Description
Book SynopsisReal Analysis and Applications starts with a streamlined, but complete approach to real analysis. It finishes with a wide variety of applications in Fourier series and the calculus of variations, including minimal surfaces, physics, economics, Riemannian geometry, and general relativity. The basic theory includes all the standard topics: limits of sequences, topology, compactness, the Cantor set and fractals, calculus with the Riemann integral, a chapter on the Lebesgue theory, sequences of functions, infinite series, and the exponential and Gamma functions. The applications conclude with a computation of the relativistic precession of Mercury's orbit, which Einstein called convincing proof of the correctness of the theory [of General Relativity].The text not only provides clear, logical proofs, but also shows the student how to come up with them. The excellent exercises come with select solutions in the back. Here is a text which makes it possible to do the full theory and significant
Table of Contents
- Part I: Real numbers and limits: Numbers and logic
- Infinity
- Sequences
- Subsequences
- Functions and limits
- Composition of functions
- Part II: Topology: Open and closed sets
- Compactness
- Existence of maximum
- Uniform continuity
- Connected sets and the intermediate value theorem
- The Cantor set and fractals
- Part III: Calculus: The derivative and the mean value theorem
- The Riemann integral
- The fundamental theorem of calculus
- Sequences of functions
- The Lebesgue theory
- Infinite series $\sum_{n=1}^\infty a_n$
- Absolute convergence
- Power series
- The exponential function
- Volumes of $n$-balls and the gamma function
- Part IV: Fourier series: Fourier series
- Strings and springs
- Convergence of Fourier series
- Part V: The calculus of variations: Euler's equation
- First integrals and the Brachistochrone problem
- Geodesics and great circles
- Variational notation, higher order equations
- Harmonic functions
- Minimal surfaces
- Hamilton's action and Lagrange's equations
- Optimal economic strategies
- Utility of consumption
- Riemannian geometry
- Noneuclidean geometry
- General relativity
- Partial solutions to exercises
- Greek letters
- Index
