Descripciones mejoradas de Syndetics:

Rigorously presents the fundamental concepts of mathematical analysis in the clearest, simplest way while using illuminating examples and stimulating exercises. The text shows that mathematical analysis is a coherent body of knowledge, rather than a collection of isolated facts.

Tabla de contenidos provista por Syndetics

• 1 Tools For Analysis
• The Completeness Axiom and Some of Its Consequences
• The Distribution of the Integers and the Rational Numbers
• Inequalities and Identities
• 2 Convergent sequences
• The Convergence of Sequences
• Sequences and Sets
• The Monotone Convergence Theorem
• The Sequential Compactness Theorem
• Covering Properties of Sets
• 3 Continuous Functions
• Continuity
• The Extreme Value Theorem
• The Intermediate Value Theorem
• Uniform Continuity
• The Epsilon-Delta Criterion for Continuity
• Images and Inverses; Monotone Functions
• Limits
• 4 Differentiation
• The Algebra of Derivatives
• Differentiating Inverses and Compositions
• The Mean Value Theorem and Its Geometric Consequences
• The Cauchy Mean Value Theorem and Its Analytic Consequences
• The Notation of Leibnitz
• 5 Elementary Functions As Solutions Of Differential Equations
• Solutions of Differential Equations
• The Natural Logarithm and the Exponential Functions
• The Trigonometric Functions
• The Inverse Trigonometric Functions
• 6 Integration: Two Fundamental Theorems
• Darboux Sums; Upper and Lower Integrals
• The Archimedes-Riemann Theorem
• Additivity, Monotonicity and Linearity
• Continuity and Integrability
• The First Fundamental Theorem: Integrating Derivatives
• The Second Fundamental Theorem: Differentiating Integrals
• 7 INTEGRATION: FURTHER TOPICS
• Solutions of Differential Equations
• Integration by Parts and by Substitution
• The Convergence of Darboux and Riemann Sums
• The Approximation of Integrals
• 8 Approximation By Taylor Polynomials
• Taylor Polynomials
• The Lagrange Remainder Theorem
• The Convergence of Taylor Polynomials
• A Power Series for the Logarithm
• The Cauchy Integral Remainder Theorem
• A Non-Analytic, Infinitely Differentiable Function
• The Weierstrass Approximation Theorem
• 9 Sequences And Series Of Functions
• Sequences and Series of Numbers
• Pointwise Convergence of Sequences of Functions
• Uniform Convergence of Sequences of Functions
• The Uniform Limit of Functions
• Power Series
• A Continuous, Nowhere Differentiable Function
• 10 The Euclidean Space Rn
• The Linear Structure of Rn and the Scalar Product
• Convergence of Sequences in Rn
• Open Sets and Closed Sets in Rn
• 11 Continuity, Compactness, And Connectedness
• Continuous Functions and Mappings
• Sequential Compactness, Extreme Values and Uniform Continuity
• Pathwise Connectedness and the Intermediate Value Theorem
• Connectedness and the Intermediate Value Property
• 12 Metric Spaces
• Open Sets, Closed Sets, and Sequential Convergence
• Completeness and the Contraction Mapping Principle
• The Existence Theorem for Nonlinear Differential Equations
• Continuous Mappings Between Metric Spaces
• Sequentially Compactness and Connectedness
• 13 Differentiating Functions Of Several Variables
• Limits
• Partial Derivatives
• The Mean Value Theorem and Directional Derivatives
• 14 Local Approximation Of Real-Valued Functions
• First-Order Approximation, Tangent Planes, and Affine Functions
• Quadratic Functions, Hessian Matrices, and Second Derivatives
• Second-Order Approximations and the Second-Derivative Test
• 15 Approximating Nonlinear Mappings By Linear Mappings
• Linear Mappings and Matrices
• The Derivative Matrix and the Differential
• The Chain Rule
• 16 Images And Inverses: The Inverse Function Theorem
• Functions of a Single Variable and Maps in the Plane
• Stability of Nonlinear Mappings
• A Minimization Principle and the General Inverse Function Theorem
• 17 The Implicit Function Theorem And Its Applications
• A Scalar Equation in Two Unknowns: Dini''s Theorem
• The General Implicit Function Theorem
• Equations of Surfaces and Curves in R 
• Constrained Extrema Problems and Lagrange Multipliers
• 18 Integrating Functions Of Several Variables
• Integration of Functions on Generalized Rectangles
• Continuity and Integrability
• Integration over Jordan Domains
• 19 Iterated Integration And Changes Of Variables
• Fubini''s Theorem
• The Change of Variables Theorem: Statements and Examples
• Proof of the Change of Variables Theorem
• 20 Line And Surface Integrals
• Arclength and Line Integrals
• Surface Area and Surface Integrals
• The Integral Formulas of Green and Stokes
• Appendix A
• The Field Axioms and Their Consequences
• The Positivity Axioms and Their Consequences
• Appendix B