understanding analysis undergraduate texts in mathematics

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Understanding Analysis

Author : Stephen Abbott
ISBN : 9781493927128
Genre : Mathematics
File Size : 77. 76 MB
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This lively introductory text exposes the student to the rewards of a rigorous study of functions of a real variable. In each chapter, informal discussions of questions that give analysis its inherent fascination are followed by precise, but not overly formal, developments of the techniques needed to make sense of them. By focusing on the unifying themes of approximation and the resolution of paradoxes that arise in the transition from the finite to the infinite, the text turns what could be a daunting cascade of definitions and theorems into a coherent and engaging progression of ideas. Acutely aware of the need for rigor, the student is much better prepared to understand what constitutes a proper mathematical proof and how to write one. Fifteen years of classroom experience with the first edition of Understanding Analysis have solidified and refined the central narrative of the second edition. Roughly 150 new exercises join a selection of the best exercises from the first edition, and three more project-style sections have been added. Investigations of Euler’s computation of ζ(2), the Weierstrass Approximation Theorem, and the gamma function are now among the book’s cohort of seminal results serving as motivation and payoff for the beginning student to master the methods of analysis.

Vektoranalysis

Author : Klaus Jänich
ISBN : 9783662107508
Genre : Mathematics
File Size : 55. 48 MB
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Die Vektoranalysis handelt, in klassischer Darstellung, von Vektorfeldern, den Operatoren Gradient, Divergenz und Rotation, von Linien-, Flächen- und Volumenintegralen und von den Integralsätzen von Gauß, Stokes und Green. In moderner Fassung ist es der Cartansche Kalkül mit dem Satz von Stokes. Das vorliegende Buch vertritt grundsätzlich die moderne Herangehensweise, geht aber auch sorgfältig auf die klassische Notation und Auffassung ein. Das Buch richtet sich an Mathematik- und Physikstudenten ab dem zweiten Studienjahr, die mit den Grundbegriffen der Differential- und Integralrechnung in einer und mehreren Variablen sowie der Topologie vertraut sind. Der sehr persönliche Stil des Autors und die aus anderen Büchern bereits bekannten Lernhilfen, wie: viele Figuren, mehr als 50 kommentierte Übungsaufgaben, über 100 Tests mit Antworten, machen auch diesen Text zum Selbststudium hervorragend geeignet.

Datenanalyse Mit Python

Author : Wes McKinney
ISBN : 9783960102144
Genre : Computers
File Size : 36. 60 MB
Format : PDF, ePub
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Erfahren Sie alles über das Manipulieren, Bereinigen, Verarbeiten und Aufbereiten von Datensätzen mit Python: Aktualisiert auf Python 3.6, zeigt Ihnen dieses konsequent praxisbezogene Buch anhand konkreter Fallbeispiele, wie Sie eine Vielzahl von typischen Datenanalyse-Problemen effektiv lösen. Gleichzeitig lernen Sie die neuesten Versionen von pandas, NumPy, IPython und Jupyter kennen.Geschrieben von Wes McKinney, dem Begründer des pandas-Projekts, bietet Datenanalyse mit Python einen praktischen Einstieg in die Data-Science-Tools von Python. Das Buch eignet sich sowohl für Datenanalysten, für die Python Neuland ist, als auch für Python-Programmierer, die sich in Data Science und Scientific Computing einarbeiten wollen. Daten und zugehöriges Material des Buchs sind auf GitHub verfügbar.Aus dem Inhalt:Nutzen Sie die IPython-Shell und Jupyter Notebook für das explorative ComputingLernen Sie Grundfunktionen und fortgeschrittene Features von NumPy kennenSetzen Sie die Datenanalyse-Tools der pandasBibliothek einVerwenden Sie flexible Werkzeuge zum Laden, Bereinigen, Transformieren, Zusammenführen und Umformen von DatenErstellen Sie interformative Visualisierungen mit matplotlibWenden Sie die GroupBy-Mechanismen von pandas an, um Datensätzen zurechtzuschneiden, umzugestalten und zusammenzufassenAnalysieren und manipulieren Sie verschiedenste Zeitreihen-DatenFür diese aktualisierte 2. Auflage wurde der gesamte Code an Python 3.6 und die neuesten Versionen der pandas-Bibliothek angepasst. Neu in dieser Auflage: Informationen zu fortgeschrittenen pandas-Tools sowie eine kurze Einführung in statsmodels und scikit-learn.

Undergraduate Analysis

Author : Serge Lang
ISBN : 9781475718010
Genre : Mathematics
File Size : 83. 69 MB
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The present volume is a text designed for a first course in analysis. Although it is logically self-contained, it presupposes the mathematical maturity acquired by students who will ordinarily have had two years of calculus. When used in this context, most of the first part can be omitted, or reviewed extremely rapidly, or left to the students to read by themselves. The course can proceed immediately into Part Two after covering Chapters o and 1. However, the techniques of Part One are precisely those which are not emphasized in elementary calculus courses, since they are regarded as too sophisticated. The context of a third-year course is the first time that they are given proper emphasis, and thus it is important that Part One be thoroughly mastered. Emphasis has shifted from computational aspects of calculus to theoretical aspects: proofs for theorems concerning continuous 2 functions; sketching curves like x e-X, x log x, xlix which are usually regarded as too difficult for the more elementary courses; and other similar matters.

