Mathematical analysis :
Schröder, Bernd S. W. 1966-
Mathematical analysis : a concise introduction / Bernd S.W. Schröder. - Hoboken, N.J. : Wiley-Interscience, c2008. - xv, 562 p. : ill. ; 25 cm.
Includes bibliographical references (p. 551-552) and index.
Table of contents -- Preface -- Analysis of functions of a single real variable -- The real numbers -- Field axioms -- Order axioms -- Lowest upper and greatest lower bounds -- Natural numbers, integers, and rational numbers -- Recursion, induction, summations, and products -- Sequences of real numbers -- Limits -- Limit laws -- Cauchy sequences -- Bounded sequences -- Infinite limits -- Continuous functions -- Limits of functions -- Limit laws -- One-sided limits and infinite limits -- Continuity -- Properties of continuous functions -- Limits at infinity -- Differentiable functions -- Differentiability -- Differentiation rules -- Rolle's theorem and the mean value theorem -- The Riemann integral 1 -- Riemann sums and the integral -- Uniform continuity and integrability of continuous functions -- The fundamental theorem of calculus -- The Darboux integral -- pt. 1. 1. 1.1. 1.2. 1.3. 1.4. 1.5. 2. 2.1. 2.2. 2.3. 2.4. 2.5. 3. 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 4. 4.1. 4.2. 4.3. 5. 5.1. 5.2. 5.3. 5.4. Series of real numbers 1 -- Series as a vehicle to define infinite sums -- Absolute convergence and unconditional convergence -- Some set theory -- The algebra of sets -- Countable sets -- Uncountable sets -- The Riemann integral 2 -- Outer Lebesgue measure -- Lebesgue's criterion for Riemann integrability -- More integral theorems -- Improper Riemann integrals -- The Lebesgue integral -- Outer Lebesgue measure -- Lebesgue measurable sets -- Lebesgue measurable functions -- Lebesgue integration -- Lebesgue integrals versus Riemann integrals-- Series of real numbers 2 -- Limits superior and inferior -- The root test and the ratio test -- Power series -- Sequences of functions -- Notions of convergence -- Uniform convergence -- Transcendental functions -- The exponential function -- Sine and cosine -- L'Hôpital's rule -- Numerical methods -- Approximation with Taylor polynomials -- Newton's method -- Numerical integration -- 6. 6.1. 6.2. 7. 7.1. 7.2. 7.3. 8. 8.1. 8.2. 8.3. 8.4. 9. 9.1. 9.2. 9.2. 9.3. 9.4. 10. 10.1. 10.2. 10.3. 11. 11.1. 11.2. 12. 12.1. 12.2. 12.3. 13. 13.1. 13.2. 13.3. Analysis in abstract spaces -- Integration on measure spaces -- Measure spaces -- Outer measures -- Measurable functions -- Integration of measurable functions -- Monotone and dominated convergence -- Convergence in mean, in measure, and almost everywhere -- Product [sigma]-algebras -- Product measures and Fubini's theorem -- The abstract venues for analysis -- Abstraction 1 : Vector spaces -- Representation of elements : bases and dimension -- Identification of spaces : isomorphism -- Abstraction 2 : inner product spaces -- Nicer representations : orthonormal sets -- Abstraction 3 : normed spaces -- Abstraction 4 : metric spaces -- L[superscript]p spaces -- Another number field : complex numbers -- The topology of metric spaces -- Convergence of sequences -- Completeness -- Continuous functions -- Open and closed sets -- Compactness -- The normed topology of R[superscript]d -- Dense subspaces -- Connectedness -- Locally compact spaces -- pt. 2. 14. 14.1. 14.2. 14.3. 14.4. 14.5. 14.6. 14.7. 14.8. 15. 15.1. 15.2. 15.3. 15.4. 15.5. 15.6. 15.7. 15.8. 15.9. 16. 16.1. 16.2. 16.3. 16.4. 16.5. 16.6. 16.7. 16.8. 