Parabola and Catenary Equations for Conductor Height Calculation
|
|
- Mavis Bryan
- 5 years ago
- Views:
Transcription
1 ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 6 (), nr. 3 9 Prbol nd Ctenr Equtions for Condutor Height Clultion Alen HATIBOVIC Abstrt This pper presents new equtions for ondutor height lultion bsed on the known miml sg. The equtions n be diretl pplied for plotting the ondutor urve fter ompleted sg-tension lultion for plnning nd designing overhed lines. The bsi differenes between the prbol nd the tenr urves re lso disussed. The vlidit of the shown formuls hs lso been proved b some numeril emples. Kewords: trnsendentl funtions, lgebri funtions, sg, overhed lines, leveled spn, inlined spn. Introdution The origin of the - oordinte sstem for sg-tension lultion is generll put t the top of the ondutor urve []. However, it is more dvntgeous to set the origin to the bottom of the left-hnd side support of the spn for defining the eqution for the ondutor height. B this w the -oordinte presents the height of the ondutor urve relted to -is. The distne of ondutor s rbitrr point from the -is is then the distne from the left-hnd side support. This pper shows both the tenr nd the prbol equtions under this ondition. The tul ondutor urve n be desribed b tenr funtion, i.e. b the hperboli osine funtion, whih belongs to the group of the trnsendentl funtions. The prbol urve n be desribed b qudrti funtion, whih belongs to the group of the lgebri funtions. The bsi differene between the lgebri nd trnsendentl funtions is in their eponent. While the eponent of the lgebri funtions is permnent, it is vring in the se of trnsendentl funtions. Despite the ft tht the the prbol nd the tenr funtions re mthemtill quite different, their urves n be ver similr. Therefore, when plnning overhed eletril lines the tenr is often pproimted b the prbol, sine it results in signifint simplifition of the lultion. It is eptble, beuse in most of the ses the differene between the tenr nd the prbol is negligible []. It is generll epted ft in the literture tht the ondutor urve n be pproimted b prbol for spns up to bout metres. For longer spns the et tnr bsed lultion shll be used, beuse the differene between the tenr nd the prbol urves nnot be ignored.. Ctenr eqution (;) Figure. Grphs of tenr urves osh(/) osh(/) - The top of the tenr urve is loted t the point (,) s it is shown in Figure, see urve. Its bsi eqution is the following [3]: ( ) h / () Alen HATIBOVIC, Senior engineer for eletril network development, EDF DÉMÁSZ Hálózt, Szeged, Hungr
2 ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 6 (), nr. 3 For the determintion of the tenr eqution it is neessr to know its prmeter ( >). It n be obtined b using the sg-tension lultion for plnning overhed lines. When the origin of the oordinte sstem is t the top of the tenr urve, the eqution beomes: ( / ) - h () After this step the origin shll be moved to the bottom of the left-hnd side support of the spn ording to Figure. This figure shows n inlined spn with verte t point. Figure. Ctenr urve in n inlined spn The eqution for ondutor height will be defined b the following dt: prmeter of the tenr urve oordinte of the verte point oordinte of the verte point The finl tenr eqution for ondutor height is (3). Its eponentil form is given b (). The intervl is lws [,], where is the spn length. (3) h e e Both of the two equtions re universl, sine these re vlid in the se of n tpe of inlined spn (h <h or h >h ) nd in the se of leveled spn (h h h), too. So b (3) or () it is possible to lulte the ondutor height t n point of the spn. It n be seen in () tht the vrible is loted t the eponent, whih is n importnt feture of trnsendentl funtions. In the se of leveled spns the equtions (3) nd () get simpler forms given in (5) nd (6) on () the bsis of the known miml sg b m. The height of the two supports is denoted with h. / h h b (5) e / e / m h b 3. Prbol eqution There is ver importnt defferene between the prbol nd the tenr onerning the miml sg of the ondutor. Sine the miml sg of the prbol is lws loted t midspn, both in the se of leveled nd inlined spns, the miml sg of the tenr in n inlined spn is slightl moved towrd the higher suspension point of the ondutor. This is one of the resons of the simpliit of the prbol bsed lultion in omprison to the tenr bsed lultion. When the miml sg of the prbol is known it is possible to obtin the prbol eqution for the ondutor height, sine the prbol is defined b n three points of its urve. In the se of the tenr it is more diffiult, beuse it is neessr to know both the prmeter of the tenr nd the oordintes of the verte point ording to the bse eqution (3). So the knowledge of the miml sg is not enough for the determintion of the tenr. The stndrd eqution of the prbol is (7), A B C where A, B nd C re the oeffiients of the prbol. The oeffiient A defines the shpe of the prbol urve. If A>, the urve hs the minimum [], [5], nd if A<, the urve hs the mimum. It will be mthemtill proved lter tht in the se of the eqution for the ondutor height the oeffiient A is positive. m 3.. Prbol eqution b three points Figure 3 demonstrtes the pplied method for n inlined spn with h <h. Points A nd B re the suspension points of the ondutor, while C is the ondutor point t mid-spn. The prmeters shown in Figure 3 re the following: spn length h height of the left-hnd side suspension point h height of the right-hnd side susp. point oordinte of the verte point oordinte of the verte point (6) (7)
