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I'm starting a course in advanced/multivariable calculus this upcoming semester.

It will cover: Series, Differential Analysis in R^n and multiple integrals

What are some rigorous books/lecture notes that cover these topics?

Frederico de Castro Oliveira
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    The lecturer will surely advise on the best book for their module, and the best notes for the module will be their notes for the module? – Paul Jan 22 '24 at 12:49
  • Some traditional advanced calculus texts are listed in this MSE answer. Regarding what I mean here by "traditional", see the first paragraph there which discusses a distinction I make between "advanced calculus" and "real analysis". Of course, many books fall into a grey area in which it is not easy (or even helpful) to classify the book as primarily "advanced calculus" or primarily "real analysis". The books I listed there are those that, to me, are fairly clearly classified as "advanced calculus". – Dave L. Renfro Jan 22 '24 at 18:10

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I will suggest here a few books that can help you but I recommend you to follow your module leader chosen book:

1 . "Principles of Mathematical Analysis" by Walter Rudin:

Why it's good: Rudin's book, often referred to as "Baby Rudin," is a classic in mathematical analysis. It is known for its clarity, rigor, and comprehensive coverage of topics. It provides a solid foundation in real analysis and includes topics on sequences, series, and multi-variable calculus.

2."Advanced Calculus" by Patrick M. Fitzpatrick:

Why it's good: Fitzpatrick's book is praised for its clear exposition and rigorous treatment of topics in advanced calculus. It covers real analysis, including sequences, series, and functions of several variables, with a good balance between theory and application.

3."Calculus" by Michael Spivak:

Why it's good: Spivak's book is known for its rigorous approach to calculus. It covers topics such as sequences, limits, continuity, and differential calculus in $\mathbb{R}^n$ in a careful and insightful manner. It is suitable for students seeking a deeper understanding of calculus.

4."Advanced Calculus" by Lynn H. Loomis and Shlomo Sternberg:

Why it's good: Loomis and Sternberg's book offers a rigorous treatment of advanced calculus, including topics in analysis and differential geometry. It provides a comprehensive and modern perspective on the subject, suitable for students interested in both analysis and geometry.

5."Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach" by John H. Hubbard and Barbara Burke Hubbard:

Why it's good: This book provides a unified treatment of vector calculus, linear algebra, and differential forms. It emphasizes the connections between these areas and offers a modern and geometric approach to the subject.

Even though i havent read all of that but used few of them as refrences before and my best advice for you is just follow the module leader's chosen book

BerserkEVA01
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