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I am a junior student majored in mechanical engineering. I have great passion for pure and applied math.Sadly, the maths disciplines in my university did not fulfill my requirements.For now, I have self learned Analysis with a single variable(chapter 1-7 of baby rudin) and Axler's entire Linear Algebra Done Right. I want to keep studying on the following topics:Multivariable Analysis / Modern Algebra (groups,rings,which didn’t covered in Axler’s textbook) Any good textbooks or website recommendations for me? The book should be rigorous and well explained. More importantly, it should be friendly to self-learners. It may also has the solution manual or at least some hints for the problem. Because it is hard for a non-math majored students to get professional assistance.

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    $\begingroup$ You can see Springer's Undergraduate Texts in Mathematics and Technology series. $\endgroup$ Commented Oct 24, 2018 at 15:18
  • $\begingroup$ I can't remember the exact title of a book I found helpful in a similar situation, but it would have been one of these, springer.com/gp/book/9783642001734 or more likely amazon.co.uk/… $\endgroup$
    – J.G.
    Commented Oct 24, 2018 at 15:27
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    $\begingroup$ take a look to Abstract algebra textbook of Herstein to see if it fit your needs for an introduction to abstract algebra. It is elementary and just 266 pages. $\endgroup$
    – Masacroso
    Commented Oct 24, 2018 at 15:35
  • $\begingroup$ Indeed, Herstein is an excellent reference. +1 $\endgroup$
    – user279515
    Commented Oct 24, 2018 at 15:38
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    $\begingroup$ I really love Pinter's book of abstract algebra. $\endgroup$
    – user296602
    Commented Oct 24, 2018 at 15:44

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For groups, rings and other topics from modern algebra, I highly recommend Dummit and Foote's Abstract Algebra. It is comprehensive and very student friendly.

For multivariable calculus, there are several good recommendations you can find here: References for multivariable calculus

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  • $\begingroup$ well, but Dummit & Foote have nearly 1000 pages!!! It is more like an encyclopedia than a textbook. Im not sure if a long text like that can be considered "introductory". $\endgroup$
    – Masacroso
    Commented Oct 24, 2018 at 15:34
  • $\begingroup$ @Masacroso The length of the textbook need not be concerning, because the presentation of the material is very gentle in Dummit and Foote. Compare it with, say, Lang's Algebra and you'll see what I mean. $\endgroup$
    – user279515
    Commented Oct 24, 2018 at 15:38
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I found the following Youtube videos by Dr.Matt Salomone really useful as well on Abstract Algebra.

https://youtu.be/N-5Av1KC11w

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