3 ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 6 (), nr. 3 h h A(;h ) C / Figure 3. Prbol urve in n inlined spn with h <h B( ;h ) Sine the urve is prbol, the miml sg b m is loted t /. The left-hnd side nd the right-hnd side suspension points A(;h ) nd B(;h ) re lws known points. The third neessr point C is obtined b the known miml sg (8). The vlue of the miml sg n be obtined from the sg-tension lultion. h h C ; b m Bsed on the three points of the prbol the sstem of three lgebri equtions (9)-() n be written b utilizing of the stndrd eqution of the prbol (8): C h (9) A B C A (8) h () ( / ) B ( / ) C ( h h )/ bm () Writing these equtions in the mtri form P Q nd using the Crmer s rule [6], [7] to find the solution, the unknown oeffiients of the prbol A nd B n be obtined b: j j det( P )/det( P) ( j,, 3 ) () h h h h bm b A (3) m B h h b h h h h b m m () After the substitution of the oeffiients A, B, C into (7), the eqution for the ondutor height gets its finl form (5): bm h h bm h (5) This is universl eqution of the prbol, sine it is usble for both leveled nd inlined spns. In the se of the leveled spn the eqution (5) hnges into (6) nd the oeffiients hnge into (7). bm bm h bm bm A B C h (6) (7) There is ver importnt onsequene from (5) nd (6): beside the sme spn length nd miml sg both in leveled nd inlined spns, the oeffiient A of the prbol does not hnge. The vlidit nd usbilit of (5) nd (6) will be proved in the following three emples, one for leveled nd two for inlined spns. Emple. Leveled spn (h h h) m; h8m; b m 8m; ()? (8) [m] [m] Figure. Prbol urve from emple. (leveled spn)
4 ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 6 (), nr. 3 Emple. Inlined spn (h <h ) m; h 8m; h 3m; b m 8m; ()? (9) [m] [m] Figure 5. Prbol urve from emple. (inlined spn with h <h ) Emple 3. Inlined spn (h >h ) m; h 3m; h 8m; b m 8m; ()? 8 3 () [m] [m] Figure 6. Prbol urve from emple 3. (inlined spn with h >h ) The verte of the urve is shown in eh lst three figures nd the will be needed in the net prgrph 3.. With the help of the previous three emples it hs been proved tht the eqution for the ondutor height (5) is orret nd universl, sine it n be used in eh tpe of the spn. Therefore, we do not p ttention to either h <h or h >h, beuse the input dt is the sme in eh se of the tsks. We hve lso seen tht in the se of the eqution for the ondutor height, the oeffiient A of the prbol is lws positive, sine the spn length nd the miml sg of the prbol re lso positive. Therefore, the ondutor urve hs the minimum. Let us mention tht the verte of the prbol nd the lowest point of the ondutor re generll the sme point (point ) nd [,], like in eh of the previous figures. Figure 7. shows one rre se of the inlined spn when the verte point is out of the spn, i.e. [,]. In this se the lowest point of the ondutor (point M) is equl with the lower suspension point of the spn. For the pproprite presenttion of this rre se [8] the prbol urve is shown in n intervl [, ]. Figure 7. Inlined spn with M 3.. Prbol eqution in the verte form Beside the stndrd eqution of the prbol (7) its verte form is lso often used (). Eqution (7) n be trnsformed into form () b using (): ( ) A B A A AC B A () () Aording to () the oordintes of the verte point n be defined s (3) nd (): B h h A bm (3) AC B h h h bm () A bm Substituting (3) nd () into (), the verte form of the eqution for the ondutor height (5) is obtined: b m h h h bm b m h h bm (5)
5 ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 6 (), nr. 3 3 In the se of the leveled spn the verte oordintes beome / nd h b m, so the previous eqution hnges into (6): b m h b (6) B using eqution (5) it is es to lulte the oordintes of the verte point. So, b equtions (6) nd (5) we n now write the equtions in the verte form of the prbol from emples nr., nd 3, s it is shown below: m Emple. Leveled spn (h h h) Verte point (;) ( ) 8 (7) Emple. Inlined spn (h <h ) Verte point (5;6) ( 5) 6 8 (8) Emple 3. Inlined spn (h >h ) Verte point (5;6) ( 5) 6 8 (9) Sine the verte point nd its oordintes re lred shown in Figures., 5. nd 6., it is es to hek the results of the numeril lultion for defining nd. It n be onluded tht the results re orret Prbol eqution with prmeter p Hereunder shll be shown how the eqution for the ondutor height shll be defined b the prmeter of the prbol p insted of the miml sg b m. Knowing the mthemtil onnetion (3) between the oeffiient A of the prbol nd its prmeter p, the identit (3) is obtined. p A (3) A p bm bm p 8p (3) The prmeter p lws hs positive sign. Generll, the oeffiient A of the prbol n be positive or negtive, but sine in our se it is lws positive, there re not n problems with signs in (3). Using identit (3), the eqution for the ondutor