...corrected/improved versions. Problem 1(a) - Fall 2008 Find parametric equations for the line L which contains A(1, 2, 3) and B(4, 6, 5). Solution: To get the parametric equations of L you need a point through which the line passes and a vector parallel to the line. −→ Take the point to be A and the vector to be the AB. The vector equation of L is −→ −→ r(t) = OA + t AB = 1, 2, 3 + t 3, 4, 2 = 1 + 3t, 2 + 4t, 3 + 2t , where O is the origin. The parametric equations are: x = 1 + 3t y = 2 + 4t, z = 3 + 2t t ∈ R. Problem 1(b) - Fall 2008 Find parametric equations for the line L of intersection of the planes x − 2y + z = 10 and 2x + y − z = 0. Solution: The vector part v of the line L of intersection is orthogonal to the normal vectors 1, −2, 1 and 2, 1, −1 . Hence v can be taken to be: i j k v = 1, −2, 1 × 2, 1, −1 = 1 −2 1 = 1i + 3j + 5k. 2 1 −1 Choose P ∈ L so the z-coordinate of P is zero. Setting z = 0, we obtain: x − 2y = 10 2x + y = 0. Solving, we find that x = 2 and y = −4. Hence, P = 2, −4, 0 lies on the line L. The parametric equations are: x =2+t y = −4 + 3t z = 0 + 5t = 5t. Problem 2(a) - Fall 2008 Find an equation of the plane which contains the points P(−1, 0, 1), Q(1, −2, 1) and R(2, 0, −1). Solution: Method 1 −→ −→ Consider the vectors...
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...Math 2321 Calculus III for Science & Engineering Summer I 2015 Instructor: Dr. Sumi Seo Office: 535 NI (Nightingale Hall) Phone: 617-373- 2080 Email: s.seo@neu.edu Office Hours: TWR 12.15 – 1.15 pm, or by appointment Textbooks: Worldwide Multivariable Calculus, by David B. Massey PDF and printed versions available at: http://www.centerofmathematics.com/wwcomstore/index.php/ The PDF is priced at $9.95, while the black and white (grayscale) soft-back printed version is $29.95. The PDF textbook contains a link, at the beginning of each section, to one or more free video lectures, by Prof. Massey, on the contents of that section. The PDF has hyperlinked Tables of Contents, Indices, and cross-references; you may need to activate the Forward and Back buttons in your PDF viewer to take full advantage of the hyperlinks. To use the textbook on an iPad, we recommend the GoodReader app. It is absolutely NOT required that you purchase a printed textbook. Web Materials: All class announcements, material, and grades will be posted on Blackboard. Homework and Quizzes: No homework will be collected. We will have quizzes on Thursdays (at the beginning of the class) starting from the first week. The quiz in the last week will be given on Wednesday. No quiz will be given in the mid-term exam week. Ordinarily there will be no make-up quizzes; instead, I will drop the one lowest quiz score. A missed quiz will be counted in the dropped lowest score. There may be exceptions...
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...1. Solve the inequality 12x + 11 > 3x - 1 2. Find the gradient of the curve with equation: 3. The sequence of values given by the Iterative formula 2x2 - 4xy + 31 = 9 at the point (2,1) + 1 = 2. (xn +_1_ ) 2 3 Xn with initial value x I = 1 converges to a. Xn (i) Use this formula to find a correct to 2 d.p, showing the result of each iteration. (ii) State an equation satisfied by a, and hence fmd the exact value of a. 4. Express sine -J3 cose =.fi in the form Rsin(e - a) where R > 0 and 0 < a < 1! 2 giving the exact value of a. Hence show that one solution of the equation sine - J3 cose = J2 is e = 7n and find 12 '1 l Cj 100 . 5.(i) 3x - 1 2. Find the gradient of the curve with equation: 3. The sequence of values given by the Iterative formula 2x2 - 4xy + 31 = 9 at the point (2,1) + 1 = 2. (xn +_1_ ) 3 xn2 with initial value Xl = 1 converges to a. Xn (i) Use this formula to find a correct to 2 d.p, showing the result of each iteration. (ii) State an equation satisfied by 4. Express sine -.J3 cose = Ct, Ji and hence fmd the exact value of a. in the form Rsin(e - a) where R > 0 and 0 < a < 1! 2 giving the exact value of a. Hence show that one solution of the equation sine '1 -.J3 cose = Ji is e = 7n and find 12 ® lljlOO' Wl ( 5.(i) Show that 10glO(x+ 5) = 2 - 10glOx may be written...
