Basic Analysis I: Functions of a Real Variable

ISBN-13: 9781138055025 / Angielski / Twarda / 2020 / 580 str.

James K. Peterson

Opis

Basic Analysis I: Functions of a Real Variable is designed for students who have completed the usual calculus and ODE sequence and a basic course in linear algebra. This is a critical course in the use of abstraction. 

 

Spis treści

I. Introduction.  II. Understanding Smoothness.  2. Proving Propositions.  3. Sequences of Real Numbers.  4. BolzanoWeierstrass Results.  5.  Topological Compactness.  6.  Function Limits.  7.  Continuity.  8. Consequences of continuity of intervals.  9. Lower Semicontinuous and Convex Functions.  10. Basic Differentiability.  11. The Properties of Derivatives. 12. Consequences of Derivatives.  13. Exponential and Logarithm Functions.  14. Extremal Theory for One Variable.  15. Differentiation in R² and R³_.16. Multivariable Extremal Theory.  III. Integration and Sequences of Functions.  17.  Uniform Continuity.  18. Cauchy Sequences of Real Numbers.  19.  Series of Real Numbers.  20. Series in Gerenal. 21.  Integration Theiry.  22. Existence of Reimann Integral Theories.  23. The Fundamental Theorem of Calculus (FTOC).  24.  Convergence of sequences of functions.  25.  Series of Functions and Power Series.  26. Riemann Integration: Discontinuities and Compositions.  27. Fourier Series.  28. Application.  IV. Summing it All Up .  29. Summary.  V.  References.  VI.  Detailed References.