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All the Mathematics You Missed

Book Description

Beginning graduate students in mathematics and other quantitative subjects are expected to have a daunting breadth of mathematical knowledge. But few have such a background. This 2002 book will help students to see the broad outline of mathematics and to fill in the gaps in their knowledge. The author explains the basic points and a few key results of all the most important undergraduate topics in mathematics, emphasizing the intuitions behind the subject. The topics include linear algebra, vector calculus, differential geometry, real analysis, point-set topology, probability, complex analysis, abstract algebra, and more. An annotated bibliography then offers a guide to further reading and to more rigorous foundations. This book will be an essential resource for advanced undergraduate and beginning graduate students in mathematics, the physical sciences, engineering, computer science, statistics, and economics who need to quickly learn some serious mathematics.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. On the Structure of Mathematics
  9. Brief Summaries of Topics
    1. 0.1 Linear Algebra
    2. 0.2 Real Analysis
    3. 0.3 Differentiating Vector-Valued Functions
    4. 0.4 Point Set Topology
    5. 0.5 Classical Stokes’ Theorems
    6. 0.6 Differential Forms and Stokes’ Theorem
    7. 0.7 Curvature for Curves and Surfaces
    8. 0.8 Geometry
    9. 0.9 Complex Analysis
    10. 0.10 Countability and the Axiom of Choice
    11. 0.11 Algebra
    12. 0.12 Lebesgue Integration
    13. 0.13 Fourier Analysis
    14. 0.14 Differential Equations
    15. 0.15 Combinatorics and Probability Theory
    16. 0.16 Algorithms
  10. 1. Linear Algebra
    1. 1.1 Introduction
    2. 1.2 The Basic Vector Space R[sup(n)]
    3. 1.3 Vector Spaces and Linear Transformations
    4. 1.4 Bases and Dimension
    5. 1.5 The Determinant
    6. 1.6 The Key Theorem of Linear Algebra
    7. 1.7 Similar Matrices
    8. 1.8 Eigenvalues and Eigenvectors
    9. 1.9 Dual Vector Spaces
    10. 1.10 Books
    11. 1.11 Exercises
  11. 2. ([E]) and δ Real Analysis
    1. 2.1 Limits
    2. 2.2 Continuity
    3. 2.3 Differentiation
    4. 2.4 Integration
    5. 2.5 The Fundamental Theorem of Calculus
    6. 2.6 Pointwise Convergence of Functions
    7. 2.7 Uniform Convergence
    8. 2.8 The Weierstrass M-Test
    9. 2.9 Weierstrass’ Example
    10. 2.10 Books
    11. 2.11 Exercises
  12. 3. Calculus for Vector-Valued Functions
    1. 3.1 Vector-Valued Functions
    2. 3.2 Limits and Continuity
    3. 3.3 Differentiation and Jacobians
    4. 3.4 The Inverse Function Theorem
    5. 3.5 Implicit Function Theorem
    6. 3.6 Books
    7. 3.7 Exercises
  13. 4. Point Set Topology
    1. 4.1 Basic Definitions
    2. 4.2 The Standard Topology on R[sup(n)]
    3. 4.3 Metric Spaces
    4. 4.4 Bases for Topologies
    5. 4.5 Zariski Topology of Commutative Rings
    6. 4.6 Books
    7. 4.7 Exercises
  14. 5. Classical Stokes’ Theorems
    1. 5.1 Preliminaries about Vector Calculus
      1. 5.1.1 Vector Fields
      2. 5.1.2 Manifolds and Boundaries
      3. 5.1.3 Path Integrals
      4. 5.1.4 Surface Integrals
      5. 5.1.5 The Gradient
      6. 5.1.6 The Divergence
      7. 5.1.7 The Curl
      8. 5.1.8 Orientability
    2. 5.2 The Divergence Theorem and Stokes’ Theorem
    3. 5.3 Physical Interpretation of Divergence Thm
    4. 5.4 A Physical Interpretation of Stokes’ Theorem
    5. 5.5 Proof of the Divergence Theorem
    6. 5.6 Sketch of a Proof for Stokes’ Theorem
    7. 5.7 Books
    8. 5.8 Exercises
  15. 6. Differential Forms and Stokes’ Thm.
