ruled surfces which re formed by the binormls two cures whose curtures nd bsolute lue of torsions re the sme re the Bonnet pirs. Moreoer the ruled miniml Bonnet surfces nd the ruled Weingrten surfces re Emil ddress: ÒÝÝÐÞºÙºØÖ c 2012 EJPAM All rights resered.
2 F. Knby/ Eur. J. Pure Appl. Mth 5 ( indicted (the deelopble surfce re not inestigted. The tngentil Bonnet surfces s the deelopble Bonnet surfces re exmined in[3]. In this work the min gol is to find some ruled orthogonl surfces by using the ruled Bonnet surfces. Using this method it is shown tht the pirs of orthogonl surfces except one pir cn not be obtined. 2. Ruled Bonnet Surfces nd Some Orthogonl Pirs 2.1. Some Ruled Bonnet Surfces A ruled surfce in three dimensionl Eucliden spce E 3 cn be gien by the ectoril eqution X(u =r(+ut( (1 where r=r( is directrix cure T=T( is unit ector with T 2 (= 2 ( 0 T ( r (= b( The first fundmentl form of the ruled surfce (1 is T( r (=cosθ( (0 θ π where f = d f d (2 ds 2 = du cosθdud+( 2 u 2 + 2bu+1d 2 (3 TThe lue of u for the centrl point; the prmeter of the distributionβ re expressed s u=α(= b( 2 ( β= 1 2(tTT (4 Here the centrl point is the limit of the point in which genertor g 1 is met by the common perpendiculr of g 1 nd neighboring genertor g 2 s g 2 pproches g 1 oer ruled surfce; the prmeter of distribution is the limit of the rtio of the shortest distnce between two genertors nd their included ngle. Now let n orthogonl trjectory to the genertors be tken s directrix nd T 2 (=1 T 2 (=1. Such ruled surfce cn be obtined from (1 under the conditions[6 1]: T = 1 β (T r T = α β β 2 +α 2 r + β 2 +α 2 r T r = αt +βt T (5 N = 1 βw [(β2 +α 2 uαt +(α ur ] w= (u α 2 +β 2 (β 0 We recll tht the director-cone is the cone formed by drwing through point lines prllel to the genertors nd it is determined by the function D(= r T =(TT T (6 β
3 F. Knby/ Eur. J. Pure Appl. Mth 5 ( Becseκ 2 = 1+D 2 ndτ= D 1+D 2 re the curture nd the torsion of the unit sphericl cure x = T( which determines the director-cone[2]. The miniml surfces nd the isothermic Weingrten ruled surfces re gien by the equtionsα =β = D= 0 ndα =β = D = 0 respectiely[2 6]. If isometric representtion between two surfces preseres the principl curtures of these surfce these surfces re sid to be Bonnet surfces. In[4] in order to find Bonnet surfce method is gien. In ccording to this A-net on surfce such tht when this net is prmetrized the conditions E=G F= 0 M= c= const. 0 re stisfied is clled n A-net where E F G re the coefficients of the first fundmentl form of the surfce nd L M N re the coefficients of the second fundmentl form. And necessry nd sufficient condition for surfce to be Bonnet surfce is tht the surfce cn he n A-net. In[2] on the ruled surfce the genertors nd orthogonl trjectories form n A-net if nd only if the prmeter of the distributionβ nd the bsciss of the centrl pointαre constnts. And the prmeter of distribution nd the bsciss of the centrl point