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I am going to start studying the analysis texts (Rudin-PMA, Apostol-MA, Pugh-RMA) on the first week of August. I have a good proof skills through working on Artin's Algebra and Hoffman/Kunze's Linear Algebra, but I unfortunately only took computational 1-variable calculus (Lang's A First Course in Calculus), and I did not take multivariable calculus, which I might take concurrently with Analysis I and Theoretical Linear Algebra on the upcoming Fall.

I am looking for a brief text which explains the key ideas from both 1-variable and multivariable calculus, one I can read and jump directly into the analysis texts. Could you recommend one?

Also will my lack of multivariable calculus be a problem when I tackle those analysis texts? I seem to understand at least the beginning chapters.

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    $\begingroup$ Calculus of Several Variables by Casper Goffman is old (1966), but might be a good fit for what you want. I don't know how easily available it is to buy (or find on the internet), but I do know that it's very common in U.S. college/university libraries. $\endgroup$
    – Dave L Renfro
    Jul 23, 2015 at 22:09

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I first learned calculus from the predecessor of the current college outline series Calculus, which you can get from Amazon for 59 cents, free shipping if you sign up for a trial 6 month trial of "Amazon Student."

http://www.amazon.com/Calculus-Harcourt-Jovanovich-College-Outline/dp/0156015560/ref=sr_1_1?ie=UTF8&qid=1438026539&sr=8-1&keywords=college+outline+series+calculus

A second edition of my analysis book will be out in September. Among other things it adds a discussion of visualizing in 4D.

Best, Charles Pugh

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    $\begingroup$ Thank you very much for the advice, Professor Pugh. Your textbook "Real Mathematical Analysis" is a very outstanding analysis text, and it beats Rudin and Apostol from the exposition to the quality of problem sets. However, should I be familiarized myself with the multivariable calculus first before studying your later chapters on multivariable analysis? If so, is Serge Lang's Calculus of Several Variables enough for the background? $\endgroup$
    – MathWanderer
    Jul 27, 2015 at 20:21
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    $\begingroup$ I had forgotten about Lang's book when I mentioned Goffman's book in my earlier comment (under your question). I think Lang's book would be great for you, and in fact I've used it myself at times when I needed to review certain topics in elementary multivariable calculus. See my comments about Lang's book in this 21 January 2011 Math Forum archived post. However, I don't think you'll need any multivariable calculus for Pugh's book until you get to the chapter on multivariable calculus, which is more than halfway through his book. $\endgroup$
    – Dave L Renfro
    Jul 28, 2015 at 15:01

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