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Mathematical analysis : a concise introduction / Bernd S.W. Schröder.

Por: Tipo de material: TextoTextoDetalles de publicación: Hoboken, N.J. : Wiley-Interscience, c2008.Descripción: xv, 562 p. : ill. ; 25 cmISBN:
  • 9780470107966
  • 0470107960
Tema(s): Clasificación CDD:
  • 515 22
Clasificación LoC:
  • QA300 .S376 2008
Recursos en línea:
Contenidos:
Table of contents -- Preface -- pt. 1. Analysis of functions of a single real variable -- 1. The real numbers -- 1.1. Field axioms -- 1.2. Order axioms -- 1.3. Lowest upper and greatest lower bounds -- 1.4. Natural numbers, integers, and rational numbers -- 1.5. Recursion, induction, summations, and products -- 2. Sequences of real numbers -- 2.1. Limits -- 2.2. Limit laws -- 2.3. Cauchy sequences -- 2.4. Bounded sequences -- 2.5. Infinite limits -- 3. Continuous functions -- 3.1. Limits of functions -- 3.2. Limit laws -- 3.3. One-sided limits and infinite limits -- 3.4. Continuity -- 3.5. Properties of continuous functions -- 3.6. Limits at infinity -- 4. Differentiable functions -- 4.1. Differentiability -- 4.2. Differentiation rules -- 4.3. Rolle's theorem and the mean value theorem -- 5. The Riemann integral 1 -- 5.1. Riemann sums and the integral -- 5.2. Uniform continuity and integrability of continuous functions -- 5.3. The fundamental theorem of calculus -- 5.4. The Darboux integral --
6. Series of real numbers 1 -- 6.1. Series as a vehicle to define infinite sums -- 6.2. Absolute convergence and unconditional convergence -- 7. Some set theory -- 7.1. The algebra of sets -- 7.2. Countable sets -- 7.3. Uncountable sets -- 8. The Riemann integral 2 -- 8.1. Outer Lebesgue measure -- 8.2. Lebesgue's criterion for Riemann integrability -- 8.3. More integral theorems -- 8.4. Improper Riemann integrals -- 9. The Lebesgue integral -- 9.1. Outer Lebesgue measure -- 9.2. Lebesgue measurable sets -- 9.2. Lebesgue measurable functions -- 9.3. Lebesgue integration -- 9.4. Lebesgue integrals versus Riemann integrals-- 10. Series of real numbers 2 -- 10.1. Limits superior and inferior -- 10.2. The root test and the ratio test -- 10.3. Power series -- 11. Sequences of functions -- 11.1. Notions of convergence -- 11.2. Uniform convergence -- 12. Transcendental functions -- 12.1. The exponential function -- 12.2. Sine and cosine -- 12.3. L'Hôpital's rule -- 13. Numerical methods -- 13.1. Approximation with Taylor polynomials -- 13.2. Newton's method -- 13.3. Numerical integration --
pt. 2. Analysis in abstract spaces -- 14. Integration on measure spaces -- 14.1. Measure spaces -- 14.2. Outer measures -- 14.3. Measurable functions -- 14.4. Integration of measurable functions -- 14.5. Monotone and dominated convergence -- 14.6. Convergence in mean, in measure, and almost everywhere -- 14.7. Product [sigma]-algebras -- 14.8. Product measures and Fubini's theorem -- 15. The abstract venues for analysis -- 15.1. Abstraction 1 : Vector spaces -- 15.2. Representation of elements : bases and dimension -- 15.3. Identification of spaces : isomorphism -- 15.4. Abstraction 2 : inner product spaces -- 15.5. Nicer representations : orthonormal sets -- 15.6. Abstraction 3 : normed spaces -- 15.7. Abstraction 4 : metric spaces -- 15.8. L[superscript]p spaces -- 15.9. Another number field : complex numbers -- 16. The topology of metric spaces -- 16.1. Convergence of sequences -- 16.2. Completeness -- 16.3. Continuous functions -- 16.4. Open and closed sets -- 16.5. Compactness -- 16.6. The normed topology of R[superscript]d -- 16.7. Dense subspaces -- 16.8. Connectedness -- 16.9. Locally compact spaces --
17. Differentiation in normed spaces -- 17.1. Continuous linear functions -- 17.2. Matrix representation of linear functions -- 17.3. Differentiability -- 17.4. The mean value theorem -- 17.5. How partial derivatives fit in -- 17.6. Multilinear functions (tensors) -- 17.7. Higher derivatives -- 17.8. The implicit function theorem -- 18. Measure, topology and differentiation -- 18.1. Lebesgue measurable sets in R[superscript]d -- 18.2. C[infinity] and approximation of integrable functions -- 18.3. Tensor algebra and determinants -- 18.4. Multidimensional substitution -- 19. Manifolds and integral theorems -- 19.1. Manifolds -- 19.2. Tangent spaces and differentiable functions -- 19.3. Differential forms, integrals over the unit cube -- 19.4. k-forms and integrals over k-chains -- 19.5. Integration on manifolds -- g 19.6. Stokes' theorem -- 20. Hilbert spaces -- 20.1. Orthonormal bases -- 20.2. Fourier series -- 20.3. The Riesz representation theorem --
pt. 3. Applied analysis -- 21. Physics background -- 21.1. Harmonic oscillators -- 21.2. Heat and diffusion -- 21.3. Separation of variables, Fourier series, and ordinary differential equations -- 21.4. Maxwell's equations -- 21.5. The Navier Stokes equation for the conservation of mass -- 22. Ordinary differential equations -- 22.1. Banach space valued differential equations -- 22.2. An existence and uniqueness theorem -- 22.3. Linear differential equations -- 23. The finite element method -- 23.1. Ritz-Galerkin approximation -- 23.2. Weakly differentiable functions -- 23.3. Sobolev spaces -- 23.4. Elliptic differential operators -- 23.5. Finite elements -- Conclusions and outlook -- Appendices -- A. Logic -- A.1. Statements -- A.2. Negations -- B. Set theory -- B.1. The Zermelo-Fraenkel axioms -- B.2. Relations and functions -- C. Natural numbers, integers, and rational numbers -- C.1. The natural numbers -- C.2. The integers -- C.3. The rational numbers -- Bibliography -- Index.
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Tipo de ítem Biblioteca actual Signatura Copia número Estado Fecha de vencimiento Código de barras Reserva de ítems
Colección general Colección general Biblioteca Yachay Tech 515 S3811m 2008 (Navegar estantería(Abre debajo)) Ej. 1 Disponible 001156
Colección general Colección general Biblioteca Yachay Tech 515 S3811m 2008 (Navegar estantería(Abre debajo)) Ej. 2 Disponible 001158
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Includes bibliographical references (p. 551-552) and index.

