Mannheim curves in the threedimensional sphere


 Emery Waters
 2 years ago
 Views:
Transcription
1 Mannheim curves in the threeimensional sphere anju Kahraman, Mehmet Öner Manisa Celal Bayar University, Faculty of Arts an Sciences, Mathematics Department, Muraiye Campus, 5, Muraiye, Manisa, urkey. s: Abstract Mannheim curves are efine for immerse curves in imensional sphere S. he efinition is given by consiering the geoesics of S. First, two special geoesics, calle principal normal geoesic an binormal geoesic, of S are efine by using Frenet vectors of a curve immerse in S. Later, the curve is calle a Mannheim curve if there exits another curve in S such that the principal normal geoesics of coincie with the binormal geoesics of. It is obtaine that if an form a Mannheim pair then there exist a constant λ an a nonconstant function µ such that λκ + µτ = where κ, τ are the curvatures of. Moreover, the relation between a Mannheim curve immerse in S an a generalize Mannheim curve in E is obtaine an a table containing comparison of Bertran an Mannheim curves in S is introuce. MSC: 5A Key wors: Spherical curves; generalize Mannheim curves; geoesics.. Introuction an Preliminaries Mannheim curves an Bertran curves are the most fascinating subject of the curve pairs efine by some relationships between two space curves. Mannheim curves are first efine by A. Mannheim in 878 [6]. In the Eucliean space E, Mannheim curves are characterize as a kin of corresponing relation between two curves, such that the binormal vector fiels of coincie with the principal normal vector fiels of. hen an are calle Mannheim curve an Mannheim partner curve, respectively [9]. he main result for a curve to have a Mannheim partner curve in E is that there exists a constant λ such that τ κ = ( + λ τ ) hols, where κ, τ an s are the curvatures an arc length parameter of, s λ respectively [9]. Mannheim curves have been stuie by many mathematicians. Blum stuie a remarkable class of Mannheim curves []. In [7], Matsua an Yorozu have given a efinition of generalize Mannheim curve in Eucliean space an introuce some characterizations an examples of generalize Mannheim curves. Later, Choi, Kang an Kim have efine Mannheim curves in  imensional Riemannian manifol []. Moreover, in the same paper, they have stuie Mannheim curves in imensional space forms. Another type of associate curves is Bertran curves which first efine by French mathematician SaintVenant in 85 by the property that two curves have common principal normal vector fiels at the corresponing points of curves [8]. Lucas an OrtegaYagües have consiere Bertran curves in the threeimensional sphere S [5]. hey have given another S
2 efinition for space curves to be Bertran curves immerse in S an they have come to the result that a curve with curvatures κ, τ immerse in S is a Bertran curve if an only if either τ an is a curve in some unit twoimensional sphere constants λ, µ such that λκ + µτ = [5]. S () or there exit two In this stuy, we efine Mannheim curves in S by efining some special geoesics relate to the Frenet vectors of a curve immerse in S. We show that the angle between the tangent vector fiels of Mannheim curves is not constant while it is constant for Bertran curves. Moreover, we obtaine that a curve with curvatures κ, τ immerse in S is a Mannheim curve if there exist a constant λ an a nonconstant function µ such that λκ + µτ =. We want to pointe out that this property hols for Bertran curves uner the conition that both λ an µ are constants.. Mannheim curves in the threeimensional sphere Before giving the main subject, first we give the following ata relate to the curves immerse in S. For this section, we refer the reaer to ref. [,5]. Let S ( r ) enote the threeimensional sphere in R of raius r, efine by S ( r) = ( x, x, x, x ) R xi = r, r >. i= Let = ( t) : I R S ( r) be an arclength parameterize immerse curve in the sphere S ( r ) an let {, N, B } an enotes the Frenet frame of an the LeviCivita connection of S ( r ), respectively. hen, Frenet formulae of is given by = κ N N = κ + τ B B = τ N where κ an τ enote the curvature an torsion of, respectively. If Civita connection of R, then the Gauss formula gives X = X X,, r for any tangent vector fiel X χ( ). In particular, we have = κ N r N = κ + τ B B = τ N. A curve ( t) in stans for the Levi S ( r ) is calle a plane curve if it lies in a totally geoesic twoimensional sphere S S which means that a curve ( t) in torsion τ is zero at all points [5]. S ( r ) is calle a plane curve if an only if its
