Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.

Amazon search for Dover Books on mathematics

There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm

Pre-Calculus / Problem-Solving

• The Art of Problem Solving - Lehoczky/Ruczyk - in two volumes, both have full solutions available separately
  
  Calculus
  
  
• Calculus - Stewart - ignore the bad reviews, it's very clear and covers all of basic calculus well - it's just the book that most intro students use and as a consequence gets a bad rap
  
  
• Calculus - Spivak - this will take your understanding of calculus to the next level
  
  
• Calculus - Apostol - there are two volumes and they are very comprehensive, albeit difficult
  
  Linear Algebra
  
  
• Linear Algebra - Lay
  
  
• Linear Algebra Done Right - Axler
  
  Differential Equations
  
  
• Elementary Differential Equations - Boyce/DePrima
  
  
• Ordinary Differential Equations - Tenenbaum
  
  Number Theory
  
  
• Number Theory - Pommersheim for a change of pace and proof skills
  
  Proof-Writing
  
  
• Transition to Higher Mathematics - Dumas/McCarthy - helps to bridge the gap between "lower" and higher math
  
  Analysis
  
  
• The Way of Analysis - Strichartz
  
• Principles of Mathematical Analysis - Rudin
  
• Real and Complex Analysis - Rudin
  
  Complex Analysis
  
  
• Complex Analysis - Ahlfors
  
  Functional Analysis
  
  
• Functional Analysis - Rudin
  
  
• Introductory Functional Analysis with Applications - Kreyszig
  
  Partial Differential Equations
  
  
• Partial Differential Equations for Scientists and Engineers - Farlow
  
  Higher-dimensional Calculus and Differential Geometry
  
  
• Calculus on Manifolds - Spivak - for a modern take on multivariable calculus
  
  
• Differential Geometry - Kreyszig
  
  Abstract Algebra
  
  
• A First Course in Abstract Algebra - Fraleigh
  
• Abtract Algebra - Dummit and Foote
  
  Geometry
  
  
• The Elements - Euclid - for culture and for understanding where math has come from, plus it's still an awesome book after all these years; don't need to go through all of it
  
• Euclidean and Non-Euclidean Geometries - Greenberg
  
  Topology
  
  
• Introduction to Topology - Mendelson
  
  
• Topology - Munkres
  
  Set Theory and Logic
  
  
• Introduction to Set Theory - Hrbacek/Jech
  
  
• Elements of Set Theory - Enderton
  
  
• Mathematical Logic - Kleene
  
  
• Set Theory and the Continuum Hypothesis - Cohen
  
  Combinatorics / Discrete Math
  
  
• Combinatorics - Cameron
  
  
• Concrete Math - Graham/Knuth/Patashnik
  
  
• Discrete Math - Biggs
  
  Graph Theory
  
  
• A First Course in Graph Theory - Chartrand/Zhang
  
  P. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"
  

i have three categories of suggestions.

advanced calculus

these are essentially precursors to smooth manifold theory. you mention you have had calculus 3, but this is likely the modern multivariate calculus course.

• advanced calculus: a differential forms approach by harold edwards
  
  
• advanced calculus: a geometric view by james callahan
  
  
• vector calculus, linear algebra, and differential forms: a unified approach by john hubbard
  
  out of these, if you were to choose one, i think the callahan book is probably your best bet to pull from. it is the most modern, in both approach and notation. it is a perfect setup for smooth manifolds (however, all of these books fit that bill). hubbard's book is very similar, but i don't particularly like its notation. however, it has some unique features and does attempt to unify the concepts, which is a nice approach. edwards book is just fantastic, albeit a bit nonstandard. at a minimum, i recommend reading the first three chapters and then the latter chapters and appendices, in particular chapter 8 on applications. the first three chapters cover the core material, where chapters 4-6 then go on to solidify the concepts presented in the first three chapters a bit more rigorously.
  