of surfce re constnt then the surfce is Bonnet surfce. The torsion of the striction line of such Bonnet surfce is constnt nd is equl to the reciprocl of the prmeter of distribution. And the ruled surfces which re formed by the binormls two cures whose curtures nd bsolute lue of torsions re the sme re the Bonnet pirs. This mens tht the ruled surfces gien by the equtions X(u =r(+ β sinhut( Y(u = r(+ β sinhut( (7 respectiely re Bonnet pirs. Hereβ 0 nd D( is n rbitrry function. In[4] the cses D(=const. nd D(=0re indicted. In the cse D(=const. the Bonnet pirs re obtined tht X(u = cos sin + ǫ sinhusin ǫ sinhucos ǫ + ( sinhu (8 Y(u = cos sin + ǫ sinh usin ǫ sinh ucos ǫ + ( sinh u
4 F. Knby/ Eur. J. Pure Appl. Mth 5 ( whereǫ= sgn(b=sgn(β. In the cse D(=0 X(u = β sinhucos β sinhusin β Y(u = (9 β sinhucos β sinhusin β re found. These solutions re lid for the non-deelopble surfces(β 0[2]. In[3] the tngentil deelopble surfces of the circulr helices(β= 0 re gien s n exmple of the ruled Bonnet pirs. These surfces re clculted s X(u = Y(u = cos sin + b + sin cos bu 2( + 2 +b rctn( cos 2( + 2 +b rctn( sin 2 +b 2 2 +b 2 b(+ 2(2 +b rctn( 2 +b sin + cos bu 2( + 2 +b rctn( 2( + 2 +b rctn( All ruled Bonnet pirs which he been known up to now cn be tken the equtions (7 (8 (9 ( Some Orthogonl Ruled Surfces If two surfces S( nd S(b gien by the equtions =(u nd b=b(u mp upon ech other such tht their liner elements re orthogonl t corresponding points d db=0 then these surfces re clled to be orthogonl surfces. Now we recll the reltionship between the orthogonl surfces nd the isometric surfces: if two surfces S(x nd S(y gien by the equtions x=x(u nd y=y(u re isometric(dx 2 = dy 2 two surfces written s S(x+y nd S(x y re orthogonl surfces d(x+y d(x y=0. By using two surfces S(x nd S(y which cn be mp isometriclly on the ech other the orthogonl surfces S( nd S(b gien by the equtions =x+y b=x y (11 cn be written. By using this method we cn get the orthogonl surfces esily. For this im we cn use the ruled surfces gien by the equtions (7 (8 (9 (10 to find orthogonl 2 +b 2 2 +b 2 (10
5 F. Knby/ Eur. J. Pure Appl. Mth 5 ( ruled surfces. But the process of using the ruled surfces is not s esy s the generl process. Since the eqution of ruled surfce consists of two prts(x(u =r(+ut( mostly one of the first(r( or second(t( prt cn nish nd the surfce becomes cure. Becse of this only one pirs of orthogonl surfces gien by x y z = = = cos + cos + 2(2 +b rctn( + sin + sin 2 + b 2 sin sin + 2(2 +b rctn( + cos + cos 2 + b 2 2b( 2 +b rctn( 2 +b b (2 +b rctn( 2 +b (2 +b rctn( 2 +b 2 2 +b 2 2bu (12 x y = = cos cos + 2(2 +b rctn( sin + sin 2 + b 2 sin + sin 2 +b 2 + 2(2 +b rctn( + 2(2 +b rctn( + cos cos 2 + b 2 2 +b (2 +b rctn( 2 +b 2 2 +b 2 z = 2b+ 2b(2 +b rctn( 2 +b 2 cn be found. Theorem 1. By using the Bonnet Ruled surfces; only one pir of orthogonl surfces cn be found. This surfces re generted by the tngentil deelopble surfces of the circulr helices.
