(Part 2) Best geometry & topology books according to redditors
We found 814 Reddit comments discussing the best geometry & topology books. We ranked the 294 resulting products by number of redditors who mentioned them. Here are the products ranked 21-40. You can also go back to the previous section.
Well, that is a terrible problem if you are entering Real Analysis I without any exposure to proofs or writing proofs. What I tell everyone is to get Velleman's How To Prove It and couple it with another book such as Spivak since "How to Prove It" is a little raw (any pure proof book is).
EDIT: For the sake of clarification, I'm talking about Spivak's Calculus.
Pick up a copy of Algebraic Geometry: A Problem Solving Approach and work through the first chapter.
It shouldn't require much more than high school algebra, with just a smidgen of understanding of partial derivatives.
The first chapter defines algebraic sets of a polynomial, which is a subset of the plane defined by a polynomial: {(x, y) | P(x, y) = 0}.
The degree of the polynomial determines the degree of the curve. Degree 1 polynomials give straight lines, as you might expect. Degree 2 polynomials give the conic sections. You might remember conic sections from your high school algebra II class, but chances are it was mostly an exercise in memorizing equations.
It goes on to classify the conics up to affine change of coordinates. In R^2, there are ellipses (including the circle), hyperbolas, parabolas, and the degenerate conics, a double-line and a pair of crossing lines.
The chapters are fairly short and filled with super easy exercises that get you thinking about the material you're reading.
The chapter builds up some of the basic notions studied in algebraic geometry. While working over R^2 is great, it is harder to study because not every polynomial will have roots. So you upgrade to C^2 instead. In C^2, though, ellipses and hyperbolas become equivalent, thanks to allowing complex numbers in our affine change of coordinates.
Lastly, it builds up to projective geometry in CP^2. Even in C^2, there are cases where two intersecting lines may fail to meet if they are parallel to each other. By moving to CP^2, we force all lines to eventually greet each other (at some point of infinity if at no finite point).
This final upgrade is a bit technical, but it is a key ingredient to world-famous Bezout's Theorem, studied in chapter 3. But one immediately awesome result is that all nondegenerate conics become equivalent: ellipses, hyperbolas, and parabolas are just three ways of looking at the same geometrical object.
Algebraic geometry is an amazing field whose roots go back to at least Desargues in the 17th century. It has intimate ties with complex analysis (Chow's Theorem says that curves in the projective plane are actually compact Riemann surfaces) and number theory (where we work over the rationals, rather than the reals or complex numbers). In the 1930s, the field was put on a rigorous algebraic basis by Hilbert and Noether (this is essentially what Commutative Algebra is). And in the 1960s, Alexandre Grothendieck went totally ham and rephrased the entire subject in terms of categories and schemes.
Your post has too little context/content for anyone to give you particularly relevant or specific advice. You should list what you know already and what you’re trying to learn. I find it’s easiest to research a new subject when I have a concrete problem I’m trying to solve.
But anyway, I’m going to assume you studied up through single variable calculus and are reasonably motivated to put some effort in with your reading. Here are some books which you might enjoy, depending on your interests. All should be reasonably accessible (to, say, a sharp and motivated undergraduate), but they’ll all take some work:
(in no particular order)
Gödel, Escher, Bach: An Eternal Golden Braid (wikipedia)
To Mock a Mockingbird (wikipedia)
Structure in Nature is a Strategy for Design
Geometry and the Imagination
Visual Group Theory (website)
The Little Schemer (website)
Visual Complex Analysis (website)
Nonlinear Dynamics and Chaos (website)
Music, a Mathematical Offering (website)
QED
Mathematics and its History
The Nature and Growth of Modern Mathematics
Proofs from THE BOOK (wikipedia)
Concrete Mathematics (website, wikipedia)
The Symmetries of Things
Quantum Computing Since Democritus (website)
Solid Shape
On Numbers and Games (wikipedia)
Street-Fighting Mathematics (website)
But also, you’ll probably get more useful response somewhere else, e.g. /r/learnmath. (On /r/math you’re likely to attract downvotes with a question like this.)
You might enjoy:
https://www.reddit.com/r/math/comments/2mkmk0/a_compilation_of_useful_free_online_math_resources/
https://www.reddit.com/r/mathbooks/top/?sort=top&t=all
This reminds me of Counterexamples in Topology which is literally a book of pathological shapes and sets that frequently serve as counterexamples.
