Unit Speed Curves. Recall that a curve Α is said to be a unit speed curve if
|
|
- Dayna Phelps
- 5 years ago
- Views:
Transcription
1 Unit Speed Curves Recall that a curve Α is said to be a unit speed curve if The reason that we like unit speed curves that the parameter t is equal to arc length; i.e. the value of t tells us how far along the curve we've traveled. For example, the curve r t cos t, sin t, the unit circle, is of unit speed and so t tells us how far around the circle we've gone; i.e. it is arc length.
2 2 Curve_EGs.nb Non-Unit Speed Curves If the norm of the tangent vector to the curve r t x t, y t, z t is not one, then the parameter does not equal arc length. In this case, if the curve is smooth (in the sense of our text), then we can "reparametrize" the curve, i.e. re-scale the parameter, so that the resulting function is of unit speed.
3 Curve_EGs.nb Reparametrizing by Arc Length Define the arc length function of the smooth curve r t, a t b, by With this function (that describes how far along the curve we are at "time" t), we can carry out the reparametrization. What we do is (when possible) evalute the integral to obtain an equation of the form Solve this equation for t in terms of s. Calling the solution t s, we obtain the new parametrization (or reparametrization). The "new" curve is One thing to note: reparametrizing has NOT changed the geometric object - the curve - but has only changed the speed that we move along the curve.
4 4 Curve_EGs.nb Example 1 of Reparametrizing by Arc Length Consider the helix r t a cos t, a sin t, b t, 0 t 4 Π. Note that r ' t a 2 b 2 and so This equation is easily solved for t: The reparametrized curve is
5 Curve_EGs.nb Example 1 Continued r' t The Unit Tangent: We can calculate the unit tangent vector in two ways. The first is to simply calculate T t. One r' t need not worry about the parameter being arc length when using this formula. For the helix, using the original parametrization, and it follows that The second way is to reparametrize by arc length as shown on the previous slide: r s a cos s, a sin s, b s. Then the same as the vector obtained above, when the relation between s and t is taken into account.
6 6 Curve_EGs.nb Curvature and Principal Normal For a curve r t parametrized by arc length, the curvature is defined by The curvature is simply the rate of change of the direction of the unit tangent vector when the curve is parametrized by arc length. One can think of the curvature as describing how the curve "bends". For a curve r t parametrized by arc length, the principal (unit) normal is defined as It is easy to show that T is perpendicular to N. (Indeed, 0 d T T T ' T T T ' 2 ΚT N and so the vectors T d s and N are orthogonal.) For a plane curve, these vectors form a "moving frame" that moves along the curve while always remaining orthogonal and so every vector in the plane can be written as a linear combination of these vectors at any point along the curve.
7 Curve_EGs.nb Curvature and Principal Normal for the Helix Returning to the helix from example 1, r t a cos t, a sin t, b t, we calculate the curvature to be using T s r ' s a cos s, a sin s, b. The principal normal is
8 8 Curve_EGs.nb Example 2 of Reparametrizing by Arc Length Consider the curve r t 2 e t 2 cos t, 2 e t 2 sin t. We have Clearly this curve is not parametrized by arc length as r ' t 5 e t. To parametrize by arc length, we integrate: 0 t t gives 5 e u u t 2. Solving the equation s t 2 for The reparametrized curve is therefore
9 Curve_EGs.nb Example 2 Continued The unit tangent is The curvature is, since we've reparametrized by arc length, the norm of T '. We get The principal normal is
10 10 Curve_EGs.nb Example 3 of Reparametrizing by Arc Length Let r t t 2, t 3, t 0. Once again, our goal is to reparametrize this curve by arc length and then calculate the unit tangent, curvature, and principal normal. To reparametrize by arc length, we have to calculate the arc length function. Note that r ' t 2 t, 3 t 2 and so r ' t 4 t 2 9 t 4 t 4 9 t 2. Integrating this norm gives The next step is to solve the equation s t2 3 2 for t. We obtain The reparametrized curve is
11 Curve_EGs.nb Example 3: Unit Tangent Vector and Curvature The unit tangent vector is, after a little work, The curvature is the norm of the derivative of the unit tangent. Again, some calculus I and calculating the norm of the derivative, i.e. some algebra, leads to
12 12 Curve_EGs.nb Example 3: Principal Normal Vector The definition of the principal normal is N 1 T ' s. The curvature is calculated above and so all that is needed is the Κ s derivative of the unit tangent vector. The principal normal vector is the above vector divided by the curvature. A little algebra leads to
Arc Length. Philippe B. Laval. Today KSU. Philippe B. Laval (KSU) Arc Length Today 1 / 12
Philippe B. Laval KSU Today Philippe B. Laval (KSU) Arc Length Today 1 / 12 Introduction In this section, we discuss the notion of curve in greater detail and introduce the very important notion of arc
More information13.3 Arc Length and Curvature
13 Vector Functions 13.3 Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. We have defined the length of a plane curve with parametric equations x = f(t),
More informationWhat is a Space Curve?