height n be written both in stndrd form (3) nd in verte form (33) of the prbol eqution: h h h (3) p p p p p p ( h h ) ( h h ) h (33) In speil se when the suspension points on two supports re on the sme elevtion, the previous two equtions get simple forms s (3) nd (35): h p p p h 8p (3) (35). Usge of the prbol equtions for the ondutor height An of the bove presented equtions re useful for lulting the ondutor height nd for plotting the grph of the ondutor line s well. For solving other tsks the pproprite one hs to be hosen from the shown equtions depending on the tul tsk. Eqution (5) s the stndrd form of the prbol eqution is ver prtil for quik finding of the first derivtive () nd the seond derivtive () of funtion [9]. The oordinte of the verte n be obtined b solving (). After tht it is es to define the oordinte of the verte b solving ( ). This method will be shown numerill in eqution () from the emple 3. We lred know tht the verte point is (5;6), but it n be heked with the method below. ( ) 8 ' ( ) ' ( ) ( )
6 ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 6 (), nr. 3 ( ; ) (5;6) As it n be seen the sme results prove the vlidit of the shown method for using eqution (5) for defining the verte point. Eqution (5) s the verte form of the prbol eqution hs n even bigger usbilit thn the eqution (5). Beside the determintion of the verte point nd the oeffiient A, it n be used to reple the ondutor urve within the - oordinte sstem. A onrete emple of suh n pplition is the following formul [] for the determintion of the ondutor length in inlined spns on the bsis of the known miml sg of the prbol. L 6b rsh m m m m rsh m ( ) m ( ) ( ) m (36) For deriving formul (36) the urve hd to be ppropritel repled within the - oordinte sstem in order to mke possible the integrl lulus for the ondutor length. In the se of the leveled spn the previous formul hs muh simpler form: m L (37) bm bm b rsh m 5. Conlusions The eqution for ondutor height defined b the prbol eqution hs more vritions thn the one obtined b the tenr eqution. The simpliit of defining the prbol eqution is prtl due to the ft tht the miml sg of the prbol is lws loted t mid-spn, either it is leveled or n inlined spn. In the se of tenr this rule is not vlid. As it is shown, the prbol gives possibilit to find solution in different ws, so it ensures n effetive w of heking the results. These re some of the resons for the frequent usge of the prbol, when the differene between the prbol nd the tenr is insignifint. Through the numeril emples we hve seen the w of obtining the eqution for the ondutor height in the se of n tpe of the spn. It hs lso been shown how to define the oordinte of the verte point. The verte is ver often equl with the lowest point of the ondutor nd in the se of the inlined spn it is one ritil point of the ondutor, so its heking is highl reommended. The speil se of the inlined spn hs lso been disussed, when the verte point is out of the spn. Using different vritions of the prbol eqution for the ondutor height the vlidit of shown equtions hs been proved through numeril emples. A ver importnt result of the shown 3 emples is tht in the se of the sme spn length nd the vlue of the miml sg b m, both in leveled nd inlined spn, the oeffiient A of the prbol does not hnge. This pper highlighted this importnt hrteristi of the prbol. Referenes [] CIGRÉ 3, Sg-tension lultion methods for overhed lines, CIGRÉ 7 [] HATIBOVIC A., Usge of Prbol Clultion for Plnning of Eletril Overhed Network, ENELKO onferene, Kolozsvár [3] PANSINI A., Eletril Distribution Engineering, The Firmont Press, In., 7 [] GUSTAFSON D., FRISK P. nd HUGHES J., College Algebr, CENGAGE Lerning [5] OBÁDOVICS GY., Mtemtik, SCOLAR Budpest [6] TURKINGTON D. A., Mátri Clulus & Zero- One Mtries, CAMBRIDGE 5 [7] GENTLE J. E., Mtri Algebr, Springer 7 [8] HATIBOVIC A., TOMIC M., Determintion of the lowest point of ondutor for inlined spns, CIGRÉ onferene, Srjevo [9] OBÁDOVICS GY., Felsőbb mtemtiki feldtgűjtemén, SCOLAR Budpest [] HATIBOVIC A., Integrl Clulus Usge for Condutor Length Determintion on the Bsis of Known Miml Sg of Prbol, Periodi Poltehni Eletril Engineering, Budpest Universit of Tehnolog nd Eonomis Alen HATIBOVIC Senior engineer for eletril network development, EDF DÉMÁSZ Hálózt, Szeged, Hungr
Core 2 Logarithms and exponentials. Section 1: Introduction to logarithms
Core Logrithms nd eponentils Setion : Introdution to logrithms Notes nd Emples These notes ontin subsetions on Indies nd logrithms The lws of logrithms Eponentil funtions This is n emple resoure from MEI
More information] dx (3) = [15x] 2 0
Leture 6. Double Integrls nd Volume on etngle Welome to Cl IV!!!! These notes re designed to be redble nd desribe the w I will eplin the mteril in lss. Hopefull the re thorough, but it s good ide to hve
More informationElectromagnetism Notes, NYU Spring 2018
Eletromgnetism Notes, NYU Spring 208 April 2, 208 Ation formultion of EM. Free field desription Let us first onsider the free EM field, i.e. in the bsene of ny hrges or urrents. To tret this s mehnil system
More informationLogarithms LOGARITHMS.
