Advanced Calculus/Real Analysis Book

In summary, an Advanced Calculus/Real Analysis book is a comprehensive guide to the advanced concepts and theories of calculus and real analysis. It covers topics such as convergence, continuity, differentiability, integrability, and sequences and series. The book also delves into more advanced topics like measure theory, Lebesgue integration, and metric spaces. It is a valuable resource for students and professionals in mathematics, physics, and engineering, providing a deep understanding of the fundamental principles and techniques of advanced calculus and real analysis.
  • #1
tarheelborn
123
0
I am currently taking an advanced calculus course and, for the most part, I really like it. However, I believe I could benefit from a supplemental textbook. My class uses Goldberg's Methods of Real Analysis. I would like to be able to find a text that offers more detailed explanations and, particularly, step-by-step examples to show the theorems. I suppose I need a "for dummies" version of advanced calculus. I am an older student, attempting to refresh/learn enough to get into graduate school in mathematics. Thanks for your help.
 
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  • #2
I recommend

https://www.amazon.com/dp/143484367X/?tag=pfamazon01-20 by Thomson, Bruckner, and Bruckner.

You can preview parts of it using the "look inside" feature at Amazon to see if it's what you are after. The price is pretty good at $27 for a paperback, or you can download a full PDF version here for a token charge (I think it was literally one dollar, with substantial chunks available for free):

http://classicalrealanalysis.com/TBB.aspx"

Caution: the same authors also published a graduate-level book called "Real Analysis."
 
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  • #3
Thank you so much!
 
  • #4
Two books published by Dover that are, in my opinion, excellent for your purpose are

https://www.amazon.com/dp/0486689220/?tag=pfamazon01-20 by Georgi Shilov

and
https://www.amazon.com/dp/0486650383/?tag=pfamazon01-20 by Maxwell Rosenslicht.

Also, if you plan on pursuing pure math, Dover is a fantastic publisher offering many cheap, well-written texts that are far better than many texts put out by publishers for 5-10x the price. To see their math catalogue, go to http://store.doverpublications.com/by-subject-science-and-mathematics-mathematics.html".

Best of luck!
 
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  • #5
Analysis: With an Introduction to Proof by Steven Lay
This book is an excellent starter to analysis. It gives many proofs that are extremely clear, and has quite a big chunk devoted to techniques of proof.

Elementary Analysis: The Theory of Calculus by Kenneth Ross

These are great introductory books that don't have an ego.
 
  • #6
Thanks for all your help!
 

1. What is the difference between advanced calculus and real analysis?

Advanced calculus is a more advanced version of traditional calculus that focuses on mathematical concepts such as limits, derivatives, and integrals. Real analysis, on the other hand, is a branch of mathematics that deals with the rigorous study of functions and their properties. It uses the tools and techniques from advanced calculus to analyze functions in a more formal and abstract manner.

2. What are the prerequisites for studying advanced calculus/real analysis?

The prerequisites for studying advanced calculus/real analysis typically include a strong foundation in single and multivariable calculus, linear algebra, and basic mathematical proofs. It is also helpful to have some exposure to abstract mathematics and mathematical notation.

3. What topics are typically covered in an advanced calculus/real analysis book?

The topics covered in an advanced calculus/real analysis book may vary, but some common topics include limits, continuity, derivatives, Riemann integration, sequences and series, and functions of several variables. The book may also cover topics such as metric spaces, Lebesgue integration, and Fourier analysis.

4. How can I improve my understanding of advanced calculus/real analysis?

To improve your understanding of advanced calculus/real analysis, it is important to practice solving problems and proofs, as well as reading and understanding mathematical notation and concepts. It can also be helpful to seek out additional resources such as online tutorials, lectures, or study groups.

5. How can advanced calculus/real analysis be applied in real-world situations?

Advanced calculus/real analysis can be applied in a variety of real-world situations, such as physics, engineering, economics, and computer science. It provides a framework for understanding and analyzing functions and their properties, which can be useful in solving real-world problems and making predictions or models. Additionally, the logical and analytical skills developed through studying advanced calculus/real analysis can be applied in many other fields and professions.

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