User-friendly -- yet rigorous -- in approach, this introduction to analysis meets readers where they are by providing extra support for those who like a slower, less detailed approach, but not getting in the way of those who want a quicker pace and deeper focus. It uses analogy and geometry to motivate and explain the theory, and precedes many complicated proofs with a "Strategy" which motivates the proof, shows why it was chosen, and why it should work. Examples follow many theorems, showing why each hypothesis is needed, allowing readers to remember the hypotheses by recalling the examples. Proofs are presented in complete detail, with each step carefully documented, and proofs are linked together in a way that teaches readers to think ahead. Physical interpretations are used to examine some concepts from a second or third point of view. Includes over 200 worked examples and over 600 exercises. Provides extensive coverage of multidimensional analysis.
Junior level course for math majors-generally required-usually for 2 terms. Chapters 1-5 are for 1st semester, chapters 6-10 for 2nd semester. Text offers strategies of proof for major theorems. This is a "friendly, baby Rudin." Covers both single and multivariable analysis.
I. ONE-DIMENSIONAL THEORY. 1. The Real Number System. 2. Continuity and Differentiability on Rn. 3. Integrability on Rn. 4. Infinite Series. II. MULTIDIMENSIONAL THEORY. 5. Euclidean Spaces. 6. Differentiability on Rn. 7. Integration on Rn. 8. Fundamental Theorems of Multivariable Calculus. 9. Fourier Series. 10. Stokes, Theorem on Manifolds. References. Index.