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Full Description
Real 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 applications in one semester. Frank Morgan is the author of six books and over one hundred articles on mathematics. He is an inaugural recipient of the Mathematical Association of America's national Haimo award for excellence in teaching. With this applied version of his Real Analysis text, Morgan brings his famous direct style to the growing numbers of potential mathematics majors who want to see applications right along with the theory.
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