I think you need to list which books you didn't understand. I'm having a hard time understanding what you have trouble with. Studying general relativity, you should be familiar with metrics and curvature, but somethings you say indicate otherwise. It is also unclear to me what you want to learn. Do you want to learn differential geometry related to QFT, e.g. Yang-Mills connections?
Since you are familiar with GR, maybe you would appreciate O'Neill's book Semi-riemannian Geometry. Jurgen Jost also has a book Geometry and Physics that may help be a bridge between the language. The book is meant to be a bridge for mathematicians, but maybe it will also be helpful for going the opposite way.
Edit: Also, without more explanation to what you want, I think it would be useless to go back to stuff like undergraduate analysis. For example, you may be put off by geometry books giving a topological definition of manifold. This is a technical detail most geometers don't actually work with. The important thing to concentrate on is the smoothness and invertibility of the transition maps. For things involving groups, you could probably go very far just thinking of linear groups, e.g. special matrix groups.
Adding on to this, we need like a workbook for working with sheaves. They are difficult for me to get a feel for
On the other hand, I know there are very concrete problems in https://www.amazon.com/Algebraic-Geometry-Problem-Approach-Mathematical/dp/0821893963
Particularly the last chapter when sheaves and cech cohomology are introduced.
However when I think of sheaves, I cannot see the trees in the forest, I just see the forest
Basic Algebra by Nathan Jacobson
Advanced Calculus by Shlomo Zvi Sternberg, Lynn Harold Loomis
Perhaps we can get the special flair users in /r/math to setup some of this (the ones with the red background in their flair)?
I know nothing about any of these topics but we could use course notes from a school's Open Courseware.
Here are the relevant ones I've found. If a cell says "none" that just means I've left a placeholder for if people find something I can put in that spot. The ones with all nones means I either wasn't sure what to look for, or if what I found was the right thing (Lie Theory = Lie Groups? for example)
Subject | Source1 | Source2 | Source3| Source4
---|---|----|----|----
Algebraic Topology | MIT Seems to have all relevant readings as PDFs | Introductory Algebraic Topology I don not know the source for this one| Algebraic Topology by Hatcher is free | A Basic Course in Algebraic Topology by Massey - Not free
Algebraic Geometry | MIT Fall 2003 Has lecture notes| MIT Spring 2009 Also has lecture notes | Vakil's course notes| Eyal Goren Syllabus and course notes
Functional Analysis | MIT Lecture notes and assignments with solutions | Nottingham 2010 | Nottingham 2008 These ones not only have lecture notes, but audio of the lecture. | none
Lie Theory | MIT - Intro to Lie Groups | MIT - Topics in Lie Theory: Tensor Categories | none | none
General Relativity | Sean Carroll's Lecture Notes | Stanford video lectures on general relativity, Leonard Susskind | Lecture notes from Nobel Laureate Gerard Hooft on GR | Semi-Riemannian geometry with Applications to Relativity - Not free
Dynamical Systems | Very applied (Strogatz style) course notes for dynamical systems | More theoretical (Perko style) course notes for dynamical systems by the same author | none | none
Numerical Analysis | MIT Spring 2012 | MIT Spring 2004 | none | none
This is obviously not an exhaustive list. I thought Stanford and their own open courseware thing but it seems to just be a list of courses they have on Coursera.
The first thing that I thought of was How to Solve it by Polya.
But I would not suggest Analysis as your first proof course since junior high school. How did you get into this situation? It can't be the normal for your school. Linear Algebra is oftentimes the first undergraduate proof course.
You would probably like these two books:
Neither of those are "popular math" books; the authors are famous mathematicians, and they explore various fields of mathematics without requiring too much advanced knowledge.
Anti-disclaimer: I do have personal experience with all the below books.
I really enjoyed Lee for Riemannian geometry, which is highly related to the Lorentzian geometry of GR. I've also heard good things about Do Carmo.
It might be advantageous to look at differential topology before differential geometry (though for your goal, it is probably not necessary). I really really liked Guillemin and Pollack. Another book by Lee is also very good.
If you really want to dig into the fundamentals, it might be worthwhile to look at a topology textbook too. Munkres is the standard. I also enjoyed Gamelin and Greene, a Dover book (cheap!). I though that the introduction to the topology of R^n in the beginning of Bartle was good to have gone through first.
