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I'm taking Calculus 3 at the moment often I like to look at the courses I am going to be taking in the coming semester. At my school, they only list Math Analysis and there are no courses for Real Analysis or Advanced Calculus. Some professors have mentioned they are different while others say they cover the same thing, so I'm a bit confused. I looked at the books at the library and it seems that the three subject cover very similar topics. I have three questions:

1. Are they the same subjects just branded differently or are there differences such as the style with which they approach the material and even the material itself covered?

2. If so which is the best of the three to tackle first? Since I cannot take the Math Analysis course at my school for another year and half( course if taught in the spring of even number years only) I may as well self study til then.

3. What is a recommended set of best books to buy for each (if they're different) or if they are more or less the same what are some recommended books that cover the topics most concisely and give the material a "modern treatment"?

Thanks for the help in advance!

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To some extent the terms, mathematical analysis, real analysis and advanced calculus are used interchangeably. Arguably, real analysis is a proper subset of mathematical analysis, as the latter can also contain complex analysis.

Note also that advanced calculus is sometimes used to refer to a course on multivariable and vector calculus, i.e. probably what you are doing in Calculus 3 at the moment. Other times it is used to refer to an introductory course on real analysis, probably covering a rigorous approach to single variable calculus, but possibly also including material from multivariable/vector calculus.

As for books, if you've had little or no exposure to real analysis, you might want to begin with Alcock's How to Think about Analysis, followed by Abbott's Understanding Analysis or an equivalent book, followed by Pugh's Real Mathematical Analysis. On the other hand, if you would like an interesting perspective on your Calculus 3 material, you might want to try Callahan's Advanced Calculus: A Geometric View.

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    $\begingroup$ I'd add that "advanced calculus" would now be dubbed "introductory real analysis." No recommendation is really complete without Spivak, I guess. $\endgroup$ – Andres Mejia Mar 26 '16 at 21:46
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It's slightly orthogonal to your question, but I would consider to take a traditional course in ordinary differential equations as the "next thing", rather than assuming a theoretical calc course (whether very theoretical [real analysis] or medium theoretical [advanced calculus]) is your natural next step.

DiffyQs has way more utility since it is needed in engineering and physics, as well as math. Whereas, you can get by, certainly at undergrad level without real analysis. And even most grad engineers and physicists don't need RA either (except for extremely math inclined theorists). So do the ODE first. If you liked integration and the tricks and techniques there, you will enjoy the tricks and techniques of diffyQs.

For that matter, I would even be inclined to take an engineering slanted PDE course and engineering slanted complex analysis course before diving into RA.

Also, if you have not done so, consider to take a linear algebra course. Can be a pretty easy one ("matrices") rather than the most difficult. But get something. (This is nothing against the harder versions or the ones with more content, but just those can be taken later.)

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