6 REFERENCES 210 The infinitesiml bending problem is reduced the finding them[5 7 8]: the eqution x t (u t=x(u + ty(u (13 gies the infinitesiml bending surfces of the surfce S(x under the condition dx t 2 (u t=(dx(u + tdy(u 2 (14 where t is rel infinitesiml prmeter. Here the condition (14 is equilent to the condition dx dy=0. Consequently we gie the following corollry: Corollry 1. One of the surfces gien by the eqution (12 cn be infinitesiml deformtion by using the other one. References [1] L Eisenhrt. "A Tretise On The Differntil Geometry Of Cures And Surfces" Doer Publictions Inc. New York p [2] F Knby. "Bonnet ruled surfces" Act Mthemtic Sinic English Series Vol 21 No: [3] I Rssos. "Tngentil deelopble surfces s Bonnet surfces". Act Mthemtic Sinic English Series 15( [4] Z Soyuc.ok. "The problem of non-triil isometries of surfces presering principl curtures" Journl of Geometry Vol [5] Z Soyuc. ok. "Infinitesiml Deformtions of surfces nd the stress distribution on some membrnes under constnt inner pressure" Int. J. Engng Sci. Vol 34 N0 9 pp [6] F Urs. "Diferensiyel Geometri II Dersleri" Yıldız Teknik Üniersitesi Fen-Edebiyt Fkultesi Mtemtik Bölümü Syı:261. Istnbul [7] F Urs. "On the net of the principl stresses relted with the infinitesiml bending of surfces" Bulletin of the Technicl Uniersity of Istnbul Vol 49 No: [8] I Veku. "Generlized Anliytic Functions. Pergmon Pres Oxford
Numerical Linear Algebra Assignment 008
Numericl Liner Algebr Assignment 008 Nguyen Qun B Hong Students t Fculty of Mth nd Computer Science, Ho Chi Minh University of Science, Vietnm emil. nguyenqunbhong@gmil.com blog. http://hongnguyenqunb.wordpress.com
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationConducting Ellipsoid and Circular Disk
1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid,
More informationSection 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationMathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
More informationMath 6455 Oct 10, Differential Geometry I Fall 2006, Georgia Tech
Mth 6455 Oct 10, 2006 1 Differentil Geometry I Fll 2006, Georgi Tech Lecture Notes 12 Riemnnin Metrics 0.1 Definition If M is smooth mnifold then by Riemnnin metric g on M we men smooth ssignment of n
More informationg i fφdx dx = x i i=1 is a Hilbert space. We shall, henceforth, abuse notation and write g i f(x) = f
1. Appliction of functionl nlysis to PEs 1.1. Introduction. In this section we give little introduction to prtil differentil equtions. In prticulr we consider the problem u(x) = f(x) x, u(x) = x (1) where
More informationIntegration of tensor fields
Integrtion of tensor fields V. Retsnoi Abstrct. The im of this pper is to introduce the ide of integrtion of tensor field s reerse process to the Lie differentition. The definitions of indefinite nd definite
More informationClassification of Spherical Quadrilaterals
Clssifiction of Sphericl Qudrilterls Alexndre Eremenko, Andrei Gbrielov, Vitly Trsov November 28, 2014 R 01 S 11 U 11 V 11 W 11 1 R 11 S 11 U 11 V 11 W 11 2 A sphericl polygon is surfce homeomorphic to
More informationElementary Linear Algebra
Elementry Liner Algebr Anton & Rorres, 9 th Edition Lectre Set Chpter : Vectors in -Spce nd -Spce Chpter Content Introdction to Vectors (Geometric Norm of Vector; Vector Arithmetic Dot Prodct; Projections
More informationLecture 3: Curves in Calculus. Table of contents
Mth 348 Fll 7 Lecture 3: Curves in Clculus Disclimer. As we hve textook, this lecture note is for guidnce nd supplement only. It should not e relied on when prepring for exms. In this lecture we set up
More informationMAC-solutions of the nonexistent solutions of mathematical physics
Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements MAC-solutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE
More informationNotes on length and conformal metrics
Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued
More informationThe Third Motivation for Spherical Geometry
The Third Motivtion for Sphericl Geometry Yoichi Med med@keykiccu-tokicjp Deprtment of Mthemtics Toki University Jpn Astrct: Historiclly, sphericl geometry hs developed minly on the terrestril gloe nd
More information1.7 Geodesics. Recall. For curve α on surface M, α can be written as components tangent and normal to M as α = α tan + α nor where
1.7 Geodesics Note. A curve α(s) on surfce M cn curve in two different wys. First, α cn bend long with surfce M (the norml curvture discussed bove). Second, α cn bend within the surfce M (the geodesic
More informationPHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS
PHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS To strt on tensor clculus, we need to define differentition on mnifold.a good question to sk is if the prtil derivtive of tensor tensor on mnifold?
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationSpace Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.
Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)
More informationSome estimates on the Hermite-Hadamard inequality through quasi-convex functions
Annls of University of Criov, Mth. Comp. Sci. Ser. Volume 3, 7, Pges 8 87 ISSN: 13-693 Some estimtes on the Hermite-Hdmrd inequlity through qusi-convex functions Dniel Alexndru Ion Abstrct. In this pper
More informationWell Centered Spherical Quadrangles
Beiträge zur Algebr und Geometrie Contributions to Algebr nd Geometry Volume 44 (003), No, 539-549 Well Centered Sphericl Qudrngles An M d Azevedo Bred 1 Altino F Sntos Deprtment of Mthemtics, University
More informationDEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS
3 DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS This chpter summrizes few properties of Cli ord Algebr nd describe its usefulness in e ecting vector rottions. 3.1 De nition of Associtive
More information4. Calculus of Variations
4. Clculus of Vritions Introduction - Typicl Problems The clculus of vritions generlises the theory of mxim nd minim. Exmple (): Shortest distnce between two points. On given surfce (e.g. plne), nd the
More informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
More informationCurves. Differential Geometry Lia Vas
Differentil Geometry Li Vs Curves Differentil Geometry Introduction. Differentil geometry is mthemticl discipline tht uses methods of multivrible clculus nd liner lgebr to study problems in geometry. In
More informationarxiv: v1 [math.ra] 1 Nov 2014
CLASSIFICATION OF COMPLEX CYCLIC LEIBNIZ ALGEBRAS DANIEL SCOFIELD AND S MCKAY SULLIVAN rxiv:14110170v1 [mthra] 1 Nov 2014 Abstrct Since Leibniz lgebrs were introduced by Lody s generliztion of Lie lgebrs,
More information. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) =
Review of some needed Trig. Identities for Integrtion. Your nswers should be n ngle in RADIANS. rccos( 1 ) = π rccos( - 1 ) = 2π 2 3 2 3 rcsin( 1 ) = π rcsin( - 1 ) = -π 2 6 2 6 Cn you do similr problems?
More informationTHREE-DIMENSIONAL KINEMATICS OF RIGID BODIES
THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES 1. TRANSLATION Figure shows rigid body trnslting in three-dimensionl spce. Any two points in the body, such s A nd B, will move long prllel stright lines if
More informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
More informationJim Lambers MAT 280 Spring Semester Lecture 17 Notes. These notes correspond to Section 13.2 in Stewart and Section 7.2 in Marsden and Tromba.
Jim Lmbers MAT 28 Spring Semester 29- Lecture 7 Notes These notes correspond to Section 3.2 in Stewrt nd Section 7.2 in Mrsden nd Tromb. Line Integrls Recll from single-vrible clclus tht if constnt force
More informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
More informationPoint Lattices: Bravais Lattices
Physics for Solid Stte Applictions Februry 18, 2004 Lecture 7: Periodic Structures (cont.) Outline Review 2D & 3D Periodic Crystl Structures: Mthemtics X-Ry Diffrction: Observing Reciprocl Spce Point Lttices:
More informationAbstract inner product spaces
WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the
More informationSummary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
More informationCONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round
More informationPlane curvilinear motion is the motion of a particle along a curved path which lies in a single plane.
Plne curiliner motion is the motion of prticle long cured pth which lies in single plne. Before the description of plne curiliner motion in n specific set of coordintes, we will use ector nlsis to describe
More informationDEFINITION The inner product of two functions f 1 and f 2 on an interval [a, b] is the number. ( f 1, f 2 ) b DEFINITION 11.1.
398 CHAPTER 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 11.1 ORTHOGONAL FUNCTIONS REVIEW MATERIAL The notions of generlized vectors nd vector spces cn e found in ny liner lger text. INTRODUCTION The concepts
More informationWeek 10: Line Integrals
Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.
More informationGeometric Probabilities for an Arbitrary Convex Body of Revolution in E 3 and Certain Lattice
Int. J. Contemp. Mth. Sciences, Vol. 4, 29, no. 25, 12-128 Geometric Probbilities for n Arbitrry Convex Body of Revolution in E nd Certin Lttice Giuseppe Cristi Diprtimento di discipline economico ziendli
More informationMath 100 Review Sheet
Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
More informationPOLYPHASE CIRCUITS. Introduction:
POLYPHASE CIRCUITS Introduction: Three-phse systems re commonly used in genertion, trnsmission nd distribution of electric power. Power in three-phse system is constnt rther thn pulsting nd three-phse
More information{ } λ are real THE COSINE RULE II FOR A SPHERICAL TRIANGLE ON THE DUAL UNIT SPHERE S % 2 1. INTRODUCTION 2. DUAL NUMBERS AND DUAL VECTORS
Mthemticl nd Computtionl Applictions, Vol 10, No, pp 1-20, 2005 Assocition for Scientific Reserch THE COSINE RULE II FOR A SPHERICAL TRIANGLE ON THE DUAL UNIT SPHERE S % 2 M Kzz 1, H H Uğurlu 2 nd A Özdemir
More informationWhen e = 0 we obtain the case of a circle.
