Arbitrary-Speed Curves
|
|
- Wilfrid Neal
- 7 years ago
- Views:
Transcription
1 Arbitrary-Speed Curves (Com S 477/577 Notes) Yan-Bin Jia Oct 18, 2016 The Frenet formulas are valid only for unit-speed curves; they tell the rate of change of the orthonormal vectors T, N, B with respect to arc length. However, most curves that arise from practice are hardly parameterized with arc length. Also for numerical computations, reparametrization with arc length is impractical, since it is rarely possible to find explicit formulas for α. When a regular curve α is not unit-speed, we can transfer to α the Frenet apparatus of a unit-speed reparametrization α of α. Explicitly, if s is an arc-length function for α, then ( ) α(t) = α s(t), for all t, or, in function notation, α = α(s). Now if κ > 0, τ, T, Ñ, and B are defined for α as we studied before. We define for α the 1 Frenet Frame Computation Curvature function : κ = κ(s) Torsion function : τ = τ(s) Unit tangent : T = T(s) Principal normal : N = Ñ(s) Binormal : B = B(s) In general κ and κ are different functions, defined on different intervals. But they give exactly the same description of the turning of the common route of α and α, since at any point α(t) = α(s(t)) the numbers κ(t) and κ(s(t)) are by the definition the same. Similarly with the rest of the Frenet apparatus; since only a change of parameterization is involved, its fundamental geometric meaning is the same as before. In particular, T, N, B again form t an orthonormal basis at every point t on α. The speed v of the curve is the proper correction factor on the rate of change of T,N,B in the general case. α α(t) s B(t) T(t) N(t) s(t) α All the materials are from [1]. 1
2 Lemma 1 If α is a regular curve in R 3 with κ > 0, then T = κvn, N = κvt + τvb, B = τvn Proof Let α be a unit-speed reparametrization of α. Then by definition, T = T(s), where s is an arc-length function for α. The chain rule as applied to differentiation gives T = T (s) ds dt. By the usual Frenet equations, T = κñ. Substituting the function s in this equation yields T (s) = κ(s)ñ(s) = κn by the definition of κ and N in the arbitrary-speed case. Since ds/dt is the speed function v of α, these two equations combine to yield T = κvn. The formulas for N and B are derived in the same way. Let us use the same letter to designate both a curve α and its unit-speed parameterization α, and similarly with the Frenet apparatus of these two curves. Differences in derivatives are handled by writing, say dt/dt for T, but dt/ds for either T or its reparametrization T (s). With these conventions, the proof above would combine the chain rule dt/dt = (dt/ds)(ds/dt) and Frenet formula dt/ds = κn to give dt/dt = κvn. Curvature is revealed from the second order derivative, i.e., the acceleration, of the curve. Only for a constant-speed curve is acceleration orthogonal to velocity, since β β constant is equivalent to (β β ) = 2β β = 0. In the general case, we analyze velocity and acceleration by expressing them in terms of the Frenet frame field. Lemma 2 If α is a regular curve with speed function v, then the velocity and acceleration of α are given by = vt, α = dv dt T + κv2 N. α Proof Since α = α(s), where s is the arc-length function of α, we find that Then a second differentiation yields α = α (s) ds dt = v T(s) = vt. α = dv dt T + vt = dv dt T + κv2 N 2
3 according to Lemma 1. α T dv dt T α = vt N α κv 2 N The formula α = vt is to be expected α and T are each tangent to the curve, and T has a unit length while α = v. The formula for acceleration is more interesting. By definition, α is the rate of change of the velocity α, and in general both the length and the direction of α are changing. The tangential component (dv/dt)t of α measures the rate of change of the length of α (that is, of the speed of α). The normal component κv 2 N measures the rate of change of the direction of α. Newton s laws of motion show that these components may be experienced as forces. For example, in a car that is speeding up or slowing down on a straight road the only force one feels is due to (dv/dt)t. If one takes an unbanked curve at speed v, the sideways force one feels is due to κv 2 N. Here κ measures how sharply the road turns; the effect of speed is given by v 2, so 60 miles per hour is four times as unsettling as 30. We now find effectively computable expressions for the Frenet apparatus. Theorem 3 Let α be a regular curve in R 3, and α α 0. Then T = α α, where det(α α α ) = (α α ) α. B = α α α α, (1) N = B T, κ = α α α 3, (2) τ = (α α ) α α α 2 = det(α α α ) α α 2, (3) Proof The equations for T and N follow from their definitions. So here we need only prove (1), (2), and (3). Since v = α > 0, the formula T = α / α is equivalent to α = vt. From the preceding lemma we get ( ) dv α α = (vt) dt T + κv2 N = v dv dt T T + κv3 T N = κv 3 B. 3