  smooth manifolds
  
  
• an introduction to manifolds by loring tu
  
  
• introduction to smooth manifolds by john m. lee
  
  
• manifolds and differential geometry by jeffrey m. lee
  
  
• first steps in differential geometry: riemannian, contact, sympletic by andrew mcinerney
  
  out of these books, i only have explicit experience with the first two. i learned the material in graduate school from john m. lee's book, which i later solidifed by reading tu's book. tu's book actually covers the same core material as lee's book, but what makes it more approachable is that it doesn't emphasize, and thus doesn't require a lot of background in, the topological aspects of manifolds. it also does a better job of showing examples and techniques, and is better written in general than john m. lee's book. although, john m. lee's book is rather good.
  
  so out of these, i would no doubt choose tu's book. i mention the latter two only to mention them because i know about them. i don't have any experience with them.
  
  conceptual books
  
  these books should be helpful as side notes to this material.
  
  
• div, grad, curl are dead by william burke [pdf]
  
  
• geometrical vectors by gabriel weinreich
  
  
• about vectors by banesh hoffmann
  
  i highly recommend all of these because they're all rather short and easy reads. the first two get at the visual concepts and intuition behind vectors, covectors, etc. they are actually the only two out of all of these books (if i remember right) that even talk about and mention twisted forms.
  
  there are also a ton of books for physicists, applied differential geometry by william burke, gauge fields, knots and gravity by john baez and javier muniain (despite its title, it's very approachable), variational principles of mechanics by cornelius lanczos, etc. that would all help with understanding the intuition and applications of this material.
  
  conclusion
  
  if you're really wanting to get right to the smooth manifolds material, i would start with tu's book and then supplement as needed from the callahan and hubbard books to pick up things like the implicit and inverse function theorems. i highly recommend reading edwards' book regardless. if you're long-gaming it, then i'd probably start with callahan's book, then move to tu's book, all the while reading edwards' book. :)
  
  i have been out of graduate school for a few years now, leaving before finishing my ph.d. i am actually going back through callahan's book (didn't know about it at the time and/or it wasn't released) for fun and its solid expositions and approach. edwards' book remains one of my favorite books (not just math) to just pick up and read.

An Introduction to Manifolds by Tu is a very approachable book that will get you up to Stokes. Might as well get the full version of Stokes on manifolds not just in analysis. From here you can go on to books by Ramanan, Michor, or Sharpe.

A Guide to Distribution Theory and Fourier Transforms by Strichartz was my introduction to Fourier analysis in undergrad. Probably helps to have some prior Fourier experience in a complex analysis or PDE course.

Bartle's Elements of Integration and Legesgue Measure is great for measure theory. Pretty short too.

Intro to Functional Analysis by Kreysig is an amazing introduction to functional analysis. Don't know why you'd learn it from any other book. Afterwards you can go on to functional books by Brezis, Lax, or Helemskii.

It's hard to give an objective answer, because any sufficiently advanced book will be bound to not appeal to everyone.

You probably want Daddy Rudin for real analysis and Dummit & Foote for general abstract algebra.

Mac Lane for category theory, of course.

I think people would agree on Hartshorne as the algebraic geometry reference.

Spanier used to be the definitive algebraic topology reference. It's hard to actually use it as a reference because of the density and generality with which it's written.

Spivak for differential geometry.

Rotman is the group theory book for people who like group theory.

As a physics person, I must have a copy of Fulton & Harris.

Learn math first. Physics is essentially applied math with experiments. Start with Calculus then Linear Algebra then Real Analysis then Complex Analysis then Ordinary Differential Equations then Partial Differential Equations then Functional Analysis. Also, if you want to pursue high energy physics and/or cosmology, Differential Geometry is also essential. Make sure you do (almost) all the exercises in every chapter. Don't just skim and memorize.

This is a lot of math to learn, but if you are determined enough you can probably master Calculus to Real Analysis, and that will give you a big head start and a deeper understanding of university-level physics.

It's a book on the math of GR (Differential Geometry by Erwin Kreyszig). Pretty great book and it's like 12 bucks on amazon.

I'm not too sure about it personally, but several friends have taught from Nakahara, and have a lot of good things to say about it. It's graduate level.

