All the math you missed : but need to know for graduate school / Thomas A. Garrity ; figures by Lori Pedersen.

Format:

Book

Author/Creator:

Garrity, Thomas A., author.

Language:

English

Subjects (All):

Mathematics.

Physical Description:

1 online resource (xxviii, 387 pages) : digital, PDF file(s).

Edition:

Second edition.

Place of Publication:

Cambridge : Cambridge University Press, 2021.

Summary:

Beginning graduate students in mathematical sciences and related areas in physical and computer sciences and engineering are expected to be familiar with a daunting breadth of mathematics, but few have such a background. This bestselling book helps students fill in the gaps in their knowledge. Thomas A. Garrity 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 explanations are accompanied by numerous examples, exercises and suggestions for further reading that allow the reader to test and develop their understanding of these core topics. Featuring four new chapters and many other improvements, this second edition of All the Math You Missed is an essential resource for advanced undergraduates and beginning graduate students who need to learn some serious mathematics quickly.

Contents:

Cover

Half-title

Title page

Copyright information

Dedication

Contents

Preface

On the Structure of Mathematics

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

1 Linear Algebra

1.1 Introduction

1.2 The Basic Vector Space R[sup(n)]

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

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

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

4 Point Set Topology

4.1 Basic Definitions

4.2 The Standard Topology on R[sup(n)]

4.3 Metric Spaces

4.4 Bases for Topologies.

4.5 Zariski Topology of Commutative Rings

4.6 Books

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

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

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

8 Geometry

8.1 Euclidean Geometry

8.2 Hyperbolic Geometry

8.3 Elliptic Geometry

8.4 Curvature

8.5 Books

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

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

11 Algebra

11.1 Groups

11.2 Representation Theory

11.3 Rings

11.4 Fields and Galois Theory

11.5 Books

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

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

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

15 Lebesgue Integration

15.1 Lebesgue Measure

15.2 The Cantor Set

15.3 Lebesgue Integration

15.4 Convergence Theorems

15.5 Books

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

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.

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

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

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

Appendix Equivalence Relations

Bibliography

Index.

Notes:

Title from publisher's bibliographic system (viewed on 11 Jun 2021).

ISBN:

1-009-00640-1

1-009-00620-7

1-108-99287-0

OCLC:

1295274039