The Math You Need

A comprehensive survey of undergraduate mathematics, compressing four years of study into one robust overview.

In The Math You Need, Thomas Mack provides a singular, comprehensive survey of undergraduate mathematics, compressing four years of math curricula into one volume. Without sacrificing rigor, this book provides a go-to resource for the essentials that any academic or professional needs. Each chapter is followed by numerous exercises to provide the reader an opportunity to practice what they learned. The Math You Need is distinguished in its use of the Bourbaki style—the gold standard for concision and an approach that mathematicians will find of particular interest. As ambitious as it is compact, this text embraces mathematical abstraction throughout, avoiding ad hoc computations in favor of general results.

Covering nine areas—group theory, commutative algebra, linear algebra, topology, real analysis, complex analysis, number theory, probability, and statistics—this thorough and highly effective overview of the undergraduate curriculum will prove to be invaluable to students and instructors alike.Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Definitions and Examples . . . . . . . . . . . . . . . . . . 1
1.2 Subgroups and Group Homomorphisms . . . . . . . . . . 4
1.3 Group Constructions . . . . . . . . . . . . . . . . . . . . 8
1.4 The Isomorphism Theorems . . . . . . . . . . . . . . . . 13
1.5 Group Actions . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7 Permutation Groups . . . . . . . . . . . . . . . . . . . . . 20
1.8 p-Groups and the Sylow Theorems . . . . . . . . . . . . . 27
1.9 Solvable and Nilpotent Groups . . . . . . . . . . . . . . . 30
1.10 Free Groups and Presentations . . . . . . . . . . . . . . . 35
1.11 Further Topics . . . . . . . . . . . . . . . . . . . . . . . . 39
1.12 Further Reading . . . . . . . . . . . . . . . . . . . . . . . 40
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2 Commutative Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.1 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.3 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.5 Module Constructions . . . . . . . . . . . . . . . . . . . . 60
2.6 Noetherian Modules . . . . . . . . . . . . . . . . . . . . . 63
2.7 Prime and Maximal Ideals . . . . . . . . . . . . . . . . . 66
2.8 Localization . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.9 Gauss’s Lemma . . . . . . . . . . . . . . . . . . . . . . . 76
2.10 Principal Ideal Domains . . . . . . . . . . . . . . . . . . 78
2.11 Field Extensions . . . . . . . . . . . . . . . . . . . . . . . 85
2.12 Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.13 Further Topics . . . . . . . . . . . . . . . . . . . . . . . . 92
2.14 Further Reading . . . . . . . . . . . . . . . . . . . . . . . 93
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . 99
viii Contents
3.2 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.3 Vector Space Constructions . . . . . . . . . . . . . . . . . 107
3.4 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . 111
3.5 The Determinant . . . . . . . . . . . . . . . . . . . . . . 115
3.6 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.7 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . 125
3.8 Inner Product Spaces . . . . . . . . . . . . . . . . . . . . 131
3.9 Matrix Decompositions . . . . . . . . . . . . . . . . . . . 138
3.10 Further Topics . . . . . . . . . . . . . . . . . . . . . . . . 142
3.11 Further Reading . . . . . . . . . . . . . . . . . . . . . . . 143
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.1 Definitions and Examples . . . . . . . . . . . . . . . . . . 149
4.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.3 Topological Space Constructions . . . . . . . . . . . . . . 156
4.4 Separation Axioms . . . . . . . . . . . . . . . . . . . . . 159
4.5 Connectedness . . . . . . . . . . . . . . . . . . . . . . . 163
4.6 Compactness . . . . . . . . . . . . . . . . . . . . . . . . 166
4.7 Tychonoff’s Theorem . . . . . . . . . . . . . . . . . . . . 170
4.8 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . 173
4.9 Completeness . . . . . . . . . . . . . . . . . . . . . . . . 179
4.10 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4.11 Further Topics . . . . . . . . . . . . . . . . . . . . . . . . 186
4.12 Further Reading . . . . . . . . . . . . . . . . . . . . . . . 188
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 189
5 Real Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.2 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . 197
5.3 Uniform Convergence . . . . . . . . . . . . . . . . . . . . 200
5.4 Differentiation on R . . . . . . . . . . . . . . . . . . . . . 204
5.5 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . 210
5.6 Measurable Spaces . . . . . . . . . . . . . . . . . . . . . 214
5.7 Measurable Functions . . . . . . . . . . . . . . . . . . . . 217
5.8 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 222
5.9 Measure Extensions . . . . . . . . . . . . . . . . . . . . . 230
5.10 Borel Measure . . . . . . . . . . . . . . . . . . . . . . . . 235
5.11 The Fundamental Theorem of Calculus . . . . . . . . . . 238
Contents ix