What is a Space Curve? A space curve is a smooth map γ : I R R 3. In our analysis of defining the curvature for space curves we will be able to take the inclusion (γ, 0) and have that the curvature of
More informationHOMEWORK 2 SOLUTIONS
HOMEWORK SOLUTIONS MA11: ADVANCED CALCULUS, HILARY 17 (1) Find parametric equations for the tangent line of the graph of r(t) = (t, t + 1, /t) when t = 1. Solution: A point on this line is r(1) = (1,,
More informationTangent and Normal Vector - (11.5)
Tangent and Normal Vector - (.5). Principal Unit Normal Vector Let C be the curve traced out by the vector-valued function rt vector T t r r t t is the unit tangent vector to the curve C. Now define N
More informationMotion in Space Parametric Equations of a Curve
Motion in Space Parametric Equations of a Curve A curve, C, inr 3 can be described by parametric equations of the form x x t y y t z z t. Any curve can be parameterized in many different ways. For example,
More informationII. Unit Speed Curves
The Geometry of Curves, Part I Rob Donnelly From Murray State University s Calculus III, Fall 2001 note: This material supplements Sections 13.3 and 13.4 of the text Calculus with Early Transcendentals,
More information3 = arccos. A a and b are parallel, B a and b are perpendicular, C a and b are normalized, or D this is always true.
Math 210-101 Test #1 Sept. 16 th, 2016 Name: Answer Key Be sure to show your work! 1. (20 points) Vector Basics: Let v = 1, 2,, w = 1, 2, 2, and u = 2, 1, 1. (a) Find the area of a parallelogram spanned
More informationSection Arclength and Curvature. (1) Arclength, (2) Parameterizing Curves by Arclength, (3) Curvature, (4) Osculating and Normal Planes.
Section 10.3 Arclength and Curvature (1) Arclength, (2) Parameterizing Curves by Arclength, (3) Curvature, (4) Osculating and Normal Planes. MATH 127 (Section 10.3) Arclength and Curvature The University
More informationMAT 272 Test 1 Review. 1. Let P = (1,1) and Q = (2,3). Find the unit vector u that has the same
11.1 Vectors in the Plane 1. Let P = (1,1) and Q = (2,3). Find the unit vector u that has the same direction as. QP a. u =< 1, 2 > b. u =< 1 5, 2 5 > c. u =< 1, 2 > d. u =< 1 5, 2 5 > 2. If u has magnitude
More informationVector equations of lines in the plane and 3-space (uses vector addition & scalar multiplication).
