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Full Description
Now in its second edition, this book provides a rigorous treatment of the foundations of differential and integral calculus. It proceeds gradually from an axiomatic characterization of the real number system to the study of differentiation and integration on m-dimensional surfaces. Proofs of theorems are given in detail, and many examples are provided to illustrate the concepts expressed in the theorems.
The book consists of three parts. Part I treats the calculus of functions of one variable. Traditional topics such as sequences, continuity, differentiability, Riemann integrability, numerical series, and the convergence of sequences and series of functions are covered. Optional sections on Stirling's formula, Riemann-Stieltjes integration, and other topics are also included.
The second part focuses on functions of several variables. It introduces the topological ideas (such as compact and connected sets) needed to describe analytical properties of multivariable functions. This part also discusses differentiability and integrability of multivariable functions, and it develops the theory of differential forms on surfaces in Rn.
Many proofs and explanations in the first edition have been revised, and details have been added to clarify the exposition. Part III contains appendices on set theory and linear algebra as well as solutions to some of the exercises are offered, whilst a full solutions manual contains complete solutions to all exercises for qualifying instructors.
Contents
Part 1: Functions of One Variable 1. The Real Number System 2. Numerical Sequences 3. Limits and Continuity on R 4. Differentiation on R 5. Riemann Integration on R 6. Numerical Infinite Series 7. Sequences and Series of Functions Part 2: Functions of Several Variables 8. Metric Spaces 9. Differentiation on Rn 10. Lebesgue Measure on Rn 11. Lebesgue Integration on Rn 12. Curves and Surfaces in Rn 13. Integration on Surfaces
