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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 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 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.Less

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Book details

List price: $44.99 Copyright year: 2002 Publisher: Cambridge University Press Publication date: 11/12/2001 Binding: Paperback Pages: 376 Size: 5.75" wide x 8.50" long x 1.00" tall Weight: 1.364 Language: English

AuthorTable of Contents

Preface

On the Structure of Mathematics

Brief Summaries of Topics

Linear Algebra

Real Analysis

Differentiating Vector-Valued Functions

Point Set Topology

Classical Stokes' Theorems

Differential Forms and Stokes' Theorem

Curvature for Curves and Surfaces

Geometry

Complex Analysis

Countability and the Axiom of Choice

Algebra

Lebesgue Integration

Fourier Analysis

Differential Equations

Combinatorics and Probability Theory

Algorithms

Linear Algebra

Introduction

The Basic Vector Space R[superscript n]

Vector Spaces and Linear Transformations

Bases and Dimension

The Determinant

The Key Theorem of Linear Algebra

Similar Matrices

Eigenvalues and Eigenvectors

Dual Vector Spaces

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Exercises

[epsilon] and [delta] Real Analysis

Limits

Continuity

Differentiation

Integration

The Fundamental Theorem of Calculus

Pointwise Convergence of Functions

Uniform Convergence

The Weierstrass M-Test

Weierstrass' Example

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Exercises

Calculus for Vector-Valued Functions

Vector-Valued Functions

Limits and Continuity

Differentiation and Jacobians

The Inverse Function Theorem

Implicit Function Theorem

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Exercises

Point Set Topology

Basic Definitions

The Standard Topology on R[superscript n]

Metric Spaces

Bases for Topologies

Zariski Topology of Commutative Rings

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Exercises

Classical Stokes' Theorems

Preliminaries about Vector Calculus

Vector Fields

Manifolds and Boundaries

Path Integrals

Surface Integrals

The Gradient

The Divergence

The Curl

Orientability

The Divergence Theorem and Stokes' Theorem

Physical Interpretation of Divergence Thm.

A Physical Interpretation of Stokes' Theorem

Proof of the Divergence Theorem

Sketch of a Proof for Stokes' Theorem

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Exercises

Differential Forms and Stokes' Thm.

Volumes of Parallelepipeds

Diff. Forms and the Exterior Derivative

Elementary [kappa]-forms

The Vector Space of [kappa]-forms

Rules for Manipulating [kappa]-forms

Differential [kappa]-forms and the Exterior Derivative

Differential Forms and Vector Fields

Manifolds

Tangent Spaces and Orientations

Tangent Spaces for Implicit and Parametric Manifolds

Tangent Spaces for Abstract Manifolds

Orientation of a Vector Space

Orientation of a Manifold and its Boundary

Integration on Manifolds

Stokes' Theorem

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Exercises

Curvature for Curves and Surfaces

Plane Curves

Space Curves

Surfaces

The Gauss-Bonnet Theorem

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Exercises

Geometry

Euclidean Geometry

Hyperbolic Geometry

Elliptic Geometry

Curvature

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Exercises

Complex Analysis

Analyticity as a Limit

Cauchy-Riemann Equations

Integral Representations of Functions

Analytic Functions as Power Series

Conformal Maps

The Riemann Mapping Theorem

Several Complex Variables: Hartog's Theorem

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Exercises

Countability and the Axiom of Choice

Countability

Naive Set Theory and Paradoxes

The Axiom of Choice

Non-measurable Sets

Godel and Independence Proofs

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Exercises

Algebra

Groups

Representation Theory

Rings

Fields and Galois Theory

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Exercises

Lebesgue Integration

Lebesgue Measure

The Cantor Set

Lebesgue Integration

Convergence Theorems

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Exercises

Fourier Analysis

Waves, Periodic Functions and Trigonometry

Fourier Series

Convergence Issues

Fourier Integrals and Transforms

Solving Differential Equations

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Exercises

Differential Equations

Basics

Ordinary Differential Equations

The Laplacian

Mean Value Principle

Separation of Variables

Applications to Complex Analysis

The Heat Equation

The Wave Equation

Derivation

Change of Variables

Integrability Conditions

Lewy's Example

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Exercises

Combinatorics and Probability

Counting

Basic Probability Theory

Independence

Expected Values and Variance

Central Limit Theorem

Stirling's Approximation for n!

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Exercises

Algorithms

Algorithms and Complexity

Graphs: Euler and Hamiltonian Circuits

Sorting and Trees

P=NP?

Numerical Analysis: Newton's Method

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Exercises

Equivalence Relations

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