I'm concerned that I don't see linear algebra in your course list. There's a saying "Linear algebra is what separates Mathematicians from everyone else" or something like that. Differential geometry is, in large part, about tensor fields on manifolds, and these are studied by looking at them as elements of a vector space, so I'd say that linear algebra is something you should get comfortable with before proceeding. (It's also great to study it before taking quantum.) I can't really recommend a great book from personal experience here; I learned from poor ones :( .
Also, there are physics GR books that contain semi-rigorous introductions to differential geometry, even if these sections are skipped over in the actual class. Carroll is such a book. If you read the introductory chapter and appendices, you'll know a lot. On the differential topology side of things, there's Schutz, which is a great book for breadth but is pretty material dense. Schwarz and Schwarz is a really good higher level intro to special relativity that introduces the mathematical machinery of GR, but sticks to flat spaces.
Finally, once you have reached the mountain top, there's Hawking and Ellis, the ultimate pinnacle of gravity textbooks. This one doesn't really fall under the anti-disclaimer from above; it sits on my shelf to impress people.
You need some grounding in foundational topics like Propositional Logic, Proofs, Sets and Functions for higher math. If you've seen some of that in your Discrete Math class, you can jump straight into Abstract Algebra, Rigorous Linear Algebra (if you know some LA) and even Real Analysis. If thats not the case, the most expository and clearly written book on the above topics I have ever seen is Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.
Some user friendly books on Real Analysis:
Some user friendly books on Linear/Abstract Algebra:
Topology(even high school students can manage the first two titles):
Some transitional books:
Plus many more- just scour your local library and the internet.
Good Luck, Dude/Dudette.
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.
157 is very different from most other first year university courses. The lectures are helpful because they illustrate the ideas, but they don't get you familiar with any particular type of problem or prepare you for the tests/exams. Also, for most first year math/science courses, textbooks are really just there to provide you practice questions. It's different for MAT157. You need to actually read it, from the first page to the last, understanding every single line of it. It's a tough book, but also an amazing one. I think you will enjoy it if you do like math.
https://www.amazon.ca/Calculus-Michael-Spivak/dp/0521867444
You have 4 months before September. Even 10 mins/day of work will be enough for you to finish this book prior to the course starts. Good luck.
As far as I know, there's no "standard" book for the rite of passage, but obviously Munkres is an alright intro to Point-Set [and that's all you should do from the book] and Hatcher is a wonderful introduction to Algebraic Topology.
Hatcher has this tendency to ramble a bit and to not be exceptionally clear, though. Moreover, there is a bit lacking in Hatcher's text [no doubt done on purpose!]. Because of this, I usually recommend Bredon's Topology and Geometry.
I'll note also that AT is currently a pretty hot-topic because of its application to Data Analysis; you may want to read Topology and Data by Carlsson and the "What is...Persistent Homology?" paper, just so you can see some of the things people are doing with AT.
It's not free, but in my opinion Semi-Riemannian geometry with Applications to Relativity by Barret O'Neill is the best introductory text for GR.
I think these days it's really important to make it to the generalized stokes theorem, not just for an honors crowd but in general. This means covering differential forms. Hubbard and Hubbard has been mentioned.
Not a book but in my mind a very nice update on H&H is Ghrist's video lecture on multivariable calculus which covered traditional integral theorems (Green, Gauss and Stokes) while showing their full relationship to generalized stokes in a very natural way. I really think this is a kind of template how modern courses on multivariable/vector calculus should be taught these days. it's not just the content but also the order of presentation that is very neat and maximizes clarity.
There are a bunch of books that had treaded this path over the years. Loomis & Sternberg, and Harold Edwards are books worth considering, though H&H is in some sense most detailed while also having a nice pace.
I actually believe that there is a dearth of really good updated and polished books in the area, and that there are so few really good options calls for some effort to develop lecture notes into books on the topic.
You are missing Abstract Algebra that usually comes before or after Real Analysis. As for that 4chan post, Rudin's book will hand anyone their ass if they havent seen proofs and dont have a proper foundation (Logic/Proofs/Sets/Functions). Transition to Higher Math courses usually cover such matters. Covering Rudin in 4 months is a stretch. It has to be the toughest intro to Real Analysis. There are tons of easier going alternatives:
Real Mathematical Analysis by Charles Pugh
Understanding Analysis by Stephen Abbot
A Primer of Real Functions by Ralph Boas
Yet Another Introduction to Analysis
Elementary Analysis: The Theory of Calculus
Real Analysis: A Constructive Approach
Introduction to Topology and Modern Analysis by George F. Simmons
...and tons more.
