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.

Machine learning is largely based on the following chain of mathematical topics

Calculus (through Vector, could perhaps leave out a subsequent integration techniques course)

Linear Algebra (You are going to be using this all, a lot)

Abstract Algebra (This isn't always directly applicable but it is good to know for computer science and the terms of groups, rings, algebras etc will show up quite a bit)

General Topology (Any time we are going to deal with construction of a probability space on some non trivial manifold, we will need this. While most situations are based on just Borel sets in R^n or C^n things like computer vision, genomics, etc are going to care about Random Elements rather than Random Variables and those are constructed in topological spaces rather than metric ones. This is also helpful for understanding definitions in well known algorithms like Manifold Training)

Real Analysis (This is where you learn proper constructive formulations and a bit of measure theory as well as bounding theorems etc)

Complex Analysis (This is where you will get a proper treatment of Hilbert Spaces, Holomorphic functions etc, honestly unless you care about QM / QFT, P-chem stuff in general like molecular dynamics, you are likely not going to need a full course in this for most ML work, but I typically just tell people to read the full Rudin: Real and Complex Analysis. You'll get the full treatment fairly briefly that way)

Probability Theory (Now that you have your Measure theory out of the way from Real Analysis, you can take up a proper course on Measure Theoretic Probability Theory. Random Variables should be defined here as measurable functions etc, if they aren't then your book isn't rigorous enough imho.)

Ah, Statistics. Statistics sits atop all of that foundational mathematics, it is divided into two main philosophical camps. The Frequentists, and the Bayesians. Any self respecting statistician learns both.

After that, there are lots, and lots, and lots, of subfields and disciplines when it comes to statistical learning.

A sample of what is on my reference shelf includes:

Real and Complex Analysis by Rudin

Functional Analysis by Rudin

A Book of Abstract Algebra by Pinter

General Topology by Willard

Machine Learning: A Probabilistic Perspective by Murphy

Bayesian Data Analysis Gelman

Probabilistic Graphical Models by Koller

Convex Optimization by Boyd

Combinatorial Optimization by Papadimitriou

An Introduction to Statistical Learning by James, Hastie, et al.

The Elements of Statistical Learning by Hastie, et al.

Statistical Decision Theory by Liese, et al.

Statistical Decision Theory and Bayesian Analysis by Berger

I will avoid listing off the entirety of my shelf, much of it is applications and algorithms for fast computation rather than theory anyway. Most of those books, though, are fairly well known and should provide a good background and reference for a good deal of the mathematics you should come across. Having a solid understanding of the measure theoretic underpinnings of probability and statistics will do you a great deal--as will a solid facility with linear algebra and matrix / tensor calculus. Oh, right, a book on that isn't a bad idea either... This one is short and extends from your vector classes

Tensor Calculus by Synge

Anyway, hope that helps.

Yet another lonely data scientist,

Tim.

hey nerdinthearena,

i too find this area to be fascinating and wish i knew more on the upper end myself. i'm just going to list off a few resources. in my opinion, graduate school will concentrate a lot on progressing your technical knowledge, but will likely not give you a lot of time to hone your intuition (at least in the first few years). so, the more time you spend in undergraduate school doing so, the better.

helpful for intuition and basic understanding

• geometrical vectors by gabriel weinreich - why there's more to vectors and how to draw them. this basically shows visual examples of differential forms and the like.
  
• advanced calculus: a differential forms approach by harold edwards - a fantastic approach to differential forms. it's a treasure, so just read it. :) it is actually well built for skimming and skipping around.
  
• vector calculus, linear algebra, and differential forms: a unified approach - this is a big book, but you could just jump ahead to the later sections covering differential forms for an interesting approach and technical skill building.
  
• div, grad, curl are dead by william burke - pure intuition and similar to geometrical vectors. don't look for anything else here, but definitely read it.
  
• applied differential geometry by william burke - a more complete work than the above and showcases a lot of the connections between geometry and physics.
  
  

  more advanced but still intuitive
  

• an introduction to manifolds - this is at the same level and covers essentially the same material as lee, but in my opinion, it is much better. it is much more concise and to the point with more focused exercises.
  
• differential forms in algebraic topology by raoul bott and loring tu - the sequel to the above, but i haven't read beyond the first chapter. it is well known as a must read classic, but having some topology background will help with this book.
  
• the laplacian on a riemannian manifold: an introduction to analysis on manifolds - an interesting introduction to geometric analysis.
  
