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| To the Reader | |
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| Mathematical Thinking | |
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| The NCAA March Madness Problem | |
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| Gauss and the Arithmetic Series | |
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| The Pythagorean Theorem | |
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| Torus Area and Volume | |
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| Einstein's Velocity Addition Law | |
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| The Birthday Problem | |
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| Fibonacci Numbers and the Golden Ratio | |
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| [characters not reproducible] [pi] in the Gaussian Integral | |
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| Function Equal to Its Derivative | |
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| Log of N Factorial for Large N | |
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| Potential and Kinetic Energies | |
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| Riemann Zeta Function and Prime Numbers | |
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| How to Solve It | |
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| A Note on Mathematical Rigor | |
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| Numbers | |
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| Integers | |
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| Primes | |
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| Divisibility | |
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| Rational Numbers | |
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| Exponential Notation | |
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| Powers of 10 | |
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| Binary Number System | |
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| Infinity | |
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| Algebra | |
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| Symbolic Variables | |
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| Legal and Illegal Algebraic Manipulations | |
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| Factor-Label Method | |
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| Powers and Roots | |
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| Logarithms | |
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| The Quadratic Formula | |
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| Imagining i | |
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| Factorials, Permutations, and Combinations | |
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| The Binomial Theorem | |
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| e Is for Euler | |
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| Trigonometry | |
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| What Use Is Trigonometry? | |
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| The Pythagorean Theorem | |
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| [pi] in the Sky | |
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| Sine and Cosine | |
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| Tangent and Secant | |
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| Trigonometry in the Complex Plane | |
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| de Moivre's Theorem | |
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| Euler's Theorem | |
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| Hyperbolic Functions | |
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| Analytic Geometry | |
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| Functions and Graphs | |
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| Linear Functions | |
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| Conic Sections | |
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| Conic Sections in Polar Coordinates | |
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| Calculus | |
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| A Little Road Trip | |
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| A Speedboat Ride | |
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| Differential and Integral Calculus | |
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| Basic Formulas of Differential Calculus | |
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| More on Derivatives | |
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| Indefinite Integrals | |
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| Techniques of Integration | |
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| Curvature, Maxima, and Minima | |
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| The Gamma Function | |
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| Gaussian and Error Functions | |
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| Series and Integrals | |
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| Some Elementary Series | |
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| Power Series | |
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| Convergence of Series | |
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| Taylor Series | |
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| L'Hopital's Rule | |
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| Fourier Series | |
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| Dirac Deltafunction | |
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| Fourier Integrals | |
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| Generalized Fourier Expansions | |
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| Asymptotic Series | |
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| Differential Equations | |
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| First-Order Differential Equations | |
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| AC Circuits | |
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| Second-Order Differential Equations | |
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| Some Examples from Physics | |
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| Boundary Conditions | |
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| Series Solutions | |
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| Bessel Functions | |
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| Second Solution | |
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| Matrix Algebra | |
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| Matrix Multiplication | |
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| Further Properties of Matrices | |
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| Determinants | |
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| Matrix Inverse | |
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| Wronskian Determinant | |
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| Special Matrices | |
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| Similarity Transformations | |
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| Eigenvalue Problems | |
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| Group Theory | |
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| Minkowski Spacetime | |
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| Multivariable Calculus | |
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| Partial Derivatives | |
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| Multiple Integration | |
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| Polar Coordinates | |
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| Cylindrical Coordinates | |
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| Spherical Polar Coordinates | |
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| Differential Expressions | |
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| Line Integrals | |
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| Green's Theorem | |
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| Vector Analysis | |
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| Scalars and Vectors | |
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| Scalar or Dot Product | |
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| Vector or Cross Product | |
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| Triple Products of Vectors | |
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| Vector Velocity and Acceleration | |
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| Circular Motion | |
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| Angular Momentum | |
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| Gradient of a Scalar Field | |
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| Divergence of a Vector Field | |
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| Curl of a Vector Field | |
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| Maxwell's Equations | |
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| Covariant Electrodynamics | |
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| Curvilinear Coordinates | |
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| Vector Identities | |
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| Partial Differential Equations and Special Functions | |
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| Partial Differential Equations | |
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| Separation of Variables | |
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| Special Functions | |
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| Leibniz's Formula | |
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| Vibration of a Circular Membrane | |
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| Bessel Functions | |
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| Laplace's Equation in Spherical Coordinates | |
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| Legendre Polynomials | |
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| Spherical Harmonics | |
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| Spherical Bessel Functions | |
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| Hermite Polynomials | |
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| Laguerre Polynomials | |
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| Complex Variables | |
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| Analytic Functions | |
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| Derivative of an Analytic Function | |
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| Contour Integrals | |
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| Cauchy's Theorem | |
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| Cauchy's Integral Formula | |
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| Taylor Series | |
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| Laurent Expansions | |
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| Calculus of Residues | |
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| Multivalued Functions | |
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| Integral Representations for Special Functions | |
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| About the Author | |
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| Index | |