4 Taking norms we find α α = κv 3 B = κv 3 because B = 1, κ 0, and v > 0. This proves (2). Indeed this equation shows that for regular curves, α α > 0 is equivalent to the usual condition κ > 0. (Thus for κ > 0, α and α are linearly independent and determine the osculating plane at each point, as do T and N.) Then B = α α κv 3 = α α α α. Now only the formula for torsion remains to be proved. To find the dot product (α α ) α, we express everything in terms of T, N, B. We already know that α α = κv 3 B. Since B T = B N = 0, we need only find the B component of α. But by Lemma 2, ( ) dv α = dt T + κv2 N = d2 v dt 2 T + dv dt κvn + d ( κv 2) N + κv 2 N ( dt d = κτv 3 2 ) ( v dv B + dt 2 κ2 v 3 T + dt κv + d ) dt (κv2 ) N following Lemma 2. Consequently (α α ) α α α = κv 3. = κ 2 v 6 τ. Equation (3) then follows since Example 1. We compute the Frenet frame of the curve α(t) = (3t t 3, 3t 2, 3t + t 3 ). The derivatives are α (t) = 3(1 t 2, 2t, 1 + t 2 ), α (t) = 6( t, 1, t), α (t) = 6( 1, 0, 1). And the velocity is Next, we have v(t) = α (t) = α (t) α (t) = 3 2(1 + t 2 ). α (t) α (t) = 18( 1 + t 2, 2t, 1 + t 2 ), α (t) α (t) = 18 2(1 + t 2 ). The expressions above for α α and α yield (α α ) α = (1 t t 2 ) = 216. It remains only to substitute this data into the formulas in Theorem 3, with N being computed by another cross product. The final results are T = (1 t2, 2t, 1 + t 2 ), 2(1 + t2 ) 4
5 N = ( 2t, 1 t2, 0) 1 + t 2, B = ( 1 + t2, 2t, 1 + t 2 ), 2(1 + t2 ) κ = τ = 1 3(1 + t 2 ) 2, 1 3(1 + t 2 ) 2. Example 2. Let us compute the torsion of the helix in its standard parameterization γ(θ) = (a cosθ, a sin θ, bθ). First, we have γ = ( a sinθ, a cosθ, b), γ = ( a cosθ, a sin θ, 0), γ = (a sin θ, a cosθ, 0). Hence, γ γ = (ab sinθ, ab cosθ, a 2 ), γ γ 2 = a 2 (a 2 + b 2 ), (γ γ ) γ = a 2 b, and so the torsion is τ = a 2 b a 2 (a 2 + b 2 ) = b a 2 + b 2. Let us summarize the situation. We now have the Frenet apparatus for an arbitrary-speed curve α. This apparatus satisfies the extended Frenet formulas (with factor v) and may be computed by Theorem 3. If v = 1, that is, if α is unit-speed curve, the Frenet formulas in Lemma 1 simplify slightly, but Theorem 3 may be replaced by the much simpler definitions that we learned before. 2 An Application Spherical Images Let us consider one application of the results in the previous section. There are a number of interesting ways in which one can assign to a given curve β a new curve β whose geometric properties illuminate some aspects of the behavior of β. For example, if β is a unit-speed curve, the curve σ(s) = T(s) = β (s) is the spherical image of β. Here σ is the curve such that each point σ(s) has the same Euclidean coordinates as the unit tangent vector T(s). Roughly speaking, σ(s) is obtained by translating T(s) to the origin. The spherical image lies entirely on the unit sphere Σ of R 3, since σ = T = 1, and the motion represents the curving of β. Example 3. Suppose β is the helix described by β(s) = ( a cos s c, a sin s c, bs c ), 5
6 Figure 1: Spherical image, Fig on p. 71 of [1]. where c = a 2 + b 2. Thus the spherical image of a helix lies on the circle cut from the unit sphere by the plane z = b c. There is no loss of generality in assuming that the original curve β has unit speed (in the case of the helix given above), but we cannot also expect σ to have unit speed. In fact, since σ = T, we have σ = T = κn. Thus σ moves always in the principal normal direction of β, with speed σ equal to the curvature κ of β. Next we assume κ > 0, and use the Frenet formulas for β to compute the curvature of σ. Now Thus σ = (κn) = dκ ds N + κn = dκ N + κ( κt + τb) ds = κ 2 T + dκ ds N + κτb. σ σ = κ 3 N T + κ 2 τn B = κ 2 (κb + τt). Therefore the curvature of the spherical image σ is κ σ = σ σ σ 3 κ2 + τ = 2 κ ( ( ) τ 2 1/2 = 1 + κ) > 1 and thus depends only on the ratio of torsion to curvature for the original curve β. References [1] B. O Neill. Elementary Differential Geometry. Academic Press, Inc.,