Logrithms LOGARITHMS www.mthletis.om.u Logrithms LOGARITHMS Logrithms re nother method to lulte nd work with eponents. Answer these questions, efore working through this unit. I used to think: In the
More informationGreen s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e
Green s Theorem. Let be the boundry of the unit squre, y, oriented ounterlokwise, nd let F be the vetor field F, y e y +, 2 y. Find F d r. Solution. Let s write P, y e y + nd Q, y 2 y, so tht F P, Q. Let
More informationSolutions to Assignment 1
MTHE 237 Fll 2015 Solutions to Assignment 1 Problem 1 Find the order of the differentil eqution: t d3 y dt 3 +t2 y = os(t. Is the differentil eqution liner? Is the eqution homogeneous? b Repet the bove
More informationIntegration. antidifferentiation
9 Integrtion 9A Antidifferentition 9B Integrtion of e, sin ( ) nd os ( ) 9C Integrtion reognition 9D Approimting res enlosed funtions 9E The fundmentl theorem of integrl lulus 9F Signed res 9G Further
More informationMath 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1)
Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte single-vrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion
More informationLecture 1 - Introduction and Basic Facts about PDEs
* 18.15 - Introdution to PDEs, Fll 004 Prof. Gigliol Stffilni Leture 1 - Introdution nd Bsi Fts bout PDEs The Content of the Course Definition of Prtil Differentil Eqution (PDE) Liner PDEs VVVVVVVVVVVVVVVVVVVV
More informationApplications of Definite Integral
Chpter 5 Applitions of Definite Integrl 5.1 Are Between Two Curves In this setion we use integrls to find res of regions tht lie between the grphs of two funtions. Consider the region tht lies between
More informationSolving Radical Equations
Solving dil Equtions Equtions with dils: A rdil eqution is n eqution in whih vrible ppers in one or more rdinds. Some emples o rdil equtions re: Solution o dil Eqution: The solution o rdil eqution is the
More informationPolynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230
Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given
More informationChapter Gauss Quadrature Rule of Integration
Chpter 7. Guss Qudrture Rule o Integrtion Ater reding this hpter, you should e le to:. derive the Guss qudrture method or integrtion nd e le to use it to solve prolems, nd. use Guss qudrture method to
More informationSECTION A STUDENT MATERIAL. Part 1. What and Why.?
SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are
More informationy z A left-handed system can be rotated to look like the following. z
Chpter 2 Crtesin Coördintes The djetive Crtesin bove refers to René Desrtes (1596 1650), who ws the first to oördintise the plne s ordered pirs of rel numbers, whih provided the first sstemti link between
More informationAQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions
Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic
More informationApplications of Definite Integral
Chpter 5 Applitions of Definite Integrl 5.1 Are Between Two Curves In this setion we use integrls to find res of regions tht lie between the grphs of two funtions. Consider the region tht lies between
More information( ) 1. 1) Let f( x ) = 10 5x. Find and simplify f( 2) and then state the domain of f(x).
Mth 15 Fettermn/DeSmet Gustfson/Finl Em Review 1) Let f( ) = 10 5. Find nd simplif f( ) nd then stte the domin of f(). ) Let f( ) = +. Find nd simplif f(1) nd then stte the domin of f(). ) Let f( ) = 8.
More informationINTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable
INTEGRATION NOTE: These notes re supposed to supplement Chpter 4 of the online textbook. 1 Integrls of Complex Vlued funtions of REAL vrible If I is n intervl in R (for exmple I = [, b] or I = (, b)) nd
More informationUniversity of Sioux Falls. MAT204/205 Calculus I/II
University of Sioux Flls MAT204/205 Clulus I/II Conepts ddressed: Clulus Textook: Thoms Clulus, 11 th ed., Weir, Hss, Giordno 1. Use stndrd differentition nd integrtion tehniques. Differentition tehniques
More informationTutorial Worksheet. 1. Find all solutions to the linear system by following the given steps. x + 2y + 3z = 2 2x + 3y + z = 4.
Mth 5 Tutoril Week 1 - Jnury 1 1 Nme Setion Tutoril Worksheet 1. Find ll solutions to the liner system by following the given steps x + y + z = x + y + z = 4. y + z = Step 1. Write down the rgumented mtrix
More informationT b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions.