Boise State Math 275 (Ultman) Worksheet 1.6: Lines and Planes From the Toolbox (what you need from previous classes) Plotting points, sketching vectors. Be able to find the component form a vector given
More informationWeek 3: Differential Geometry of Curves
Week 3: Differential Geometry of Curves Introduction We now know how to differentiate and integrate along curves. This week we explore some of the geometrical properties of curves that can be addressed
More informationTangent and Normal Vector - (11.5)
Tangent and Normal Vector - (.5). Principal Unit Normal Vector Let C be the curve traced out bythe vector-valued function r!!t # f!t, g!t, h!t $. The vector T!!t r! % r! %!t!t is the unit tangent vector
More informationMath 317 M1A, October 8th, 2010 page 1 of 7 Name:
Math 317 M1A, October 8th, 2010 page 1 of 7 Name: Problem 1 (5 parts, 30 points): Consider the curve r(t) = 3 sin(t 2 ), 4t 2 + 7, 3 cos(t 2 ), 0 t < a) (5 points) Find the arclength function s(t) giving
More informatione 2 = e 1 = e 3 = v 1 (v 2 v 3 ) = det(v 1, v 2, v 3 ).
3. Frames In 3D space, a sequence of 3 linearly independent vectors v 1, v 2, v 3 is called a frame, since it gives a coordinate system (a frame of reference). Any vector v can be written as a linear combination
More informationSELECTED SAMPLE FINAL EXAM SOLUTIONS - MATH 5378, SPRING 2013
SELECTED SAMPLE FINAL EXAM SOLUTIONS - MATH 5378, SPRING 03 Problem (). This problem is perhaps too hard for an actual exam, but very instructional, and simpler problems using these ideas will be on the
More informationGeometric approximation of curves and singularities of secant maps Ghosh, Sunayana
University of Groningen Geometric approximation of curves and singularities of secant maps Ghosh, Sunayana IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish
More informationChapter Four. Derivatives. in the interior of a set S of real numbers means there is an interval centered at t 0
Chapter Four Derivatives 4 Derivatives Suppose f is a vector function and t 0 is a point in the interior of the domain of f ( t 0 in the interior of a set S of real numbers means there is an interval centered
More informationIntroduction to Vector Functions
Introduction to Vector Functions Differentiation and Integration Philippe B. Laval KSU Today Philippe B. Laval (KSU) Vector Functions Today 1 / 14 Introduction In this section, we study the differentiation
More informationFigure 10: Tangent vectors approximating a path.
3 Curvature 3.1 Curvature Now that we re parametrizing curves, it makes sense to wonder how we might measure the extent to which a curve actually curves. That is, how much does our path deviate from being
More informationThere is a function, the arc length function s(t) defined by s(t) = It follows that r(t) = p ( s(t) )
MATH 20550 Acceleration, Curvature and Related Topics Fall 2016 The goal of these notes is to show how to compute curvature and torsion from a more or less arbitrary parametrization of a curve. We will
More informationAPPM 2350, Summer 2018: Exam 1 June 15, 2018
APPM 2350, Summer 2018: Exam 1 June 15, 2018 Instructions: Please show all of your work and make your methods and reasoning clear. Answers out of the blue with no supporting work will receive no credit
More informationVectors, dot product, and cross product
MTH 201 Multivariable calculus and differential equations Practice problems Vectors, dot product, and cross product 1. Find the component form and length of vector P Q with the following initial point
More informationCHAPTER 6 VECTOR CALCULUS. We ve spent a lot of time so far just looking at all the different ways you can graph
CHAPTER 6 VECTOR CALCULUS We ve spent a lot of time so far just looking at all the different ways you can graph things and describe things in three dimensions, and it certainly seems like there is a lot
More informationArbitrary-Speed Curves
Arbitrary-Speed Curves (Com S 477/577 Notes) Yan-Bin Jia Oct 12, 2017 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
More informationMA 351 Fall 2007 Exam #1 Review Solutions 1
MA 35 Fall 27 Exam # Review Solutions THERE MAY BE TYPOS in these solutions. Please let me know if you find any.. Consider the two surfaces ρ 3 csc θ in spherical coordinates and r 3 in cylindrical coordinates.