Parametric Curves. (Com S 477/577 Notes) Yan-Bin Jia. Oct 8, 2015
Parametric Curves (Com S 477/577 Notes) Yan-Bin Jia Oct 8, 2015 1 Introduction A curve in R 2 (or R 3 ) is a differentiable function α : [a,b] R 2 (or R 3 ). The initial point is α[a] and the final point
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 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 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 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 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 informationPROBLEM SET. Practice Problems for Exam #1. Math 2350, Fall 2004. Sept. 30, 2004 ANSWERS
PROBLEM SET Practice Problems for Exam #1 Math 350, Fall 004 Sept. 30, 004 ANSWERS i Problem 1. The position vector of a particle is given by Rt) = t, t, t 3 ). Find the velocity and acceleration vectors
More informationSmarandache Curves in Minkowski Space-time
International J.Math. Combin. Vol.3 (2008), 5-55 Smarandache Curves in Minkowski Space-time Melih Turgut and Süha Yilmaz (Department of Mathematics of Buca Educational Faculty of Dokuz Eylül University,
More information14.11. Geodesic Lines, Local Gauss-Bonnet Theorem
14.11. Geodesic Lines, Local Gauss-Bonnet Theorem Geodesics play a very important role in surface theory and in dynamics. One of the main reasons why geodesics are so important is that they generalize
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationLS.6 Solution Matrices
LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions
More information1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.
.(6pts Find symmetric equations of the line L passing through the point (, 5, and perpendicular to the plane x + 3y z = 9. (a x = y + 5 3 = z (b x (c (x = ( 5(y 3 = z + (d x (e (x + 3(y 3 (z = 9 = y 3
More informationVector has a magnitude and a direction. Scalar has a magnitude
Vector has a magnitude and a direction Scalar has a magnitude Vector has a magnitude and a direction Scalar has a magnitude a brick on a table Vector has a magnitude and a direction Scalar has a magnitude
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 informationThe small increase in x is. and the corresponding increase in y is. Therefore
Differentials For a while now, we have been using the notation dy to mean the derivative of y with respect to. Here is any variable, and y is a variable whose value depends on. One of the reasons that
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More information2.2. Instantaneous Velocity
2.2. Instantaneous Velocity toc Assuming that your are not familiar with the technical aspects of this section, when you think about it, your knowledge of velocity is limited. In terms of your own mathematical
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More information( 1)2 + 2 2 + 2 2 = 9 = 3 We would like to make the length 6. The only vectors in the same direction as v are those
1.(6pts) Which of the following vectors has the same direction as v 1,, but has length 6? (a), 4, 4 (b),, (c) 4,, 4 (d), 4, 4 (e) 0, 6, 0 The length of v is given by ( 1) + + 9 3 We would like to make
More informationLecture L5 - Other Coordinate Systems
S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L5 - Other Coordinate Systems In this lecture, we will look at some other common systems of coordinates. We will present polar coordinates
More informationRelations among Frenet Apparatus of Space-like Bertrand W-Curve Couples in Minkowski Space-time
International Mathematical Forum, 3, 2008, no. 32, 1575-1580 Relations among Frenet Apparatus of Space-like Bertrand W-Curve Couples in Minkowski Space-time Suha Yilmaz Dokuz Eylul University, Buca Educational
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More information1 3 4 = 8i + 20j 13k. x + w. y + w
) Find the point of intersection of the lines x = t +, y = 3t + 4, z = 4t + 5, and x = 6s + 3, y = 5s +, z = 4s + 9, and then find the plane containing these two lines. Solution. Solve the system of equations
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationSection 4.4 Inner Product Spaces
Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer
More informationDifferentiation of vectors
Chapter 4 Differentiation of vectors 4.1 Vector-valued functions In the previous chapters we have considered real functions of several (usually two) variables f : D R, where D is a subset of R n, where
More informationDefinition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point.