3.4 Conic sections Circles belong to specil clss of cures clle conic sections. Other such cures re the ellipse, prbol, n hyperbol. We will briefly escribe the stnr conics. These re chosen to he simple
More information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the
More informationKINEMATICS OF RIGID BODIES
KINEMTICS OF RIGID ODIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. Description
More informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
More informationCalculus of Variations
Clculus of Vritions Com S 477/577 Notes) Yn-Bin Ji Dec 4, 2017 1 Introduction A functionl ssigns rel number to ech function or curve) in some clss. One might sy tht functionl is function of nother function
More informationChapter 3. Vector Spaces
3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce
More informationLecture XVII. Vector functions, vector and scalar fields Definition 1 A vector-valued function is a map associating vectors to real numbers, that is
Lecture XVII Abstrct We introduce the concepts of vector functions, sclr nd vector fields nd stress their relevnce in pplied sciences. We study curves in three-dimensionl Eucliden spce nd introduce the
More informationAPPLICATIONS OF THE DEFINITE INTEGRAL
APPLICATIONS OF THE DEFINITE INTEGRAL. Volume: Slicing, disks nd wshers.. Volumes by Slicing. Suppose solid object hs boundries extending from x =, to x = b, nd tht its cross-section in plne pssing through
More informationIntegrals along Curves.
Integrls long Curves. 1. Pth integrls. Let : [, b] R n be continuous function nd let be the imge ([, b]) of. We refer to both nd s curve. If we need to distinguish between the two we cll the function the
More information(4.1) D r v(t) ω(t, v(t))
1.4. Differentil inequlities. Let D r denote the right hnd derivtive of function. If ω(t, u) is sclr function of the sclrs t, u in some open connected set Ω, we sy tht function v(t), t < b, is solution
More informationAbsolute values of real numbers. Rational Numbers vs Real Numbers. 1. Definition. Absolute value α of a real
Rtionl Numbers vs Rel Numbers 1. Wht is? Answer. is rel number such tht ( ) =. R [ ( ) = ].. Prove tht (i) 1; (ii). Proof. (i) For ny rel numbers x, y, we hve x = y. This is necessry condition, but not
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl o Inequlities in Pure nd Applied Mthemtics http://jipm.vu.edu.u/ Volume 6, Issue 4, Article 6, 2005 MROMORPHIC UNCTION THAT SHARS ON SMALL UNCTION WITH ITS DRIVATIV QINCAI ZHAN SCHOOL O INORMATION
More informationSolution to Fredholm Fuzzy Integral Equations with Degenerate Kernel
Int. J. Contemp. Mth. Sciences, Vol. 6, 2011, no. 11, 535-543 Solution to Fredholm Fuzzy Integrl Equtions with Degenerte Kernel M. M. Shmivnd, A. Shhsvrn nd S. M. Tri Fculty of Science, Islmic Azd University
More informationPlane curvilinear motion is the motion of a particle along a curved path which lies in a single plane.
Plne curiliner motion is the motion of prticle long cured pth which lies in single plne. Before the description of plne curiliner motion in n specific set of coordintes, we will use ector nlsis to describe
More informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
More information2. THE HEAT EQUATION (Joseph FOURIER ( ) in 1807; Théorie analytique de la chaleur, 1822).
