All the Math You Missed: But Need to Know for Graduate School

Contents
Preface
On the Structure of Mathematics
	Equivalence Problems
	The Study of Functions
	Equivalence Problems in Physics
Brief Summaries of Topics
	0.1 Linear Algebra
	0.2 Real Analysis
	0.3 Differentiating Vector-Valued Functions
	0.4 Point Set Topology
	0.5 Classical Stokes’ Theorems
	0.6 Differential Forms and Stokes’ Theorem
	0.7 Curvature for Curves and Surfaces
	0.8 Geometry
	0.9 Countability and the Axiom of Choice
	0.10 Elementary Number Theory
	0.11 Algebra
	0.12 Algebraic Number Theory
	0.13 Complex Analysis
	0.14 Analytic Number Theory
	0.15 Lebesgue Integration
	0.16 Fourier Analysis
	0.17 Differential Equations
	0.18 Combinatorics and Probability Theory
	0.19 Algorithms
	0.20 Category Theory
Chapter 1: Linear Algebra
	1.1 Introduction
	1.2 The Basic Vector Space Rn
	1.3 Vector Spaces and Linear Transformations
	1.4 Bases, Dimension, and Linear Transformations as Matrices
	1.5 The Determinant
	1.6 The Key Theorem of Linear Algebra
	1.7 Similar Matrices
	1.8 Eigenvalues and Eigenvectors
	1.9 Dual Vector Spaces
	1.10 Books
	Exercises
Chapter 2: ϵ and δ Real Analysis
	2.1 Limits
	2.2 Continuity
	2.3 Differentiation
	2.4 Integration
	2.5 The Fundamental Theorem of Calculus
	2.6 Pointwise Convergence of Functions
	2.7 Uniform Convergence
	2.8 The Weierstrass M-Test
	2.9 Weierstrass’ Example
	2.10 Books
	Exercises
Chapter 3: Calculus for Vector-Valued Functions
	3.1 Vector-Valued Functions
	3.2 Limits and Continuity of Vector-Valued Functions
	3.3 Differentiation and Jacobians
	3.4 The Inverse Function Theorem
	3.5 The Implicit Function Theorem
	3.6 Books
	Exercises
Chapter 4: Point Set Topology
	4.1 Basic Definitions
	4.2 The Standard Topology on Rn
	4.3 Metric Spaces
	4.4 Bases for Topologies
	4.5 Zariski Topology of Commutative Rings
	4.6 Books
	Exercises
Chapter 5: Classical Stokes’ Theorems
	5.1 Preliminaries about Vector Calculus
		5.1.1 Vector Fields
		5.1.2 Manifolds and Boundaries
		5.1.3 Path Integrals
		5.1.4 Surface Integrals
		5.1.5 The Gradient
		5.1.6 The Divergence
		5.1.7 The Curl
		5.1.8 Orientability
	5.2 The Divergence Theorem and Stokes’ Theorem
	5.3 A Physical Interpretation of the Divergence Theorem
	5.4 A Physical Interpretation of Stokes’ Theorem
	5.5 Sketch of a Proof of the Divergence Theorem
	5.6 Sketch of a Proof of Stokes’ Theorem
	5.7 Books
	Exercises
Chapter 6: Differential Forms and Stokes’ Theorem
	6.1 Volumes of Parallelepipeds
	6.2 Differential Forms and the Exterior Derivative
		6.2.1 Elementary k-Forms
		6.2.2 The Vector Space of k-Forms
		6.2.3 Rules for Manipulating k-Forms
		6.2.4 Differential k-Forms and the Exterior Derivative
	6.3 Differential Forms and Vector Fields
	6.4 Manifolds
	6.5 Tangent Spaces and Orientations
		6.5.1 Tangent Spaces for Implicit and Parametric Manifolds
		6.5.2 Tangent Spaces for Abstract Manifolds
		6.5.3 Orientation of a Vector Space
		6.5.4 Orientation of a Manifold and Its Boundary
	6.6 Integration on Manifolds
	6.7 Stokes’ Theorem
	6.8 Books
	Exercises
Chapter 7: Curvature for Curves and Surfaces
	7.1 Plane Curves
	7.2 Space Curves
	7.3 Surfaces
