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is there a book that covers linear algebra and multivariable calculus, preferably computationally but with hard (maybe even proof based) exercises. I tried Apostol's Calculus volumes but they seem to adopt a rigorous approach. I am mainly trying to prepare for math 25 which I'm planning to take next year (deferred entry) at harvard.

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  • $\begingroup$ I know of no book that does both at the same time. Is there any reason you would want a book that does so? $\endgroup$ – Omnomnomnom Sep 24 '15 at 12:04
  • $\begingroup$ @Omnomnomnom: Not really but I figured that since textbooks usually have a size limit of around 500 pages, then if I could find a book that did both I would be better off than reading a 500 page treatment of multivariable calculus followed by a 500 page treatment of linear algebra. I am fine with the 500 page treatment of both together (if one exists) being more concise, since I am mathematically mature enough for rigor but at the same time need to learn these topics quickly at a computational level. If you know of any brief treatments of either of these topics I would highly appreciate it. $\endgroup$ – user45220 Sep 24 '15 at 12:48
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In the late 1960s through the mid 1970s there was a slight tendency (in the U.S.) to combine calculus and linear algebra, at least for honors level courses. Two books that I'm fairly familiar with (because I worked through parts of each while in high school) are [1] and [2] below. While looking up the amazon web pages for these two books I also came across [3], but I don't know anything about this book.

[1] Calculus of Vector Functions by Richard E. Williamson and Richard H. Crowell and Hale F. Trotter

[2] Calculus and Linear Algebra: An Integrated Approach by Mary R. Embry and Joseph F. Schell and J. Pelham Thomas

[3] Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds by Theodore Shifrin

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The two I know of are Ted Shifrin's Multivariable Mathematics and Hubbard and Hubbard's Vector Calculus, Linear Algebra, and Differential Forms. I've heard good things of both, but I haven't actually read either so I can only recommend them as texts I know others have enjoyed.

Or if you just want short, geometrically motivated treatments of linear algebra and multivariable calculus, try Alan Macdonald's Linear and Geometric Algebra and Vector and Geometric Calculus. Each one is only about $200$ pages so reading them back to back is entirely an option. I have read both of these and they are excellent. Neither one goes quite as far into the material as I might like, but what they do cover, they do very well. Then again because you're just trying to prepare for future courses, you don't need books that cover everything.

A note though, Macdonald's book do take a little bit different approach than the normal curriculum. He does everything from a geometric (Clifford) algebra perspective. I really loved that, and it's certainly not going to hurt to learn some geometric algebra, but it will require a little mental adjustment when you learn it the standard (matrix-based) way at university.

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  • $\begingroup$ I can vouch for Shifrin's book, it's quite nice! $\endgroup$ – pjs36 Sep 24 '15 at 18:13
  • $\begingroup$ @pjs36 but it's very expensive though $\endgroup$ – TheQuantumMan Dec 12 '15 at 15:49

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