More informationIntroduction to Algebraic and Geometric Topology Week 14
Introduction to Algebraic and Geometric Topology Week 14 Domingo Toledo University of Utah Fall 2016 Computations in coordinates I Recall smooth surface S = {f (x, y, z) =0} R 3, I rf 6= 0 on S, I Chart
More informationarxiv: v1 [math.dg] 11 Nov 2007
Length of parallel curves arxiv:711.167v1 [math.dg] 11 Nov 27 E. Macías-Virgós Abstract We prove that the length difference between a closed periodic curve and its parallel curve at a sufficiently small
More information(1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3
Math 127 Introduction and Review (1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3 MATH 127 Introduction to Calculus III
More informationName Period. Date: Topic: 9-2 Circles. Standard: G-GPE.1. Objective:
Name Period Date: Topic: 9-2 Circles Essential Question: If the coefficients of the x 2 and y 2 terms in the equation for a circle were different, how would that change the shape of the graph of the equation?
More informationMTH 277 Test 4 review sheet Chapter , 14.7, 14.8 Chalmeta
MTH 77 Test 4 review sheet Chapter 13.1-13.4, 14.7, 14.8 Chalmeta Multiple Choice 1. Let r(t) = 3 sin t i + 3 cos t j + αt k. What value of α gives an arc length of 5 from t = 0 to t = 1? (a) 6 (b) 5 (c)
More informationTangent and Normal Vectors
Tangent and Normal Vectors MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Navigation When an observer is traveling along with a moving point, for example the passengers in
More informationMath 32A Discussion Session Week 5 Notes November 7 and 9, 2017
Math 32A Discussion Session Week 5 Notes November 7 and 9, 2017 This week we want to talk about curvature and osculating circles. You might notice that these notes contain a lot of the same theory or proofs
More informationMATH 12 CLASS 5 NOTES, SEP
MATH 12 CLASS 5 NOTES, SEP 30 2011 Contents 1. Vector-valued functions 1 2. Differentiating and integrating vector-valued functions 3 3. Velocity and Acceleration 4 Over the past two weeks we have developed
More informationLater in this chapter, we are going to use vector functions to describe the motion of planets and other objects through space.
10 VECTOR FUNCTIONS VECTOR FUNCTIONS Later in this chapter, we are going to use vector functions to describe the motion of planets and other objects through space. Here, we prepare the way by developing
More informationSOLUTIONS TO SECOND PRACTICE EXAM Math 21a, Spring 2003
SOLUTIONS TO SECOND PRACTICE EXAM Math a, Spring 3 Problem ) ( points) Circle for each of the questions the correct letter. No justifications are needed. Your score will be C W where C is the number of
More informationIntroduction to Vector Functions
Introduction to Vector Functions Limits and Continuity Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) Introduction to Vector Functions Spring 2012 1 / 14 Introduction In this section, we study
More information9.1 Mean and Gaussian Curvatures of Surfaces
Chapter 9 Gauss Map II 9.1 Mean and Gaussian Curvatures of Surfaces in R 3 We ll assume that the curves are in R 3 unless otherwise noted. We start off by quoting the following useful theorem about self
More informationEngineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Kinematics
Engineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Kinematics Module 10 - Lecture 24 Kinematics of a particle moving on a curve Today,
More informationChapter 14: Vector Calculus
Chapter 14: Vector Calculus Introduction to Vector Functions Section 14.1 Limits, Continuity, Vector Derivatives a. Limit of a Vector Function b. Limit Rules c. Component By Component Limits d. Continuity
More informationMechanics, Heat, Oscillations and Waves Prof. V. Balakrishnan Department of Physics Indian Institute of Technology, Madras
Mechanics, Heat, Oscillations and Waves Prof. V. Balakrishnan Department of Physics Indian Institute of Technology, Madras Lecture 08 Vectors in a Plane, Scalars & Pseudoscalers Let us continue today with
More informationFaculty of Engineering, Mathematics and Science School of Mathematics
Faculty of Engineering, Mathematics and Science School of Mathematics GROUPS Trinity Term 06 MA3: Advanced Calculus SAMPLE EXAM, Solutions DAY PLACE TIME Prof. Larry Rolen Instructions to Candidates: Attempt
More informationChapter 12: Differentiation. SSMth2: Basic Calculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M.