There is an excellent series of Counterexamples in ... books which might be relevant to this thread:
counterexamples in...
Or just run a search on amazon for "Counterexamples in".
So many of the counterexamples in here or here (no link to the 2nd).
I think the worst for me is that some convergent series can be rearranged to be divergent. Fucking conditional convergence...
There's a great book by Hartshorne called Geometry: Euclid and Beyond, in which the author develops Hilbert's axioms for plane geometry in a very accessible way, given that you know a little bit of abstract algebra. You don't need much algebra.
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.
For historical treatments, Bressoud has written textbooks on Real analysis [1] and measure theory/lesbegue integration [2]. There is also a historical treatment of freshman calculus by Toeplitz [3], first published in the 1960s. And Leo Corry's history of Modern Algebra [4] is not really an algebra textbook, but is so interesting I can't help but advertise it!
[1] https://www.amazon.com/Approach-Analysis-Mathematical-Association-Textbooks/dp/0883857472
[2] https://www.amazon.com/Lebesgues-Integration-Mathematical-Association-Textbooks/dp/0521711835
[3] https://www.amazon.com/Calculus-Genetic-Approach-Otto-Toeplitz/dp/0226806685
[4] https://www.amazon.com/Modern-Algebra-Rise-Mathematical-Structures/dp/3764370025
I own this book
https://www.amazon.com/Introduction-Topology-Modern-Analysis-Simmons/dp/1575242389/ref=tmm_hrd_swatch_0?_encoding=UTF8&qid=&sr=
Which seems to be free here.
http://susanka.org/HSforQM/[Simmons]_Introduction_to_Topology_and_Modern_Analysis.pdf
It depends. Consider Advanced Calculus by Sternberg/Loomis. You need to be comfortable with something like baby Rudin to tackle this book.
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.
This one is geared towards students who want to do competition mathematics and have a future career in Math/Physical Sciences.
AOPS Intro to Geometry
This one is used by home schooled students a lot. It is a classic. Geometry
I haven't read this one but I hear it is really good for high end students. Geometry for Enjoyment and Challenge
If you feel like you have the time, I could recommend http://www.amazon.com/Algebraic-Geometry-Problem-Approach-Mathematical/dp/0821893963 which is a very gentle introduction to the subject using classical curves. Only in the last chapter does it introduce sheaves and cohomology. I suspect something like this might be helpful to place everything in a concrete context, and also build up motivation for all the modern machinery that you'll find in Hartshorne.
Try this book for help with understanding Algebra. My uncle had left a copy at my grandparents house, and I picked it up when I was there when I was in the third grade (we were working on multiplication and division). I made a perfect score in the state tests for Algebra 1, Algebra 2, Geometry, and Trigonometry.
I read this book in high school, and it really helped me figure out how to think about breaking down more complex problems.
This book made math very clear for me as well.
I think these books may help you because you could do the math he read to you. These books helps give you an understanding of what is actually happening. Foe example, most people do not understand that multiplication is nothing more than extended addition, until you explain it to them. If you can think about the problems and understand what the problem is saying, it will be easier to figure out. I did a lot of math in my head that would have taken several pages to write it out the way I did it, but if you wrote it the way they expect would only take a few lines.
I am very happy for you for finally finding someone who knew what was going on with you. I had a similar problem in elementary school, but my parents did not trust the school and had me tested on their own. They decided that I had a "social communication disorder, kind of like a really weak autism" (This is what my parents ended up telling me anyway). The school thought I was "developmentally challenged" ("borderline retarded" was the phrase that was bandied about) but when my parents had my IQ tested, it was a 141, which is not quite what was expected, they decided that the problems were elsewhere.
One thing that is very important in math is that if you do not understand, you can go back and work on fundamentals and build up your foundation, and the more advanced stuff will be easier.
Good luck, and I believe you really are an adept writer. What you wrote grabbed my interest and was compelling.
Try What is Mathematics?, by Courant & Robbins. It's a good overview of mathematics beyond the elementary level you've completed. Another good book like that is Geometry and the Imagination, by Hilbert & Cohn-Vesson.
