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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 Hardback – 4 iun 2006

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: 9780387305301
ISBN-10: 0387305300
Pagini: 432
Dimensiuni: 156 x 235 x 26 mm
Greutate: 0.77 kg
Ediția:2006
Editura: Springer
Colecția Springer
Seria Undergraduate Texts in Mathematics

Locul publicării:New York, NY, United States

Public țintă

Graduate

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.

Descriere

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.

Cuprins

Numbers and Functions.- Sequences.- Continuity and Limits.- Differentiation.- Applications of Differentiation.- Integration.- Elementary Transcendental Functions.- Applications and Approximations of Riemann Integrals.- Infinite Series and Improper Integrals.

Recenzii

“Ghorpade and Limaye (both, Indian Institute of technology, Bombay) have written a very coherent and rigorous advanced calculus text on functions of one variable. … Theorems are carefully proven and exercises (without solutions are provided. … Recommended. Upper-division undergraduates.” ( R. L. Pour, Choice, February, 2018)



“A Course in Calculus and Real Analysis by S.R. Ghorpade and B.V. Limaye (Springer UTM) is a rigorous, well-presented and original introduction to the core of undergraduate mathematics — first-year calculus. … This book is a tour de force, and a necessary addition to the library of anyone involved in teaching calculus, or studying it seriously.” (N.J. Wildberger, Gazette of the Australian Mathematical Society, Vol. 35 (3), 2008)



From the reviews:
"The book is a self-contained and rigorous introduction to calculus of one variable. It covers the programme of the first-year in mathematical analysis. The main goal of the book is a clear and successful presentation of the basic ideas of analysis. The reader can see an attention of the authors to help students in understanding and further developing of the basic notions of the subject. … are illustrated by a number of interesting examples and remarks. In some sense it is a book for self-education." (Sergi V. Rogosin, Zentralblatt MATH, Vol. 1100 (2), 2007)

Textul de pe ultima copertă

This book provides a self-contained and rigorous introduction to calculus of functions of one variable. The presentation and sequencing of topics emphasizes the structural development of calculus. At the same time, due importance is given to computational techniques and applications. The authors have strived to make a distinction between the intrinsic definition of a geometric notion and its analytic characterization. Throughout the book, the authors highlight the fact that calculus provides a firm foundation to several concepts and results that are generally encountered in high school and accepted on faith. For example, one can find here a proof of the classical result that the ratio of the circumference of a circle to its diameter is the same for all circles. Also, this book helps students get a clear understanding of the concept of an angle and the definitions of the logarithmic, exponential and trigonometric functions together with a proof of the fact that these are not algebraic functions. A number of topics that may have been inadequately covered in calculus courses and glossed over in real analysis courses are treated here in considerable detail. As such, this book provides a unified exposition of calculus and real analysis.
The only prerequisites for reading this book are topics that are normally covered in high school; however, the reader is expected to possess some mathematical maturity and an ability to understand and appreciate proofs. This book can be used as a textbook for a serious undergraduate course in calculus, while parts of the book can be used for advanced undergraduate and graduate courses in real analysis. Each chapter contains several examples and a large selection of exercises, as well as "Notes and Comments" describing salient features of the exposition, related developments and references to relevant literature.

Caracteristici

Introduces the principles of real analysis, as a formidable counterpart to calculus
Offers self-contained introduction to the calculus of functions of one variable
The text is sequenced to emphasize the structural development of calculus
Places appropriate emphasis on computational techniques and applications