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A Course in Calculus and Real Analysis de Sudhir R. Ghorpade

A Course in Calculus and Real Analysis: Undergraduate Texts in Mathematics

Autor Sudhir R. Ghorpade, Balmohan V. Limaye
en Limba Engleză Paperback – 28 noi 2021

Notăm cu interes apariția acestui volum în prestigioasa serie Undergraduate Texts in Mathematics a editurii Springer. Ediția de față, A Course in Calculus and Real Analysis, reprezintă un efort pedagogic de a unifica intuiția geometrică a calculului diferențial și integral cu rigoarea formală a analizei reale. Structura cărții urmărește o progresie logică impecabilă: debutează cu axiomatizarea numerelor și studiul secvențelor, trece prin continuitate și derivabilitate, și acordă o atenție deosebită integralei Riemann, aplicațiilor acesteia și seriilor de funcții.

Subliniem faptul că acest manual nu se mulțumește cu prezentarea tehnicilor de calcul, ci prioritizează demonstrația. De exemplu, în capitolele dedicate integrării, autorii tratează funcțiile transcendentale elementare și aproximările într-un mod care pregătește terenul pentru concepte avansate. Credem că abordarea lor reușește să umple golul dintre calculul învățat în liceu și analiza abstractă de nivel universitar. Această lucrare completează perspectiva oferită de Calculus de Amber Habib, adăugând un strat suplimentar de rigoare teoretică și demonstrații complete acolo unde alte texte preferă o abordare bazată pe algoritmi.

Poziționată în contextul operei autorilor, cartea servește drept fundament pentru volumul lor ulterior, A Course in Multivariable Calculus and Analysis. Dacă lucrarea ulterioară extinde conceptele la mai multe dimensiuni, acest prim curs rafinează înțelegerea funcțiilor de o singură variabilă, oferind inclusiv anexe valoroase despre construcția numerelor reale și Teorema Fundamentală a Algebrei. Tonul este unul academic, precis, iar absența soluțiilor pentru exerciții indică clar destinația volumului: un suport de curs pentru studenții care doresc să stăpânească fundamentele analizei.

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Specificații

ISBN-13: 9783030827410
ISBN-10: 3030827410
Pagini: 538
Ilustrații: IX, 538 p.
Dimensiuni: 155 x 235 x 33 mm
Greutate: 0.91 kg
Ediția:2nd ed. 2018
Editura: Springer International Publishing
Colecția Springer
Seria Undergraduate Texts in Mathematics

Locul publicării:Cham, Switzerland

De ce să citești această carte

Această carte este ideală pentru studenții la matematică sau fizică care doresc să treacă de la aplicarea mecanică a formulelor la înțelegerea profundă a analizei reale. Cititorul câștigă o bază teoretică solidă, susținută de demonstrații riguroase și o structură coerentă. Este un instrument esențial pentru pregătirea examenelor de profil, oferind o claritate conceptuală superioară manualelor standard de calcul.

Despre autor

Sudhir R. Ghorpade și Balmohan V. Limaye sunt profesori reputați în cadrul Institutului Indian de Tehnologie din Bombay (IIT Bombay). Experiența lor vastă în predarea analizei matematice la nivel universitar este reflectată în claritatea expunerii și în selecția riguroasă a materialului didactic. Specializați în matematică pură, cei doi autori au colaborat strâns pentru a crea o serie de manuale care au devenit referințe în bibliografia multor facultăți de profil din lume. Stilul lor pedagogic se remarcă prin echilibrul dintre rigoarea teoretică și relevanța aplicațiilor matematice clasice.

Cuprins

1. Numbers and Functions.- 2. Sequences.- 3. Continuity and Limits.- 4. Differentiation.- 5. Applications of Differentiation.- 6. Integration.- 7. Elementary Transcendental Functions.- 8. Applications and Approximations of Riemann Integrals.- 9. Infinite Series and Improper Integrals.- 10. Sequences and Series of Functions, Integrals Depending on a Parameter.- A. Construction of the Real Numbers.- B. Fundamental Theorem of Algebra.- References.- List of Symbols and Abbreviations.- Index.

