دانلود کتاب Exploring Mathematics With CAS Assistance: Topics in Geometry, Algebra, Univariate Calculus, and Probability

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Front Cover Title page Copyright Contents Introduction Acknowledgments Part 1. Algebra & Geometry Chapter 1. Computer Algebra Systems and Elements of Algorithmics 1.1. Terms and notation 1.2. About data types and data structures 1.3. Elements of algorithmics and algorithmic problem solving 1.4. Glossary Chapter 2. Topics in Classical Geometry 2.1. Review: Matrices, vectors, and lines 2.2. Rigid transformations of the plane 2.3. Complex numbers in classical geometry 2.4. Three centers of a triangle 2.5. Glossary Chapter 3. More Topics in Classical Geometry 3.1. Lab 2: The Euler line 3.2. The Simson line 3.3. Conics 3.4. Glossary Chapter 4. Topics in Elementary Number Theory 4.1. Number of primes and the Riemann Hypothesis 4.2. Algorithms from elementary number theory 4.3. Pythagorean triples 4.4. Lab 4: Plotting legs of primitive Pythagorean triples 4.5. Linear Diophantine equations in two variables 4.6. Lab 5: Industrial application of an LDE in three variables 4.7. Glossary Chapter 5. Topics in Algebra: Solving Univariate Algebraic Equations 5.1. Roots of univariate polynomials 5.2. Geometry of cubic equations: Counting the number of real roots 5.3. Lab 6: Solving cubic equations using Vieta’s substitution 5.4. Nonnegative univariate polynomials 5.5. Glossary Chapter 6. Topics in Algebra: Bivariate Systems of Polynomial Equations 6.1. Linear systems of two equations 6.2. Nonlinear systems of polynomial equations: Motivating example 6.3. Solving nonlinear polynomial systems 6.4. Implicitization of plane curves 6.5. Glossary Part 2. Calculus and Numerics Chapter 7. Derivatives 7.1. Review: Definitions, notation, and terminology 7.2. Convexity of a univariate function 7.3. Some facts about functions and derivatives 7.4. Lab 8: Constructing a square circumscribed about ellipse 7.5. Glossary Chapter 8. Definite Integrals 8.1. Review: Some basic concepts and facts of univariate integral calculus 8.2. Area of a region bounded by a simple closed curve 8.3. Lab 9: Submergence depth of a body of revolution in equilibrium 8.4. Solving some ordinary differential equations 8.5. Glossary Chapter 9. Approximating Zeros of Functions by Iteration Methods 9.1. Fixed point iteration method 9.2. Newton’s method 9.3. Lab 11: Kepler’s Equation and deriving Kepler’s Second Law 9.4. Lab 12: Exploration of the logistic maps 9.5. Glossary Chapter 10. Polynomial Approximations 10.1. Taylor polynomials 10.2. Interpolating polynomials in the Lagrange form 10.3. Piecewise polynomial interpolation: Splines 10.4. Approximating large data sets: Regression 10.5. Two real-life applications of the LS method 10.6. Glossary Chapter 11. Trigonometric Approximation 11.1. Short review of trigonometric functions 11.2. Fourier series 11.3. About the accuracy of trigonometric approximations 11.4. Celebrated classical application of Fourier series 11.5. Glossary Chapter 12. Fourier Analysis in Music and Signal Processing 12.1. Introduction and background 12.2. Fourier series and periodic signals 12.3. The Fourier transform for non-periodic signals 12.4. The Discrete Fourier Transform 12.5. Fourier series in signal processing 12.6. Glossary Part 3. Probability and Statistics Chapter 13. Probability and Statistics Basics 13.1. Review: Some basic concepts of probability 13.2. Some discrete probability distributions 13.3. About continuous probability distributions 13.4. Law of Large Numbers 13.5. Central Limit Theorem 13.6. Glossary Chapter 14. Computer Simulation of Statistical Sampling 14.1. Random number generation 14.2. Lab 17: CLT and LLN in action: Life expectancy in the world population 14.3. Sampling from non-uniform distributions (optional) 14.4. Monte Carlo methods for finding integrals and areas 14.5. Lab 18: Buffon’s needle problem 14.6. Glossary Chapter 15. Simple Random Walks 15.1. Simple random walks on integers 15.2. Lab 19: The gambler’s ruin problem 15.3. Random walk on the square lattice 15.4. Lab 20: Drunken sailor problem 15.5. Glossary Appendix A. Data for Lab 17 in Chapter 14 Bibliography Index Back Cover