Analysis By Its History

Author : Ernst Hairer
ISBN : 9780387770314
Genre : Mathematics
File Size : 80. 96 MB
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This book presents first-year calculus roughly in the order in which it was first discovered. The first two chapters show how the ancient calculations of practical problems led to infinite series, differential and integral calculus and to differential equations. The establishment of mathematical rigour for these subjects in the 19th century for one and several variables is treated in chapters III and IV. Many quotations are included to give the flavor of the history. The text is complemented by a large number of examples, calculations and mathematical pictures and will provide stimulating and enjoyable reading for students, teachers, as well as researchers.

The Real Numbers

Author : John Stillwell
ISBN : 9783319015774
Genre : Mathematics
File Size : 62. 67 MB
Format : PDF
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While most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. Analysis needs the real numbers to model the line, and to support the concepts of continuity and measure. But these seemingly simple requirements lead to deep issues of set theory—uncountability, the axiom of choice, and large cardinals. In fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself. By focusing on the set-theoretic aspects of analysis, this text makes the best of two worlds: it combines a down-to-earth introduction to set theory with an exposition of the essence of analysis—the study of infinite processes on the real numbers. It is intended for senior undergraduates, but it will also be attractive to graduate students and professional mathematicians who, until now, have been content to "assume" the real numbers. Its prerequisites are calculus and basic mathematics. Mathematical history is woven into the text, explaining how the concepts of real number and infinity developed to meet the needs of analysis from ancient times to the late twentieth century. This rich presentation of history, along with a background of proofs, examples, exercises, and explanatory remarks, will help motivate the reader. The material covered includes classic topics from both set theory and real analysis courses, such as countable and uncountable sets, countable ordinals, the continuum problem, the Cantor–Schröder–Bernstein theorem, continuous functions, uniform convergence, Zorn's lemma, Borel sets, Baire functions, Lebesgue measure, and Riemann integrable functions.

Real Mathematical Analysis

Author : Charles Chapman Pugh
ISBN : 9783319177717
Genre : Mathematics
File Size : 31. 98 MB
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Based on an honors course taught by the author at UC Berkeley, this introduction to undergraduate real analysis gives a different emphasis by stressing the importance of pictures and hard problems. Topics include: a natural construction of the real numbers, four-dimensional visualization, basic point-set topology, function spaces, multivariable calculus via differential forms (leading to a simple proof of the Brouwer Fixed Point Theorem), and a pictorial treatment of Lebesgue theory. Over 150 detailed illustrations elucidate abstract concepts and salient points in proofs. The exposition is informal and relaxed, with many helpful asides, examples, some jokes, and occasional comments from mathematicians, such as Littlewood, Dieudonné, and Osserman. This book thus succeeds in being more comprehensive, more comprehensible, and more enjoyable, than standard introductions to analysis. New to the second edition of Real Mathematical Analysis is a presentation of Lebesgue integration done almost entirely using the undergraph approach of Burkill. Payoffs include: concise picture proofs of the Monotone and Dominated Convergence Theorems, a one-line/one-picture proof of Fubini's theorem from Cavalieri’s Principle, and, in many cases, the ability to see an integral result from measure theory. The presentation includes Vitali’s Covering Lemma, density points — which are rarely treated in books at this level — and the almost everywhere differentiability of monotone functions. Several new exercises now join a collection of over 500 exercises that pose interesting challenges and introduce special topics to the student keen on mastering this beautiful subject.

Mathematical Analysis

Author : Andrew Browder
ISBN : 9781461207153
Genre : Mathematics
File Size : 90. 42 MB
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Among the traditional purposes of such an introductory course is the training of a student in the conventions of pure mathematics: acquiring a feeling for what is considered a proof, and supplying literate written arguments to support mathematical propositions. To this extent, more than one proof is included for a theorem - where this is considered beneficial - so as to stimulate the students' reasoning for alternate approaches and ideas. The second half of this book, and consequently the second semester, covers differentiation and integration, as well as the connection between these concepts, as displayed in the general theorem of Stokes. Also included are some beautiful applications of this theory, such as Brouwer's fixed point theorem, and the Dirichlet principle for harmonic functions. Throughout, reference is made to earlier sections, so as to reinforce the main ideas by repetition. Unique in its applications to some topics not usually covered at this level.

Paradoxien Des Unendlichen

Author : Bernard Bolzano
ISBN : RMS:RMS45IST000004534$$$-
Genre :
File Size : 68. 68 MB
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Introduction To Calculus And Classical Analysis

Author : Omar Hijab
ISBN : 9783319284002
Genre : Mathematics
File Size : 88. 32 MB
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This text is intended for an honors calculus course or for an introduction to analysis. Involving rigorous analysis, computational dexterity, and a breadth of applications, it is ideal for undergraduate majors. This third edition includes corrections as well as some additional material. Some features of the text include: The text is completely self-contained and starts with the real number axioms; The integral is defined as the area under the graph, while the area is defined for every subset of the plane; There is a heavy emphasis on computational problems, from the high-school quadratic formula to the formula for the derivative of the zeta function at zero; There are applications from many parts of analysis, e.g., convexity, the Cantor set, continued fractions, the AGM, the theta and zeta functions, transcendental numbers, the Bessel and gamma functions, and many more; Traditionally transcendentally presented material, such as infinite products, the Bernoulli series, and the zeta functional equation, is developed over the reals; and There are 385 problems with all the solutions at the back of the text.

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