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S.E friends,

I am a college sophomore with a major in mathematics. I am trying to self-study multivariable and vector calculus (they means the same, right?) and prepare for Summer course on multivariable calculus. Our university uses a course packet for the multivariable calculus which is not theoretical enough to satisfy my curiosity. I am seeking a textbook that covers both theories and applications, with more emphasis on theories.

I have been searching the forum and it seems there are sook good books on multivariable calculus: Hubbard/Hubbard's Vector Calculus, Linear Algebra, and Differential Forms; Marsden/Tromba's Vector Calculus, Collier's Vector Calculus, and Lang's Multivariable Calculus. I want to pick only one from them. Could you help me?


marked as duplicate by Did calculus Jan 2 '18 at 13:36

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  • $\begingroup$ How about Stewart Calculus, the staple calc book in North America. Reasonably good book, can be had for very little if you pick an outdated edition. Solution manuals are also widely available $\endgroup$ – imranfat Mar 20 '15 at 21:05

The best introductory textbook on multivariable calculus for the rank beginner that I know is Vector Calculus by Peter Baxandall and Hans Liebeck. I stumbled across this terrific and very underrated book while searching for a modern treatment of functions of several variables that could be used by bright undergraduates without the use of manifolds or differential forms.

The overwhelming majority of such books are basically plug and chug books that may as well have been written in the 19th century and avoid any hard theory like Lyme Disease. Most of them are afraid to discuss linear transformations and vector spaces, for God’s sake. Such books are obviously mostly written for engineering and non-mathematics students, assuming that mathematics and serious physics majors would opt into courses and texts based in differentiable manifolds. Like I said above, this is a highly questionable assumption to say the least and even if it were true, a careful study of vector valued functions in R2 and R3 first might allow students to transition directly to modern differential topology and geometry without the “forms for dummies” approach.

In any event, this wonderful textbook is rooted in the UK university system and therefore begins at a higher level them most American texts. The authors presume the students have strong backgrounds in linear algebra and a careful study of calculus using $\epsilon-\delta$ limit definitions. This makes a world of difference as it allows them to present the elements of several variable calculus as the study of certain linear transformations (the general derivative, the differential) between subspaces of $R^n$. The book has a “spiral ascent” structure-it begins with the simplest kinds of functions of several variables, namely the real valued maps of $R^2$ and proceeds through vector valued maps in $R^2$ then maps from $R^2$ to $R^3$, etc., culminating with a careful study of vector valued functions, derivatives and differentials, and line and surface integrals in general $R^n$. This way the presentation begins in the simplest manner and gradually achieves full generality. The standard concepts-such as chain rule, the inverse and implicit function theorems and multiple integrals-are presented several times at different levels of generality. The language of linear algebra is used freely and without reservation, careful definitions are given and the presentation is still extremely visual as each concept is given with several graphs. Even better, the presentation is extremely example driven-there are literally hundreds of examples throughout-from both mathematics and classical physics. And it’s all topped off with lots of equally terrific exercises, none too hard.

This is the kind of book you wish you’d had when you learned vector calculus. I remember reading it and thinking how much easier Barrett O’Neill’s differential geometry book would have been if I’d mastered Baxandall and Liebeck first. An absolute must for any student trying to master multivariable calculus and it’ll make very helpful collateral or prior reading for any student about to take a course in differentiable manifolds or differential geometry. That would be the book I would begin with before moving on to more sophisticated texts on manifolds.

  • $\begingroup$ Could you parse the above into a more readable/organized format (e.g. New paragraphs would go a long way to better organize and present what you simply spit out in one block of text.) I started reading your answer couldn't catch my breath nor figure out the point you were trying to make. Is it a rant on other texts or approaches? Is it a recommendation for a book? $\endgroup$ – Namaste Mar 20 '15 at 19:51
  • $\begingroup$ @Mathemagician1234 If what you call “forms for dummies” approach are the books by Hubbard, Tromba, etc., what book do you recommend learning differential forms after learning vector calculus from baxandall? Specially for physics majors. $\endgroup$ – mathemania Feb 4 '18 at 16:29
  • $\begingroup$ @mathemania The best book on forms for physics students (and would be terrific for mathematics students interested in physics) is T.Frankel's THE GEOMETRY OF PHYSICS. The prerequisites for it is a good course in multivariable calculus and some linear algebra-in other words,basically Bandaxall/Liebeck. Darling's GEOMETRY FROM A DIFFERENTIABLE STANDPOINT is pitched at about the same level and has wonderful discussions of applications of differential forms to physics. Both will do very nicely for math-inclined physics students. $\endgroup$ – Mathemagician1234 Feb 5 '18 at 23:51
  • $\begingroup$ @Mathemagician1234 I can't find the author Darling for the book GEOMETRY FROM A DIFFERENTIABLE STANDPOINT. Although there is one by John McCleary. $\endgroup$ – mathemania Feb 6 '18 at 4:30
  • $\begingroup$ @mathmania Sigh. I made a huge mistake and confused the 2 books. My bad. The book by Darling I was referring to mistakenly up there is DIFFERENTIAL FORMS AND CONNECTIONS. McCleary is one of the best undergraduate introductions to differential geometry there is,but it's not the book you want for this purpose. Again,my bad. $\endgroup$ – Mathemagician1234 Feb 6 '18 at 5:39

Best book: Calculus, by Ron Larson and Bruce Edwards 10th edition. By the way YouTube is great resource. This is a great resource, I had his calss

  • $\begingroup$ I own 8th and 9th editions of this book and it is wonderful. I've recommended to several friends as well who all agree that it's great. $\endgroup$ – Lanier Freeman May 29 '16 at 22:13

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