6.1 Vectors in the Plane PreCalculus 6.1 VECTORS IN THE PLANE Learning Targets: 1. Find the component form and the magnitude of a vector.. Perform addition and scalar multiplication of two vectors. 3.
More informationL 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has
The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:
More information3 Contour integrals and Cauchy s Theorem
3 ontour integrals and auchy s Theorem 3. Line integrals of complex functions Our goal here will be to discuss integration of complex functions = u + iv, with particular regard to analytic functions. Of
More informationDERIVATIVES AS MATRICES; CHAIN RULE
DERIVATIVES AS MATRICES; CHAIN RULE 1. Derivatives of Real-valued Functions Let s first consider functions f : R 2 R. Recall that if the partial derivatives of f exist at the point (x 0, y 0 ), then we
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationEuler-Savary s Formula for the Planar Curves in Two Dimensional Lightlike Cone
International J.Math. Combin. Vol.1 (010), 115-11 Euler-Saary s Formula for the Planar Cures in Two Dimensional Lightlike Cone Handan BALGETİR ÖZTEKİN and Mahmut ERGÜT (Department of Mathematics, Fırat
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More informationEikonal Slant Helices and Eikonal Darboux Helices In 3-Dimensional 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 informationChapter 4 One Dimensional Kinematics
Chapter 4 One Dimensional Kinematics 41 Introduction 1 4 Position, Time Interval, Displacement 41 Position 4 Time Interval 43 Displacement 43 Velocity 3 431 Average Velocity 3 433 Instantaneous Velocity
More information5.3 The Cross Product in R 3
53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or
More informationSection 11.1: Vectors in the Plane. Suggested Problems: 1, 5, 9, 17, 23, 25-37, 40, 42, 44, 45, 47, 50
Section 11.1: Vectors in the Plane Page 779 Suggested Problems: 1, 5, 9, 17, 3, 5-37, 40, 4, 44, 45, 47, 50 Determine whether the following vectors a and b are perpendicular. 5) a = 6, 0, b = 0, 7 Recall
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 informationMath 21a Curl and Divergence Spring, 2009. 1 Define the operator (pronounced del ) by. = i
Math 21a url and ivergence Spring, 29 1 efine the operator (pronounced del by = i j y k z Notice that the gradient f (or also grad f is just applied to f (a We define the divergence of a vector field F,
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 informationThe Math Circle, Spring 2004
The Math Circle, Spring 2004 (Talks by Gordon Ritter) What is Non-Euclidean Geometry? Most geometries on the plane R 2 are non-euclidean. Let s denote arc length. Then Euclidean geometry arises from the
More informationNote on growth and growth accounting
CHAPTER 0 Note on growth and growth accounting 1. Growth and the growth rate In this section aspects of the mathematical concept of the rate of growth used in growth models and in the empirical analysis
More information= δx x + δy y. df ds = dx. ds y + xdy ds. Now multiply by ds to get the form of the equation in terms of differentials: df = y dx + x dy.
ERROR PROPAGATION For sums, differences, products, and quotients, propagation of errors is done as follows. (These formulas can easily be calculated using calculus, using the differential as the associated
More informationSecond Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.
Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard
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 informationTo define concepts such as distance, displacement, speed, velocity, and acceleration.
Chapter 7 Kinematics of a particle Overview In kinematics we are concerned with describing a particle s motion without analysing what causes or changes that motion (forces). In this chapter we look at
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationv w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors.