3 Let ( t) an ( t) be two immerse curve in {, N, B } S ( r ) with Frenet frames {, N, B } an, respectively. A geoesic curve in S ( r ) starting at any point ( t) of an efine as u u t ( u) = cos ( t) + r sin B ( t), u R, r r is calle the binormal geoesic of an, similarly, a geoesic curve in S ( r ) starting at any point ( t) of an efine as u u t ( u) = cos ( t) + r sin N ( t), u R, r r is calle the principal normal geoesic of in S ( r ). Let be a regular smooth curve in Eucliean space E efine by arclength parameter s. he curve is calle a special Frenet curve if there exist three smooth functions k, k, k on e, e, e, e along the curve such that these satisfy the following an smooth frame fiel { } properties: i) he formulas of FrenetSerret hols: e = e = k e e = k e + k e e = k e + k e e = k e where the prime (') enotes ifferentiation with respect to s. ii) he frame fiel {,,, } e e e e is orthonormal an has positive orientation. iii) he functions k an k are positive an the function k oesn t vanish. iv) he functions k, k an k are calle the first, the secon an the thir curvature e, e, e, e is calle the Frenet frame fiel functions of, respectively. he frame fiel { } on []. A special Frenet curve in E is a generalize Mannheim curve if there exists a special Frenet curve ˆ in E such that the first normal line at each point of is inclue in the plane generate by the secon normal line an thir normal line of ˆ at corresponing point uner a bijection φ from to ˆ. he curve ˆ is calle the generalize Mannheim mate curve of [7]. Now, we can introuce the main subject. First, we give the following efinition. Definition.. A curve in S ( r ) with nonzero curvature κ is sai to be a Mannheim curve if there exists another immerse curve = ( σ ) : J IR S ( r) an a onetoone corresponence between an such that the principal normal geoesics of coincie with
4 the binormal geoesics of at corresponing points. We will say that is a Mannheim partner curve of ; the curves an are calle a pair of Mannheim curves. From this efinition it is clear that Mannheim partner curve can not a plane curve since the efinition given by consiering the binormal vector fiel of. For simplicity, we consier that the raius of the sphere S is an the curves taken on S are parameterize by the arclength parameter. Let an ( σ ) be a pair of Mannheim curves. From Definition., we have a ifferentiable function a( s ) such that ( ) ( ) ( ) s( σ ) = cos a ( σ ) sin a B ( σ ) () where {, N, B } enotes the Frenet frame along ( ) σ, ( s( )) σ is the point in corresponing to ( σ ) an a( s ) is calle the angle function between the irection vectors an ( σ ). he function ( s ) is calle istance function in S an measures the istance between the points ( s( σ )) an ( σ ). Now, we can give the following proposition. Proposition.. Let an be a pair of Mannheim curves in S. In that case, we have the followings a) he angle function a( s ) is constant. b) he istance function ( s ) is constant. c) he angle θ between the tangent vector fiels at corresponing points is not constant. ) he angle between the binormal vectors fiels at corresponing points is constant. Proof. a) Since principal normal geoesic of an binormal geoesic of are common at corresponing points, we have s ( u) = B an s ( u) = N () u u u= u= a an then we obtain N = sin a ( σ ) + cos a B ( σ ) () ( ) ( ) where {, N, B } enotes the Frenet frame of. From (), the tangent vector to is given by ( s( σ )) = a sin ( a ) ( σ ) + cos ( a ) ( σ ) s () a cos a B ( σ ) + sin a τ N ( σ ) ( ) ( ) where s enotes ifferentiation with respect to s. Since, ( s( σ )) = s ( σ ) ( s( σ )) (5) σ we get = ( s( σ )), N = a ( sin ( a ) cos ( a ) ) (6) σ which gives that a =, i.e., a( s ) is constant.
5 b) Without loss of generality, for the angle function it can be taken as a π. hen the istance function ( s ) is given by = min{ a( s ), π a}, which is a constant function since a( s ) is constant. c) Let θ = θ ( σ ) enotes the angle between tangent vectors an, i.e., ( ) s( σ ), ( σ ) = cos θ ( σ ). Differentiating the left sie of this equality, it follows ( s ( σ )), ( σ ) = s ( σ ) κ N, +, κ N. (7) σ Moreover, from () an (5), we have ( s( σ )) = ( cos a + τ sin a N ) (8) s ( σ ) Writing (), () an (8) in (7), it follows κ τ sin a θ = s ( σ )sin θ ( σ ). (9) Since is not a plane curve from (9), it is clear that θ is not a constant. ) Using equality () an (), we can write B = sin a s( σ ) + cos a N ( σ ). () ( ) Since B, B =, the angle between binormal vectors is constant. σ heorem.. Let an be a pair of Mannheim curves in hol: tanθ a) τ = tan a τ sin a cosθ = cos a κ sin a sinθ b) ( ) c) ) θ a κ a a cos = cos sin cos sin θ = τ τ sin a S. hen the following equalities Proof. a) aking the covariant erivative in () an using (5), we obtain ( s( σ )) = cos θ s ( σ ) ( σ ) + sin θ s ( σ ) N ( σ ). () σ On the other han, by using Frenet equations we have ( s( σ )) = cos a ( σ ) + τ sin a N ( σ ) () σ where a = a is constant. he last two equations lea to s ( σ ) cosθ = cos a () s ( σ )sinθ = τ sin a () from which we conclue (a). b) We nee to write the Frenet frame of in terms of the Frenet frame of : ( ) ( ) ( σ ) = cos a s( σ ) + sin a N s( σ ) (5) 5
