1. Números reais: introdução axiomática. Intervalos encaixantes. Sequências numéricas. Sequências de Cauchy. Limite superior e inferior. Sequências monótona limitadas. 2. Continuidade: teoremas do anulamento, do máximo e do mínimo, preservação da conexidade. Continuidade por sequências. Continuidade uniforme. 3. Derivabilidade: diferencial e teorema do valor médio. 4. Integral de Riemann: definição e exemplos especiais. Integrabilidade de funções contínuas e teorema fundamental do Cálculo. Critérios de Integrabilidade. 5. Séries numéricas e critérios de convergência. 6. Sequências e séries de funções: convergência pontual e uniforme, teste M de Weierstrass. Continuidade, integrabilidade e derivabilidade com convergência uniforme. Séries de potências e propriedades.
Tenho um livro aqui intitulado Undergraduate analysis do lang segue os topicos do livro
Preface 0 Sets and Mappings 0.2 Mappings 0.3 Natural Numbers and Induction 0.4 Denumerable Sets 0.5 Equivalence Relations I Real Numbers I.1 Algebraic Axioms I.2 Ordering Axioms I.3 Integers and Rational Numbers I.4 The Completeness Axiom II Limits and Continuous Functions II.1 Sequences of Numbers II.2 Functions and Limits II.3 Limits with Infinity II.4 Continuous Functions III Differentiation III.1 Properties of the Derivative III.2 Mean Value Theorem III.3 Inverse Functions IV Elementary Functions IV.1 Exponential IV.2 Logarithm IV.3 Sine and Cosine IV.4 Complex Numbers V The Elementary Real Integral V.2 Properties of the Integral V.3 Taylor's Formula V.4 Asymptotic Estimates and Stirling's Formula VI Normed Vector Spaces VI.2 Normed Vector Spaces VI.3 n-Space and Function Spaces VI.4 Completeness VI.5 Open and Closed Sets VII Limits VII.1 Basic Properties VII.2 Continuous Maps VII.3 Limits in Function Spaces VIII Compactness VIII.1 Basic Properties of Compact Sets VIII.2 Continuous Maps on Compact Sets VIII.4 Relation with Open Coverings IX Series IX.2 Series of Positive Numbers IX.3 Non-Absolute Convergence IX.5 Absolute and Uniform Convergence IX.6 Power Series IX.7 Differentiation and Integration of Series X The Integral in One Variable X.3 Approximation by Step Maps X.4 Properties of the Integral X.6 Relation Between the Integral and the Derivative XI Approximation with Convolutions XI.1 Dirac Sequences XI.2 The Weierstrass Theorem XII Fourier Series XII.1 Hermitian Products and Orthogonality XII.2 Trigonometric Polynomials as a Total Family XII.3 Explicit Uniform Approximation XII.4 Pointwise Convergence XIII Improper Integrals XIII.1 Definition XIII.2 Criteria for Convergence XIII.3 Interchanging Derivatives and Integrals XIV The Fourier Integral XIV.1 The Schwartz Space XIV.2 The Fourier Inversion Formula XIV.3 An Example of Fourier Transform Not in the Schwartz Space XV Functions on n-Space XV.1 Partial Derivatives XV.2 Differentiability and the Chain Rule XV.3 Potential Functions XV.4 Curve Integrals XV.5 Taylor's Formula XV.6 Maxima and the Derivative XVI The Winding Number and Global Potential Functions XVI.2 The Winding Number and Homology XVI.5 The Homotopy Form of the Integrability Theorem XVI.6 More on Homotopies XVII Derivatives in Vector Spaces XVII.1 The Space of Continuous Linear Maps XVII.2 The Derivative as a Linear Map XVII.3 Properties of the Derivative XVII.4 Mean Value Theorem XVII.5 The Second Derivative XVII.6 Higher Derivatives and Taylor's Formula XVIII Inverse Mapping Theorem XVIII.1 The Shrinking Lemma XVIII.2 Inverse Mappings, Linear Case XVIII.3 The Inverse Mapping Theorem XVIII.5 Product Decompositions XIX Ordinary Differential Equations XIX.1 Local Existence and Uniqueness XIX.3 Linear Differential Equations XX Multiple Integrals XX.1 Elementary Multiple Integration XX.2 Criteria for Admissibility XX.3 Repeated Integrals XX.4 Change of Variables XX.5 Vector Fields on Spheres XXI Differential Forms XXI.1 Definitions XXI.2 Inverse Image of a Form XXI.4 Stokes' Formula for Simplices
Pergunto, o livro é bom para essa materia que vou ter? tem coisa a mais? coisa a menos?
obrigado
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