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...CHAPTER 0 Contents Preface v vii Problems Solved in Student Solutions Manual 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Matrices, Vectors, and Vector Calculus Newtonian Mechanics—Single Particle Oscillations 79 127 1 29 Nonlinear Oscillations and Chaos Gravitation 149 Some Methods in The Calculus of Variations 165 181 Hamilton’s Principle—Lagrangian and Hamiltonian Dynamics Central-Force Motion 233 277 333 Dynamics of a System of Particles Motion in a Noninertial Reference Frame Dynamics of Rigid Bodies Coupled Oscillations 397 435 461 353 Continuous Systems; Waves Special Theory of Relativity iii iv CONTENTS CHAPTER 0 Preface This Instructor’s Manual contains the solutions to all the end-of-chapter problems (but not the appendices) from Classical Dynamics of Particles and Systems, Fifth Edition, by Stephen T. Thornton and Jerry B. Marion. It is intended for use only by instructors using Classical Dynamics as a textbook, and it is not available to students in any form. A Student Solutions Manual containing solutions to about 25% of the end-of-chapter problems is available for sale to students. The problem numbers of those solutions in the Student Solutions Manual are listed on the next page. As a result of surveys received from users, I continue to add more worked out examples in the text and add additional problems. There are now 509 problems, a significant number over the 4th edition. The instructor will find a large...
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...This page intentionally left blank Physical Constants Quantity Electron charge Electron mass Permittivity of free space Permeability of free space Velocity of light Value e = (1.602 177 33 ± 0.000 000 46) × 10−19 C m = (9.109 389 7 ± 0.000 005 4) × 10−31 kg �0 = 8.854 187 817 × 10−12 F/m µ0 = 4π10−7 H/m c = 2.997 924 58 × 108 m/s Dielectric Constant (�r� ) and Loss Tangent (� �� /� � ) Material Air Alcohol, ethyl Aluminum oxide Amber Bakelite Barium titanate Carbon dioxide Ferrite (NiZn) Germanium Glass Ice Mica Neoprene Nylon Paper Plexiglas Polyethylene Polypropylene Polystyrene Porcelain (dry process) Pyranol Pyrex glass Quartz (fused) Rubber Silica or SiO2 (fused) Silicon Snow Sodium chloride Soil (dry) Steatite Styrofoam Teflon Titanium dioxide Water (distilled) Water (sea) Water (dehydrated) Wood (dry) � r �� / � 1.0005 25 8.8 2.7 4.74 1200 1.001 12.4 16 4–7 4.2 5.4 6.6 3.5 3 3.45 2.26 2.25 2.56 6 4.4 4 3.8 2.5–3 3.8 11.8 3.3 5.9 2.8 5.8 1.03 2.1 100 80 1 1.5–4 0.1 0.000 6 0.002 0.022 0.013 0.000 25 0.002 0.05 0.000 6 0.011 0.02 0.008 0.03 0.000 2 0.000 3 0.000 05 0.014 0.000 5 0.000 6 0.000 75 0.002 0.000 75 0.5 0.000 1 0.05 0.003 0.000 1 0.000 3 0.001 5 0.04 4 0 0.01 Conductivity (� ) Material Silver Copper Gold Aluminum Tungsten Zinc Brass Nickel Iron Phosphor bronze Solder Carbon steel German silver Manganin Constantan Germanium Stainless steel , S/m 6.17 × 107 4.10 × 107 3.82 × 107 1.82 × 107 1.67 × 107 1.5 × 107 1.45 × 107 1.03...