3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with
More informationMechanics 1: Vectors
Mechanics 1: Vectors roadly speaking, mechanical systems will be described by a combination of scalar and vector quantities. scalar is just a (real) number. For example, mass or weight is characterized
More informationSolutions to Practice Problems for Test 4
olutions to Practice Problems for Test 4 1. Let be the line segmentfrom the point (, 1, 1) to the point (,, 3). Evaluate the line integral y ds. Answer: First, we parametrize the line segment from (, 1,
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationExam 1 Sample Question SOLUTIONS. y = 2x
Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can
More informationScalar Valued Functions of Several Variables; the Gradient Vector
Scalar Valued Functions of Several Variables; the Gradient Vector Scalar Valued Functions vector valued function of n variables: Let us consider a scalar (i.e., numerical, rather than y = φ(x = φ(x 1,
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 information28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z
28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product 1.4.1 Definitions The cross product is the second multiplication operation between vectors we will study. The goal behind the definition
More informationAbout the Gamma Function
About the Gamma Function Notes for Honors Calculus II, Originally Prepared in Spring 995 Basic Facts about the Gamma Function The Gamma function is defined by the improper integral Γ) = The integral is
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 informationLecture L3 - Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
More information1 of 7 9/5/2009 6:12 PM
1 of 7 9/5/2009 6:12 PM Chapter 2 Homework Due: 9:00am on Tuesday, September 8, 2009 Note: To understand how points are awarded, read your instructor's Grading Policy. [Return to Standard Assignment View]
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 informationA First Course in Elementary Differential Equations. Marcel B. Finan Arkansas Tech University c All Rights Reserved
A First Course in Elementary Differential Equations Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 Contents 1 Basic Terminology 4 2 Qualitative Analysis: Direction Field of y = f(t, y)
More information3.1 MAXIMUM, MINIMUM AND INFLECTION POINT & SKETCHING THE GRAPH. In Isaac Newton's day, one of the biggest problems was poor navigation at sea.
BA01 ENGINEERING MATHEMATICS 01 CHAPTER 3 APPLICATION OF DIFFERENTIATION 3.1 MAXIMUM, MINIMUM AND INFLECTION POINT & SKETCHING THE GRAPH Introduction to Applications of Differentiation In Isaac Newton's
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 3-space. This time the outcome will be a vector in 3-space. Definition
More informationSolutions to Practice Problems
Higher Geometry Final Exam Tues Dec 11, 5-7:30 pm Practice Problems (1) Know the following definitions, statements of theorems, properties from the notes: congruent, triangle, quadrilateral, isosceles
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationWHEN DOES A CROSS PRODUCT ON R n EXIST?
WHEN DOES A CROSS PRODUCT ON R n EXIST? PETER F. MCLOUGHLIN It is probably safe to say that just about everyone reading this article is familiar with the cross product and the dot product. However, what
More informationImplicit Differentiation
Revision Notes 2 Calculus 1270 Fall 2007 INSTRUCTOR: Peter Roper OFFICE: LCB 313 [EMAIL: roper@math.utah.edu] Standard Disclaimer These notes are not a complete review of the course thus far, and some
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More information6 Further differentiation and integration techniques
56 6 Further differentiation and integration techniques Here are three more rules for differentiation and two more integration techniques. 6.1 The product rule for differentiation Textbook: Section 2.7
More informationSection 13.5 Equations of Lines and Planes
Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines - specifically, tangent lines.
More informationBasic numerical skills: EQUATIONS AND HOW TO SOLVE THEM. x + 5 = 7 2 + 5-2 = 7-2 5 + (2-2) = 7-2 5 = 5. x + 5-5 = 7-5. x + 0 = 20.