16.9. Differentiation in normed spaces -- Continuous linear functions -- Matrix representation of linear functions -- Differentiability -- The mean value theorem -- How partial derivatives fit in -- Multilinear functions (tensors) -- Higher derivatives -- The implicit function theorem -- Measure, topology and differentiation -- Lebesgue measurable sets in R[superscript]d -- C[infinity] and approximation of integrable functions -- Tensor algebra and determinants -- Multidimensional substitution -- Manifolds and integral theorems -- Manifolds -- Tangent spaces and differentiable functions -- Differential forms, integrals over the unit cube -- k-forms and integrals over k-chains -- Integration on manifolds -- Stokes' theorem -- Hilbert spaces -- Orthonormal bases -- Fourier series -- The Riesz representation theorem -- 17. 17.1. 17.2. 17.3. 17.4. 17.5. 17.6. 17.7. 17.8. 18. 18.1. 18.2. 18.3. 18.4. 19. 19.1. 19.2. 19.3. 19.4. 19.5. g 19.6. 20. 20.1. 20.2. 20.3. Applied analysis -- Physics background -- Harmonic oscillators -- Heat and diffusion -- Separation of variables, Fourier series, and ordinary differential equations -- Maxwell's equations -- The Navier Stokes equation for the conservation of mass -- Ordinary differential equations -- Banach space valued differential equations -- An existence and uniqueness theorem -- Linear differential equations -- The finite element method -- Ritz-Galerkin approximation -- Weakly differentiable functions -- Sobolev spaces -- Elliptic differential operators -- Finite elements -- Conclusions and outlook -- Appendices -- Logic -- Statements -- Negations -- Set theory -- The Zermelo-Fraenkel axioms -- Relations and functions -- Natural numbers, integers, and rational numbers -- The natural numbers -- The integers -- The rational numbers -- Bibliography -- Index. pt. 3. 21. 21.1. 21.2. 21.3. 21.4. 21.5. 22. 22.1. 22.2. 22.3. 23. 23.1. 23.2. 23.3. 23.4. 23.5. A. A.1. A.2. B. B.1. B.2. C. C.1. C.2. C.3.
9780470107966 0470107960
Mathematical analysis.
QA300 / .S376 2008
515
Mathematical analysis : a concise introduction / Bernd S.W. Schröder. - Hoboken, N.J. : Wiley-Interscience, c2008. - xv, 562 p. : ill. ; 25 cm.
Includes bibliographical references (p. 551-552) and index.
Table of contents -- Preface -- Analysis of functions of a single real variable -- The real numbers -- Field axioms -- Order axioms -- Lowest upper and greatest lower bounds -- Natural numbers, integers, and rational numbers -- Recursion, induction, summations, and products -- Sequences of real numbers -- Limits -- Limit laws -- Cauchy sequences -- Bounded sequences -- Infinite limits -- Continuous functions -- Limits of functions -- Limit laws -- One-sided limits and infinite limits -- Continuity -- Properties of continuous functions -- Limits at infinity -- Differentiable functions -- Differentiability -- Differentiation rules -- Rolle's theorem and the mean value theorem -- The Riemann integral 1 -- Riemann sums and the integral -- Uniform continuity and integrability of continuous functions -- The fundamental theorem of calculus -- The Darboux integral -- pt. 1. 1. 1.1. 1.2. 1.3. 1.4. 1.5. 2. 2.1. 2.2. 2.3. 2.4. 2.5. 3. 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 4. 4.1. 4.2. 4.3. 5. 5.1. 5.2. 5.3. 5.4. Series of real numbers 1 -- Series as a vehicle to define infinite sums -- Absolute convergence and unconditional convergence -- Some set theory -- The algebra of sets -- Countable sets -- Uncountable sets -- The Riemann integral 2 -- Outer Lebesgue measure -- Lebesgue's criterion for Riemann integrability -- More integral theorems -- Improper Riemann integrals -- The Lebesgue integral -- Outer Lebesgue measure -- Lebesgue measurable sets -- Lebesgue measurable functions -- Lebesgue integration -- Lebesgue integrals versus Riemann integrals-- Series of real numbers 2 -- Limits superior and inferior -- The root test and the ratio test -- Power series -- Sequences of functions -- Notions of convergence -- Uniform convergence -- Transcendental functions -- The exponential function -- Sine and cosine -- L'Hôpital's rule -- Numerical methods -- Approximation with Taylor polynomials -- Newton's method -- Numerical integration -- 6. 