Chapter 12: Differentiation SSMth2: Basic Calculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Mendoza Chapter 12: Differentiation Lecture 12.1: The Derivative Lecture
More informationOn constant isotropic submanifold by generalized null cubic
On constant isotropic submanifold by generalized null cubic Leyla Onat Abstract. In this paper we shall be concerned with curves in an Lorentzian submanifold M 1, and give a characterization of each constant
More informationArc Length and Curvature
Arc Length and Curvature. Last time, we saw that r(t) = cos t, sin t, t parameteried the pictured curve. (a) Find the arc length of the curve between (, 0, 0) and (, 0, π). (b) Find the unit tangent vector
More informationMath 114, Section 003 Fall 2011 Practice Exam 1 with Solutions
Math 11, Section 003 Fall 2011 Practice Exam 1 with Solutions Contents 1 Problems 2 2 Solution key 8 3 Solutions 9 1 1 Problems Question 1: Let L be the line tangent to the curve r (t) t 2 + 3t + 2, e
More informationGeometry of Cylindrical Curves over Plane Curves
Applied Mathematical Sciences, Vol 9, 015, no 113, 5637-5649 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/101988/ams01556456 Geometry of Cylindrical Curves over Plane Curves Georgi Hristov Georgiev, Radostina
More informationMATH Final Review
MATH 1592 - Final Review 1 Chapter 7 1.1 Main Topics 1. Integration techniques: Fitting integrands to basic rules on page 485. Integration by parts, Theorem 7.1 on page 488. Guidelines for trigonometric
More informationAPPM 2350 Section Exam points Wednesday September 26, 6:00pm 7:30pm, 2018
APPM 2350 Section Exam 1 140 points Wednesday September 26, 6:00pm 7:30pm, 2018 ON THE FRONT OF YOUR BLUEBOOK write: (1) your name, (2) your student ID number, (3) lecture section/time (4) your instructor
More informationCurves - A lengthy story
MATH 2401 - Harrell Curves - A lengthy story Lecture 4 Copyright 2007 by Evans M. Harrell II. Reminder What a lonely archive! Who in the cast of characters might show up on the test? Curves r(t), velocity
More informationGLOBAL PROPERTIES OF PLANE AND SPACE CURVES
GLOBAL PROPERTIES OF PLANE AND SPACE CURVES KEVIN YAN Abstract. The purpose of this paper is purely expository. Its goal is to explain basic differential geometry to a general audience without assuming
More informationExam. There are 6 problems. Your 5 best answers count. Please pay attention to the presentation of your work! Best 5
Department of Mathematical Sciences Instructor: Daiva Pucinskaite Calculus III June, 06 Name: Exam There are 6 problems. Your 5 best answers count. Please pay attention to the presentation of your work!
More informationMATH 423/ Note that the algebraic operations on the right hand side are vector subtraction and scalar multiplication.
MATH 423/673 1 Curves Definition: The velocity vector of a curve α : I R 3 at time t is the tangent vector to R 3 at α(t), defined by α (t) T α(t) R 3 α α(t + h) α(t) (t) := lim h 0 h Note that the algebraic
More informationMathematics 2203, Test 1 - Solutions
Mathematics 220, Test 1 - Solutions F, 2010 Philippe B. Laval Name 1. Determine if each statement below is True or False. If it is true, explain why (cite theorem, rule, property). If it is false, explain
More informationMA 323 Geometric Modelling Course Notes: Day 20 Curvature and G 2 Bezier splines
MA 323 Geometric Modelling Course Notes: Day 20 Curvature and G 2 Bezier splines David L. Finn Yesterday, we introduced the notion of curvature and how it plays a role formally in the description of curves,
More informationMATH 332: Vector Analysis Summer 2005 Homework
MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,
More informationMATH 280 Multivariate Calculus Fall Integrating a vector field over a curve
MATH 280 Multivariate alculus Fall 2012 Definition Integrating a vector field over a curve We are given a vector field F and an oriented curve in the domain of F as shown in the figure on the left below.
