Real Analysis and Applications: Including Fourier Series and the Calculus of Variations

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American Mathematical Soc. - 197 σελίδες
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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 derive them. The excellent exercises come with select solutions in the back. This is a text that 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 along with the theory. The book is suitable for undergraduates interested in real analysis.

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Σελίδα 65 - *-n , \*-*i/ each of the four variables s/, «„", tn', tn" will be steadily increasing in the sense of section 52. But, from (10), and the fact that the absolute value of a sum is less than or equal to the sum of the absolute values...
Σελίδα 59 - If f is differentiable at x, then f is continuous at x. PROOF.
Σελίδα 35 - ... is closed if and only if it contains all of its...
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Σελίδα 38 - Prove that every point of S is either an interior point or a boundary point.
Σελίδα 174 - Metric" : (1) dr2 = -(l-2GMr-lr1dr2 -r2 d92 + (\-2GMr-1) dt2: where M is the mass of the sun and G is the gravitational constant.
Σελίδα 6 - A are elements of B, we say that A is a subset of B (ACB) or that B contains A (BD .A). You have to be careful, because the word "contains" is used both for subsets and for elements, as determined by context.
Σελίδα 3 - The natural numbers N = {1, 2, 3,...}. The integers Z = {..., -3, -2, -1, 0, 1, 2, 3, ...} (Z stands for the German word Zahl for number).
Σελίδα 6 - If it happens that the implication and its converse are both true, ie, if A =>• B and B => A, then we say that A and B are logically equivalent or that A if and only if B (A&B).

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