I'm a math tutor and I use these books with almost all my students. They go into pretty good detail about the why's and have quite challenging problems. They include chapter tests, chapter reviews, tons of word problems, challenging multiple SAT-type questions every other chapter, and cumulative reviews. They also thoroughly prepare you for every calculus topic, as well as probability and statistics. They cover Algebra through Pre-calc:
Algebra
Geometry
Algebra 2
Precalculus
To supplement those, you could also use this British math series. It should fill in any possible gaps or clarify certain topics:
9th grade
10th/11th grade
12th grade
My (Real) Analysis textbook started out with this example.
If you love calculus it might be worth brushing up on your real and complex analysis. It's the pure maths side of calculus (or at least can usefully be regarded as such. That's not a 100% accurate description). I'm not sure what to offer for that that isn't very clearly in the class of a textbook and expensive though. :-) Tom Korner's "A Companion to Analysis" is a good intro though.
One book which I really enjoyed is (don't laugh) Schaum's Outlines on Advanced Calculus. It's got a lot of really good exercises and teaches you some unconventional approaches to solving integrals.
I'll also second the recommendations for "proofs from the book" and "proofs and refutations"
One little maths book which I loved (and you have to be slightly odd kind of person to say that about it) is Counter examples in topology. It's an entire book devoted to perverse examples. I really enjoy the thought process that goes into such things. But if you don't know any topology that's not a very good recommendation. :-)
My topology textbooks were Munkres, Hatcher, and Bredon.
The following book is good at the undergraduate level:
https://www.amazon.com/Geometry-David-Brannan/dp/1107647835/ref=sr_1_1?ie=UTF8&qid=1519204111&sr=8-1&keywords=geometry+brannan
It covers Euclidean, Projective, Inversive, Elliptic, Hyperbolic, and Spherical geometries from the point of view of group theory (Klein's Erlangen program: a geometry is a space along with a group of transformations of the space.)
The book is undergraduate level and contains complete solutions for the problems at the end. It is used in the Open University in the UK.
A typical, medium-level-rigor undergraduate geometry textbook is Hartshorne. I found a .pdf of reasonable quality for you here.
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
This book is really really good about this topic.
I took calculus my senior year in high school, too, then took a calculus course in college in the 80s. (I graduated from high school in '65.) I took whatever the algebra class was the was the precursor for calculus, but I'm not sure you have time for that.
Ross Finney gets extremely high marks for his work, so his high school text may be of value to you to brush up on your skills.
Good luck and have fun. If I can make it with a 20-something year gap, I have no doubts about you.
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.
http://www.amazon.com/Geometry-Grades-9-11-Mcdougal-Littell/dp/0395977274/ref=sr_1_1?ie=UTF8&qid=1324530687&sr=8-1
I'd be interested in moderating. I could do the winter school page (though I haven't watched much of them, but they seem to be strong lectures), or if anyone is interested in reading https://www.amazon.com/Geometry-Physics-Introduction-Theodore-Frankel/dp/1107602602 I'd run a page for that.
I have quals/prelims coming up in a few weeks, so I won't be able to be too active until after I'm done, but I think this sub is a great idea.
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?
A Discipline of Programming. Classic but expensive. Read the first few chapters at your library before buying.
Polya's How to Solve It!
The Mythical Man-Month
Finally, The Pragmatic Programmer.
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.
If you need to brush up on some of the more basic topics, there's a series of books by IM Gelfand:
Algebra
Trigonometry
Functions and Graphs
The Method of Coordinates
https://www.amazon.com/Geometry-Euclid-Beyond-Undergraduate-Mathematics/dp/0387986502
Yes that's the book I ended up doing catch up with when I took graduate differential topology with this book.
For me, a "good read" in mathematics should be 1) clear, 2) interestingly written, and 3) unique. I dislike recommending books that have, essentially, the same topics in pretty much the same order as 4-5 other books.
I guess I also just disagree with a lot of people about the
"best" way to learn topology. In my opinion, knowing all the point-set stuff isn't really that important when you're just starting out. Having said that, if you want to read one good book on topology, I'd recommend taking a look at Kinsey's excellent text Topology of Surfaces.
If you're interested in a sequence of books...keep reading.