• differentiable manifolds: forms, currents, harmonic forms by georges de rham - a classic by one of the greats and also one of the only sources i've found that covers de rham currents (besides research papers that just use them).
  
• riemannian manifolds: an introduction to curvature by john lee - the sequel to lee's smooth manifolds book.
  
  hopefully this helps. if i were to revisit geometric analysis, i would basically use the above books to help bone up my understanding, intuition, and technical skill before moving on. these are also mainly geometry books, so learning analysis (like functional analysis) would be good as well. i mainly have three suggestions there.
  
  

  three general analysis favorites
  

• introductory functional analysis with applications by erwin kreyszig - used everywhere.
  
• calculus of variations by gelfand and fomin
  
• linear differential operators by cornelius lanczos - a geometrical look at the subject. also take a look at the variational principles of mechanics by him as well.

I'm in a similar boat with you. I went through calculus in high school, graduated university with a B.A. in music, but have recently taken a keen interest in developing an actual understanding of math.

Aside from music, I have a strong background in philosophy, and from philosophy, so do the natural sciences extend and I've taken advantage of that. Math was discovered through raw observation of the world and through the concourse of logic, and so I have designed for myself the study of math through the source works of where the math originated, for practical and ontological purposes. Here's a few books that I've picked up and began reading:


A History of Greek Mathematics, Vol. 1: From Thales to Euclid https://www.amazon.com/dp/0486240738/ref=cm_sw_r_cp_apa_RljGybYRSB723

The Mathematical Principles of Natural Philosophy: The Principia https://www.amazon.com/dp/1512245844/ref=cm_sw_r_cp_apa_AmjGyb14R4B2V

Euclid's Elements https://www.amazon.com/dp/1888009187/ref=cm_sw_r_cp_apa_7mjGybZ97DBR7


Introduction to Mathematical Philosophy https://www.amazon.com/dp/1420938401/ref=cm_sw_r_cp_apa_OnjGybQ0078ZX

The Fractal Geometry of Nature https://www.amazon.com/dp/0716711869/ref=cm_sw_r_cp_apa_lojGybPPY25P4

The study of equations and formulas had been unfulfilling and unengaging until I framed it with the historical context of the natural sciences. I'm still a novice to this approach, but I believe it to be of merit- Ive also see some indication (when researching my own self-study method) that this is more similar to the method which Waldorf schools teach math and science as opposed to the traditional American Public school classroom, which as I grow older and reflect upon the majority of my experiences in classrooms, were uninspired, with the exception of very few memorable educators.

You could even base your study on other, less abstract interests than the interest of learning mathematics, such as an interest in modern physics or economy (or Comp sci, anything that utilizes math). Using that interest as a guide, you would be more clear minded to reverse-engineer your own individually purposed self-study. Such a direction of interest would certainly help for you to be able to design your course and keep you engaged. I hate how I've worded most of this Frankenstein of a comment; it's unnecessarily verbose and unorganized, but it's late and I'm tired to I'm not gonna edit it, nevertheless, hopefully you'll get the point(s).

Anyway, I'm curious what other people have to say about this approach, and especially I am open for people to suggest in response here to additional and essential sourcebooks!

I hope others will chime in here, but I'll answer as well as I can.

Euclidean and Non-Euclidean Geometry

Euclidean and non-Euclidean geometries are interesting and important for various reasons, so I certainly wouldn't say it's a bad idea to study them in depth.

If you want to study these subjects first because you find them interesting and you have plenty of years to spend, then go for it! However, it's not necessary (more on this below).

Multivariable Calculus and Linear Algebra

Before attempting even an elementary treatment of differential geometry, you'll want to have a working knowledge of calculus (single and multivariable) and linear algebra.

Elementary Differential Geometry

You could potentially skip the elementary treatments of differential geometry, but these might be useful for tackling more advanced treatments. Studying elementary differential geometry first is perhaps similar to taking a calculus class (with an emphasis on computation and hopefully on intuition) before taking a class in real analysis (with an emphasis on abstraction and rigorous proofs).

If you do want to work through an elementary treatment, then you have options. One well reviewed book, and the one I learned from as an undergraduate, is Elementary Differential Geometry by Barrett O'Neill.

Note that O'Neill lists calculus and linear algebra as prerequisites, but not Euclidean and Non-Euclidean geometry. Experience with Euclidean geometry is definitely relevant, but if you understand calculus and linear algebra, then you already know enough geometry to get started.