Rel Vribles, Fll 2014 Problem set 5 Solution suggestions Exerise 1. Let f be bsolutely ontinuous on [, b] Show tht nd T b (f) P b (f) f (x) dx [f ] +. Conlude tht if f is in AC then it is the differene
More informationSection 3.6. Definite Integrals
The Clulus of Funtions of Severl Vribles Setion.6 efinite Integrls We will first define the definite integrl for funtion f : R R nd lter indite how the definition my be extended to funtions of three or
More informationThe study of dual integral equations with generalized Legendre functions
J. Mth. Anl. Appl. 34 (5) 75 733 www.elsevier.om/lote/jm The study of dul integrl equtions with generlized Legendre funtions B.M. Singh, J. Rokne,R.S.Dhliwl Deprtment of Mthemtis, The University of Clgry,
More informationA Mathematical Model for Unemployment-Taking an Action without Delay
Advnes in Dynmil Systems nd Applitions. ISSN 973-53 Volume Number (7) pp. -8 Reserh Indi Publitions http://www.ripublition.om A Mthemtil Model for Unemployment-Tking n Ation without Dely Gulbnu Pthn Diretorte
More informationAP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals
AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into
More informationSIMPLE NONLINEAR GRAPHS
S i m p l e N o n l i n e r G r p h s SIMPLE NONLINEAR GRAPHS www.mthletis.om.u Simple SIMPLE Nonliner NONLINEAR Grphs GRAPHS Liner equtions hve the form = m+ where the power of (n ) is lws. The re lle
More information6.5 Improper integrals
Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =
More informationElectromagnetic-Power-based Modal Classification, Modal Expansion, and Modal Decomposition for Perfect Electric Conductors
LIAN: EM-BASED MODAL CLASSIFICATION EXANSION AND DECOMOSITION FOR EC 1 Eletromgneti-ower-bsed Modl Clssifition Modl Expnsion nd Modl Deomposition for erfet Eletri Condutors Renzun Lin Abstrt Trditionlly
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1. 1 [(y ) 2 + yy + y 2 ] dx,
MATH3403: Green s Funtions, Integrl Equtions nd the Clulus of Vritions 1 Exmples 5 Qu.1 Show tht the extreml funtion of the funtionl I[y] = 1 0 [(y ) + yy + y ] dx, where y(0) = 0 nd y(1) = 1, is y(x)
More informationThe Ellipse. is larger than the other.
The Ellipse Appolonius of Perg (5 B.C.) disovered tht interseting right irulr one ll the w through with plne slnted ut is not perpendiulr to the is, the intersetion provides resulting urve (oni setion)
More informationMatrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix
tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri
More informationGRAND PLAN. Visualizing Quaternions. I: Fundamentals of Quaternions. Andrew J. Hanson. II: Visualizing Quaternion Geometry. III: Quaternion Frames
Visuliing Quternions Andrew J. Hnson Computer Siene Deprtment Indin Universit Siggrph Tutoril GRAND PLAN I: Fundmentls of Quternions II: Visuliing Quternion Geometr III: Quternion Frmes IV: Clifford Algers
More informationapproaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below
. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.
More informationH (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a.
Chpter Review 89 IGURE ol hord GH of the prol 4. G u v H (, ) (A) Use the distne formul to show tht u. (B) Show tht G nd H lie on the line m, where m ( )/( ). (C) Solve m for nd sustitute in 4, otining
More informationThe Double Integral. The Riemann sum of a function f (x; y) over this partition of [a; b] [c; d] is. f (r j ; t k ) x j y k
The Double Integrl De nition of the Integrl Iterted integrls re used primrily s tool for omputing double integrls, where double integrl is n integrl of f (; y) over region : In this setion, we de ne double
More information(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.
Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time
More informationReflection Property of a Hyperbola
Refletion Propert of Hperol Prefe The purpose of this pper is to prove nltill nd to illustrte geometrill the propert of hperol wherein r whih emntes outside the onvit of the hperol, tht is, etween the
More informationarxiv: v1 [math.ca] 21 Aug 2018
rxiv:1808.07159v1 [mth.ca] 1 Aug 018 Clulus on Dul Rel Numbers Keqin Liu Deprtment of Mthemtis The University of British Columbi Vnouver, BC Cnd, V6T 1Z Augest, 018 Abstrt We present the bsi theory of
More information18.06 Problem Set 4 Due Wednesday, Oct. 11, 2006 at 4:00 p.m. in 2-106
8. Problem Set Due Wenesy, Ot., t : p.m. in - Problem Mony / Consier the eight vetors 5, 5, 5,..., () List ll of the one-element, linerly epenent sets forme from these. (b) Wht re the two-element, linerly
More informationHyers-Ulam stability of Pielou logistic difference equation
vilble online t wwwisr-publitionsom/jns J Nonliner Si ppl, 0 (207, 35 322 Reserh rtile Journl Homepge: wwwtjnsom - wwwisr-publitionsom/jns Hyers-Ulm stbility of Pielou logisti differene eqution Soon-Mo
More informationTable of Content. c 1 / 5
Tehnil Informtion - t nd t Temperture for Controlger 03-2018 en Tble of Content Introdution....................................................................... 2 Definitions for t nd t..............................................................
More informationReference : Croft & Davison, Chapter 12, Blocks 1,2. A matrix ti is a rectangular array or block of numbers usually enclosed in brackets.
