Introduction To Mathematical Analysis

Author: Beatriz Lafferriere
Publisher: Lulu.com
ISBN: 1312742844
Size: 43.54 MB
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Our goal in this set of lecture notes is to provide students with a strong foundation in mathematical analysis. Such a foundation is crucial for future study of deeper topics of analysis. Students should be familiar with most of the concepts presented here after completing the calculus sequence. However, these concepts will be reinforced through rigorous proofs. The lecture notes contain topics of real analysis usually covered in a 10-week course: the completeness axiom, sequences and convergence, continuity, and differentiation. The lecture notes also contain many well-selected exercises of various levels. Although these topics are written in a more abstract way compared with those available in some textbooks, teachers can choose to simplify them depending on the background of the students. For instance, rather than introducing the topology of the real line to students, related topological concepts can be replaced by more familiar concepts such as open and closed intervals. Some other topics such as lower and upper semicontinuity, differentiation of convex functions, and generalized differentiation of non-differentiable convex functions can be used as optional mathematical projects. In this way, the lecture notes are suitable for teaching students of different backgrounds.

Introduction To Mathematical Analysis

Author: Igor Kriz
Publisher: Springer Science & Business Media
ISBN: 3034806361
Size: 28.21 MB
Format: PDF, Kindle
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The book begins at the level of an undergraduate student assuming only basic knowledge of calculus in one variable. It rigorously treats topics such as multivariable differential calculus, Lebesgue integral, vector calculus and differential equations. After having built on a solid foundation of topology and linear algebra, the text later expands into more advanced topics such as complex analysis, differential forms, calculus of variations, differential geometry and even functional analysis. Overall, this text provides a unique and well-rounded introduction to the highly developed and multi-faceted subject of mathematical analysis, as understood by a mathematician today.​

An Introduction To Mathematical Analysis For Economic Theory And Econometrics

Author: Dean Corbae
Publisher: Princeton University Press
ISBN: 1400833086
Size: 56.62 MB
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Providing an introduction to mathematical analysis as it applies to economic theory and econometrics, this book bridges the gap that has separated the teaching of basic mathematics for economics and the increasingly advanced mathematics demanded in economics research today. Dean Corbae, Maxwell B. Stinchcombe, and Juraj Zeman equip students with the knowledge of real and functional analysis and measure theory they need to read and do research in economic and econometric theory. Unlike other mathematics textbooks for economics, An Introduction to Mathematical Analysis for Economic Theory and Econometrics takes a unified approach to understanding basic and advanced spaces through the application of the Metric Completion Theorem. This is the concept by which, for example, the real numbers complete the rational numbers and measure spaces complete fields of measurable sets. Another of the book's unique features is its concentration on the mathematical foundations of econometrics. To illustrate difficult concepts, the authors use simple examples drawn from economic theory and econometrics. Accessible and rigorous, the book is self-contained, providing proofs of theorems and assuming only an undergraduate background in calculus and linear algebra. Begins with mathematical analysis and economic examples accessible to advanced undergraduates in order to build intuition for more complex analysis used by graduate students and researchers Takes a unified approach to understanding basic and advanced spaces of numbers through application of the Metric Completion Theorem Focuses on examples from econometrics to explain topics in measure theory

An Introduction To Mathematical Analysis

Author: Robert A. Rankin
Publisher: Elsevier
ISBN: 1483137309
Size: 52.16 MB
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An Introduction to Mathematical Analysis is an introductory text to mathematical analysis, with emphasis on functions of a single real variable. Topics covered include limits and continuity, differentiability, integration, and convergence of infinite series, along with double series and infinite products. This book is comprised of seven chapters and begins with an overview of fundamental ideas and assumptions relating to the field operations and the ordering of the real numbers, together with mathematical induction and upper and lower bounds of sets of real numbers. The following chapters deal with limits of real functions; differentiability and maxima, minima, and convexity; elementary properties of infinite series; and functions defined by power series. Integration is also considered, paying particular attention to the indefinite integral; interval functions and functions of bounded variation; the Riemann-Stieltjes integral; the Riemann integral; and area and curves. The final chapter is devoted to convergence and uniformity. This monograph is intended for mathematics students.

An Interactive Introduction To Mathematical Analysis Paperback With Cd Rom

Author: Jonathan Lewin
Publisher: Cambridge University Press
ISBN: 9780521017183
Size: 61.13 MB
Format: PDF, Docs
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This book provides a rigorous course in the calculus of functions of a real variable. Its gentle approach, particularly in its early chapters, makes it especially suitable for students who are not headed for graduate school but, for those who are, this book also provides the opportunity to engage in a penetrating study of real analysis.The companion onscreen version of this text contains hundreds of links to alternative approaches, more complete explanations and solutions to exercises; links that make it more friendly than any printed book could be. In addition, there are links to a wealth of optional material that an instructor can select for a more advanced course, and that students can use as a reference long after their first course has ended. The on-screen version also provides exercises that can be worked interactively with the help of the computer algebra systems that are bundled with Scientific Notebook.