If you are confident with calculus (I'm assuming through multivariable or vector calculus) and linear algebra, then I'd suggest picking up a copy of Edwards' Advanced Calculus: A Differential Forms Approach. Read that at about the same time as Spivak's Calculus on Manifolds. Next up is Milnor Topology from a Differentiable Viewpoint, Kinsey's book, and then Fulton's Algebraic Topology. At this point, you might have to supplement with some point-set topology nonsense, but there are decent Dover books that you can reference for that. You also might be needing some more algebra, maybe pick up a copy of Axler's already-mentioned-and-excellent Linear Algebra Done Right and, maybe, one of those big, dumb algebra books like Dummit and Foote.
Finally, the books I really want to recommend. Spivak's A Comprehensive Introduction to Differential Geometry, Guillemin and Pollack Differential Topology (which is a fucking steal at 30 bucks...the last printing cost at least $80) and Bott & Tu Differential Forms in Algebraic Topology. I like to think of Bott & Tu as "calculus for grown-ups". You will have to supplement these books with others of the cookie-cutter variety in order to really understand them. Oh, and it's going to take years to read and fully understand them, as well :) My advisor once claimed that she learned something new every time she re-read Bott & Tu...and I'm starting to agree with her. It's a deep book. But when you're done reading these three books, you'll have a real education in topology.
Counterexamples in topology
Counterexamples in analysis
Read this: https://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773 and you're more than set for algebraic manipulation.
And if you're looking to get super fancy, then some of that: https://www.amazon.com/Method-Coordinates-Dover-Books-Mathematics/dp/0486425657/
And some of this for graphing practice: https://www.amazon.com/Functions-Graphs-Dover-Books-Mathematics/dp/0486425649/
And if you're looking to be a sage, these: https://www.amazon.com/Kiselevs-Geometry-Book-I-Planimetry/dp/0977985202/ + https://www.amazon.com/Kiselevs-Geometry-Book-II-Stereometry/dp/0977985210/
If you're uncomfortable with mental manipulation of geometric objects, then, before anything else, have a crack at this: https://www.amazon.com/Introduction-Graph-Theory-Dover-Mathematics/dp/0486678709/
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.
Wald does introduce the necessary DG, but if you're interested in a more mathematical perspective then O'Neill may suit you.
Bredon's book is an algebraic topology book also it has something about manifolds and something about smooth manifolds. Honestly the book is rind of ridiculous. Just look at the table of contents.
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. :)
These are, in my opinion, some of the best books for learning high school level math:
These are all 1900's Russian math text books (probably the type that /u/oneorangehat was thinking of) edited by I.M. Galfand, who was something like the head of the Russian School for Correspondence. I basically lived off them during my first years of high school. They are pretty much exactly what you said you wanted; they have no pictures (except for graphs and diagrams), no useless information, and lots of great problems and explanations :) There is also I.M Gelfand Trigonometry {[.pdf] (http://users.auth.gr/~siskakis/GelfandSaul-Trigonometry.pdf) | Amazon} (which may be what you mean when you say precal, I'm not sure), but I do not own this myself and thus cannot say if it is as good as the others :)
I should mention that these books start off with problems and ideas that are pretty easy, but quickly become increasingly complicated as you progress. There are also a lot of problems that require very little actual math knowledge, but a lot of ingenuity.
Sorry for bad Englando, It is my native language but I haven't had time to learn it yet.
There is no programmer thinking. There's maths problem thinking. That's my life advice based on my very intense maths education based on problem solving (I had 9-10 hours of math a week of in grade 9-12). For example, sir_eeps top voted comment is a very condensed version of this book https://www.amazon.com/How-Solve-Aspect-Mathematical-Method/dp/0691023565 -- once you are through with this very short book the eerie similarity will hit you too.
This has turned out to be a much more interesting question than I had thought it would be. It seems to be unexpectedly hard to find a good, short book on Euclidean geometry. Most of the really good books are advanced treatments that have a lot more to say than what you probably want. Anyway, there is a good discussion of this question on mathoverflow. It appears that Kiselev is a pretty good choice. Hartshorne might be good as a guide to learning straight from Euclid (and lots more besides). I don't know how far you really want to go with this project. It might be enough to just get a taste of how the whole synthetic geometry program is organized.
By the way, you know about libary.nu, right?