6 ( ) ( ) ( ) ( ) ( ) ( ) ( σ ) = cos θ s( σ ) + sin θ B s( σ ) (6) N ( σ ) = sin θ s( σ ) cos θ B s( σ ) (7) B ( σ ) = sin a s( σ ) + cos a N s( σ ). (8) From (a), it follows σ (s) cosθ = cos a κ sin a (9) σ (s)sinθ = τ sin a () which gives us (b). c) It is a consequence of Eqs. () an (9); cos θ = cos a κ sin a cos a. () ) Similarly, from Eqs. () an (), we have esire equality sin θ = τ τ sin a. () From (), one can consier that is a plane curve if an only if θ = or σ (s) =. Since σ is arc length parameter, it cannot be constant. Similarly, from Proposition.. (c), θ is not a constant. hen we can give the following corollary. Corollary.. A Mannheim curve cannot be a plane curve. heorem.. Relationship between arclength parameters of curves an is given by s cos a cosθ τ sin a sinθ σ = +. () Proof. If equations (6) an (7) are written in (), we get equation (). heorem.. Let an be a pair of Mannheim curves in an a nonconstant function µ such that S. hen there exists a constant λ = λκ + µτ. () Proof. Multiplying (9) an () by sinθ an cosθ, respectively, an equalizing obtaine results, gives that sinθ cos a = sinθ sin aκ + sin a cosθτ. (5) Writing λ = tan a an µ = tan a cotθ, from last equality we have = λκ + µτ. Corollary.. Both curvature κ an torsion τ of a Mannheim curve cannot be constant. Proof. Since is a Mannheim curve Eq. () hols. Let now assume that the curvature κ be a n constant. hen, from () we have τ = which is not a constant since µ is not a constant, µ where n = λκ is a real constant. Similarly, if it is assume that the torsion τ is constant, then by a similar way it is seen that κ is not constant. 6
7 heorem.. Let ( t) be a Mannheim curve in regular ifferentiable mapping s = s( t) with s ( t) S with constant curvature. hen there exists a > such that the curve ( ) t ( ( )) = t B s u u is an arclength parametrize generalize Mannheim curve. Proof. By ifferentiating the curve ( t) three times an using Frenet formulae, we have κ = ε s τ, κ = s κ, κ = ε s, ε = ±. (6) (See [5]). From heorem., there exist constant λ an a nonconstant µ such that = λκ + µτ hols. Let signs of λ an τ be same an consier a function s( t ) such that λτ s ( t) =, (7) τ + κ where λ is a nonzero real constant. Defining a constant by ε c =, λ from Eqs. ()(6), we see that κ = c( κ + κ ) (8) hols for all s. hen from [7, heorem.], we have that ( t) is a generalize Mannheim curve. By consiering these characterizations an results obtaine for Bertran curves in [,5], we can give the following table giving the comparison of Bertran an Mannheim curves. t Characterizations Bertran Curves Mannheim Curves Angel between tangent vector fiels constant nonconstant Angel between binormal vector fiels constant constant Main Characterization λκ + µτ = is a Bertran curve in S if an only if there exits two constant λ an µ such that λκ + µτ =. is a Bertran curve in S if there exit a constant λ an a nonconstant function µ such that λκ + µτ = Curves, Curvatures an can be plane curves. an cannot be plane curves. Both κ an τ can be Both κ an τ cannot be constants. constants able. Comparison of Bertran an Mannheim curves in S. Some examples Example.. (ccrcurves) A C special Frenet curve on S is sai to be a ccrcurve on S if κ its intrinsic curvature ratio is a constant number []. Let now etermine special ccrcurve τ on S which is also Mannheim curve. First assume that is a ccrcurve with nonconstant 7
8 curvature an nonconstant torsion. hen, we have κ = cτ for a nonzero constant c. Writing this equality in () gives us c κ =, τ =. tan a( c + cot θ ) tan a( c + cot θ ) hen we conclue that the ccrcurve on curve. Example.. (Conical helix) A twiste curve in S given by curvatures as given above is a Mannheim S with nonconstant curvatures is sai to be conical helix if both the curvature raius κ an the torsion raius τ evolve linearly along the curve [5]. hen the curvature an torsion of curve are given by δ κ =, τ = s + r s + r, respectively, where r, r, an δ are constants. aking = δ, we see that () hols for a constant λ = δ an a nonconstant function Mannheim curve. µ = s + r + r s + r r r ( ) ( ) s + r, i.e., is a Example.. (General helix) A twiste curve in S is a general helix if there exists a constant b such that τ = bκ ± []. Let now etermine general helices in S which are also µ Mannheim curves. Writing the conition τ = bκ ± in (), it follows κ =. hen we λ + bµ µ b( µ ) have that a general helix in S with curvatures κ =, τ = ± is a Mannheim λ + bµ λ + bµ curve. Example.. (Curve with constant curvatures) Consier C curve on S () given by the parametrization = cos s, sin s, cos s, sin s for all s R. he curvatures are compute as κ = an τ = ε, ε = ± []. hen from Corollary., is not a Mannheim curve. References [] Barros, M., General helices an a theorem of Lancret, Proceeings of the American Mathematical Society 5 (997) [] Blum, R., A Remarkable class of Mannheimcurves, Cana. Math. Bull., 9(966), 8. [] Choi, J., Kang,. an Kim, Y., Mannheim Curves in Dimensional Space Forms, Bull. Korean Math. Soc. 5() ()
9 [] Kim, C.Y., Park, J.H., Yorozu, S., Curves on the unit sphere S () in the Eucliean space R, Bull. Korean Math. Soc., 5(5) () [5] Lucas, P., OrtegaYagües, J., Bertran Curves in the threeimensional sphere, Journal of Geometry an Physics, 6 () 9 9. [6] Mannheim, A., Paris C.R. 86 (878) [7] Matsua, H., Yorozu, S., On Generalize Mannheim Curves in Eucliean space, Nihonkai Math. J., (9) 56. [8] SaintVenant, J.C., Mémoire sur les lignes courbes non planes, Journal Ecole Polytechnique (85) 76. [9] Wang, F., Liu, H., Mannheim partner curves in Eucliean space, Mathematics in Practice an heory, 7() (7) . [] Wong, Y.C., Lai, H.F., A critical examination of the theory of curves in three imensional ifferential geometry, ohoku Math. J. 9 (967). 9
