A Course in Calculus and Real Analysis [Ghorpade]

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A Course in Calculus and Real Analysis
Authors: Sudhir R. Ghorpade & Balmohan V. Limaye
Publication date: 2018
Publisher: Springer Cham
Edition: 2nd edition
Series:Undergraduate Texts in Mathematics
Pages: IX + 538

Summary
A Course in Calculus and Real Analysis is a rigorous undergraduate textbook designed to bridge the gap between elementary calculus and formal real analysis. Instead of treating differentiation and integration primarily as computational techniques, Ghorpade and Limaye carefully develop the underlying mathematical foundations, emphasizing definitions, proofs, and the distinction between geometric intuition and analytic characterization. The book begins with real numbers and functions, proceeds through sequences, limits and continuity, and then develops differentiation, applications of derivatives, Riemann integration, and elementary transcendental functions. 

The later chapters move significantly closer to a standard course in real analysis, covering applications and approximations of Riemann integrals, infinite series, improper integrals, and—particularly in this second edition—sequences and series of functions together with integrals depending on a parameter. The second edition also adds appendices constructing the real numbers using Cauchy sequences and providing a self-contained proof of the Fundamental Theorem of Algebra. Numerous examples, exercises, and chapter-ending notes connect the theory with additional literature and mathematical context. 

The book is therefore more demanding than a conventional first calculus textbook. It assumes the reader is comfortable following mathematical proofs and is particularly suitable for an honors-calculus course, mathematics majors preparing for real analysis, or teachers and students who want to understand why the standard theorems of calculus actually work. Springer explicitly describes it as suitable either for a rigorous undergraduate calculus course or as a supplement to a later course in real analysis.

Key takeaways

• Calculus taught as rigorous mathematics: limits, continuity, derivatives and integrals are developed from precise definitions rather than primarily through computational rules.
  
• Strong bridge to Real Analysis: sequences, convergence, infinite series, improper integrals and sequences/series of functions prepare the reader for more advanced analysis.
  
• Proof-oriented: it is best suited to students with some mathematical maturity rather than someone looking only for a standard computational calculus text.
  
• Especially useful for mathematics students and teachers: the authors emphasize foundations and proofs of results that introductory courses often state without justification.
  

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