Advanced Calculus is designed for the two-semester course on functions of one and several variables. The text provides a rigorous treatment of the fundamental concepts of mathematical analysis, yet it does so in a clear, direct way. The author wants students to leave the course with an appreciation of the subject's coherence and significance, and an understanding of the ideas that underlie mathematical analysis.

1. THE REAL NUMBERS The Complete Axiom: The Natural, Rational, and Irrational Numbers / The Archimedian Property and the Density of the Rationals and the Irrationals / Three Inequalities and Three Algebraic Identities 2. SEQUENCES OF REAL NUMBERS The Convergence of Sequences / Monotone Sequences, the Bolzano-Weierstrass Theorem, and the Nested Interval Theorem 3. CONTINUOUS FUNCTIONS AND LIMITS Continuity / The Extreme Value Theorem / The Intermediate Value Theorem / Images and Inverses / An Equivalent Definition of Continuity: Uniform Continuity / Limits 4. DIFFERENTIATION The Algebra of Derivatives / Differentiating Inverses and Compositions / The Lagrange Mean Value Theorem and Its Geometric Consequences / The Cauchy Mean Value Theorem and Its Analytic Consequences / A Fundamental Differential Equation / The Notation of Leibnitz 5. THE ELEMENTARY FUNCTIONS AS SOLUTIONS OF DIFFERENTIAL EQUATIONS The Natural Logarithm and the Exponential Functions / The Trigonometric Functions / The Inverse Trigonometric Functions 6. INTEGRATION Motivation for the Definition / The Definition of the Integral and Criteria for Integrability / The First Fundamental Theorem of Calculus / The Convergence of Darboux Sums and Riemann Sums / Linearity, Monotonicity, and Additivity over Intervals 7. THE SECOND FUNDAMENTAL THEOREM AND ITS CONSEQUENCES The Second Fundamental Theorem of Calculus / The Existence of Solutions of Differential Equations / The Verification of Two Classical Integration Methods / The Approximation of Integrals 8. APPROXIMATION BY TAYLOR POLYNOMIALS Taylor Polynomials and Order of Contact / The Lagrange Remainder Theorem / The Convergence of Taylor Polynomials / Power Series for the Logarithm / The Cauchy Integral Remainder Formula and the Binomial Expansion / An Infinitely Differentiable Function That Is Not Analytic / The Weierstrass Approximation Theorem 9. THE CONVERGENCE OF SEQUENCES AND SERIES OF FUNCTIONS Sequences and Series of Numbers / Pointwise Convergence and Uniform Convergence of Sequences and Functions / The Uniform Limit of Continuous Functions, of Integrable Functions, and of Differentiable Functions / Power Series / A Continuous Function That Fails at Each Point to Be Differentiable 10. THE EUCLIDEAN SPACE R^n The Linear Structure of R^n and the Inner Product / Convergence of Sequences in R^n / Interiors, Exteriors, and Boundaries of Subsets of R^n 11. CONTINUITY, COMPACTNESS, AND CONNECTEDNESS Continuity of Functions and Mappings / Compactness and the Extreme Value Theorem / Connectedness and the Intermediate Value Theorem 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 / Compactness and Connectedness 13. PARTIAL DIFFERENTIABILITY OF REAL-VALUED 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, The Differential and First-Order Approximation / 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 The Solutions of a Scalar Equation in Two Unknowns: Dini's Theorem / Underdetermined Systems of Nonlinear Equations: The General Implicit Function Theorem / Equations of Surfaces and Curves in R / Constrained Extrema Problems and Lagrange Multipliers 18. INTEGRATION FOR FUNCTIONS OF SEVERAL VARIABLES Integration over G