Eikonal Slant Helices and Eikonal Darboux Helices In 3Dimensional Riemannian Manifolds
Eikonal Slant Helices and Eikonal Darboux Helices In Dimensional Riemannian Manifolds Mehmet Önder a, Evren Zıplar b, Onur Kaya a a Celal Bayar University, Faculty of Arts and Sciences, Department of
More informationOn the Developable Mannheim Offsets of Timelike Ruled Surfaces
On the Developable Mannheim Offsets of Timelike Ruled Surfaces Mehmet Önder *, H Hüseyin Uğurlu ** * Celal Bayar University, Faculty of Science and Arts, Department of Mathematics, Muradiye Campus, Muradiye,
More informationCURVES: VELOCITY, ACCELERATION, AND LENGTH
CURVES: VELOCITY, ACCELERATION, AND LENGTH As examples of curves, consier the situation where the amounts of ncommoities varies with time t, qt = q 1 t,..., q n t. Thus, the amount of the commoities are
More information88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a
88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small
More informationInverse Trig Functions
Inverse Trig Functions c A Math Support Center Capsule February, 009 Introuction Just as trig functions arise in many applications, so o the inverse trig functions. What may be most surprising is that
More informationChapter 2. Parameterized Curves in R 3
Chapter 2. Parameterized Curves in R 3 Def. A smooth curve in R 3 is a smooth map σ : (a, b) R 3. For each t (a, b), σ(t) R 3. As t increases from a to b, σ(t) traces out a curve in R 3. In terms of components,
More informationRelations among Frenet Apparatus of Spacelike Bertrand WCurve Couples in Minkowski Spacetime
International Mathematical Forum, 3, 2008, no. 32, 15751580 Relations among Frenet Apparatus of Spacelike Bertrand WCurve Couples in Minkowski Spacetime Suha Yilmaz Dokuz Eylul University, Buca Educational
More informationLecture L253D Rigid Body Kinematics
J. Peraire, S. Winall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L253D Rigi Boy Kinematics In this lecture, we consier the motion of a 3D rigi boy. We shall see that in the general threeimensional
More informationGiven three vectors A, B, andc. We list three products with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B);
1.1.4. Prouct of three vectors. Given three vectors A, B, anc. We list three proucts with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B); a 1 a 2 a 3 (A B) C = b 1 b 2 b 3 c 1 c 2 c 3 where the
More informationSmarandache Curves in Minkowski Spacetime
International J.Math. Combin. Vol.3 (2008), 555 Smarandache Curves in Minkowski Spacetime Melih Turgut and Süha Yilmaz (Department of Mathematics of Buca Educational Faculty of Dokuz Eylül University,
More informationHere the units used are radians and sin x = sin(x radians). Recall that sin x and cos x are defined and continuous everywhere and
Lecture 9 : Derivatives of Trigonometric Functions (Please review Trigonometry uner Algebra/Precalculus Review on the class webpage.) In this section we will look at the erivatives of the trigonometric
More informationarcsine (inverse sine) function
Inverse Trigonometric Functions c 00 Donal Kreier an Dwight Lahr We will introuce inverse functions for the sine, cosine, an tangent. In efining them, we will point out the issues that must be consiere
More informationExponential Functions: Differentiation and Integration. The Natural Exponential Function
46_54.q //4 :59 PM Page 5 5 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions Section 5.4 f () = e f() = ln The inverse function of the natural logarithmic function is the natural eponential
More informationTHE HELICOIDAL SURFACES AS BONNET SURFACES
Tδhoku Math. J. 40(1988) 485490. THE HELICOIDAL SURFACES AS BONNET SURFACES loannis M. ROUSSOS (Received May 11 1987) 1. Introduction. In this paper we deal with the following question: which surfaces
More informationMeasures of distance between samples: Euclidean
4 Chapter 4 Measures of istance between samples: Eucliean We will be talking a lot about istances in this book. The concept of istance between two samples or between two variables is funamental in multivariate
More informationSOME NOTES ON THE HYPERBOLIC TRIG FUNCTIONS SINH AND COSH
SOME NOTES ON THE HYPERBOLIC TRIG FUNCTIONS SINH AND COSH Basic Definitions In homework set # one of the questions involves basic unerstaning of the hyperbolic functions sinh an cosh. We will use this
More informationM147 Practice Problems for Exam 2
M47 Practice Problems for Exam Exam will cover sections 4., 4.4, 4.5, 4.6, 4.7, 4.8, 5., an 5.. Calculators will not be allowe on the exam. The first ten problems on the exam will be multiple choice. Work
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More informationNotes on tangents to parabolas