Many people here have suggested Munkres, an excellent resource in introductory Topology (especially for self-teaching).

I would also like to suggest J. M. Lee's Introduction to Topological Manifolds. It's not nearly as pricey as Munkres, but maintains the same solid and clear exposition. I also enjoy how Lee emphasizes manifolds throughout the text as a type of motivation for introductory topology.

Do Carmo's Differential Geometry of Curves and Surfaces

Gallian's Contemporary Abstract Algebra

For differential geometry a great book is: [Analysis and Algebra on Differentiable Manifolds] (https://www.amazon.com/Analysis-Algebra-Differentiable-Manifolds-Mathematics/dp/9400793308/ref=sr_1_6?ie=UTF8&qid=1484591942&sr=8-6&keywords=problems+differential+geometry+and+manifolds+springer). It maybe doesn't have tremendous creativity required to solve the problems, but it'll give you lots of good practice.

For linear algebra and abstract, if you're not satisfied using Dummit and Foote (with easily accessible solutions online) or Lang Algebra (with harder to find solutions), Lang has some great exercises by the way, then I recommend the [Berkeley Problems in Mathematics book] (https://www.amazon.com/Berkeley-Problems-Mathematics-Problem-Books/dp/0387008926/ref=sr_1_sc_2?ie=UTF8&qid=1484591875&sr=8-2-spell&keywords=problems+in+abstracct+algebra).

Both of these books have complete solutions for problems and should be very useful for you.

Axiomatic "non-euclidean geometry" is something that was studied hundreds of years ago but isn't really an area of math that one studies or learns about anymore. Today, "non-euclidean geometry" (like the geometry of spheres or the hyperbolic plane) is part of differential geometry. There are many undergraduate-level books on manifolds and differential geometry, but I've never really looked at these. One you could try is Elementary Differential Geometry by Pressley.

the only thing that comes to mind is Frankel's geometry of physics

http://www.amazon.com/The-Geometry-Physics-An-Introduction/dp/1107602602

it's not really a math book as such (not the most rigorous proofs, and few at that) and it has way more.

i'm no expert though.

I've always preferred the 1st edition of Barrett O'Neill's book to doCarmo's. Struik's book is another good one at that level.

Actually there are some books which aren't reprints which are published by Dover. See: Curvature in Mathematics and Physics - Sternberg

Modern Differential Geometry for Physicists seems the most legit. It covers everything important (though it could have more cohomology) and is written well. It also navigates the line between physics and math, which is what you seem to be doing.

The links between topology, geometry and classical mechanics are fairly well documented in the other comments. Geometry and topology are fairly important in modern physics, at least what I've seen of it. General Relativity is the main example of where geometric ideas began to enter into physics. A good resource for this is Sean Carroll's GR notes and corresponding book. There are more advanced GR texts as well, like Wald's book.

There are also some books which deal directly with the links between physics and geometry, such as Frankels book, Szekeres, Agricola and Friedrich and Sternberg. Of these I own Szekeres book which is very good, and Frankels looks very good as well. The other two I am not sure about.

Geometric ideas do raise their head in more areas, as an example it is possible to formulate electromagnetism in terms of tensors or the hodge dual (see here). Additionally, and this is a bit beyond my knowledge, a friend of mine is working on topics in quantum field theory involving knot theory. I'm not exactly sure how this works but the links are certainly there.

Sorry if this all has more of a differential geometry flavour to it rather than a topological one, the diff geo side is what I know better. Hope that all helps. :)

TL;DR Start here

_

I think the classic introduction to the topic is Do Carmo's Riemannian Geometry. One that my colleagues use a lot (and is always taken out of the library, grrr) is Jurgen Jost's Riemannian Geomery and Geometric Analysis this second book is more recent and put out by springer.