mpc2w4.tex Week 4. 2.11.2011 2. THE HEAT EQUATION (Joseph FOURIER (1768-1830) in 1807; Théorie nlytique de l chleur, 1822). One dimension. Consider uniform br (of some mteril, sy metl, tht conducts het),
More informationA Note on Feng Qi Type Integral Inequalities
Int Journl of Mth Anlysis, Vol 1, 2007, no 25, 1243-1247 A Note on Feng Qi Type Integrl Inequlities Hong Yong Deprtment of Mthemtics Gungdong Business College Gungzhou City, Gungdong 510320, P R Chin hongyong59@sohucom
More informationUS01CMTH02 UNIT Curvature
Stu mteril of BSc(Semester - I) US1CMTH (Rdius of Curvture nd Rectifiction) Prepred by Nilesh Y Ptel Hed,Mthemtics Deprtment,VPnd RPTPScience College US1CMTH UNIT- 1 Curvture Let f : I R be sufficiently
More informationProblem Set 3 Solutions
Msschusetts Institute of Technology Deprtment of Physics Physics 8.07 Fll 2005 Problem Set 3 Solutions Problem 1: Cylindricl Cpcitor Griffiths Problems 2.39: Let the totl chrge per unit length on the inner
More information3.4 Conic sections. In polar coordinates (r, θ) conics are parameterized as. Next we consider the objects resulting from
3.4 Conic sections Net we consier the objects resulting from + by + cy + + ey + f 0. Such type of cures re clle conics, becuse they rise from ifferent slices through cone In polr coorintes r, θ) conics
More information( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that
Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we
More informationSet Integral Equations in Metric Spaces
Mthemtic Morvic Vol. 13-1 2009, 95 102 Set Integrl Equtions in Metric Spces Ion Tişe Abstrct. Let P cp,cvr n be the fmily of ll nonempty compct, convex subsets of R n. We consider the following set integrl
More information. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. cos(2θ) = sin(2θ) =.
Review of some needed Trig Identities for Integrtion Your nswers should be n ngle in RADIANS rccos( 1 2 ) = rccos( - 1 2 ) = rcsin( 1 2 ) = rcsin( - 1 2 ) = Cn you do similr problems? Review of Bsic Concepts
More informationMath Advanced Calculus II
Mth 452 - Advnced Clculus II Line Integrls nd Green s Theorem The min gol of this chpter is to prove Stoke s theorem, which is the multivrible version of the fundmentl theorem of clculus. We will be focused
More informationON THE FIXED POINTS OF AN AFFINE TRANSFORMATION: AN ELEMENTARY VIEW
ON THE FIXED POINTS OF AN AFFINE TRANSFORMATION: AN ELEMENTARY VIEW Abstrct. This note shows how the fixed points of n ffine trnsformtion in the plne cn be constructed by n elementry geometric method.
More informationFourier series. Preliminary material on inner products. Suppose V is vector space over C and (, )
Fourier series. Preliminry mteril on inner products. Suppose V is vector spce over C nd (, ) is Hermitin inner product on V. This mens, by definition, tht (, ) : V V C nd tht the following four conditions
More informationChapter 14. Matrix Representations of Linear Transformations
Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationTotal Score Maximum
Lst Nme: Mth 8: Honours Clculus II Dr. J. Bowmn 9: : April 5, 7 Finl Em First Nme: Student ID: Question 4 5 6 7 Totl Score Mimum 6 4 8 9 4 No clcultors or formul sheets. Check tht you hve 6 pges.. Find
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More informationDiophantine Steiner Triples and Pythagorean-Type Triangles
Forum Geometricorum Volume 10 (2010) 93 97. FORUM GEOM ISSN 1534-1178 Diophntine Steiner Triples nd Pythgoren-Type Tringles ojn Hvl bstrct. We present connection between Diophntine Steiner triples (integer
More informationPhysics 207 Lecture 5
Phsics 07 Lecture 5 Agend Phsics 07, Lecture 5, Sept. 0 Chpter 4 Kinemtics in or 3 dimensions Independence of, nd/or z components Circulr motion Cured pths nd projectile motion Frmes of reference dil nd
More informationPractice final exam solutions
University of Pennsylvni Deprtment of Mthemtics Mth 26 Honors Clculus II Spring Semester 29 Prof. Grssi, T.A. Asher Auel Prctice finl exm solutions 1. Let F : 2 2 be defined by F (x, y (x + y, x y. If
More informationLecture Outline. Dispersion Relation Electromagnetic Wave Polarization 8/7/2018. EE 4347 Applied Electromagnetics. Topic 3c
