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All the Mathematics You Missed But Need to Know for Graduate School

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ISBN-10: 0521797071

ISBN-13: 9780521797078

Edition: 2002

Authors: Thomas A. Garrity, Lori Pedersen

List price: $44.99
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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…    
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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

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
Books
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
Books
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