I MATRIX ALGEBRA INTRODUCTION TO MATRICES Referene : Croft & Dvison, Chpter, Blos, A mtri ti is retngulr rr or lo of numers usull enlosed in rets. A m n mtri hs m rows nd n olumns. Mtri Alger Pge If the
More information21.1 Using Formulae Construct and Use Simple Formulae Revision of Negative Numbers Substitution into Formulae
MEP Jmi: STRAND G UNIT 1 Formule: Student Tet Contents STRAND G: Alger Unit 1 Formule Student Tet Contents Setion 1.1 Using Formule 1. Construt nd Use Simple Formule 1.3 Revision of Negtive Numers 1.4
More informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
More informationECE Microwave Engineering. Fall Prof. David R. Jackson Dept. of ECE. Notes 8. Waveguides Part 5: Coaxial Cable
ECE 5317-6351 Mirowve Engineering Fll 17 Prof. Dvid R. Jkson Dept. of ECE Notes 8 Wveguides Prt 5: Coil Cle 1 Coil Line: TEM Mode To find the TEM mode fields, we need to solve: ( ρφ) Φ, ; Φ ( ) V Φ ( )
More informationVIBRATION ANALYSIS OF AN ISOLATED MASS WITH SIX DEGREES OF FREEDOM Revision G
B Tom Irvine Emil: tom@virtiondt.om Jnur 8, 3 VIBRATION ANALYSIS OF AN ISOLATED MASS WITH SIX DEGREES OF FREEDOM Revision G Introdution An vionis omponent m e mounted with isoltor grommets, whih t s soft
More information(h+ ) = 0, (3.1) s = s 0, (3.2)
Chpter 3 Nozzle Flow Qusistedy idel gs flow in pipes For the lrge vlues of the Reynolds number typilly found in nozzles, the flow is idel. For stedy opertion with negligible body fores the energy nd momentum
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More informationBefore we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!
Nme: Algebr II Honors Pre-Chpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble
More informationf (x)dx = f(b) f(a). a b f (x)dx is the limit of sums
Green s Theorem If f is funtion of one vrible x with derivtive f x) or df dx to the Fundmentl Theorem of lulus, nd [, b] is given intervl then, ording This is not trivil result, onsidering tht b b f x)dx
More informationm m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r
CO-ORDINTE GEOMETR II I Qudrnt Qudrnt (-.+) (++) X X - - - 0 - III IV Qudrnt - Qudrnt (--) - (+-) Region CRTESIN CO-ORDINTE SSTEM : Retngulr Co-ordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr
More informationLecture Summaries for Multivariable Integral Calculus M52B
These leture summries my lso be viewed online by liking the L ion t the top right of ny leture sreen. Leture Summries for Multivrible Integrl Clulus M52B Chpter nd setion numbers refer to the 6th edition.
More informationCalculus Cheat Sheet. Integrals Definitions. where F( x ) is an anti-derivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx
Clulus Chet Sheet Integrls Definitions Definite Integrl: Suppose f ( ) is ontinuous Anti-Derivtive : An nti-derivtive of f ( ) on [, ]. Divide [, ] into n suintervls of is funtion, F( ), suh tht F = f.
More informationMATH Final Review
MATH 1591 - Finl Review November 20, 2005 1 Evlution of Limits 1. the ε δ definition of limit. 2. properties of limits. 3. how to use the diret substitution to find limit. 4. how to use the dividing out
More informationMore Properties of the Riemann Integral
More Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologil Sienes nd Deprtment of Mthemtil Sienes Clemson University Februry 15, 2018 Outline More Riemnn Integrl Properties The Fundmentl
More informationLesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.
27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we
More information16z z q. q( B) Max{2 z z z z B} r z r z r z r z B. John Riley 19 October Econ 401A: Microeconomic Theory. Homework 2 Answers
John Riley 9 Otober 6 Eon 4A: Miroeonomi Theory Homework Answers Constnt returns to sle prodution funtion () If (,, q) S then 6 q () 4 We need to show tht (,, q) S 6( ) ( ) ( q) q [ q ] 4 4 4 4 4 4 Appeling
More informationMathematics SKE: STRAND F. F1.1 Using Formulae. F1.2 Construct and Use Simple Formulae. F1.3 Revision of Negative Numbers
Mthemtis SKE: STRAND F UNIT F1 Formule: Tet STRAND F: Alger F1 Formule Tet Contents Setion F1.1 Using Formule F1. Construt nd Use Simple Formule F1.3 Revision of Negtive Numers F1.4 Sustitution into Formule
More informationTHE EXTREMA OF THE RATIONAL QUADRATIC FUNCTION f(x)=(x 2 +ax+b)/(x 2 +cx+d) Larry Larson & Sally Keely
THE EXTREMA OF THE RATIONAL QUADRATIC FUNCTION f(x)(x +x+b)/(x +x+d) Lrry Lrson & Slly Keely Rtionl funtions nd their grphs re explored in most prelulus ourses. Mny websites provide intertive investigtions
More informationTopic 1 Notes Jeremy Orloff
Topic 1 Notes Jerem Orloff 1 Introduction to differentil equtions 1.1 Gols 1. Know the definition of differentil eqution. 2. Know our first nd second most importnt equtions nd their solutions. 3. Be ble
More informationLogarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100.