	7.4 The Gauss–Bonnet Theorem
	7.5 Books
	Exercises
Chapter 8: Geometry
	8.1 Euclidean Geometry
	8.2 Hyperbolic Geometry
	8.3 Elliptic Geometry
	8.4 Curvature
	8.5 Books
	Exercises
Chapter 9: Countability and the Axiom of Choice
	9.1 Countability
	9.2 Naive Set Theory and Paradoxes
	9.3 The Axiom of Choice
	9.4 Non-measurable Sets
	9.5 Gödel and Independence Proofs
	9.6 Books
	Exercises
Chapter 10: Elementary Number Theory
	10.1 Types of Numbers
	10.2 Prime Numbers
	10.3 The Division Algorithm and the Euclidean Algorithm
	10.4 Modular Arithmetic
	10.5 Diophantine Equations
	10.6 Pythagorean Triples
	10.7 Continued Fractions
	10.8 Books
	Exercises
Chapter 11: Algebra
	11.1 Groups
	11.2 Representation Theory
	11.3 Rings
	11.4 Fields and Galois Theory
	11.5 Books
	Exercises
Chapter 12: Algebraic Number Theory
	12.1 Algebraic Number Fields
	12.2 Algebraic Integers
	12.3 Units
	12.4 Primes and Problems with Unique Factorization
	12.5 Books
	Exercises
Chapter 13: Complex Analysis
	13.1 Analyticity as a Limit
	13.2 Cauchy–Riemann Equations
	13.3 Integral Representations of Functions
	13.4 Analytic Functions as Power Series
	13.5 Conformal Maps
	13.6 The Riemann Mapping Theorem
	13.7 Several Complex Variables: Hartog’s Theorem
	13.8 Books
	Exercises
Chapter 14: Analytic Number Theory
	14.1 The Riemann Zeta Function
	14.2 Riemann’s Insight
	14.3 The Gamma Function
	14.4 The Functional Equation: A Hidden Symmetry
	14.5 Linking π(x) with the Zeros of ζ(s)
	14.6 Books
	Exercises
Chapter 15: Lebesgue Integration
	15.1 Lebesgue Measure
	15.2 The Cantor Set
	15.3 Lebesgue Integration
	15.4 Convergence Theorems
	15.5 Books
	Exercises
Chapter 16: Fourier Analysis
	16.1 Waves, Periodic Functions and Trigonometry
	16.2 Fourier Series
	16.3 Convergence Issues
	16.4 Fourier Integrals and Transforms
	16.5 Solving Differential Equations
	16.6 Books
	Exercises
Chapter 17: Differential Equations
	17.1 Basics
	17.2 Ordinary Differential Equations
	17.3 The Laplacian
		17.3.1 Mean Value Principle
		17.3.2 Separation of Variables
		17.3.3 Applications to Complex Analysis
	17.4 The Heat Equation
	17.5 The Wave Equation
		17.5.1 Derivation
		17.5.2 Change of Variables
	17.6 The Failure of Solutions: Integrability Conditions
	17.7 Lewy’s Example
	17.8 Books
	Exercises
Chapter 18: Combinatorics and Probability Theory
	18.1 Counting
	18.2 Basic Probability Theory
	18.3 Independence
	18.4 Expected Values and Variance
	18.5 Central Limit Theorem
	18.6 Stirling’s Approximation for n!
	18.7 Books
	Exercises
Chapter 19: Algorithms
	19.1 Algorithms and Complexity
	19.2 Graphs: Euler and Hamiltonian Circuits
	19.3 Sorting and Trees
	19.4 P=NP?
	19.5 Numerical Analysis: Newton’s Method
	19.6 Books
	Exercises
Chapter 20: Category Theory
	20.1 The Basic Definitions
	20.2 Examples
	20.3 Functors
		20.3.1 Link with Equivalence Problems
		20.3.2 Definition of Functor
		20.3.3 Examples of Functors
	20.4 Natural Transformations
	20.5 Adjoints
	20.6 “There Exists” and “For All” as Adjoints
	20.7 Yoneda Lemma
	20.8 Arrow, Arrows, Arrows Everywhere
	20.9 Books
	Exercises
Appendix: Equivalence Relations
	Exercises
Bibliography
Index