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...CHEM 1035 | GENERAL CHEMISTRY | 4 | A | ENGE 1024 | ENGINEERING EXPLORATION | 2 | A | ENGL 1105 | FRESHMAN ENGLISH | 3 | A | MATH 1205 | CALCULUS | 3 | A- | ECON 2005 | MICRO PRINCIPLES OF ECONOMICS | 3 | A- | ACIS 1504 | INTRODUCTION TO BUSINESS INFORMATION SYSTEMS | 3 | B+ | ENGE 1114 | EXPLORATION OF ENGINEERING DESIGN | 2 | A | MATH 1206 | CALCULUS | 3 | B+ | PHYS 2305 | FOUNDATIONS OF PHYSICS | 4 | A | AOE 2074 | COMPUTATIONAL METHODS | 3 | B+ | ECON 2006 | MACRO PRINCIPLES OF ECONOMICS | 3 | A- | ECON 3104 | MICRO ECONOMICS THEORY | 3 | B+ | AOE 2104 | INTRODUCTION TO AEROSPACE ENGINEERING | 3 | A- | ESM 2104 | STATICS | 3 | B | MATH 2224 | MULTIVARIABLE CALCULUS | 3 | A- | AOE 3094 | MATERIALS FOR AEROSPACE AND OCEAN ENGINEERING | 3 | B+ | ACIS 2115 | PRINCIPLES OF ACCOUNTING | 3 | A | BIT 2405 | QUANTITATIVE METHODS | 3 | A | AOE 3104 | AIRCRAFT PERFORMANCE | 3 | B+ | ESM 2204 | MECHANICS OF DEFORMABLE BODIES | 3 | B+ | ESM 2304 | DYNAMICS | 3 | A- | ECON 3204 | MACRO ECONOMICS THEORY | 3 | B+ | MGT 3304 | MANAGEMENT THEORY AND LEADERSHIP PRACTICE | 3 | A | AOE 3054 | AEROSPACE EXPERIMENTAL METHODS | 3 | B+ | AOE 3114 | COMPRESSIBLE AERODYNAMICS | 3 | B | AOE 3124 | AEROSPACE STRUCTURES | 3 | A- | AOE 3134 | STABILITY AND CONTROL | 3 | A | MKT 3104 | MARKETING MANAGEMENT | 3 | B+ | FIN 3104 | INTRODUCTION TO FINANCE | 3 | A | AOE 3044 | BOUNDARY LAYER THEORY | 3 | B | AOE 4154 | AEROSPACE ENGINEERING LAB | 1 |...
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... Calculus From Wikipedia, the free encyclopedia Jump to: navigation, search This article is about the branch of mathematics. For other uses, see Calculus (disambiguation). | It has been suggested that Infinitesimal calculus be merged into this article or section. (Discuss) Proposed since May 2011. | Topics in Calculus | Fundamental theorem Limits of functions Continuity Mean value theorem [show]Differential calculus | Derivative Change of variables Implicit differentiation Taylor's theorem Related rates Rules and identities:Power rule, Product rule, Quotient rule, Chain rule | [show]Integral calculus | IntegralLists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution, partial fractions, changing order | [show]Vector calculus | Gradient Divergence Curl Laplacian Gradient theorem Green's theorem Stokes' theorem Divergence theorem | [show]Multivariable calculus | Matrix calculus Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian | | Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the...
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...Calculus From Wikipedia, the free encyclopedia This article is about the branch of mathematics. For other uses, see Calculus (disambiguation). Topics in Calculus Fundamental theorem Limits of functions Continuity Mean value theorem [show]Differential calculus [show]Integral calculus [show]Vector calculus [show]Multivariable calculus Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits,functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modernmathematics education. It has two major branches,differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change,[1] in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science,economics, and engineering and can solve many problems for which algebra alone is insufficient. Historically, calculus was called "the calculus of infinitesimals", or "infinitesimal calculus". More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus...
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...courses and I decide to take financial related courses, especially Actuarial Studies as the direction of my master’s study. I have been interested in numbers since I was a high school student. I felt satisfied even though I had to contribute more than one hour to solve a mathematic problem. I often spent time to think about other methods to solve mathematic problems that my teacher had provided answers. My enthusiasm about mathematics was inspired again when I began my college study. I took some basic mathematics concepts, such as limit, series, calculus and differential coefficient. I also learned some basic theories and the application of related concepts, such as differential coefficient of function of one variable, calculus, partial derivative of function of many variables, differential equation, and Taylor's formula, intermediate value theorem and infinite series which help me to know the nature of function, and the independent vector algebra and space analytic geometry. To be honest, I even made more efforts in the study of mathematics than that in my academic courses. Therefore, I believe that my mathematics achievements are pretty competitive in pursuing Master of Actuarial Studies. I took multi-directed development when I was in college since I believe that enhancing my learning and surviving ability accounts even...