More informationThe Frenet Serret formulas
The Frenet Serret formulas Attila Máté Brooklyn College of the City University of New York January 19, 2017 Contents Contents 1 1 The Frenet Serret frame of a space curve 1 2 The Frenet Serret formulas
More informationGeometric Modelling Summer 2016
Geometric Modelling Summer 2016 Exercises Benjamin Karer M.Sc. http://gfx.uni-kl.de/~gm Benjamin Karer M.Sc. Geometric Modelling Summer 2016 1 Dierential Geometry Benjamin Karer M.Sc. Geometric Modelling
More informationCHAPTER TWO: THE GEOMETRY OF CURVES
CHAPTER TWO: THE GEOMETRY OF CURVES Thi material i for June 7, 8 (Tueday to Wed.) 2.1 Parametrized Curve Definition. A parametrized curve i a map α : I R n (n = 2 or 3), where I i an interval in R. We
More informationENGI 4430 Parametric Vector Functions Page dt dt dt
ENGI 4430 Parametric Vector Functions Page 2-01 2. Parametric Vector Functions (continued) Any non-zero vector r can be decomposed into its magnitude r and its direction: r rrˆ, where r r 0 Tangent Vector:
More information1 Vectors and 3-Dimensional Geometry
Calculus III (part ): Vectors and 3-Dimensional Geometry (by Evan Dummit, 07, v..55) Contents Vectors and 3-Dimensional Geometry. Functions of Several Variables and 3-Space..................................
More informationCurves from the inside
MATH 2401 - Harrell Curves from the inside Lecture 5 Copyright 2008 by Evans M. Harrell II. Who in the cast of characters might show up on the test? Curves r(t), velocity v(t). Tangent and normal lines.
More information+ 2gx + 2fy + c = 0 if S
CIRCLE DEFINITIONS A circle is the locus of a point which moves in such a way that its distance from a fixed point, called the centre, is always a constant. The distance r from the centre is called the
More informationPhysics 6303 Lecture 3 August 27, 2018
Physics 6303 Lecture 3 August 27, 208 LAST TIME: Vector operators, divergence, curl, examples of line integrals and surface integrals, divergence theorem, Stokes theorem, index notation, Kronecker delta,
More informationExample 2.1. Draw the points with polar coordinates: (i) (3, π) (ii) (2, π/4) (iii) (6, 2π/4) We illustrate all on the following graph:
Section 10.3: Polar Coordinates The polar coordinate system is another way to coordinatize the Cartesian plane. It is particularly useful when examining regions which are circular. 1. Cartesian Coordinates
More information12.3 Curvature, torsion and the TNB frame
1.3 Curvature, torsion and the TNB frame Acknowledgments: Material from a Georgia Tech worksheet by Jim Herod, School of Mathematics, herod@math.gatech.edu, is incorporated into the section on curvature,
More informationMATH 32A: MIDTERM 2 REVIEW. sin 2 u du z(t) = sin 2 t + cos 2 2
MATH 3A: MIDTERM REVIEW JOE HUGHES 1. Curvature 1. Consider the curve r(t) = x(t), y(t), z(t), where x(t) = t Find the curvature κ(t). 0 cos(u) sin(u) du y(t) = Solution: The formula for curvature is t
More informationTHE FUNDAMENTAL THEOREM OF SPACE CURVES
THE FUNDAMENTAL THEOREM OF SPACE CURVES JOSHUA CRUZ Abstract. In this paper, we show that curves in R 3 can be uniquely generated by their curvature and torsion. By finding conditions that guarantee the
More informationf : R 2 R (x, y) x 2 + y 2
Chapter 2 Vector Functions 2.1 Vector-Valued Functions 2.1.1 Definitions Until now, the functions we studied took a real number as input and gave another real number as output. Hence, when defining a function,
More informationAbsolute and Local Extrema
Extrema of Functions We can use the tools of calculus to help us understand and describe the shapes of curves. Here is some of the data that derivatives f (x) and f (x) can provide about the shape of the
More informationIntroduction to Vector Functions
