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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.

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Edition: 1st Copyright year: 1996 Publisher: Brooks/Cole Binding: Cloth Text Pages: 576 Size: 7.50" wide x 9.50" long x 1.00" tall Weight: 2.288 Language: English

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

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