There's another set of books that, from what I understand, approaches much more the algebraic aspects of this topic, but I have no experience with it. But I've read a lot of people in that area think it's the bee's knees. This is the 4 volume work by Spivak, A Comprehensive Introduction to Differential Geometry

Don't know your background, but I'd look at Pollack's https://www.amazon.com/Differential-Topology-AMS-Chelsea-Publishing/dp/0821851934 and, of course, Spivak's https://www.amazon.com/Comprehensive-Introduction-Differential-Geometry-Vol/dp/0914098705

MIT has lectures on OCW, as well.

For elementary differential geometry, just calculus and linear algebra should be sufficient. You can use a book like this for that purpose. For more advanced differential geometry, you will need to know topology and analysis and maybe some algebra as well.

I heard good things about it, but honestly as an applied mathematician I found its table of contents too lackluster. Its coverage appears to be in a weird spot between "for physicists" and "for mathematicians" and I don't know who its target audience is. I think the standard recommendation for classical mechanics from the physics side is Goldstein, which is a perfectly good book with plenty of math!

For an actual mathematicians' take on classical mechanics, you'll have to wait until you take more advanced math, namely real analysis and differential geometry. Common references are Spivak and Tu. When you have that background, I think Arnold has the best mathematical treatment of classical mechanics.

Sorry, Lorring Tu. The book is http://www.amazon.com/An-Introduction-Manifolds-Universitext-Loring/dp/1441973990

The content of the upper level math courses tend to vary depending on the professor and what they feel like teaching on any given year. I took fundamentals of Geometry with prof. Piper a few years ago. We covered most everything in this book (you can read through the index to get a good idea of what the course contained)

http://www.amazon.com/Elementary-Differential-Geometry-Undergraduate-Mathematics/dp/184882890X/ref=sr_1_2?ie=UTF8&qid=1320607881&sr=8-2

We also did a bit with the more computational side of things, representing geometric transformations with quaternions or matrices, did Maple projects, etc.

Thanks, knowing what to look for makes a huge difference. I'm leaning towards Elementary Differential Geometry (Christian Bär)

• The highly relevant M.C. Escher hyperbolic tessellation on the cover draws me straight in
  
• 'The only prerequistes are one year of undergraduate calculus and linear algebra'
  
• 'The word "elementary" should not tbe understood be understood as "particularly easy", but indicates that the development of formalism, which would be necessary for a deeper study of differential geometry, is avoided as much as possible'
  
• Relatively recent publication date
  
  I can see from your recent commenting history that we also share an interest in JavaScript, so I'll ping you here if I manage to get any hyperbolic high-jinx projected onto the html5 canvas. (I do know about the existing NonEuclid site by the way.)

I'll second Spivak's two calculus texts. Apostol and Courant are good alternatives if you have some reservations about Spivak.

I'd go with Do Carmo's Differential Geometry of Curves and Surfaces instead of Spivak's five volume sequence.

The mathematics necessary for theoretical physics varies based upon what type of theoretical physics you want to work in.

I assume you are a rising senior?

Long term the best book I've seen for an overview of what you want is Geometry, Topology and Physics by Nakahara link

Although this book is really suited for graduate students with extensive mathematical background. But think of this book as a goal!

For you you'll want to read up a good deal on Abstract Algebra. That paves the way for understanding Lie Algebras, Topology etc. And you'll also want to do some analysis at the same time.

I am not a mathematical physicist but representation theory has a lot of applications in physics so I know a good bit of literature if you have more specific questions about books for self-studying some of the courses that people have listed below. (Understand though that you will have to retake them once you get to college).

I've become greatly interested in geometric concepts in physics. I would like some opinions on these text for self study. If there are better options, please share.

For a differential geometry approach for Classical Mechanics:
Saletan?

For a General self study or reference book:
Frankel or Nakahara?

For applications in differential geometry:
Fecko or Burke?



Also, what are good texts for Geometric Electrodynamics that includes spin geometry?

Geometry, Topology, and Physics isn't a complete overview of math (as suggested by the title, it focuses on, well, geometry and topology), but if you're interested in learning about those specific subfields and their application to physics, I'd definitely recommend it.

Some of my physics major friends liked Nakahara. If you want to instead just do Riemannian geometry computations like a physicist you can try a general relativity book like Wald or Carroll.

