A Programmer’s Introduction to Mathematics

Description

Our Goal......Page 9 Like Programming, Mathematics has a Culture......Page 13 Polynomials, Java, and Definitions......Page 17 A Little More Notation......Page 25 Existence & Uniqueness......Page 26 Realizing it in Code......Page 34 Application: Sharing Secrets......Page 36 Cultural Review......Page 39 Exercises......Page 40 Chapter Notes......Page 43 On Pace and Patience......Page 47 Sets......Page 51 Sets, Functions, and Their -Jections......Page 52 Clever Bijections and Counting......Page 60 Proof by Induction and Contradiction......Page 63 Application: Stable Marriages......Page 66 Cultural Review......Page 70 Exercises......Page 71 Chapter Notes......Page 73 Variable Names, Overloading, and Your Brain......Page 75 The Definition of a Graph......Page 81 Graph Coloring......Page 83 Register Allocation and Hardness......Page 85 Planarity and the Euler Characteristic......Page 87 Application: the Five Color Theorem......Page 89 Approximate Coloring......Page 94 Cultural Review......Page 95 Exercises......Page 96 Chapter Notes......Page 97 The Many Subcultures of Mathematics......Page 101 Calculus with One Variable......Page 107 Lines and Curves......Page 108 Limits......Page 113 The Derivative......Page 119 Taylor Series......Page 123 Remainders......Page 128 Application: Finding Roots......Page 130 Exercises......Page 137 On Types and Tail Calls......Page 141 Linear Algebra......Page 147 Linear Maps and Vector Spaces......Page 148 Linear Maps, Formally This Time......Page 153 The Basis and Linear Combinations......Page 155 Dimension......Page 159 Matrices......Page 161 Conjugations and Computations......Page 167 One Vector Space to Rule Them All......Page 169 Geometry of Vector Spaces......Page 171 Application: Singular Value Decomposition......Page 176 Exercises......Page 191 Chapter Notes......Page 193 Live and Learn Linear Algebra (Again)......Page 197 Eigenvectors and Eigenvalues......Page 203 Eigenvalues of Graphs......Page 205 Limiting the Scope: Symmetric Matrices......Page 207 Inner Products......Page 210 Orthonormal Bases......Page 214 Computing Eigenvalues......Page 217 The Spectral Theorem......Page 219 Application: Waves......Page 222 Cultural Review......Page 237 Exercises......Page 238 Chapter Notes......Page 241 Rigor and Formality......Page 243 Generalizing the Derivative......Page 249 Linear Approximations......Page 252 Multivariable Functions and the Chain Rule......Page 257 Computing the Total Derivative......Page 258 The Geometry of the Gradient......Page 262 Optimizing Multivariable Functions......Page 263 The Chain Rule: a Reprise and a Proof......Page 272 Gradient Descent: an Optimization Hammer......Page 275 Gradients of Computation Graphs......Page 276 Application: Automatic Differentiation and a Simple Neural Network......Page 279 Exercises......Page 295 Chapter Notes......Page 298 The Argument for Big-O Notation......Page 301 Groups......Page 311 The Geometric Perspective......Page 313 The Interface Perspective......Page 317 Homomorphisms: Structure Preserving Functions......Page 319 Building Blocks of Groups......Page 322 Geometry as the Study of Groups......Page 324 The Symmetry Group of the Poincaré Disk......Page 333 The Hyperbolic Isometry Group as a Group of Matrices......Page 339 Application: Drawing Hyperbolic Tessellations......Page 340 Exercises......Page 356 Chapter Notes......Page 361 A New Interface......Page 363 About the Author and Cover......Page 373 Index......Page 375