Notes on tangents to parabolas (These are notes for a talk I gave on 2007 March 30.) The point of this talk is not to publicize new results. The most recent material in it is the concept of Bézier curves,
More informationOn the Frenet Frame and a Characterization of Spacelike InvoluteEvolute Curve Couple in Minkowski Spacetime
International Mathematical Forum, 3, 008, no. 16, 793801 On the Frenet Frame and a Characterization of Spacelike InvoluteEvolute Curve Couple in Minkowski Spacetime Melih Turgut Dokuz Eylul University,
More informationLagrange s equations of motion for oscillating centralforce field
Theoretical Mathematics & Applications, vol.3, no., 013, 99115 ISSN: 1799687 (print), 1799709 (online) Scienpress Lt, 013 Lagrange s equations of motion for oscillating centralforce fiel A.E. Eison
More informationThe Quick Calculus Tutorial
The Quick Calculus Tutorial This text is a quick introuction into Calculus ieas an techniques. It is esigne to help you if you take the Calculus base course Physics 211 at the same time with Calculus I,
More informationW CURVES IN MINKOWSKI SPACE TIME
Novi Sad J Math Vol 3, No, 00, 5565 55 W CURVES IN MINKOWSKI SPACE TIME Miroslava Petrović Torgašev 1, Emilija Šućurović 1 Abstract In this paper we complete a classification of W curves in Minkowski
More informationFactoring Dickson polynomials over finite fields
Factoring Dickson polynomials over finite fiels Manjul Bhargava Department of Mathematics, Princeton University. Princeton NJ 08544 manjul@math.princeton.eu Michael Zieve Department of Mathematics, University
More informationPythagorean Triples Over Gaussian Integers
International Journal of Algebra, Vol. 6, 01, no., 5564 Pythagorean Triples Over Gaussian Integers Cheranoot Somboonkulavui 1 Department of Mathematics, Faculty of Science Chulalongkorn University Bangkok
More informationarxiv:1309.1857v3 [grqc] 7 Mar 2014
Generalize holographic equipartition for FriemannRobertsonWalker universes WenYuan Ai, Hua Chen, XianRu Hu, an JianBo Deng Institute of Theoretical Physics, LanZhou University, Lanzhou 730000, P.
More informationCHAPTER 5 : CALCULUS
Dr Roger Ni (Queen Mary, University of Lonon)  5. CHAPTER 5 : CALCULUS Differentiation Introuction to Differentiation Calculus is a branch of mathematics which concerns itself with change. Irrespective
More informationOberwolfach Preprints
Oberwolfach Preprints OWP 200701 Gerhar Huisken Geometric Flows an 3Manifols Mathematisches Forschungsinstitut Oberwolfach ggmbh Oberwolfach Preprints (OWP) ISSN 18647596 Oberwolfach Preprints (OWP)
More informationElliptic Functions sn, cn, dn, as Trigonometry W. Schwalm, Physics, Univ. N. Dakota
Elliptic Functions sn, cn, n, as Trigonometry W. Schwalm, Physics, Univ. N. Dakota Backgroun: Jacobi iscovere that rather than stuying elliptic integrals themselves, it is simpler to think of them as inverses
More informationnparameter families of curves
1 nparameter families of curves For purposes of this iscussion, a curve will mean any equation involving x, y, an no other variables. Some examples of curves are x 2 + (y 3) 2 = 9 circle with raius 3,
More information19.2. First Order Differential Equations. Introduction. Prerequisites. Learning Outcomes
First Orer Differential Equations 19.2 Introuction Separation of variables is a technique commonly use to solve first orer orinary ifferential equations. It is socalle because we rearrange the equation
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:1512:05. Exam 1 will be based on: Sections 12.112.5, 14.114.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationNEARFIELD TO FARFIELD TRANSFORMATION WITH PLANAR SPIRAL SCANNING
Progress In Electromagnetics Research, PIER 73, 49 59, 27 NEARFIELD TO FARFIELD TRANSFORMATION WITH PLANAR SPIRAL SCANNING S. Costanzo an G. Di Massa Dipartimento i Elettronica Informatica e Sistemistica
More informationMath 230.01, Fall 2012: HW 1 Solutions
Math 3., Fall : HW Solutions Problem (p.9 #). Suppose a wor is picke at ranom from this sentence. Fin: a) the chance the wor has at least letters; SOLUTION: All wors are equally likely to be chosen. The
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationBV has the bounded approximation property
The Journal of Geometric Analysis volume 15 (2005), number 1, pp. 17 [version, April 14, 2005] BV has the boune approximation property G. Alberti, M. Csörnyei, A. Pe lczyński, D. Preiss Abstract: We prove
More information9.3. Diffraction and Interference of Water Waves
Diffraction an Interference of Water Waves 9.3 Have you ever notice how people relaxing at the seashore spen so much of their time watching the ocean waves moving over the water, as they break repeately
More information2 HYPERBOLIC FUNCTIONS
HYPERBOLIC FUNCTIONS Chapter Hyperbolic Functions Objectives After stuying this chapter you shoul unerstan what is meant by a hyperbolic function; be able to fin erivatives an integrals of hyperbolic functions;
More informationLecture L6  Intrinsic Coordinates
S. Widnall, J. Peraire 16.07 Dynamics Fall 2009 Version 2.0 Lecture L6  Intrinsic Coordinates In lecture L4, we introduced the position, velocity and acceleration vectors and referred them to a fixed