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...Divergence and Curl of a Vector Field Recall that a vector field F on R3 can be expressed as F(x, y, z) =< F1 (x, y, z), F2 (x, y, z), F3 (x, y, z) >, where the component functions are scalar functions from R3 to R. For the sake of brevity, we will often use x to denote the vector < x, y, z > and write F(x) =< F1 (x), F2 (x), F3 (x) > or even drop the variables altogether and write F =< F1 , F2 , F3 >. Let F =< F1 , F2 , F3 > : R3 → R3 be a differentiable vector field. The divergence of F is the scalar function div F = and the curl of F is the vector field curl F = ∂F3 ∂F2 − ∂y ∂z i+ ∂F1 ∂F3 − ∂z ∂x j+ ∂F2 ∂F1 − ∂x ∂y k ∂F1 ∂F2 ∂F3 + + . ∂x ∂y ∂z Note that the divergence can be defined for any vector field F : Rn → Rn , while the curl is only defined on vector fields in 3-space. There is a nice second way to deal with these functions. Define the operator , called the “del” operator, by = ∂ ∂ ∂ i+ j+ k= ∂x ∂y ∂z ∂ ∂ ∂ , , ∂x ∂y ∂z The action of the del operator on a scalar function f : R3 → R is just the gradient: f= ∂f ∂f ∂f i+ j+ k= ∂x ∂y ∂z ∂f ∂f ∂f , , ∂x ∂y ∂z = grad(f ) The divergence and the curl of a vector field are given by the two vector products that we have: the curl is the dot product and the curl is the cross product. div(F) = ·F curl(F ) = × F. Theorem 1 Let f be a differentiable scalar function and F be a differentiable vector field on R3 . 1. curl(grad f ) = 0 or 2. div(curl F) = 0 or × ( f ) = 0. ·( × F) = 0. The Laplacian operator on a scalar function is...
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...VECTOR FUNCTIONS VECTOR FUNCTIONS Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal vectors. MOTION IN SPACE: VELOCITY AND ACCELERATION Here, we show how the ideas of tangent and normal vectors and curvature can be used in physics to study: The motion of an object, including its velocity and acceleration, along a space curve. VELOCITY AND ACCELERATION In particular, we follow in the footsteps of Newton by using these methods to derive Kepler’s First Law of planetary motion. VELOCITY Suppose a particle moves through space so that its position vector at time t is r(t). VELOCITY Vector 1 Notice from the figure that, for small values of h, the vector r(t h) r(t ) h approximates the direction of the particle moving along the curve r(t). VELOCITY Its magnitude measures the size of the displacement vector per unit time. VELOCITY The vector 1 gives the average velocity over a time interval of length h. VELOCITY VECTOR Equation 2 Its limit is the velocity vector v(t) at time t : r(t h) r(t ) v(t ) lim h 0 h r '(t ) VELOCITY VECTOR Thus, the velocity vector is also the tangent vector and points in the direction of the tangent line. SPEED The speed of the particle at time t is the magnitude of the velocity vector, that is, |v(t)|. SPEED This is appropriate because, from Equation 2 ...
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...from Mathematics, which they need to pursue their Engineering degree in different disciplines. Course Contents: Module I: Differential Calculus Successive differentiation, Leibnitz’s theorem (without proof), Mean value theorem, Taylor’s theorem (proof), Remainder terms, Asymptote & Curvature, Partial derivatives, Chain rule, Differentiation of Implicit functions, Exact differentials, Tangents and Normals, Maxima, Approximations, Differentiation under integral sign, Jacobians and transformations of coordinates. Module II: Integral Calculus Fundamental theorems, Reduction formulae, Properties of definite integrals, Applications to length, area, volume, surface of revolution, improper integrals, Multiple Integrals-Double integrals, Applications to areas, volumes. Module III: Ordinary Differential Equations Formation of ODEs, Definition of order, degree & solutions, ODE of first order : Method of separation of variables, homogeneous and non homogeneous equations, Exactness & integrating factors, Linear equations & Bernoulli equations, General linear ODE of nth order, Solution of homogeneous equations, Operator method, Method of undetermined coefficients, Solution of simple simultaneous ODE. Module IV: Vector Calculus Scalar and Vector Field, Derivative of a Vector, Gradient, Directional Derivative, Divergence and Curl and their Physical Significance, Arc Length, Tangent, Directional Derivative, Evaluation...
