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Advanced CalculusMathematics
Real AnalysisMathematics

Designed for a onesemester undergraduate analysis course, this new text
is written in a conversational, accessible style offering a great deal of examples. It
gradually ascends in difficulty to help the student avoid hitting a wall.
Moves rapidly into the substance of analysis to avoid extensive talk about real numbers beyond what is immediately needed.
Offers a simplified presentation of limits, based on the approximation
idea and the use of “for n >>1” and “for x …¢ a.” Helped along by the Limit
Demon, students can write correct and readable limit arguments from the beginning. Warnings about common pitfalls to avoid are provided.
Uses Applications to show how the basic theorems are used in differential
equations, Fourier analysis, advanced calculus, numerical analysis, number theory,
inscribing equilateral triangles in closed convex curves, and slicing Danish ham sandwiches.
Presents analysis in a unified way as the mathematics based on inequalities,
estimations, and approximations. The pointset topology viewpoint is presented late and used minimally.
Offers a flexible structure: after the basic material, there are different goals at which the course may be aimed— e.g., differentiation of power series, an introduction to the Lebesgue integral, pointset topology, differentiation of improper integrals (e.g., Laplace transform), the
existence and uniqueness theorem for differential equations.
Includes simplified and meaningful proofs. Proofs, plausibility arguments,
and intuitive explanations are written and arranged on the page so as to try for maximum
clarity.
Features Exercises and Problems at the end of each chapter, as well as
Questions at the end of each section with full answers/solutions at the end of each chapter.
1. Real Numbers and Monotone Sequences.
2. Estimations and Approximations.
3. The Limit of a Sequence.
4. The Error Term.
5. Limit Theorems for Sequences.
6. The Completeness Principle.
7. Infinite Series.
8. Power Series.
9. Functions of One Variable.
10. Local and Global Behavior.
11. Continuity and Limits of Functions.
12. The Intermediate Value Theorem.
13. Continuous Functions on Compact Intervals.
14. Differentiation: Local Properties.
15. Differentiation: Global Properties.
16. Linearization and Convexity.
17. Taylor Approximation.
18. Integrability.
19. The Riemann Integral.
20. Derivatives and Integrals.
21. Improper Integrals.
22. Sequences and Series of Functions.
23. Infinite Sets and the Lebesgue Integral.
24. Continuous Functions on the Plane.
25. Pointsets in the Plane.
26. Integrals with a Parameter.
27. Differentiating Improper Integrals.
Appendix.
A. Sets, Numbers, and Logic.
B. Quantifiers and Negation.
C. Picard's Method.
D. Applications to Differential Equations.
E. Existence and Uniqueness of ODE Solutions.
