Does Spivak and Artin or Hoffman/Kuzne suffice?
Loomis is a very difficult book. The title isn't really accurate too, it's not really calculus, but more differential geometry and analysis.
But if you managed to get through Spivak, then I'm really sure you can handle Loomis and Sternberg. Artin is not necessary. Hoffman & Kunze is also not necessary, but it does help.
Thanks for replying so quickly. It did say that some knowledge of Linear algebra is necessary(I read the preface but was unsure of the level for LA) where should I get that?
Also would I get a good understanding of MV calc from it? Thanks for your time!
Check the free book "Linear Algebra done wrong": http://www.math.brown.edu/~treil/papers/LADW/LADW.html If you know things like that, then you should be ready for Loomis and Sternberg.
Do you already know some MV calc? Loomis and Sternberg is really a bad book for a first exposure. But sure, if you manage to complete the book, you will have a very deep understanding of MV calc.
I looked at the ToC(for LA) and it wasn't too familiar. Would the MIT OCW LA course be good enough?
I do know some MV, but not to much. If I went through the MIT OCW version of MV would that be sufficent?
I guess it suffices. But I'm not a fan of MIT OCW and other online learning programs. Maybe it works for you though.
I know what you mean. They are quick though. What would you suggest otherwise?
I think you should try MIT OCW and then try Loomis. Maybe it will work out for you. Or maybe you will find that you need a deeper knowledge of LA and MV Calc.
Anyway, I highly recommend the "Linear Algebra done wrong book" for LA. Another good books is "Linear Algebra" by Lang or his "Intro to Linear Algebra" also by Lang.
For multivariable calculus, there's the Lang book "Calculus of Several Variables" or Spivak's "Calculus on Manifolds".
There is also this book by Hubbard which unites LA and MV Calc: https://www.amazon.com/Vector-Calculus-Linear-Algebra-Differential/dp/0130414085
That sounds perfect! Thank You!
You mean Loomis and Sternberg? That is a very good book. Maybe just shy of being must read, but highly recommended. It is especially worthwhile for good sections on some topics that are bad in other book, for being challenging, and for being free on Sternberg's website and recently going back in print for a good price if you want a hard copy.
For prerequisites it is a second year calculus book, so you should know first year calculus. There can be a gap between knowing what is needed and knowing enough to comfortably continue. I would suggest only a very well prepared student follow an easy first year book with Loomis and Sternberg most people should read an easy second year book or a hard first year book first. The authors suggest one of Differential and Integral Calculus by R Courant, Calculus by T Apostol, Calculus by M Spivak, or Pure Mathematics by G Hardy. Calculus of several variables (another arbitrary border, I draw the line after partial derivatives) is explicitly not required. In the past it was common to see several variables for the first time in second year calculus, now it is usual to have some at the end of the introduction or for use in applied subjects like electricity. A preview of several variables would be helpful, but something like Calculus On Manifolds (Spivak 2) or Analysis On Manifolds by Munkress (a bad title in my opinion) would be more than what is needed. For linear algebra very little is required and there is some linear algebra review included. Still some linear algebra is helpful as it is easier if it is not the first time you have seen it. It is worthwhile to learn linear algebra for its own sake, but Loomis and Sternberg uses only a small amount.
Calculus, analysis, and differential geometry and analysis are closely related. The title does not really matter. The level matters more, but it is not useful to try to be precise about level. A given book may omit some topics or present different topics at different levels.
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