    1. 6.1 Volumes of Parallelepipeds
    2. 6.2 Diff. Forms and the Exterior Derivative
      1. 6.2.1 Elementary k-forms
      2. 6.2.2 The Vector Space of k-forms
      3. 6.2.3 Rules for Manipulating k-forms
      4. 6.2.4 Differential k-forms and the Exterior Derivative
    3. 6.3 Differential Forms and Vector Fields
    4. 6.4 Manifolds
    5. 6.5 Tangent Spaces and Orientations
      1. 6.5.1 Tangent Spaces for Implicit and Parametric Manifolds
      2. 6.5.2 Tangent Spaces for Abstract Manifolds
      3. 6.5.3 Orientation of a Vector Space
      4. 6.5.4 Orientation of a Manifold and its Boundary
    6. 6.6 Integration on Manifolds
    7. 6.7 Stokes’ Theorem
    8. 6.8 Books
    9. 6.9 Exercises
  16. 7. Curvature for Curves and Surfaces
    1. 7.1 Plane Curves
    2. 7.2 Space Curves
    3. 7.3 Surfaces
    4. 7.4 The Gauss-Bonnet Theorem
    5. 7.5 Books
    6. 7.6 Exercises
  17. 8. Geometry
    1. 8.1 Euclidean Geometry
    2. 8.2 Hyperbolic Geometry
    3. 8.3 Elliptic Geometry
    4. 8.4 Curvature
    5. 8.5 Books
    6. 8.6 Exercises
  18. 9. Complex Analysis
    1. 9.1 Analyticity as a Limit
    2. 9.2 Cauchy-Riemann Equations
    3. 9.3 Integral Representations of Functions
    4. 9.4 Analytic Functions as Power Series
    5. 9.5 Conformal Maps
    6. 9.6 The Riemann Mapping Theorem
    7. 9.7 Several Complex Variables: Hartog’s Theorem
    8. 9.8 Books
    9. 9.9 Exercises
  19. 10. Countability and the Axiom of Choice
    1. 10.1 Countability
    2. 10.2 Naive Set Theory and Paradoxes
    3. 10.3 The Axiom of Choice
    4. 10.4 Non-measurable Sets
    5. 10.5 Gödel and Independence Proofs
    6. 10.6 Books
    7. 10.7 Exercises
  20. 11. Algebra
    1. 11.1 Groups
    2. 11.2 Representation Theory
    3. 11.3 Rings
    4. 11.4 Fields and Galois Theory
    5. 11.5 Books
    6. 11.6 Exercises
  21. 12. Lebesgue Integration
    1. 12.1 Lebesgue Measure
    2. 12.2 The Cantor Set
    3. 12.3 Lebesgue Integration
    4. 12.4 Convergence Theorems
    5. 12.5 Books
    6. 12.6 Exercises
  22. 13. Fourier Analysis
    1. 13.1 Waves, Periodic Functions and Trigonometry
    2. 13.2 Fourier Series
    3. 13.3 Convergence Issues
    4. 13.4 Fourier Integrals and Transforms
    5. 13.5 Solving Differential Equations
    6. 13.6 Books
    7. 13.7 Exercises
  23. 14. Differential Equations
    1. 14.1 Basics
    2. 14.2 Ordinary Differential Equations
    3. 14.3 The Laplacian
      1. 14.3.1 Mean Value Principle
      2. 14.3.2 Separation of Variables
      3. 14.3.3 Applications to Complex Analysis
    4. 14.4 The Heat Equation
    5. 14.5 The Wave Equation
      1. 14.5.1 Derivation
      2. 14.5.2 Change of Variables
    6. 14.6 Integrability Conditions
    7. 14.7 Lewy’s Example
    8. 14.8 Books
    9. 14.9 Exercises
  24. 15. Combinatorics and Probability
    1. 15.1 Counting
    2. 15.2 Basic Probability Theory
    3. 15.3 Independence
    4. 15.4 Expected Values and Variance
    5. 15.5 Central Limit Theorem
    6. 15.6 Stirling’s Approximation for n!
    7. 15.7 Books
    8. 15.8 Exercises
  25. 16. Algorithms
    1. 16.1 Algorithms and Complexity
    2. 16.2 Graphs: Euler and Hamiltonian Circuits
    3. 16.3 Sorting and Trees
    4. 16.4 P=NP?
    5. 16.5 Numerical Analysis: Newton’s Method
    6. 16.6 Books
    7. 16.7 Exercises
  26. Appendix A: Equivalence Relations
  27. Bibliography
  28. Index