Introduction to Vector Functions Limits and Continuity Philippe B. Laval KSU Today Philippe B. Laval (KSU) Vector Functions Today 1 / 14 Introduction Until now, the functions we studied took a real number
More informationBROWN UNIVERSITY MATH 0350 MIDTERM 19 OCTOBER 2017 INSTRUCTOR: SAMUEL S. WATSON. a b. Name: Problem 1
BROWN UNIVERSITY MATH 0350 MIDTERM 19 OCTOBER 2017 INSTRUCTOR: SAMUEL S. WATSON Name: Problem 1 In this problem, we will use vectors to show that an angle formed by connecting a point on a circle to two
More informationLecture 4: Partial and Directional derivatives, Differentiability
Lecture 4: Partial and Directional derivatives, Differentiability Rafikul Alam Department of Mathematics IIT Guwahati Differential Calculus Task: Extend differential calculus to the functions: Case I:
More informationWorksheet 1.4: Geometry of the Dot and Cross Products
Boise State Math 275 (Ultman) Worksheet 1.4: Geometry of the Dot and Cross Products From the Toolbox (what you need from previous classes): Basic algebra and trigonometry: be able to solve quadratic equations,
More informationParametric Curves. (Com S 477/577 Notes) Yan-Bin Jia. Oct 3, 2017
Parametric Curves (Com S 477/577 Notes) Yan-Bin Jia Oct 3, 2017 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 informationMidterm 1 Review. Distance = (x 1 x 0 ) 2 + (y 1 y 0 ) 2.
Midterm 1 Review Comments about the midterm The midterm will consist of five questions and will test on material from the first seven lectures the material given below. No calculus either single variable
More informationPhysics 6010, Fall 2016 Constraints and Lagrange Multipliers. Relevant Sections in Text:
Physics 6010, Fall 2016 Constraints and Lagrange Multipliers. Relevant Sections in Text: 1.3 1.6 Constraints Often times we consider dynamical systems which are defined using some kind of restrictions
More informationPreface.
This document was written and copyrighted by Paul Dawkins. Use of this document and its online version is governed by the Terms and Conditions of Use located at. The online version of this document is
More informationParametric Functions and Vector Functions (BC Only)
Parametric Functions and Vector Functions (BC Only) Parametric Functions Parametric functions are another way of viewing functions. This time, the values of x and y are both dependent on another independent
More informationName: ID: Math 233 Exam 1. Page 1
Page 1 Name: ID: This exam has 20 multiple choice questions, worth 5 points each. You are allowed to use a scientific calculator and a 3 5 inch note card. 1. Which of the following pairs of vectors are
More informationDIFFERENTIAL GEOMETRY HW 4. Show that a catenoid and helicoid are locally isometric.
DIFFERENTIAL GEOMETRY HW 4 CLAY SHONKWILER Show that a catenoid and helicoid are locally isometric. 3 Proof. Let X(u, v) = (a cosh v cos u, a cosh v sin u, av) be the parametrization of the catenoid and
More informationMath 3c Solutions: Exam 2 Fall 2017
Math 3c Solutions: Exam Fall 07. 0 points) The graph of a smooth vector-valued function is shown below except that your irresponsible teacher forgot to include the orientation!) Several points are indicated
More informationLagrange Multipliers
Optimization with Constraints As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each
More informationHOMEWORK 2 - SOLUTIONS
HOMEWORK 2 - SOLUTIONS - 2012 ANDRÉ NEVES Exercise 15 of Chapter 2.3 of Do Carmo s book: Okay, I have no idea why I set this one because it is similar to another one from the previous homework. I might
More informationMath 233 Calculus 3 - Fall 2016
Math 233 Calculus 3 - Fall 2016 2 12.1 - Three-Dimensional Coordinate Systems 12.1 - THREE-DIMENSIONAL COORDINATE SYSTEMS Definition. R 3 means By convention, we graph points in R 3 using a right-handed
More informationCHAPTER 3. Gauss map. In this chapter we will study the Gauss map of surfaces in R 3.