Depending on the curriculum indicators you need to hit, it might be beneficial to talk with your cohorts in your department. This might not be helpful since the new common core is rolling out but here are my books I'd recommend:
Geometry - Ray C Jurgensen http://www.amazon.com/Geometry-McDougal-Littell-Jurgensen/dp/0395977274
It maybe 14 years old but it does an amazing job of starting easy and cranking up the difficulty. There is no need to have any prior geometry knowledge because it starts you with the very basics to complexities of Geometry. There are certain things I would change in the book, but you can't go wrong with having it as a resource.
And everything else Algebra/Calculus Related, just look for Ron Larson and it's gold. GOLD I SAY!
Geometry: Euclid and Beyond by Hartshorne is a really good book. It starts out by walking you through Euclid's elements (you need a separate copy of that). Then it goes on to Hilberts's axiomatization of geometry, while discussing the ways in which Euclid's was lacking. Then it moves on to other things, like the relationship between algebra and geometry, and non-Euclidean geometries. It's not an easy book, but it does not really have any prerequisites, and it's a lot of fun. It makes Euclid a lot more clear.
If you want a gentler introduction to calculus, with many examples, lots of intuition, diagrams, and nicer explanations, take any edition of James Stewart's Calculus - Early Transcendentals.
If you feel up to a serious challenge and want to study it as a mathematician would, get Michael Spivak's Calculus.
First, please make sure everyone understands they are capable of teaching the entire subject without a textbook. "What am I to teach?" is answered by the Common Core standards. I think it's best to free teachers from the tyranny of textbooks and the entire educational system from the tyranny of textbook publishers. If teachers never address this, it'll likely never change.
Here are a few I think are capable to being used but are not part of a larger series to adopt beyond one course:
Most any book by Serge Lang, books written by mathematicians and without a host of co-writers and editors are more interesting, cover the same topics, more in depth, less bells, whistles, fluff, and unneeded pictures and other distracting things, and most of all, tell a coherent story and argument:
Geometry and solutions
Basic Mathematics is a precalculus book, but might work with some supplementary work for other classes.
A First Course in Calculus
For advanced students, and possibly just a good teacher with all students, the Art of Problem Solving series are very good books:
Middle & high school:
and elementary linked from their main page. I have seen the latter myself.
Some more very good books that should be used more, by Gelfand:
The Method of Coordinates
Functions and Graphs
Algebra
Trigonometry
Lines and Curves: A Practical Geometry Handbook
Maybe this book?
Or a standard Riemannian geometry textbook like do Carmo might suit your needs.
https://www.amazon.com/Geometry-Euclid-Beyond-Undergraduate-Mathematics/dp/0387986502
Is a solid introduction to synthetic geometry, covers some basics of abstract algebra as well
Thank you for the reply. I think you are right in that assumption, however, I think I still might be slightly hindered by not knowing any physics at all. Do you think I would only be wasting my time by reading through conceptual physics, or would it still be a useful thing to do which would only strengthen and solidly my knowledge for studying Y&F?
As far as mathematics is concerned, I have that covered I believe, I am reading through an algebra textbook currently, then hoping to go through a number theory and pre-calculus textbook. Eventually calculus and by that time I would think I should then start studying Y&F. I believe the calculus book would cover anything I need in the Y&F book? or is there other mathematics which is not specifically calculus I would need to learn from the Mary Boas book?
I would either be using this calculus textbook which is from the series of mathematics textbooks I have been reading: https://artofproblemsolving.com/store/item/calculus
or maybe, Spivak’s calculus if I am confident enough to tackle it by then: https://www.amazon.co.uk/Calculus-Michael-Spivak/dp/0521867444
I haven't read the following books, but they're supposed to be ultra simple (in this case, easy).
Algebraic Geometry for Scientists and Engineers by Abhyankar
Algebraic Geometry: A Problem Solving Approach by Garrity et al
I am not sure there are AG books more elementary than those listed.
Instructor solution manual of Calculus and Analytic Geometry by Thomas and Finney
This is the book https://www.amazon.com/Calculus-Analytic-Geometry-George-Thomas/dp/0201531747
I want a file which contains solutions to problems of all 14 chapters.
$5
This book is what you're looked for. It's a rigorous calculus book. You'll learn why things are true.
There probably isn't that much more new you can read about problem solving, so I guess the next step would be just to solve problems and learn basic tools and techniques. For this I'd suggest some book that has math olympiad type (or any other math competition) problems and solutions for high school students. That kind of problems don't require much knowledge of advanced math, but plenty of creativity and analytic thinking. I'm afraid I don't have any title to recommend, it's been years since I last read any and don't remember the names. You'll probably find some just by googling.