More informationThe Inverse Trigonometric Functions
The Inverse Trigonometric Functions These notes amplify on the book s treatment of inverse trigonometric functions an supply some neee practice problems. Please see pages 543 544 for the graphs of sin
More informationChapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis
Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >
More informationLecture 13: Differentiation Derivatives of Trigonometric Functions
Lecture 13: Differentiation Derivatives of Trigonometric Functions Derivatives of the Basic Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the
More informationA Generalization of Sauer s Lemma to Classes of LargeMargin Functions
A Generalization of Sauer s Lemma to Classes of LargeMargin Functions Joel Ratsaby University College Lonon Gower Street, Lonon WC1E 6BT, Unite Kingom J.Ratsaby@cs.ucl.ac.uk, WWW home page: http://www.cs.ucl.ac.uk/staff/j.ratsaby/
More information4. Important theorems in quantum mechanics
TFY4215 Kjemisk fysikk og kvantemekanikk  Tillegg 4 1 TILLEGG 4 4. Important theorems in quantum mechanics Before attacking threeimensional potentials in the next chapter, we shall in chapter 4 of this
More information10.2 Systems of Linear Equations: Matrices
SECTION 0.2 Systems of Linear Equations: Matrices 7 0.2 Systems of Linear Equations: Matrices OBJECTIVES Write the Augmente Matrix of a System of Linear Equations 2 Write the System from the Augmente Matrix
More informationIntegral Regular Truncated Pyramids with Rectangular Bases
Integral Regular Truncate Pyramis with Rectangular Bases Konstantine Zelator Department of Mathematics 301 Thackeray Hall University of Pittsburgh Pittsburgh, PA 1560, U.S.A. Also: Konstantine Zelator
More information1.7 Cylindrical and Spherical Coordinates
56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a twodimensional coordinate system in which the
More informationImage compression predicated on recurrent iterated function systems **
1 Image compression preicate on recurrent iterate function systems ** W. Metzler a, *, C.H. Yun b, M. Barski a a Faculty of Mathematics University of Kassel, Kassel, F. R. Germany b Faculty of Mathematics
More informationDigital barrier option contract with exponential random time
IMA Journal of Applie Mathematics Avance Access publishe June 9, IMA Journal of Applie Mathematics ) Page of 9 oi:.93/imamat/hxs3 Digital barrier option contract with exponential ranom time Doobae Jun
More informationPHY101 Electricity and Magnetism I Course Summary
TOPIC 1 ELECTROSTTICS PHY11 Electricity an Magnetism I Course Summary Coulomb s Law The magnitue of the force between two point charges is irectly proportional to the prouct of the charges an inversely
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS Contents 1. Moment generating functions 2. Sum of a ranom number of ranom variables 3. Transforms
More informationA neurogeometrical model for image completion and visual illusion
A neurogeometrical model for image completion and visual illusion Benedetta Franceschiello Advisors: A. Sarti, G. Citti CAMS (Unité Mixte CNRSEHESS), University of Bologna Midterm review meeting of MAnET
More informationEVOLUTION OF CONVEX LENSSHAPED NETWORKS UNDER CURVE SHORTENING FLOW
EVOLUTION OF CONVEX LENSSHAPED NETWORKS UNDER CURVE SHORTENING FLOW OLIVER C. SCHNÜRER, ABDERRAHIM AZOUANI, MARC GEORGI, JULIETTE HELL, NIHAR JANGLE, AMOS KOELLER, TOBIAS MARXEN, SANDRA RITTHALER, MARIEL
More informationMATH 125: LAST LECTURE
MATH 5: LAST LECTURE FALL 9. Differential Equations A ifferential equation is an equation involving an unknown function an it s erivatives. To solve a ifferential equation means to fin a function that
More informationLagrange Multipliers without Permanent Scarring
Lagrange Multipliers without Permanent Scarring Dan Klein Introuction This tutorial assumes that you want to know what Lagrange multipliers are, but are more intereste in getting the intuitions an central
More informationA new viewpoint on geometry of a lightlike hypersurface in a semieuclidean space
A new viewpoint on geometry of a lightlike hypersurface in a semieuclidean space Aurel Bejancu, Angel Ferrández Pascual Lucas Saitama Math J 16 (1998), 31 38 (Partially supported by DGICYT grant PB970784
More informationWitt#5e: Generalizing integrality theorems for ghostwitt vectors [not completed, not proofread]
Witt vectors. Part 1 Michiel Hazewinkel Sienotes by Darij Grinberg Witt#5e: Generalizing integrality theorems for ghostwitt vectors [not complete, not proofrea In this note, we will generalize most of
More informationLecture 17: Implicit differentiation
Lecture 7: Implicit ifferentiation Nathan Pflueger 8 October 203 Introuction Toay we iscuss a technique calle implicit ifferentiation, which provies a quicker an easier way to compute many erivatives we
More informationf(x + h) f(x) h as representing the slope of a secant line. As h goes to 0, the slope of the secant line approaches the slope of the tangent line.