CHAPTER 3 Gauss map In this chapter we will study the Gauss map of surfaces in R 3. 3.1. Surfaces in R 3 Let S R 3 be a submanifold of dimension 2. Let {U i, ϕ i } be a DS on S. For any p U i we have a
More information1 The Differential Geometry of Surfaces
1 The Differential Geometry of Surfaces Three-dimensional objects are bounded by surfaces. This section reviews some of the basic definitions and concepts relating to the geometry of smooth surfaces. 1.1
More informationLimits, Rates of Change, and Tangent Lines
Limits, Rates of Change, and Tangent Lines jensenrj July 2, 2018 Contents 1 What is Calculus? 1 2 Velocity 2 2.1 Average Velocity......................... 3 2.2 Instantaneous Velocity......................
More informationJust what is curvature, anyway?
MATH 2401 - Harrell Just what is curvature, anyway? Lecture 5 Copyright 2007 by Evans M. Harrell II. The osculating plane Bits of curve have a best plane. stickies on wire. Each stickie contains T and
More informationThe moving trihedron and all that
MATH 2411 - Harrell The moving trihedron and all that B Lecture 5 T N Copyright 2013 by Evans M. Harrell II. This week s learning plan You will be tested on the mathematics of curves. You will think about
More information8. THE FARY-MILNOR THEOREM
Math 501 - Differential Geometry Herman Gluck Tuesday April 17, 2012 8. THE FARY-MILNOR THEOREM The curvature of a smooth curve in 3-space is 0 by definition, and its integral w.r.t. arc length, (s) ds,
More informationMath S1201 Calculus 3 Chapters , 14.1
Math S1201 Calculus 3 Chapters 13.2 13.4, 14.1 Summer 2015 Instructor: Ilia Vovsha h@p://www.cs.columbia.edu/~vovsha/calc3 1 Outline CH 13.2 DerivaIves of Vector FuncIons Extension of definiion Tangent
More informationCURVATURE AND RADIUS OF CURVATURE
CHAPTER 5 CURVATURE AND RADIUS OF CURVATURE 5.1 Introduction: Curvature is a numerical measure of bending of the curve. At a particular point on the curve, a tangent can be drawn. Let this line makes an
More informationBROWN UNIVERSITY MATH 0350 MIDTERM 19 OCTOBER 2017 INSTRUCTOR: SAMUEL S. WATSON
BROWN UNIVERSITY MATH 0350 MIDTERM 19 OCTOBER 2017 INSTRUCTOR: SAMUEL S. WATSON Name: Problem 1 In this problem, we will use vectors to show that an angle formed by connecting a point on a circle to two
More informationMATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, Elementary tensor calculus
MATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, 205 Elementary tensor calculus We will study in this section some basic multilinear algebra and operations on tensors. Let
More informationMath 426H (Differential Geometry) Final Exam April 24, 2006.
Math 426H Differential Geometry Final Exam April 24, 6. 8 8 8 6 1. Let M be a surface and let : [0, 1] M be a smooth loop. Let φ be a 1-form on M. a Suppose φ is exact i.e. φ = df for some f : M R. Show
More informationLength of parallel curves and rotation index
Length of parallel curves and rotation index E. Macías-Virgós 1 Institute of Mathematics. Department of Geometry and Topology. University of Santiago de Compostela. 15782- SPAIN Abstract We prove that
More informationDot product. The dot product is an inner product on a coordinate vector space (Definition 1, Theorem
Dot product The dot product is an inner product on a coordinate vector space (Definition 1, Theorem 1). Definition 1 Given vectors v and u in n-dimensional space, the dot product is defined as, n v u v
More informationEQUATIONS OF MOTION: NORMAL AND TANGENTIAL COORDINATES
EQUATIONS OF MOTION: NORMAL AND TANGENTIAL COORDINATES Today s Objectives: Students will be able to: 1. Apply the equation of motion using normal and tangential coordinates. In-Class Activities: Check
More information