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...Mathematics Syllabus Algebra: Algebra of complex numbers, addition, multiplication, conjugation, polar representation, properties of modulus and principal argument, triangle inequality, cube roots of unity, geometric interpretations. Quadratic equations with real coefficients, relations between roots and coefficients, formation of quadratic equations with given roots, symmetric functions of roots. Arithmetic, geometric and harmonic progressions, arithmetic, geometric and harmonic means, sums of finite arithmetic and geometric progressions, infinite geometric series, sums of squares and cubes of the first n natural numbers. Logarithms and their properties. Permutations and combinations, Binomial theorem for a positive integral index, properties of binomial coefficients. Matrices as a rectangular array of real numbers, equality of matrices, addition, multiplication by a scalar and product of matrices, transpose of a matrix, determinant of a square matrix of order up to three, inverse of a square matrix of order up to three, properties of these matrix operations, diagonal, symmetric and skew-symmetric matrices and their properties, solutions of simultaneous linear equations in two or three variables. Addition and multiplication rules of probability, conditional probability, Bayes Theorem, independence of events, computation of probability of events using permutations and combinations. Trigonometry: Trigonometric functions, their periodicity and graphs, addition...
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...MONEERAH INTEGRATED SCHOOL Merila, Ubaldo Laya, Iligan City S.Y. 2013- 2014 A Requirement in Mathematics IV: Calculus Controversy: Leibniz vs. Newton by Noronsalih Ali, Jra Submitted to Ms. Moneerah A. Bint- Usman Dedication I would like to dedicate this research to my adviser, teacher Monie. And to all of the people who inspired me, especially my parents for their support and to God for giving me enough knowledge to make this study successful. Acknowledgement Abstract This research explores more about the history of the two Mathematicians and how did they invent calculus with the same idea. This is a study about a controversy in Mathematics where Sir Isaac and Gottfried von Leibniz were involved. It tackles about who was the real father of calculus and who gets the credit of inventing it. Inside this paper, the researcher will also discuss a brief summary about Calculus, and short biography of the Mathematicians that were involved in this matter. Many people debates about this matter and we will also tackle some of it in this study. Introduction: So who really invented calculus first? Was it Sir Isaac Newton or Gottfried von Leibniz? Well let's do some investigation. There is no doubt about it that Newton and Leibniz made great mathematical breakthroughs but even before they began studying Calculus there were other people such as Archimedes and Euclid who discovered the infinite and infinitesimal. Much of Newton and Leibniz's work...
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...Math 5616H Midterm 1 with solutions Spring 2013 March 8, 2013 Total 80 points 1. (15 points) Let f (x) and g(x) be real continuous functions on an interval [a, b], such that b b f 2 (x) dx = a a b g 2 (x) dx = 1. Prove that a f (x)g(x) dx ≥ −1, and that a b f (x)g(x) dx = −1 if and only if f ≡ −g on [a, b]. Answer: Since f and g are continuous, so is (f + g)2 , which is therefore integrable. We compute: b b b b b 0≤ a b [f (x)+g(x)]2 dx = a f (x)2 dx+2 a f (x)g(x) dx+ a g(x)2 dx = 1+2 a f (x)g(x) dx+1, so a f (x)g(x) dx ≥ −1. If it is = −1, then the first “≤” must be “=”, so the continuous function [f (x) + g(x)]2 ≡ 0, and f ≡ −g on [a, b]. 2. (25 points) Let α(x) be a strictly increasing function on the interval [0, 1], such that α(0) = 0 and α(1) = 1. Show that the Riemann-Stieltjes integral 1 α(x) dα(x), 0 exists if and only if α is continuous on [0, 1], and evaluate this integral if it is continuous. Answer: Consider any partition P of [0, 1] : P = {0 = x0 , . . . , 1 = xn }. Since α is increasing, Mi := supx∈[xi−1 ,xi ] α(x) = α(xi ) and mi = α(xi−1 ). Then n n U (P, α, α) − L(P, α, α) = i=1 (Mi − mi )∆αi = i=1 (∆αi )2 . Suppose α is continuous; then since [0, 1] is compact, α is uniformly continuous. Thus, for any given ε > 0 there is δ > 0 so that if |x − y| < δ then |α(x) − α(y)| < ε. Hence if P ∗ is a refinement of P which satisfies xi − xi−1 < δ for all i = 1, . . . , n, we have n U (P ∗ ...
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