Course Instructor Dr. Rymond C. Rumpf Office: A 337 Phone: (915) 747 6958 E Mil: rcrumpf@utep.edu EE 4347 Applied Electromgnetics Topic 3c Wve Dispersion & Polriztion Wve Dispersion These notes & Polriztion
More informationKRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION
Fixed Point Theory, 13(2012), No. 1, 285-291 http://www.mth.ubbcluj.ro/ nodecj/sfptcj.html KRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION FULI WANG AND FENG WANG School of Mthemtics nd
More informationThree Wave Hypothesis, Gear Model and the Rest Mass
Three We Hypothesis, Ger Model nd the est Mss M. I. Snduk School of Engineering, Fculty of Engineering nd Physicl Science, OA, Uniersity of Surrey, Guildford Surrey GU 7XH, UK m.snduk@surrey.c.uk Abstrct:
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More informationConsequently, the temperature must be the same at each point in the cross section at x. Let:
HW 2 Comments: L1-3. Derive the het eqution for n inhomogeneous rod where the therml coefficients used in the derivtion of the het eqution for homogeneous rod now become functions of position x in the
More informationFirst midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009
Mth 3B Review Steve 18 Mrch 2009 About the finl Fridy Mrch 20, 3pm-6pm, Lkretz 110 No notes, no book, no clcultor Ten questions Five review questions (Chpters 6,7,8) Five new questions (Chpters 9,10) No
More informationKINEMATICS OF RIGID BODIES
KINEMTICS OF RIGI OIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. escription
More information1 Line Integrals in Plane.
MA213 thye Brief Notes on hpter 16. 1 Line Integrls in Plne. 1.1 Introduction. 1.1.1 urves. A piece of smooth curve is ssumed to be given by vector vlued position function P (t) (or r(t) ) s the prmeter
More informationMA Handout 2: Notation and Background Concepts from Analysis
MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More informationalong the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate
L8 VECTOR EQUATIONS OF LINES HL Mth - Sntowski Vector eqution of line 1 A plne strts journey t the point (4,1) moves ech hour long the vector. ) Find the plne s coordinte fter 1 hour. b) Find the plne
More information( ) ( ) Chapter 5 Diffraction condition. ρ j
Grdute School of Engineering Ngo Institute of Technolog Crstl Structure Anlsis Tkshi Id (Advnced Cermics Reserch Center) Updted Nov. 3 3 Chpter 5 Diffrction condition In Chp. 4 it hs been shown tht the
More informationThe Hadamard s inequality for quasi-convex functions via fractional integrals
Annls of the University of Criov, Mthemtics nd Computer Science Series Volume (), 3, Pges 67 73 ISSN: 5-563 The Hdmrd s ineulity for usi-convex functions vi frctionl integrls M E Özdemir nd Çetin Yildiz
More informationLYAPUNOV-TYPE INEQUALITIES FOR NONLINEAR SYSTEMS INVOLVING THE (p 1, p 2,..., p n )-LAPLACIAN
Electronic Journl of Differentil Equtions, Vol. 203 (203), No. 28, pp. 0. ISSN: 072-669. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu LYAPUNOV-TYPE INEQUALITIES FOR
More informationContinuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationLine Integrals. Partitioning the Curve. Estimating the Mass
Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to
More informationMATH , Calculus 2, Fall 2018
MATH 36-2, 36-3 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly
More informationVariational problems of some second order Lagrangians given by Pfaff forms
Vritionl problems of some second order Lgrngins given by Pfff forms P. Popescu M. Popescu Abstrct. In this pper we study the dynmics of some second order Lgrngins tht come from Pfff forms i.e. differentil
More information4 VECTORS. 4.0 Introduction. Objectives. Activity 1
4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply
More informationSome Results on Cubic Residues
Interntionl Journl of Algebr, Vol. 9, 015, no. 5, 45-49 HIKARI Ltd, www.m-hikri.com htt://dx.doi.org/10.1988/ij.015.555 Some Results on Cubic Residues Dilek Nmlı Blıkesir Üniversiresi Fen-Edebiyt Fkültesi
More informationNUMERICAL INTEGRATION
NUMERICAL INTEGRATION How do we evlute I = f (x) dx By the fundmentl theorem of clculus, if F (x) is n ntiderivtive of f (x), then I = f (x) dx = F (x) b = F (b) F () However, in prctice most integrls
More information