I could be interested in reading that paper, however I might need a discussion on the Atiyah-Singer Index Theorem first - It's something I haven't really had to use, but something I'd like to know.

My own personal interests lie in manifolds with special holonomy, and I'd be particularly interested in discussing G2 manifolds, if anyone else is.

Another, more basic, option would be Frenkel's 'Geometry of Physics' book, which has a lot of nice physics formulated in the language of differential geometry. This may be a good option for people with physics backgrounds with little formal DG training, as it does all of DG from scratch while being sure to tie all the math to physics (E&M, Lagrangian/Hamiltonian Mechanics, Relativity, Yang-Mills Theory etc.) Check it out here: https://www.amazon.com/Geometry-Physics-Introduction-Theodore-Frankel/dp/1107602602

Νι διαβάζω αυτόν τον καιρό, αυτό http://www.amazon.com/Comprehensive-Introduction-Differential-Geometry-Edition/dp/0914098705, μόνο που δεν το αγόρασα το βρήκα σε djvu. Δε με λες βούρλο, αυτό εσύ το βγάζεις;

Πέρα απ την πλακά, προτιμώ κάποιος να μην έχει ανοίξει βιβλίο στην ζωή του παρά να έχει attitude "your opinions are not worth discussing". Αυτό είναι παταγωδώς γελοίο, και μιας και έχουμε στο θέμα αυτούς τους μαλέες είναι κόντρα στην φιλοσοφία των Α Ελλήνων.

Είναι anti intellectualism και πνευματικός μεσαίωνας να λες διάβασε βιβλία η γνώμη σου δεν αξίζει σχολίου. Ντροπή σου χοντροκέφαλε ps: μας ζάλισες τα αρχίδια με την επαρχία.

is it that one? https://www.amazon.com/Differential-Geometry-Connections-Curvature-Mathematics/dp/0199605874

how close is this to the graduate level? because sometimes i forgot definitions and would need to look them up. does provide basics sometimes?

not a big deal with the exercises, i learned to make my own.
thanks!
maybe we can add this to sidebar!

Agnostic here. I'm afraid it is not so easy to rule out the presence of brilliance and religion in a scientist or mathematician. Here is a list of living scientists who are christians (it is only a part of a much larger list going back several centuries).

Here are some examples with whose work I am more or less directly familiar.

John Polkinghorne was a student of Paul Dirac, and he has written a couple of books that are very lucid introductions to Quantum Mechanics.

Christopher Isham has written books on

• Quantum Theory,
  
• Modern Differential Geometry for Physicists and
  
• Lectures On Groups And Vector Spaces For Physicists.
  
  I have only actually read the first one on Quantum Theory.
  
  Edward Nelson is a mathematical physicist who has also contributed to quantum theory.
  
  Don Knuth is a mathematician and computer scientist who created the computer typesetting system, TEX, as well as a number of important theoretical developments, (e.g. the Robinson–Schensted–Knuth correspondence, and the Knuth–Bendix completion algorithm)
  
  Maddening, ain't it? But that's just how it is.

I suggest either Tu or (easy) Lee.

This is one is the best textbook for self-study I've find: Elementary Differential Geometry - A.N. Pressley.
Every self-study book should be like this one, well written and with answers to every exercises.

Here is the mobile version of your link

Both Lee's and Tu's books are on my reading list. They both seem excellent.

However, my vote is for Professor Tu's book, mainly because it manages to get to some of the big results more quickly, and he evidently does so without a loss of clarity. In the preface to the first edition, he writes "I discuss only the irreducible minimum of manifold theory that I think every mathematician should know. I hope that the modesty of the scope allows the central ideas to emerge more clearly." Consequently, his book is roughly half the length of Lee's.

I'd rather hit the most essential points first, and then if I want a more expansive view, I'd pick up Lee.

Disclaimer: I may not participate very frequently, as I have some other irons in the fire, so you might want to weigh my vote accordingly. If your sub sticks around for a while, I'd definitely like to join in when I can.

