A Programmer’s Introduction to Mathematics

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