5.12 Further Topics . . . . . . . . . . . . . . . . . . . . . . . . 242
5.13 Further Reading . . . . . . . . . . . . . . . . . . . . . . . 244
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 246
6 Multivariable Analysis . . . . . . . . . . . . . . . . . . . . . . . . 249
6.1 Multivariable Differentiation . . . . . . . . . . . . . . . . 249
6.2 Multivariable Integration . . . . . . . . . . . . . . . . . . 256
6.3 The Change of Variables Formula . . . . . . . . . . . . . 259
6.4 Differential Equations . . . . . . . . . . . . . . . . . . . . 265
6.5 Common Derivatives and Integrals . . . . . . . . . . . . . 268
6.6 The Gaussian Integral . . . . . . . . . . . . . . . . . . . . 272
6.7 The Weierstrass Approximation Theorem . . . . . . . . . 276
6.8 The Constant Rank Theorem . . . . . . . . . . . . . . . . 284
6.9 Further Topics . . . . . . . . . . . . . . . . . . . . . . . . 290
6.10 Further Reading . . . . . . . . . . . . . . . . . . . . . . . 292
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 293
7 Complex Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
7.1 Contour Integrals . . . . . . . . . . . . . . . . . . . . . . 296
7.2 The Jordan Curve Theorem . . . . . . . . . . . . . . . . . 302
7.3 The Topology of Contours . . . . . . . . . . . . . . . . . 308
7.4 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . 316
7.5 The Cauchy–Riemann Equations . . . . . . . . . . . . . . 321
7.6 Cauchy’s Integral Formula . . . . . . . . . . . . . . . . . 324
7.7 Consequences of Cauchy’s Integral Formula . . . . . . . . 327
7.8 Meromorphic Functions . . . . . . . . . . . . . . . . . . . 332
7.9 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
7.10 The Open Mapping Theorem . . . . . . . . . . . . . . . . 341
7.11 Tauberian Theorems . . . . . . . . . . . . . . . . . . . . 345
7.12 Further Topics . . . . . . . . . . . . . . . . . . . . . . . . 348
7.13 Further Reading . . . . . . . . . . . . . . . . . . . . . . . 349
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 351
8 Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
8.1 Galois Theory . . . . . . . . . . . . . . . . . . . . . . . . 353
8.2 Algebraic Integers . . . . . . . . . . . . . . . . . . . . . . 359
8.3 Prime Factorization in Ok . . . . . . . . . . . . . . . . . . 363
8.4 Quadratic Fields . . . . . . . . . . . . . . . . . . . . . . . 368
8.5 Cyclotomic Extensions . . . . . . . . . . . . . . . . . . . 371
8.6 Diophantine Equations . . . . . . . . . . . . . . . . . . . 373
8.7 Quadratic Reciprocity . . . . . . . . . . . . . . . . . . . . 378
8.8 Solvability by Radicals . . . . . . . . . . . . . . . . . . . 381
8.9 The Riemann ζ-Function . . . . . . . . . . . . . . . . . . 386
8.10 The Prime Number Theorem . . . . . . . . . . . . . . . . 390
8.11 Further Topics . . . . . . . . . . . . . . . . . . . . . . . . 394
8.12 Further Reading . . . . . . . . . . . . . . . . . . . . . . . 396
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 398
9 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
9.1 Definitions and Constructions . . . . . . . . . . . . . . . . 401
9.2 Densities . . . . . . . . . . . . . . . . . . . . . . . . . . 404
9.3 Lp spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 408
9.4 The Radon–Nikodym Theorem . . . . . . . . . . . . . . . 413
9.5 Mean and Variance . . . . . . . . . . . . . . . . . . . . . 417
9.6 Joint Density Functions . . . . . . . . . . . . . . . . . . . 422
9.7 Common Probability Distributions . . . . . . . . . . . . . 425
9.8 Convergence of Distributions . . . . . . . . . . . . . . . . 432
9.9 Higher Moments and Characteristic Functions . . . . . . . 438
9.10 The Central Limit Theorem . . . . . . . . . . . . . . . . . 444
9.11 Further Topics . . . . . . . . . . . . . . . . . . . . . . . . 445
9.12 Further Reading . . . . . . . . . . . . . . . . . . . . . . . 447
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 448
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
A.1 Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 451
A.2 The Axiom of Choice . . . . . . . . . . . . . . . . . . . . 455
A.3 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . 459
A.4 Real and Complex Numbers . . . . . . . . . . . . . . . . 463
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473“This is the ultimate math major’s study guide. Freshmen should pursue it to see where they are going; seniors should study it to review where they’ve been.”
—Jay Cummings, Associate Professor, CSU Sacramento; author of Proofs: A Long-Form Mathematics Textbook and Real Analysis: A Long-Form Mathematics Textbook
 
"This degree-in-a-book covers nine textbooks' worth of advanced undergraduate mathematics in one well-written book, with over 300 exercises included. As both a reference and a resource, it's indispensable for every student and teacher of advanced mathematics."
—Oscar E. Fernandez, Associate Professor of Mathematics, Wellesley College; author of Calculus Simplified, The Calculus of Happiness, and Everyday CalculusThomas Mack is a mathematician, who earned his PhD in the field of combinatorial group theory and hyperbolic groups from the California Institute of Technology. He has worked for more than fifteen years as a researcher in the areas of defense, robotics, and finance.