Reddit reviews Advanced Calculus of Several Variables (Dover Books on Mathematics)
We found 6 Reddit comments about Advanced Calculus of Several Variables (Dover Books on Mathematics). Here are the top ones, ranked by their Reddit score.
The answer is "virtually all of mathematics." :D
Although lots of math degrees are fairly linear, calculus is really the first big branch point for your learning. Broadly speaking, the three main pillars of contemporary mathematics are:
You might also think of these as the three main "mathematical mindsets" — mathematicians often talk about "thinking like an algebraist" and so on.
Calculus is the first tiny sliver of analysis and Spivak's Calculus is IMO the best introduction to calculus-as-analysis out there. If you thought Spivak's textbook was amazing, well, that's bread-n-butter analysis. I always thought of Spivak as "one-dimensional analysis" rather than calculus.
Spivak also introduces a bit of algebra, BTW. The first few chapters are really about abstract algebra and you might notice they feel very different from the latter chapters, especially after he introduces the least-upper-bound property. Spivak's "properties of numbers" (P1-P9) are actually the 9 axioms which define an algebraic object called a field. So if you thought those first few chapters were a lot of fun, well, that's algebra!
There isn't that much topology in Spivak, although I'm sure he hides some topology exercises throughout the book. Topology is sometimes called the study of "shape" and is where our most general notions of "continuous function" and "open set" live.
Here are my recommendations.
Analysis If you want to keep learning analysis, check out Introductory Real Analysis by Kolmogorov & Fomin, Principles of Mathematical Analysis by Rudin, and/or Advanced Calculus of Several Variables by Edwards.
Algebra If you want to check out abstract algebra, check out Dummit & Foote's Abstract Algebra and/or Pinter's A Book of Abstract Algebra.
Topology There's really only one thing to recommend here and that's Topology by Munkres.
If you're a high-school student who has read through Spivak in your own, you should be fine with any of these books. These are exactly the books you'd get in a more advanced undergraduate mathematics degree.
I might also check out the Chicago undergraduate mathematics bibliography, which contains all my recommendations above and more. I disagree with their elementary/intermediate/advanced categorization in many cases, e.g., Rudin's Principles of Mathematical Analysis is categorized as "elementary" but it's only "elementary" if your idea of doing math is pursuing a PhD. Baby Rudin (as it's called) is to first-year graduate analysis as Spivak is to first-year undergraduate calculus — Rudin says as much right in the introduction.
>what is the difference really between 'calculus' and 'real analysis'
At the undergraduate level, "calculus" typically means the what. For example: what is this limit? What is the derivative of a given function? What is the value of this integral?
"Analysis" more typically gets into the why behind calculus. Why does this function have a limit? Justify why the typical rules for differentiation—product rule, chain rule, etc.—are valid. Define what it means for a function to be integrable over a given interval, and justify your computation of a given integral.
There's a lot more going on than just that, but to first approximation, making the distinction between the what of calculus and the why of analysis is a good starting point.
---
I don't have a copy of Kolmogorov's text, so I'm at a disadvantage. I assume you mean something like this book in the Dover series? If so, then the table of contents suggests it's a pretty ambitious book, at least for typical undergraduates—and especially if it's one's introduction to the subject matter. That text by Kolmogorov covers some of both metric space topology and point-set topology, as well as linear algebra, measure theory, integration, and differentiation (itself in the context of Lebesgue integration). I'm no expert on the matter, but Kolmogorov's (and Fomin's) text seems more representative of what's often called "functional analysis" rather than just "real analysis". I suspect that pedagogically, you might benefit from a more "concrete" introduction to real analysis before tackling something like this textbook.
As for the inverse and implicit function theorems, there are a handful of ways to approach those results. One way is to show that the two theorems are equivalent: the inverse function theorem is true if and only if the implicit function theorem is true. The way a lot of books proceed is to establish the inverse function theorem by making some suitable simplifications—e.g., that the derivative map is being evaluated at the origin, and that this derivative map is the identity map—then apply the contraction mapping theorem. (Of course, the two theorems are equivalent, so one could instead prove the implicit function theorem first, instead.)
Rudin is emphatically not the only suitable textbook for something like this, but nearly any such "suitable" textbook will inevitably be challenging. It will help you considerably to have already had linear algebra, at least, especially if you turn to a textbook that presupposes linear algebra as a prerequisite. I'm not sure what to recommend to you, but here are a few textbooks I've used over the years (in addition to those already mentioned above):
Stromberg and Browder are challenging, on the general level of Rudin in that respect. Marsden and Hoffman has an unusual structure (at least in my volume), where the statements of the theorems and propositions are separated from the proofs of those assertions. (And, if memory serves, M&H may have suffered from a number of typos.) Edwards may be the most accessible of the books above, and it covers quite a bit from both a "classical" and higher-order "theoretical" perspective. But which of these books—if any—would be the best fit for you would necessarily be a matter of speculation for me.
I hope this was nonetheless helpful. Good luck!
Would something like this be up your alley? Or would you prefer an analysis book? Edit: or of course something less rigourous.
For single variable calculus, like everyone else I would recommend Calculus - Spivak. If you have already seen mechanical caluculus, mechanical meaning plug and chug type problems, this is a great book. It will teach you some analysis on the real line and get your proof writing chops up to speed.
For multivariable calculus, I have three books that I like. Despite the bad reviews on amazon, I think Vector Calculus - Marsden & Tromba is a good text. Lots of it is plug and chug, but the problems are nice.
One book which is proofed based, but still full of examples is Advanced Calculus of Several Variables - Edwards Jr.. This is a nice book and is very cheap.
Lastly, I would like to give a bump to Calculus on Manifolds - Spivak. This book is very proofed based, so if you are not comfortable with this, I would sit back and learn from of the others first.
I've heard that, while Spivak's Calculus may be difficult because of proofs, it is good. However, his Manifolds is basically a graduate level reference book, and isn't the best multivariable calculus book for rebuilding/reteaching the basics of it. I've read that this is good in that regard.
I'd also hope to find a book that goes into the physics side. I've heard this is good for that.
Have you heard anything on these? Have other suggestions?