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AuthorProtter, Murray H. author
TitleA First Course in Real Analysis [electronic resource] / by Murray H. Protter, Charles B. Morrey
ImprintNew York, NY : Springer New York : Imprint: Springer, 1991
Edition Second Edition
Connect tohttp://dx.doi.org/10.1007/978-1-4419-8744-0
Descript XVIII, 536 p. online resource

SUMMARY

Many changes have been made in this second edition of A First Course in Real Analysis. The most noticeable is the addition of many problems and the inclusion of answers to most of the odd-numbered exercises. The book's readability has also been improved by the further clarification of many of the proofs, additional explanatory remarks, and clearer notation


CONTENT

1 The Real Number System -- 1.1 Axioms for a Field -- 1.2 Natural Numbers and Sequences -- 1.3 Inequalities -- 1.4 Mathematical Induction -- 2 Continuity And Limits -- 2.1 Continuity -- 2.2 Limits -- 2.3 One-Sided Limits -- 2.4 Limits at Infinity; Infinite Limits -- 2.5 Limits of Sequences -- 3 Basic Properties of Functions on ?1 -- 3.1 The Intermediate-Value Theorem -- 3.2 Least Upper Bound; Greatest Lower Bound -- 3.3 The Bolzanoโ{128}{148}Weierstrass Theorem -- 3.4 The Boundedness and Extreme-Value Theorems -- 3.5 Uniform Continuity -- 3.6 The Cauchy Criterion -- 3.7 The Heine-Borel and Lebesgue Theorems -- 4 Elementary Theory of Differentiation -- 4.1 The Derivative in ?1 -- 4.2 Inverse Functions in ?1 -- 5 Elementary Theory of Integration -- 5.1 The Darboux Integral for Functions on ?1 -- 5.2 The Riemann Integral -- 5.3 The Logarithm and Exponential Functions -- 5.4 Jordan Content and Area -- 6 Elementary Theory of Metric Spaces -- 6.1 The Schwarz and Triangle Inequalities; Metric Spaces -- 6.2 Elements of Point Set Topology -- 6.3 Countable and Uncountable Sets -- 6.4 Compact Sets and the Heineโ{128}{148}Borel Theorem -- 6.5 Functions on Compact Sets -- 6.6 Connected Sets -- 6.7 Mappings from One Metric Space to Another -- 7 Differentiation in ?N -- 7.1 Partial Derivatives and the Chain Rule -- 7.2 Taylorโ{128}{153}s Theorem; Maxima and Minima 178 -- 7.3 The Derivative in ?N -- 8 Integration in ?N -- 8.1 Volume in ?N -- 8.2 The Darboux Integral in ?N -- 8.3 The Riemann Integral in ?N -- 9 Infinite Sequences and Infinite Series -- 9.1 Tests for Convergence and Divergence -- 9.2 Series of Positive and Negative Terms; Power Series -- 9.3 Uniform Convergence of Sequences -- 9.4 Uniform Convergence of Series; Power Series -- 9.5 Unordered Sums -- 9.6 The Comparison Test for Unordered Sums; Uniform Convergence -- 9.7 Multiple Sequences and Series -- 10 Fourier Series -- 10.1 Expansions of Periodic Functions -- 10.2 Sine Series and Cosine Series; Change of Interval -- 10.3 Convergence Theorems -- 11 Functions Defined by Integrals; Improper Integrals -- 11.1 The Derivative of a Function Defined by an Integral; the Leibniz Rule -- 1l.2 Convergence and Divergence of Improper Integrals -- 11.3 The Derivative of Functions Defined by Improper Integrals; the Gamma Function -- 12 The Riemannโ{128}{148}Stieltjes Integral and Functions of Bounded Variation -- 12.1 Functions of Bounded Variation -- 12.2 The Riemannโ{128}{148}Stieltjes Integral -- 13 Contraction Mappings, Newtonโ{128}{153}s Method, and Differential Equations -- 13.1 A Fixed Point Theorem and Newtonโ{128}{153}s Method -- 13.2 Application of the Fixed Point Theorem to Differential Equations -- 14 Implicit Function Theorems and Lagrange Multipliers -- 14.1 The Implicit Function Theorem for a Single Equation -- 14.2 The Implicit Function Theorem for Systems -- 14.3 Change of Variables in a Multiple Integral -- 14.4 The Lagrange Multiplier Rule -- 15 Functions on Metric Spaces; Approximation -- 15.1 Complete Metric Spaces -- 15.2 Convex Sets and Convex Functions -- 15.3 Arzelaโ{128}{153}s Theorem; the Tietze Extension Theorem -- 15.4 Approximations and the Stoneโ{128}{148}Weierstrass Theorem -- 16 Vector Field Theory; the Theorems of Green and Stokes -- 16.1 Vector Functions on ?1 -- 16.2 Vector Functions and Fields on ?N -- 16.3 Line Integrals in ?N -- 16.4 Greenโ{128}{153}s Theorem in the Plane -- 16.5 Surfaces in ?3; Parametric Representation -- 16.6 Area of a Surface in ?3; Surface Integrals -- 16.7 Orientable Surfaces -- 16.8 The Stokes Theorem -- 16.9 The Divergence Theorem -- Appendixes -- Appendix 1 Absolute Value -- Appendix 2 Solution of Algebraic Inequalities -- Appendix 3 Expansions of Real Numbers in Any Base -- Answers to Odd-Numbered Problems


Mathematics Functions of real variables Mathematics Real Functions



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