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

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