Basic numerical skills: EQUATIONS AND HOW TO SOLVE THEM 1. Introduction (really easy) An equation represents the equivalence between two quantities. The two sides of the equation are in balance, and solving
More informationGeometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi
Geometry of Vectors Carlo Tomasi This note explores the geometric meaning of norm, inner product, orthogonality, and projection for vectors. For vectors in three-dimensional space, we also examine the
More informationVector Algebra II: Scalar and Vector Products
Chapter 2 Vector Algebra II: Scalar and Vector Products We saw in the previous chapter how vector quantities may be added and subtracted. In this chapter we consider the products of vectors and define
More informationNumerical Solution of Differential
Chapter 13 Numerical Solution of Differential Equations We have considered numerical solution procedures for two kinds of equations: In chapter 10 the unknown was a real number; in chapter 6 the unknown
More informationSeparable First Order Differential Equations
Separable First Order Differential Equations Form of Separable Equations which take the form = gx hy or These are differential equations = gxĥy, where gx is a continuous function of x and hy is a continuously
More informationOn Motion of Robot End-Effector using the Curvature Theory of Timelike Ruled Surfaces with Timelike Directrix
Malaysian Journal of Mathematical Sciences 8(2): 89-204 (204) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: http://einspem.upm.edu.my/journal On Motion of Robot End-Effector using the Curvature
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationCopyright 2011 Casa Software Ltd. www.casaxps.com
Table of Contents Variable Forces and Differential Equations... 2 Differential Equations... 3 Second Order Linear Differential Equations with Constant Coefficients... 6 Reduction of Differential Equations
More informationOn the Frenet Frame and a Characterization of Space-like Involute-Evolute Curve Couple in Minkowski Space-time
International Mathematical Forum, 3, 008, no. 16, 793-801 On the Frenet Frame and a Characterization of Space-like Involute-Evolute Curve Couple in Minkowski Space-time Melih Turgut Dokuz Eylul University,
More informationFigure 2.1: Center of mass of four points.
Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would
More information= f x 1 + h. 3. Geometrically, the average rate of change is the slope of the secant line connecting the pts (x 1 )).
Math 1205 Calculus/Sec. 3.3 The Derivative as a Rates of Change I. Review A. Average Rate of Change 1. The average rate of change of y=f(x) wrt x over the interval [x 1, x 2 ]is!y!x ( ) - f( x 1 ) = y
More informationReview Sheet for Test 1
Review Sheet for Test 1 Math 261-00 2 6 2004 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And
More informationClass Meeting # 1: Introduction to PDEs
MATH 18.152 COURSE NOTES - CLASS MEETING # 1 18.152 Introduction to PDEs, Fall 2011 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u = u(x
More informationcircular motion & gravitation physics 111N
circular motion & gravitation physics 111N uniform circular motion an object moving around a circle at a constant rate must have an acceleration always perpendicular to the velocity (else the speed would
More informationMannheim curves in the three-dimensional sphere
Mannheim curves in the three-imensional sphere anju Kahraman, Mehmet Öner Manisa Celal Bayar University, Faculty of Arts an Sciences, Mathematics Department, Muraiye Campus, 5, Muraiye, Manisa, urkey.
More informationThe Matrix Elements of a 3 3 Orthogonal Matrix Revisited
Physics 116A Winter 2011 The Matrix Elements of a 3 3 Orthogonal Matrix Revisited 1. Introduction In a class handout entitled, Three-Dimensional Proper and Improper Rotation Matrices, I provided a derivation
More informationSolutions to Math 51 First Exam January 29, 2015
Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More informationcorrect-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:
Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that
More information1. (from Stewart, page 586) Solve the initial value problem.
. (from Stewart, page 586) Solve the initial value problem.. (from Stewart, page 586) (a) Solve y = y. du dt = t + sec t u (b) Solve y = y, y(0) = 0., u(0) = 5. (c) Solve y = y, y(0) = if possible. 3.
More information11. Rotation Translational Motion: Rotational Motion:
11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationCalculus with Parametric Curves
Calculus with Parametric Curves Suppose f and g are differentiable functions and we want to find the tangent line at a point on the parametric curve x f(t), y g(t) where y is also a differentiable function
More informationThe Black-Scholes Formula
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Black-Scholes Formula These notes examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they are based on the
More informationResearch Article On Motion of Robot End-Effector Using the Curvature Theory of Timelike Ruled Surfaces with Timelike Rulings
Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2008, Article ID 6278, 19 pages doi:10.1155/2008/6278 Research Article On Motion of Robot End-Effector Using the Curvature Theory
More informationIncreasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.
1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.
More information