Math Books Organized by Area of Mathematics(math-blog.com)
math-blog.com
Math Books Organized by Area of Mathematics
http://math-blog.com/mathematics-books/
13 comments
I no longer buy math books that do not have solutions to the exercises at the back of the book. None of the listed books satisfy this criterion. If you are going to learn math then reading is not enough, you need to do countless exercises to get all the subtleties of the theorems and definitions and you won't know if you're doing them right unless you have a template answer to compare to.
Relevant HN discussions from the past:
-http://news.ycombinator.com/item?id=108723
-http://news.ycombinator.com/item?id=201913
-http://news.ycombinator.com/item?id=108723
-http://news.ycombinator.com/item?id=201913
See also, the Chicago undergraduate mathematics bibliography: http://www.ocf.berkeley.edu/~abhishek/chicmath.htm
Robert Geroch's _Mathematical Physics_ (despite the title it's really just a broad, shallow overview of graduate-level math and covers virtually none of the traditional "mathematical physics" curriculum). Best math book I have ever read (the emphasis on motivation by example and visual, intuitive presentation of concepts, theorems, and proofs is second to none). This is really just a bunch of lecture notes and reads like one - except Geroch is likely the best teacher you will ever have.
For those with more knowledge: in what way is Sheldon Axler's "Linear Algebra Done Right" (quoted in the list above) superior to Strang's "Linear Algebra" for the complete novice? As I can only pick one to work through, I would like to make a (somewhat) informed decision.
Edit: Thank you plinkplonk and acangiano. I'll stick with Strang for the time being then. Any recommendations for Real Analysis? The "Baby Rudin" looks to be just a bit out of my range.
Edit: Thank you plinkplonk and acangiano. I'll stick with Strang for the time being then. Any recommendations for Real Analysis? The "Baby Rudin" looks to be just a bit out of my range.
"For those with more knowledge: in what way is Sheldon Axler's "Linear Algebra Done Right" (quoted in the list above) superior to Strang's "Linear Algebra" for the complete novice?"
A complete novice should go for Strang. Axler says upfront (in the preface iirc it has been a while since I worked through it) his book is intended as the second Linear Algebra book and assumes you have a base in Linear Algebra (matrix manipulations and so forth) already. Axler depends more on a classical "theorems and proofs" approach. Strang doesn't involve proofs etc and is more ocncerned with funamental operations and building intuition and so (imo) is more suited to the complete beginner.
A complete novice should go for Strang. Axler says upfront (in the preface iirc it has been a while since I worked through it) his book is intended as the second Linear Algebra book and assumes you have a base in Linear Algebra (matrix manipulations and so forth) already. Axler depends more on a classical "theorems and proofs" approach. Strang doesn't involve proofs etc and is more ocncerned with funamental operations and building intuition and so (imo) is more suited to the complete beginner.
Axler and Strang are really very different books with very different intents. The most striking example is that Gaussian elimination is practically the first thing that Strang covers, but Axler doesn't cover it at all.
I can't speak in general because I've only skimmed Strang, whereas I worked through nearly all of Axler. But if your interest in Linear Algebra is as preparation for quantum mechanics, then Axler is a great choice, perhaps even for a novice (though it does require a bit of mathematical maturity). Axler really hammers you with the idea that a matrix is just a way to represent a linear transformation, and that the numbers in the matrix depend on the basis you choose for the underlying vector space. This way of thinking is very helpful when you learn quantum mechanics.
EDIT: Axler also has the advantage that it is short. Personally, I find it much easier to work independently through short math/physics books than long books.
I can't speak in general because I've only skimmed Strang, whereas I worked through nearly all of Axler. But if your interest in Linear Algebra is as preparation for quantum mechanics, then Axler is a great choice, perhaps even for a novice (though it does require a bit of mathematical maturity). Axler really hammers you with the idea that a matrix is just a way to represent a linear transformation, and that the numbers in the matrix depend on the basis you choose for the underlying vector space. This way of thinking is very helpful when you learn quantum mechanics.
EDIT: Axler also has the advantage that it is short. Personally, I find it much easier to work independently through short math/physics books than long books.
Actually, if you are a complete novice, Linear Algebra Done Right won't serve you too well. You need to understand basics before approaching it. I should probably include Strang's book as a first introduction, before Linear Algebra Done Right. On a side note, Axler's book is also $120 cheaper than Strang's (which may or may not matter to you). You could also watch the free lectures by Strang online, and then use Axler's book as a supplement. I personally think that Axler's exposition is far more concise and clear.
For real analysis, I'd recommend that you pick up a copy of "baby Rudin" regardless, but for learning real analysis, I'd recommend a book like "A First Course in Mathematical Analysis" by Burkill. Other viable options include "Calculus" by Spivak (it's really a book on real analysis at a level somewhere between a typical freshman calculus book and "baby Rudin"), "Real Mathematical Analysis" by Pugh, or "Yet Another Introduction to Analysis" by Bryant.
Agreed, Calculus by Spivak is essentially a bridge to Real Analysis.
I'm currently enrolled in a class that uses this book:
Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by Hubbard and Hubbard.
http://matrixeditions.com/
This book should totally be listed. Though it doesn't fit particularly well in any category, it provides a pretty good intro to a number of higher math topics.
Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by Hubbard and Hubbard.
http://matrixeditions.com/
This book should totally be listed. Though it doesn't fit particularly well in any category, it provides a pretty good intro to a number of higher math topics.
Totally missing any sort of reference for Partial Differential Equations (except in one section of the mathematical methods book), Dynamical Systems (incl. Chaos Theory), Calculus of Variations, and pretty much the entire Applied side of the mathematics spectrum.