Derivative of f(z) Dr. E. Jacobs Te erivative of a function is efine as a limit: f (x) 0 f(x + ) f(x) We can visualize te expression f(x+) f(x) as representing te slope of a secant line. As goes to 0,
More informationChapter 2 Kinematics of Fluid Flow
Chapter 2 Kinematics of Flui Flow The stuy of kinematics has flourishe as a subject where one may consier isplacements an motions without imposing any restrictions on them; that is, there is no nee to
More informationSolutions for Review Problems
olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector
More informationEulerSavary s Formula for the Planar Curves in Two Dimensional Lightlike Cone
International J.Math. Combin. Vol.1 (010), 11511 EulerSaary s Formula for the Planar Cures in Two Dimensional Lightlike Cone Handan BALGETİR ÖZTEKİN and Mahmut ERGÜT (Department of Mathematics, Fırat
More informationThe Math Circle, Spring 2004
The Math Circle, Spring 2004 (Talks by Gordon Ritter) What is NonEuclidean Geometry? Most geometries on the plane R 2 are noneuclidean. Let s denote arc length. Then Euclidean geometry arises from the
More informationPhysics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus
Chapter 1 Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. The main concepts that will be covered are: Coordinate transformations
More informationPurpose of the Experiments. Principles and Error Analysis. ε 0 is the dielectric constant,ε 0. ε r. = 8.854 10 12 F/m is the permittivity of
Experiments with Parallel Plate Capacitors to Evaluate the Capacitance Calculation an Gauss Law in Electricity, an to Measure the Dielectric Constants of a Few Soli an Liqui Samples Table of Contents Purpose
More informationFraternity & Sorority Academic Report Spring 2016
Fraternity & Sorority Academic Report Organization Overall GPA Triangle 1717 1 Delta Chi 88 12 100 2 Alpha Epsilon Pi 77 3 80 3 Alpha Delta Chi 28 4 32 4 Alpha Delta Pi 190190 4 Phi Gamma Delta 85 3
More informationAnswers to the Practice Problems for Test 2
Answers to the Practice Problems for Test 2 Davi Murphy. Fin f (x) if it is known that x [f(2x)] = x2. By the chain rule, x [f(2x)] = f (2x) 2, so 2f (2x) = x 2. Hence f (2x) = x 2 /2, but the lefthan
More informationIntrinsic Geometry. 2 + g 22. du dt. 2 du. Thus, very short distances ds on the surface can be approximated by
Intrinsic Geometry The Fundamental Form of a Surface Properties of a curve or surface which depend on the coordinate space that curve or surface is embedded in are called extrinsic properties of the curve.
More informationDepartment of Mathematical Sciences, University of Copenhagen. Kandidat projekt i matematik. Jens Jakob Kjær. Golod Complexes
F A C U L T Y O F S C I E N C E U N I V E R S I T Y O F C O P E N H A G E N Department of Mathematical Sciences, University of Copenhagen Kaniat projekt i matematik Jens Jakob Kjær Golo Complexes Avisor:
More informationSprings, Shocks and your Suspension
rings, Shocks an your Suspension y Doc Hathaway, H&S Prototype an Design, C. Unerstaning how your springs an shocks move as your race car moves through its range of motions is one of the basics you must
More informationFraternity & Sorority Academic Report Fall 2015
Fraternity & Sorority Academic Report Organization Lambda Upsilon Lambda 11 1 Delta Chi 77 19 96 2 Alpha Delta Chi 30 1 31 3 Alpha Delta Pi 134 62 196 4 Alpha Sigma Phi 37 13 50 5 Sigma Alpha Epsilon
More informationComponents of Acceleration
Components of Acceleration Part 1: Curvature and the Unit Normal In the last section, we explored those ideas related to velocity namely, distance, speed, and the unit tangent vector. In this section,
More information3. Right Triangle Trigonometry
. Right Triangle Trigonometry. Reference Angle. Radians and Degrees. Definition III: Circular Functions.4 Arc Length and Area of a Sector.5 Velocities . Reference Angle Reference Angle Reference angle
More informationThe wave equation is an important tool to study the relation between spectral theory and geometry on manifolds. Let U R n be an open set and let
1. The wave equation The wave equation is an important tool to stuy the relation between spectral theory an geometry on manifols. Let U R n be an open set an let = n j=1 be the Eucliean Laplace operator.