Recenzii

“This book would be a valuable asset to a university library and that many instructors would do well to have a copy of this book in their personal libraries. In addition, I believe that some students would benefit if they possessed a copy of this book to use for reference purposes.” (Jonathan Lewin, MAA Reviews, April 15, 2019)

Notă biografică

Sudhir R. Ghorpade is Institute Chair Professor in the Department of Mathematics at the Indian Institute of Technology (IIT) Bombay. He has received several awards, including the All India Council for Technical Education (AICTE) Career Award for Young Teachers and the Prof. S.C. Bhattacharya Award for Excellence in Pure Sciences. His research interests lie in algebraic geometry, combinatorics, coding theory, and commutative algebra.
Balmohan V. Limaye is Professor Emeritus in the Department of Mathematics at the Indian Institute of Technology (IIT) Bombay. He is the author of several research monographs and textbooks, including Linear Functional Analysis for Scientists and Engineers (Springer, 2016). He worked at IIT Bombay for more than 40 years and has twice received the Award for Excellence in Teaching from IIT Bombay. His research interests include Banach algebras, approximation theory, numerical functional analysis, and linear algebra.
The authors’ companion volume A Course in Multivariable Calculus and Analysis (2010) is also in the UTM series.

Textul de pe ultima copertă

Offering a unified exposition of calculus and classical real analysis, this textbook presents a meticulous introduction to single‐variable calculus. Throughout, the exposition makes a distinction between the intrinsic geometric definition of a notion and its analytic characterization, establishing firm foundations for topics often encountered earlier without proof. Each chapter contains numerous examples and a large selection of exercises, as well as a “Notes and Comments” section, which highlights distinctive features of the exposition and provides additional references to relevant literature.
This second edition contains substantial revisions and additions, including several simplified proofs, new sections, and new and revised figures and exercises. A new chapter discusses sequences and series of real‐valued functions of a real variable, and their continuous counterpart: improper integrals depending on a parameter. Two new appendices cover a construction of the real numbers using Cauchy sequences, and a self‐contained proof of the Fundamental Theorem of Algebra. In addition to the usual prerequisites for a first course in single‐variable calculus, the reader should possess some mathematical maturity and an ability to understand and appreciate proofs. This textbook can be used for a rigorous undergraduate course in calculus, or as a supplement to a later course in real analysis. The authors’ A Course in Multivariable Calculus is an ideal companion volume, offering a natural extension of the approach developed here to the multivariable setting.

From reviews:
[The first edition is] a rigorous, well-presented and original introduction to the core of undergraduate mathematics — first-year calculus. It develops this subject carefully from a foundation of high-school algebra, with interesting improvements and insights rarely found in other books. […] This book is a tour de force, and a necessary addition to thelibrary of anyone involved in teaching calculus, or studying it seriously.  N.J. Wildberger, Aust. Math. Soc. Gaz.

Caracteristici

Offers a unified exposition of single-variable calculus and classical real analysis Contains a chapter on sequences and series of real-valued functions of a real variable Features two new appendices that offer a construction of real numbers Includes supplementary material: sn.pub/extras

Descriere

Descriere de la o altă ediție sau format:
Calculus is one of the triumphs of the human mind. It emerged from inv- tigations into such basic questions as ?nding areas, lengths and volumes. In the third century B. C. , Archimedes determined the area under the arc of a parabola. In the early seventeenth century, Fermat and Descartes studied the problem of ?nding tangents to curves. But the subject really came to life in the hands of Newton and Leibniz in the late seventeenth century. In part- ular, they showed that the geometric problems of ?nding the areas of planar regions and of ?nding the tangents to plane curves are intimately related to one another. In subsequent decades, the subject developed further through the work of several mathematicians, most notably Euler, Cauchy, Riemann, and Weierstrass. Today,calculus occupies a centralplacein mathematics and is an essential component of undergraduate education. It has an immense number of app- cations both within and outside mathematics. Judged by the sheer variety of the concepts and results it has generated, calculus can be rightly viewed as a fountainhead of ideas and disciplines in mathematics. Real analysis, often called mathematical analysis or simply analysis, may be regarded as a formidable counterpart of calculus. It is a subject where one revisits notionsencountered in calculus, but with greaterrigor and sometimes with greater generality. Nonetheless, the basic objects of study remain the same, namely, real-valued functions of one or several real variables. This book attempts to give a self-contained and rigorous introduction to calculusoffunctionsofonevariable.