Also, the Amazon page for How to Solve It has a long list of books in "customers who bought this also bought", I can't say don't really say I know any of those but maybe you can find something interesting. Link
Edit: This title caught my eye when browsing those books in the list.
Once you've gotten started with Munkres, I highly recommend Bredon's Topology and Geometry for algebraic topology. It's very well written, covers a ton of material, and has some nice pictures.
I enjoyed this textbook. A bit pricey though https://www.amazon.com/Geometry-David-Brannan/dp/1107647835
A bit of a linear algebra background will be useful (though I think this book is actually what motivated me to learn linear algebra in the first place)
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.
http://www.amazon.ca/Geometry-David-Brannan/dp/1107647835/ref=sr_1_2?ie=UTF8&qid=1375486426&sr=8-2&keywords=geometry
If you like the online course lectures, you should definately look at those. I know tons of great schools such as Yale, UCLA, MIT, Stanford etc. etc. offer full lecture series on youtube. Usually the syllabi are online for you to look at so you can get a feel for it.
I am more of a book learner myself so I will try to make some recommends, but when looking for books try googling, reading stackexchange posts and Amazon reviews.
I'm going to disagree with /u/Orion952 on Fraleigh's book, its an alright book but I have seen much better. For Abstract Algebra, I would recommend Nicholson's book. Its a very gentle introduction to the subject. There are lots of computation problems as well as proofs you can work through so you can get a nice feel for the subject. I would also hunt down the pdf for Dummit and Foote's book as well, I thought it was pretty gentle for the most part as well as comprehensive.
For analysis and topology, I have encountered some decent books.
Strichartz for analysis is very wordy and conversational, so I didn't care for it myself hence didn't read very much of it (I much prefer the style of Walter Rudin) but it might be good for starting out.
Bhatt has written a very nice book for analysis and covers a lot of material on metric space topology. I actually know the author pretty well so if you are interested in the book I may be able to hook you up.
Simmons has written a book that has a pretty conversational style, but I wasn't a big fan of his style. Bhatt's book will have a more "traditional" approach, but thats not to say it isn't readable. The first half of the book will cover the same stuff Bhatt's book does and the second half will be more advanced stuff including some concepts from Functional Analysis (which is a pretty interesting topic).
For Topology, if you have read some of the analysis books above, I would say Munkres' book is nice and it has tons of examples. But try googling beginner topology books if you want to get into the subject sooner, I know I have seen a few stackexchange threads on this.
These are really the topics one needs to know to really dive into mathematics beyond rote computation. I'm sure there are more books out there but these come off my head at this moment.
I took a course in geometry recently. We used http://www.amazon.com/Geometry-Euclid-Beyond-Undergraduate-Mathematics/dp/1441931457 The first chapter is the only one to cover Euclid, and it only reviews books 1 - 4 but if you read it and work through the problems it'll give a good foundation to cover the rest of Euclid as you see fit. The real reason I mention this book is that almost all the problems are constructions with straightedge/compass. They give a par (the minimum number of steps a construction can be completed in with a reasonable amount of time spent thinking). You could give out the problems you are interested in without the par so that your students could compare construction methods. When I began the course I had no geometric intuition. I spent a lot of time trying to find the best possible constructions and felt my (euclidean) geometric intuition bloom.
Finally, after chapter 1 the book goes into Hilbert's Axioms to show how we develop modern geometry and develops a number of interesting geometries. I can't speak of most of this as we only covered Euclid, Hilbert's Axioms, and a quick bit of non-Euclidean geometry. But I found it very interesting and think the book can be used to study geometry in a number of ways. Either way good luck.
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.
I would assume that if you're a music major and "been good at math", you might be referring to the math of high school. In any case, it would help if you spend some time doing/reviewing calculus in parallel while you go through some introductory physics book. So here's what you could do:
Other than that, feel free to google your question. You'll find good info on websites like physicsforums.com, physics.stackexchange.com, as well as past threads on this subreddit where others have asked similar questions.
Once you're past the intro (i.e., solid grasp of calculus, and solid grasp of mechanics, which could take up to a year), you are ready to venture further into math and physics territory. In that regard, I recommend you look at posts by Gerard 't Hooft and John Baez.