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More informationHigh accuracy approximation of helices by quintic curves
High accuracy approximation of helices by quintic curves Xunnian Yang Department of Mathematics,Zhejiang University Yuquan, Hangzhou 3127, People s Republic of China Submitted May 25; Revised November
More informationChapter 2 Review of Classical Action Principles
Chapter Review of Classical Action Principles This section grew out of lectures given by Schwinger at UCLA aroun 1974, which were substantially transforme into Chap. 8 of Classical Electroynamics (Schwinger
More informationIntroduction to Integration Part 1: AntiDifferentiation
Mathematics Learning Centre Introuction to Integration Part : AntiDifferentiation Mary Barnes c 999 University of Syney Contents For Reference. Table of erivatives......2 New notation.... 2 Introuction
More informationON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE
i93 c J SYSTEMS OF CURVES 695 ON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE BY C H. ROWE. Introduction. A system of co 2 curves having been given on a surface, let us consider a variable curvilinear
More informationMAT 1341: REVIEW II SANGHOON BAEK
MAT 1341: REVIEW II SANGHOON BAEK 1. Projections and Cross Product 1.1. Projections. Definition 1.1. Given a vector u, the rectangular (or perpendicular or orthogonal) components are two vectors u 1 and
More informationSome Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000)
Some Comments on the Derivative of a Vector with applications to angular momentum and curvature E. L. Lady (October 18, 2000) Finding the formula in polar coordinates for the angular momentum of a moving
More informationThe meanfield computation in a supermarket model with server multiple vacations
DOI.7/s66375 The meanfiel computation in a supermaret moel with server multiple vacations QuanLin Li Guirong Dai John C. S. Lui Yang Wang Receive: November / Accepte: 8 October 3 SpringerScienceBusinessMeiaNewYor3
More informationFINAL EXAM SOLUTIONS Math 21a, Spring 03
INAL EXAM SOLUIONS Math 21a, Spring 3 Name: Start by printing your name in the above box and check your section in the box to the left. MW1 Ken Chung MW1 Weiyang Qiu MW11 Oliver Knill h1 Mark Lucianovic
More informationTrigonometry LESSON TWO  The Unit Circle Lesson Notes
(cosθ, sinθ) Trigonometry Example 1 Introduction to Circle Equations. a) A circle centered at the origin can be represented by the relation x 2 + y 2 = r 2, where r is the radius of the circle. Draw each
More informationComputing Euler angles from a rotation matrix
Computing Euler angles from a rotation matrix Gregory G. Slabaugh Abstract This document discusses a simple technique to find all possible Euler angles from a rotation matrix. Determination of Euler angles
More informationDerivation of the Laplace equation
Derivation of the Laplace equation Svein M. Skjæveland October 19, 2012 Abstract This note presents a derivation of the Laplace equation which gives the relationship between capillary pressure, surface
More informationMathematics. Circles. hsn.uk.net. Higher. Contents. Circles 119 HSN22400
hsn.uk.net Higher Mathematics UNIT OUTCOME 4 Circles Contents Circles 119 1 Representing a Circle 119 Testing a Point 10 3 The General Equation of a Circle 10 4 Intersection of a Line an a Circle 1 5 Tangents
More informationVector Math Computer Graphics Scott D. Anderson
Vector Math Computer Graphics Scott D. Anderson 1 Dot Product The notation v w means the dot product or scalar product or inner product of two vectors, v and w. In abstract mathematics, we can talk about
More informationSensor Network Localization from Local Connectivity : Performance Analysis for the MDSMAP Algorithm
Sensor Network Localization from Local Connectivity : Performance Analysis for the MDSMAP Algorithm Sewoong Oh an Anrea Montanari Electrical Engineering an Statistics Department Stanfor University, Stanfor,
More informationMath 497C Sep 17, Curves and Surfaces Fall 2004, PSU
Math 497C Sep 17, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 4 1.9 Curves of Constant Curvature Here we show that the only curves in the plane with constant curvature are lines and circles.
More informationMATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3
MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................
More informationThe University of Kansas
All Greek Summary Rank Chapter Name Total Membership Chapter GPA 1 Beta Theta Pi 3.57 2 Chi Omega 3.42 3 Kappa Alpha Theta 3.36 4 Kappa Kappa Gamma 3.28 *5 Pi Beta Phi 3.27 *5 Gamma Phi Beta 3.27 *7 Alpha
More informationFOURIER TRANSFORM TERENCE TAO
FOURIER TRANSFORM TERENCE TAO Very broaly speaking, the Fourier transform is a systematic way to ecompose generic functions into a superposition of symmetric functions. These symmetric functions are usually
More information1 HighDimensional Space
Contents HighDimensional Space. Properties of HighDimensional Space..................... 4. The HighDimensional Sphere......................... 5.. The Sphere an the Cube in Higher Dimensions...........
More information9 Multiplication of Vectors: The Scalar or Dot Product
Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation
More informationv 1 v 3 u v = (( 1)4 (3)2, [1(4) ( 2)2], 1(3) ( 2)( 1)) = ( 10, 8, 1) (d) u (v w) = (u w)v (u v)w (Relationship between dot and cross product)
0.1 Cross Product The dot product of two vectors is a scalar, a number in R. Next we will define the cross product of two vectors in 3space. This time the outcome will be a vector in 3space. Definition
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationHow to Avoid the Inverse Secant (and Even the Secant Itself)
How to Avoi the Inverse Secant (an Even the Secant Itself) S A Fulling Stephen A Fulling (fulling@mathtamue) is Professor of Mathematics an of Physics at Teas A&M University (College Station, TX 7783)
More informationFACTORING IN THE HYPERELLIPTIC TORELLI GROUP
FACTORING IN THE HYPERELLIPTIC TORELLI GROUP TARA E. BRENDLE AND DAN MARGALIT Abstract. The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially
More information13.4. Curvature. Introduction. Prerequisites. Learning Outcomes. Learning Style
Curvature 13.4 Introduction Curvature is a measure of how sharply a curve is turning as it is traversed. At a particular point along the curve a tangent line can be drawn; this line making an angle ψ with
More information