There are two classes you might have slept through with that name. The "classical" differential geometry of curves and surfaces (I think the standard is do Carmo), or a class on Riemannian geometry (I can recommend Lee).

for the applications you seem to need, probably this is good enough: http://www.amazon.com/Projective-Geometry-H-S-M-Coxeter/dp/0387406239

Before it was re-published by Dover, Differential Geometry of Curves and Surfaces was green too; now it's blue, and the only green book by do Carmo still in publication is Riemannian Geometry.

Differential geometry track. I'll try to link to where a preview is available. Books are listed in something like an order of perceived difficulty. Check Amazon for reviews.

Calculus

Thompson, Calculus Made Easy. Probably a good first text, well suited for self-study but doesn't cover as much as the next two and the problems are generally much simpler. Legally available for free online.

Stewart, Calculus. Really common in college courses, a great book overall. I should also note that there is a "Stewart lite" called Calculus: Early Transcendentals, but you're better off with regular Stewart. Huh, it looks like there's a new series called Calculus: Concepts and Contexts which may be a good substitute for regular Stewart. Dunno.

Spivak, Calculus. More difficult, probably better than Stewart in some sense.

Linear Algebra

Poole, Linear Algebra. I haven't read this one but it has great reviews so I might as well include it.

Strang, Introduction to Linear Algebra. I think the Amazon reviews summarize how I feel about this book. Good for self-study.

Differential Geometry

Pressley, Elementary Differential Geometry. Great text covering curves and surfaces. Used this one in my undergrad course.

Do Carmo, Differential Geometry of Curves and Surfaces. Probably better left for a second course, but this one is the standard (for good reason).

Lee, Riemannian Manifolds: An Introduction to Curvature. After you've got a grasp on two and three dimensions, take a look at this. A great text on differential geometry on manifolds of arbitrary dimension.

------

Start with calculus, studying all the single-variable stuff. After that, you can either switch to linera algebra before doing multivariable calculus or do multivariable calculus before doing linear algebra. I'd probably stick with calculus. Pay attention to what you learn about vectors along the way. When you're ready, jump into differential geometry.

Hopefully someone can give you a good track for the other geometric subjects.

When I was in your position I learned some representation theory of finite groups, from this book. It was at the perfect level for somebody who only has one semester's background in group theory. It'll gently introduce you to some things that you'll constantly need when you get further into algebra, like tensor products. Also, it's a topic which doesn't get covered at all in most undergrad abstract algebra courses, so it's a good thing to learn by yourself.

On the other hand, if you liked topology more than you liked group theory, you'd probably like Tu's Introduction to Manifolds.

Maybe this book?

Or a standard Riemannian geometry textbook like do Carmo might suit your needs.

Let me point you to my friend Nakahara.

Manifolds that are Euclidean locally are called Riemann manifolds, but in general, not all manifolds have that property.

My only experience with manifolds is from shape analysis, so I used a Riemann manifold to measure differences in 2-d closed curves by geodesics. I still don't 'get' them, but you might want to check out the book by Do Carmo on Differential Geometry

http://www.amazon.com/Differential-Geometry-Curves-Surfaces-Manfredo/dp/0132125897

From my limited understanding, a Riemann manifold is a kind of generic space to compare curves in other spaces that might not normally be comparable because of curvature. Like if you want to compare a line in Euclidean coordinates to a 'line' in spherical coordinates, you'd transform each curve using the xyz or R,theta, phi, plop them on a manifold and calculate the difference using an inner product on the tangent space.

I'm actually doing differential geometry, with some branches going out to algebraic geometry (modern version). It's just that this branching seems to haven taken over quite a lot. Not that I object, I think it's actually a very natural way to think of things. It just requires more background than I currently possess.

Thanks for the link though! I was fortunate enough to have experienced this transition at least in part in my first algebraic geometry course, were we went from some more classical examples and connected them with the modern theory.

I agree with your last statement. I have recently discovered this book. If you have already studied some differential geometry or topology, it creates really good intuition.

I like Lee's Introduction to Topological Manifolds.