Logrithms. Logrithm is nother word for n inde or power. THIS IS A POWER STATEMENT BASE POWER FOR EXAMPLE : We lred know tht; = NUMBER 10² = 100 This is the POWER Sttement OR 2 is the power to which the
More information1 Part II: Numerical Integration
Mth 4 Lb 1 Prt II: Numericl Integrtion This section includes severl techniques for getting pproimte numericl vlues for definite integrls without using ntiderivtives. Mthemticll, ect nswers re preferble
More informationCo-ordinated s-convex Function in the First Sense with Some Hadamard-Type Inequalities
Int. J. Contemp. Mth. Sienes, Vol. 3, 008, no. 3, 557-567 Co-ordinted s-convex Funtion in the First Sense with Some Hdmrd-Type Inequlities Mohmmd Alomri nd Mslin Drus Shool o Mthemtil Sienes Fulty o Siene
More informationIntroduction to Olympiad Inequalities
Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd Am-Gm inequlity 2. Elementry inequlities......................
More informationMAT 403 NOTES 4. f + f =
MAT 403 NOTES 4 1. Fundmentl Theorem o Clulus We will proo more generl version o the FTC thn the textook. But just like the textook, we strt with the ollowing proposition. Let R[, ] e the set o Riemnn
More informationMathematics Number: Logarithms
plce of mind F A C U L T Y O F E D U C A T I O N Deprtment of Curriculum nd Pedgogy Mthemtics Numer: Logrithms Science nd Mthemtics Eduction Reserch Group Supported y UBC Teching nd Lerning Enhncement
More informationMath 131. Numerical Integration Larson Section 4.6
Mth. Numericl Integrtion Lrson Section. This section looks t couple of methods for pproimting definite integrls numericlly. The gol is to get good pproimtion of the definite integrl in problems where n
More informationUnit 1 Exponentials and Logarithms
HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 1 Unit 1 Eponentils nd Logrithms (2) Eponentil Functions (3) The number e (4) Logrithms (5) Specil Logrithms (7) Chnge of Bse Formul (8) Logrithmic Functions (10)
More information8.3 THE HYPERBOLA OBJECTIVES
8.3 THE HYPERBOLA OBJECTIVES 1. Define Hperol. Find the Stndrd Form of the Eqution of Hperol 3. Find the Trnsverse Ais 4. Find the Eentriit of Hperol 5. Find the Asmptotes of Hperol 6. Grph Hperol HPERBOLAS
More informationSECTION 9-4 Translation of Axes
9-4 Trnsltion of Aes 639 Rdiotelescope For the receiving ntenn shown in the figure, the common focus F is locted 120 feet bove the verte of the prbol, nd focus F (for the hperbol) is 20 feet bove the verte.
More informationPolynomials. Polynomials. Curriculum Ready ACMNA:
Polynomils Polynomils Curriulum Redy ACMNA: 66 www.mthletis.om Polynomils POLYNOMIALS A polynomil is mthemtil expression with one vrile whose powers re neither negtive nor frtions. The power in eh expression
More informationLine Integrals and Entire Functions
Line Integrls nd Entire Funtions Defining n Integrl for omplex Vlued Funtions In the following setions, our min gol is to show tht every entire funtion n be represented s n everywhere onvergent power series
More informationPart 4. Integration (with Proofs)
Prt 4. Integrtion (with Proofs) 4.1 Definition Definition A prtition P of [, b] is finite set of points {x 0, x 1,..., x n } with = x 0 < x 1
More informationDorf, R.C., Wan, Z. T- Equivalent Networks The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
orf, R.C., Wn,. T- Equivlent Networks The Eletril Engineering Hndook Ed. Rihrd C. orf Bo Rton: CRC Press LLC, 000 9 T P Equivlent Networks hen Wn University of Cliforni, vis Rihrd C. orf University of
More informationThis enables us to also express rational numbers other than natural numbers, for example:
Overview Study Mteril Business Mthemtis 05-06 Alger The Rel Numers The si numers re,,3,4, these numers re nturl numers nd lso lled positive integers. The positive integers, together with the negtive integers
More informationILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS
ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS Dvid Miller West Virgini University P.O. BOX 6310 30 Armstrong Hll Morgntown, WV 6506 millerd@mth.wvu.edu
More informationLearning Objectives of Module 2 (Algebra and Calculus) Notes:
67 Lerning Ojetives of Module (Alger nd Clulus) Notes:. Lerning units re grouped under three res ( Foundtion Knowledge, Alger nd Clulus ) nd Further Lerning Unit.. Relted lerning ojetives re grouped under
More informationThe Algebra (al-jabr) of Matrices
Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he lgebr (l-jbr) of Mtrices lgebr s brnch of mthemtics is much broder thn elementry lgebr ll of us studied in our high school dys. In sense
More informationThe Riemann-Stieltjes Integral
Chpter 6 The Riemnn-Stieltjes Integrl 6.1. Definition nd Eistene of the Integrl Definition 6.1. Let, b R nd < b. ( A prtition P of intervl [, b] is finite set of points P = { 0, 1,..., n } suh tht = 0
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationAP Calculus AB Unit 4 Assessment
Clss: Dte: 0-04 AP Clulus AB Unit 4 Assessment Multiple Choie Identify the hoie tht best ompletes the sttement or nswers the question. A lultor my NOT be used on this prt of the exm. (6 minutes). The slope
More informationANALYSIS AND MODELLING OF RAINFALL EVENTS
Proeedings of the 14 th Interntionl Conferene on Environmentl Siene nd Tehnology Athens, Greee, 3-5 Septemer 215 ANALYSIS AND MODELLING OF RAINFALL EVENTS IOANNIDIS K., KARAGRIGORIOU A. nd LEKKAS D.F.