Advanced Calculus

Author: S Zaidman
Publisher: World Scientific
ISBN: 9814498831
Size: 66.81 MB
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This book is an introduction to mathematical analysis (i.e real analysis) at a fairly elementary level. A great (unusual) emphasis is given to the construction of rational and then of real numbers, using the method of equivalence classes and of Cauchy sequences. The text includes the usual presentation of: sequences of real numbers, infinite numerical series, continuous functions, derivatives and Riemann-Darboux integration. There are also two “special” sections: on convex functions and on metric spaces, as well as an elementary appendix on Logic, Set Theory and Functions. We insist on a rigorous presentation throughout in the framework of the classical, standard, analysis. Contents: NumbersSequences of Real NumbersInfinite Numerical SeriesContinuous FunctionsDerivativesConvex FunctionsMetric SpacesIntegrationIndexIndex of NotationsAppendix (Logic, Set Theory and Functions)Bibliography Readership: Undergraduate students of calculus and real analysis. keywords:Numbers;Sequences;Series;Continuous Functions;Derivatives;Convex Functions;Metric Spaces;Integration

Mathematical Analysis

Author: Andrew Browder
Publisher: Springer Science & Business Media
ISBN: 1461207150
Size: 58.63 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.

Introduction To Mathematical Analysis

Author: C. Clapham
Publisher: Springer Science & Business Media
ISBN: 9401165726
Size: 50.87 MB
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I have tried to provide an introduction, at an elementary level, to some of the important topics in real analysis, without avoiding reference to the central role which the completeness of the real numbers plays throughout. Many elementary textbooks are written on the assumption that an appeal to the complete ness axiom is beyond their scope; my aim here has been to give an account of the development from axiomatic beginnings, without gaps, while keeping the treatment reasonably simple. Little previous knowledge is assumed, though it is likely that any reader will have had some experience of calculus. I hope that the book will give the non-specialist, who may have considerable facility in techniques, an appreciation of the foundations and rigorous framework of the mathematics that he uses in its applications; while, for the intending mathe matician, it will be more of a beginner's book in preparation for more advanced study of analysis. I should finally like to record my thanks to Professor Ledermann for the suggestions and comments that he made after reading the first draft of the text.

Mathematical Analysis

Author: Bernd S. W. Schröder
Publisher: John Wiley & Sons
ISBN: 9780470226766
Size: 71.23 MB
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A self-contained introduction to the fundamentals of mathematical analysis Mathematical Analysis: A Concise Introduction presents the foundations of analysis and illustrates its role in mathematics. By focusing on the essentials, reinforcing learning through exercises, and featuring a unique "learn by doing" approach, the book develops the reader's proof writing skills and establishes fundamental comprehension of analysis that is essential for further exploration of pure and applied mathematics. This book is directly applicable to areas such as differential equations, probability theory, numerical analysis, differential geometry, and functional analysis. Mathematical Analysis is composed of three parts: ?Part One presents the analysis of functions of one variable, including sequences, continuity, differentiation, Riemann integration, series, and the Lebesgue integral. A detailed explanation of proof writing is provided with specific attention devoted to standard proof techniques. To facilitate an efficient transition to more abstract settings, the results for single variable functions are proved using methods that translate to metric spaces. ?Part Two explores the more abstract counterparts of the concepts outlined earlier in the text. The reader is introduced to the fundamental spaces of analysis, including Lp spaces, and the book successfully details how appropriate definitions of integration, continuity, and differentiation lead to a powerful and widely applicable foundation for further study of applied mathematics. The interrelation between measure theory, topology, and differentiation is then examined in the proof of the Multidimensional Substitution Formula. Further areas of coverage in this section include manifolds, Stokes' Theorem, Hilbert spaces, the convergence of Fourier series, and Riesz' Representation Theorem. ?Part Three provides an overview of the motivations for analysis as well as its applications in various subjects. A special focus on ordinary and partial differential equations presents some theoretical and practical challenges that exist in these areas. Topical coverage includes Navier-Stokes equations and the finite element method. Mathematical Analysis: A Concise Introduction includes an extensive index and over 900 exercises ranging in level of difficulty, from conceptual questions and adaptations of proofs to proofs with and without hints. These opportunities for reinforcement, along with the overall concise and well-organized treatment of analysis, make this book essential for readers in upper-undergraduate or beginning graduate mathematics courses who would like to build a solid foundation in analysis for further work in all analysis-based branches of mathematics.