6.1. 6.2. 7. 7.1. 7.2. 7.3. 8. 8.1. 8.2. 8.3. 8.4. 9. 9.1. 9.2. 9.2. 9.3. 9.4. 10. 10.1. 10.2. 10.3. 11. 11.1. 11.2. 12. 12.1. 12.2. 12.3. 13. 13.1. 13.2. 13.3. Analysis in abstract spaces -- Integration on measure spaces -- Measure spaces -- Outer measures -- Measurable functions -- Integration of measurable functions -- Monotone and dominated convergence -- Convergence in mean, in measure, and almost everywhere -- Product [sigma]-algebras -- Product measures and Fubini's theorem -- The abstract venues for analysis -- Abstraction 1 : Vector spaces -- Representation of elements : bases and dimension -- Identification of spaces : isomorphism -- Abstraction 2 : inner product spaces -- Nicer representations : orthonormal sets -- Abstraction 3 : normed spaces -- Abstraction 4 : metric spaces -- L[superscript]p spaces -- Another number field : complex numbers -- The topology of metric spaces -- Convergence of sequences -- Completeness -- Continuous functions -- Open and closed sets -- Compactness -- The normed topology of R[superscript]d -- Dense subspaces -- Connectedness -- Locally compact spaces -- pt. 2. 14. 14.1. 14.2. 14.3. 14.4. 14.5. 14.6. 14.7. 14.8. 15. 15.1. 15.2. 15.3. 15.4. 15.5. 15.6. 15.7. 15.8. 15.9. 16. 16.1. 16.2. 16.3. 16.4. 16.5. 16.6. 16.7. 16.8. 16.9. Differentiation in normed spaces -- Continuous linear functions -- Matrix representation of linear functions -- Differentiability -- The mean value theorem -- How partial derivatives fit in -- Multilinear functions (tensors) -- Higher derivatives -- The implicit function theorem -- Measure, topology and differentiation -- Lebesgue measurable sets in R[superscript]d -- C[infinity] and approximation of integrable functions -- Tensor algebra and determinants -- Multidimensional substitution -- Manifolds and integral theorems -- Manifolds -- Tangent spaces and differentiable functions -- Differential forms, integrals over the unit cube -- k-forms and integrals over k-chains -- Integration on manifolds -- Stokes' theorem -- Hilbert spaces -- Orthonormal bases -- Fourier series -- The Riesz representation theorem -- 17. 17.1. 17.2. 17.3. 17.4. 17.5. 17.6. 17.7. 17.8. 18. 18.1. 18.2. 18.3. 18.4. 19. 19.1. 19.2. 19.3. 19.4. 19.5. g 19.6. 20. 20.1. 20.2. 20.3. Applied analysis -- Physics background -- Harmonic oscillators -- Heat and diffusion -- Separation of variables, Fourier series, and ordinary differential equations -- Maxwell's equations -- The Navier Stokes equation for the conservation of mass -- Ordinary differential equations -- Banach space valued differential equations -- An existence and uniqueness theorem -- Linear differential equations -- The finite element method -- Ritz-Galerkin approximation -- Weakly differentiable functions -- Sobolev spaces -- Elliptic differential operators -- Finite elements -- Conclusions and outlook -- Appendices -- Logic -- Statements -- Negations -- Set theory -- The Zermelo-Fraenkel axioms -- Relations and functions -- Natural numbers, integers, and rational numbers -- The natural numbers -- The integers -- The rational numbers -- Bibliography -- Index. pt. 3. 21. 21.1. 21.2. 21.3. 21.4. 21.5. 22. 22.1. 22.2. 22.3. 23. 23.1. 23.2. 23.3. 23.4. 23.5. A. A.1. A.2. B. B.1. B.2. C. C.1. C.2. C.3.
9780470107966 0470107960
Mathematical analysis.
QA300 / .S376 2008
515