More informationProving the Pythagorean Theorem
Proving the Pythgoren Theorem W. Bline Dowler June 30, 2010 Astrt Most people re fmilir with the formul 2 + 2 = 2. However, in most ses, this ws presented in lssroom s n solute with no ttempt t proof or
More informationSection 4.4. Green s Theorem
The Clulus of Funtions of Severl Vriles Setion 4.4 Green s Theorem Green s theorem is n exmple from fmily of theorems whih onnet line integrls (nd their higher-dimensionl nlogues) with the definite integrls
More informationCHENG Chun Chor Litwin The Hong Kong Institute of Education
PE-hing Mi terntionl onferene IV: novtion of Mthemtis Tehing nd Lerning through Lesson Study- onnetion etween ssessment nd Sujet Mtter HENG hun hor Litwin The Hong Kong stitute of Edution Report on using
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 informationON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs
ON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs A.I. KECHRINIOTIS AND N.D. ASSIMAKIS Deprtment of Eletronis Tehnologil Edutionl Institute of Lmi, Greee EMil: {kehrin,
More informationfor all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx
Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion
More information13.3. The Area Bounded by a Curve. Introduction. Prerequisites. Learning Outcomes
The Are Bounded b Curve 3.3 Introduction One of the importnt pplictions of integrtion is to find the re bounded b curve. Often such n re cn hve phsicl significnce like the work done b motor, or the distnce
More informationIntroduction. Definition of Hyperbola
Section 10.4 Hperbols 751 10.4 HYPERBOLAS Wht ou should lern Write equtions of hperbols in stndrd form. Find smptotes of nd grph hperbols. Use properties of hperbols to solve rel-life problems. Clssif
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 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 informationEllipses. The second type of conic is called an ellipse.
Ellipses The seond type of oni is lled n ellipse. Definition of Ellipse An ellipse is the set of ll points (, y) in plne, the sum of whose distnes from two distint fied points (foi) is onstnt. (, y) d
More informationFundamental Theorem of Calculus
Fundmentl Theorem of Clculus Recll tht if f is nonnegtive nd continuous on [, ], then the re under its grph etween nd is the definite integrl A= f() d Now, for in the intervl [, ], let A() e the re under
More informationWaveguide Circuit Analysis Using FDTD
11/1/16 EE 533 Eletromgneti nlsis Using Finite Differene Time Domin Leture # Wveguide Ciruit nlsis Using FDTD Leture These notes m ontin oprighted mteril obtined under fir use rules. Distribution of these
More informationUNCORRECTED SAMPLE PAGES. Australian curriculum NUMBER AND ALGEBRA
7A 7B 7C 7D 7E 7F 7G 7H 7I 7J 7K Chpter Wht ou will lern 7Prols nd other grphs Eploring prols Skething prols with trnsformtions Skething prols using ftoristion Skething ompleting the squre Skething using
More informationTHE INFLUENCE OF MODEL RESOLUTION ON AN EXPRESSION OF THE ATMOSPHERIC BOUNDARY LAYER IN A SINGLE-COLUMN MODEL
THE INFLUENCE OF MODEL RESOLUTION ON AN EXPRESSION OF THE ATMOSPHERIC BOUNDARY LAYER IN A SINGLE-COLUMN MODEL P3.1 Kot Iwmur*, Hiroto Kitgw Jpn Meteorologil Ageny 1. INTRODUCTION Jpn Meteorologil Ageny
More informationF / x everywhere in some domain containing R. Then, + ). (10.4.1)
0.4 Green's theorem in the plne Double integrls over plne region my be trnsforme into line integrls over the bounry of the region n onversely. This is of prtil interest beuse it my simplify the evlution
More information5. Every rational number have either terminating or repeating (recurring) decimal representation.
CHAPTER NUMBER SYSTEMS Points to Rememer :. Numer used for ounting,,,,... re known s Nturl numers.. All nturl numers together with zero i.e. 0,,,,,... re known s whole numers.. All nturl numers, zero nd
More information