TY  - BOOK
AU  - Schröder,Bernd S.W.
TI  - Mathematical analysis: a concise introduction 
SN  - 9780470107966 (cloth)
AV  - QA300 .S376 2008
PY  - 2008///
CY  - Hoboken, N.J.
PB  - Wiley-Interscience
KW  - Mathematical analysis
N1  - 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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