Table of contents -- Preface -- pt. 1. Analysis of functions of a single real variable -- 1. The real numbers -- 1.1. Field axioms -- 1.2. Order axioms -- 1.3. Lowest upper and greatest lower bounds -- 1.4. Natural numbers, integers, and rational numbers -- 1.5. Recursion, induction, summations, and products -- 2. Sequences of real numbers -- 2.1. Limits -- 2.2. Limit laws -- 2.3. Cauchy sequences -- 2.4. Bounded sequences -- 2.5. Infinite limits -- 3. Continuous functions -- 3.1. Limits of functions -- 3.2. Limit laws -- 3.3. One-sided limits and infinite limits -- 3.4. Continuity -- 3.5. Properties of continuous functions -- 3.6. Limits at infinity -- 4. Differentiable functions -- 4.1. Differentiability -- 4.2. Differentiation rules -- 4.3. Rolle's theorem and the mean value theorem -- 5. The Riemann integral 1 -- 5.1. Riemann sums and the integral -- 5.2. Uniform continuity and integrability of continuous functions -- 5.3. The fundamental theorem of calculus -- 5.4. The Darboux integral --

6. Series of real numbers 1 -- 6.1. Series as a vehicle to define infinite sums -- 6.2. Absolute convergence and unconditional convergence -- 7. Some set theory -- 7.1. The algebra of sets -- 7.2. Countable sets -- 7.3. Uncountable sets -- 8. The Riemann integral 2 -- 8.1. Outer Lebesgue measure -- 8.2. Lebesgue's criterion for Riemann integrability -- 8.3. More integral theorems -- 8.4. Improper Riemann integrals -- 9. The Lebesgue integral -- 9.1. Outer Lebesgue measure -- 9.2. Lebesgue measurable sets -- 9.2. Lebesgue measurable functions -- 9.3. Lebesgue integration -- 9.4. Lebesgue integrals versus Riemann integrals-- 10. Series of real numbers 2 -- 10.1. Limits superior and inferior -- 10.2. The root test and the ratio test -- 10.3. Power series -- 11. Sequences of functions -- 11.1. Notions of convergence -- 11.2. Uniform convergence -- 12. Transcendental functions -- 12.1. The exponential function -- 12.2. Sine and cosine -- 12.3. L'Hôpital's rule -- 13. Numerical methods -- 13.1. Approximation with Taylor polynomials -- 13.2. Newton's method -- 13.3. Numerical integration --

pt. 2. Analysis in abstract spaces -- 14. Integration on measure spaces -- 14.1. Measure spaces -- 14.2. Outer measures -- 14.3. Measurable functions -- 14.4. Integration of measurable functions -- 14.5. Monotone and dominated convergence -- 14.6. Convergence in mean, in measure, and almost everywhere -- 14.7. Product [sigma]-algebras -- 14.8. Product measures and Fubini's theorem -- 15. The abstract venues for analysis -- 15.1. Abstraction 1 : Vector spaces -- 15.2. Representation of elements : bases and dimension -- 15.3. Identification of spaces : isomorphism -- 15.4. Abstraction 2 : inner product spaces -- 15.5. Nicer representations : orthonormal sets -- 15.6. Abstraction 3 : normed spaces -- 15.7. Abstraction 4 : metric spaces -- 15.8. L[superscript]p spaces -- 15.9. Another number field : complex numbers -- 16. The topology of metric spaces -- 16.1. Convergence of sequences -- 16.2. Completeness -- 16.3. Continuous functions -- 16.4. Open and closed sets -- 16.5. Compactness -- 16.6. The normed topology of R[superscript]d -- 16.7. Dense subspaces -- 16.8. Connectedness -- 16.9. Locally compact spaces --

17. Differentiation in normed spaces -- 17.1. Continuous linear functions -- 17.2. Matrix representation of linear functions -- 17.3. Differentiability -- 17.4. The mean value theorem -- 17.5. How partial derivatives fit in -- 17.6. Multilinear functions (tensors) -- 17.7. Higher derivatives -- 17.8. The implicit function theorem -- 18. Measure, topology and differentiation -- 18.1. Lebesgue measurable sets in R[superscript]d -- 18.2. C[infinity] and approximation of integrable functions -- 18.3. Tensor algebra and determinants -- 18.4. Multidimensional substitution -- 19. Manifolds and integral theorems -- 19.1. Manifolds -- 19.2. Tangent spaces and differentiable functions -- 19.3. Differential forms, integrals over the unit cube -- 19.4. k-forms and integrals over k-chains -- 19.5. Integration on manifolds -- g 19.6. Stokes' theorem -- 20. Hilbert spaces -- 20.1. Orthonormal bases -- 20.2. Fourier series -- 20.3. The Riesz representation theorem --

pt. 3. Applied analysis -- 21. Physics background -- 21.1. Harmonic oscillators -- 21.2. Heat and diffusion -- 21.3. Separation of variables, Fourier series, and ordinary differential equations -- 21.4. Maxwell's equations -- 21.5. The Navier Stokes equation for the conservation of mass -- 22. Ordinary differential equations -- 22.1. Banach space valued differential equations -- 22.2. An existence and uniqueness theorem -- 22.3. Linear differential equations -- 23. The finite element method -- 23.1. Ritz-Galerkin approximation -- 23.2. Weakly differentiable functions -- 23.3. Sobolev spaces -- 23.4. Elliptic differential operators -- 23.5. Finite elements -- Conclusions and outlook -- Appendices -- A. Logic -- A.1. Statements -- A.2. Negations -- B. Set theory -- B.1. The Zermelo-Fraenkel axioms -- B.2. Relations and functions -- C. Natural numbers, integers, and rational numbers -- C.1. The natural numbers -- C.2. The integers -- C.3. The rational numbers -- Bibliography -- Index.

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