If you want to brush op on some of your math:
http://www.khanacademy.org/
All Youtube video's clearly explaining different theories. Helped me relearn a few topics..
I'm annoyed that "Pre-calculus," "Calculus I," and "Calculus II" are deemed distinct areas of mathematics. Do additional dimensions necessitate a new category?
Precalculus, at least in US high schools, means something like “trigonomety and limits, plus applications”. “Calculus I” means single variable calculus, though the Spivak book they list is much more rigorous than a typical first-year calculus course. “Calculus II/III” in this case means vector calculus, which is enough of its own material to get at least its own semester course. I don’t think I’d quite put Spivak’s Calculus on Manifolds in the same category – maybe more like basic differential geometry/differential topology – but hey, this isn’t my list.
They are not distinct areas. However, I feel that dividing them up helps the reader compare apples to apples, rather than having a list of 10 calculus related books.
If you take that sort of approach, I would much rather you present the topics in some sort of directed graph format to clearly show which areas depend on knowledge of other areas.
I realize that's a non-trivial exercise and no matter what you did you'd likely catch flack from people that don't agree with your prereqs. However, I think that would be particularly valuable to people who don't know what they don't know.
(edit) I mean non trivial outside of relationships like basic math -> algrebra -> precalc -> calc1 -> calc2
I realize that's a non-trivial exercise and no matter what you did you'd likely catch flack from people that don't agree with your prereqs. However, I think that would be particularly valuable to people who don't know what they don't know.
(edit) I mean non trivial outside of relationships like basic math -> algrebra -> precalc -> calc1 -> calc2
> (edit) I mean non trivial outside of relationships like basic math -> algrebra -> precalc -> calc1 -> calc2
What do you mean by Algebra? In university we had Analysis and Linear Algebra starting at the same time, and Algebra only came later on as a more advanced topic.
What do you mean by Algebra? In university we had Analysis and Linear Algebra starting at the same time, and Algebra only came later on as a more advanced topic.
At some point they get mixed up and interdependent. Go read until you run into some concept that gets you stuck, and then switch topics and work through until you understand it. Unless you’re taking university classes, no one cares what order you follow.
If you look at the divisions being made, its pretty clearly based on how people would look for books. For calculus, lots of people buy books based on classes they take. Hence the division.
No one of any mathematical repute or who took anything beyond calculus would be confused by this so I'm okay with the menu as is.
No one of any mathematical repute or who took anything beyond calculus would be confused by this so I'm okay with the menu as is.
Missing the absolutely fantastic The Higher Arithmetic (Davenport) in the number theory section.
Another notable book missing from that list is "Topology" by Munkres.
Interesting. You're the second recommendation I've been given for Munkres. This list of recommended books in topology:
http://www.math.cornell.edu/~hatcher/Other/topologybooks.pdf
suggests some other books over it, however.
Personally, I've just started reading Hocking & Young (Shocking and Fun!) and it seems quite good so far.
I've made some substantial progress in _Counterexamples in topology_ and it's really good... It's not really a textbook, just a thing booklet that goes over general topology, then goes through a lot of examples and provides all these really nice charts of topological spaces based on properties. I actually made a graph, mostly based off it: http://christopherolah.wordpress.com/2010/03/09/compactness-...
Oh, and Needham's _Visual Complex Analysis_ (in list) is awesome! Best math book I've ever read.
Personally, I've just started reading Hocking & Young (Shocking and Fun!) and it seems quite good so far.
I've made some substantial progress in _Counterexamples in topology_ and it's really good... It's not really a textbook, just a thing booklet that goes over general topology, then goes through a lot of examples and provides all these really nice charts of topological spaces based on properties. I actually made a graph, mostly based off it: http://christopherolah.wordpress.com/2010/03/09/compactness-...
Oh, and Needham's _Visual Complex Analysis_ (in list) is awesome! Best math book I've ever read.
Munkres is okay – pretty readable, gets the job done, has enough interesting problems. I like Hatcher’s book though, so his recommendations (your first link) are probably solid.
I'm missing:
"Understanding Analysis" S. Abbott - great book on the foundations of Real Analysis, esp. for an honors course
"Introductory Functional Analysis" E. Kreyzig
Anyone know a good book for non-introductory statistics?
To quote justokay quoting Mike Jordan
http://news.ycombinator.com/item?id=1055389
I've read a number of books from that sequence and they are definitely the core you need to start hitting for advanced stat.
http://news.ycombinator.com/item?id=1055389
I've read a number of books from that sequence and they are definitely the core you need to start hitting for advanced stat.
Much appreciated! Now I know how I'll be spending my free time.
Edit: I also never realized how good basketball players were at statistics.
Edit: I also never realized how good basketball players were at statistics.
Not sure whether you were joking. If you were, pls ignore the rest of the comment :)
This Mike Jordan is not whom you think he is :) (as can be seen from http://news.ycombinator.com/item?id=1055823)
This Mike Jordan is not whom you think he is :) (as can be seen from http://news.ycombinator.com/item?id=1055823)
Indeed I was. However, that is probably the best math/sports quote of all time.
Some Listmania and So You'd Like to . . . lists: http://www.amazon.com/gp/richpub/syltguides/fullview/20JWVDE...
http://www.amazon.com/Discrete-Mathematics-Combinatorics-Gra...
http://www.amazon.com/gp/richpub/syltguides/fullview/34SIY10...
http://www.amazon.com/gp/richpub/syltguides/fullview/34SIY10...