Abstract Algebra, Real Analysis, and Topology

The next step would probably be to study a semester's worth of abstract algebra, a year's worth of real analysis, and optionally, a semester's worth of point-set topology. These are the prerequisites for the introduction to manifolds listed below.

Manifolds

An Introduction to Manifolds by Loring W. Tu will give you the prerequisites to take on graduate-level differential geometry.

Note: the point-set topology is optional, since Tu doesn't assume it; he expects readers to learn it from his appendix, but a course in topology certainly wouldn't hurt.

Differential Geometry

After working through the book by Tu listed above, you'd be ready to tackle Differential Geometry: Connections, Curvature, and Characteristic Classes, also by Loring W. Tu. There may be more you want to learn, but after this second book by Tu, it should be easier to start picking up other books as needed.

Caveat

I myself have a lot left to learn. In case you want to ask me about other subjects, I've studied all the prerequisites (multivariable calculus, linear algebra, abstract algebra, real analysis, and point-set topology) and I've tutored most of that material. I've completed an elementary differential geometry course using O'Neill, another course using Calculus on Manifolds by Spivak, and I've studied some more advanced differential geometry and related topics. However, I haven't worked through Tu's books yet (not much). The plan I've outlined is basically the plan I've set for myself. I hope it helps you too!

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.

There are essentially "two types" of math: that for mathematicians and everyone else. When you see the sequence Calculus(1, 2, 3) -> Linear Algebra -> DiffEq (in that order) thrown around, you can be sure they are talking about non-rigorous, non-proof based kind that's good for nothing, imo of course. Calculus in this sequence is Analysis with all its important bits chopped off, so that everyone not into math can get that outta way quick and concentrate on where their passion lies. The same goes for Linear Algebra. LA in the sequence above is absolutely butchered so that non-math majors can pass and move on. Besides, you don't take LA or Calculus or other math subjects just once as a math major and move on: you take a rigorous/proof-based intro as an undergrad, then more advanced kind as a grad student etc.

To illustrate my point:

Linear Algebra:

1. Here's Linear Algebra described in the sequence above: I'll just leave it blank because I hate pointing fingers.
   
   
2. Here's a more serious intro to Linear Algebra:
   
   Linear Algebra Through Geometry by Banchoff and Wermer
   
   3. Here's more rigorous/abstract Linear Algebra for undergrads:
   
   Linear Algebra Done Right by Axler
   
   4. Here's more advanced grad level Linear Algebra:
   
   Advanced Linear Algebra by Steven Roman
   
   -----------------------------------------------------------
   
   Calculus:
   
   
3. Here's non-serious Calculus described in the sequence above: I won't name names, but I assume a lot of people are familiar with these expensive door-stops from their freshman year.
   
   
4. Here's an intro to proper, rigorous Calculus:
   
   Calulus by Spivak
   
   3. Full-blown undergrad level Analysis(proof-based):
   
   Analysis by Rudin
   
   4. More advanced Calculus for advance undergrads and grad students:
   
   Advanced Calculus by Sternberg and Loomis
   
   The same holds true for just about any subject in math. Btw, I am not saying you should study these books. The point and truth is you can start learning math right now, right this moment instead of reading lame and useless books designed to extract money out of students. Besides, there are so many more math subjects that are so much more interesting than the tired old Calculus: combinatorics, number theory, probability etc. Each of those have intros you can get started with right this moment.
   
   Here's how you start studying real math NOW:
   
   Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers. Essentially, this book is about the language that you need to be able to understand mathematicians, read and write proofs. It's not terribly comprehensive, but the amount of info it packs beats the usual first two years of math undergrad 1000x over. Books like this should be taught in high school. For alternatives, look into
   
   Discrete Math by Susanna Epp
   
   How To prove It by Velleman
   
   Intro To Category Theory by Lawvere and Schnauel
   
   There are TONS great, quality books out there, you just need to get yourself a liitle familiar with what real math looks like, so that you can explore further on your own instead of reading garbage and never getting even one step closer to mathematics.
   
   If you want to consolidate your knowledge you get from books like those of Rodgers and Velleman and take it many, many steps further:
   
   Basic Language of Math by Schaffer. It's a much more advanced book than those listed above, but contains all the basic tools of math you'll need.
   
   I'd like to say soooooooooo much more, but I am sue you're bored by now, so I'll stop here.
   
   Good Luck, buddyroo.

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.

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.

For compsci you need to study tons and tons and tons of discrete math. That means you don't need much of analysis business(too continuous). Instead you want to study combinatorics, graph theory, number theory, abstract algebra and the like.

Intro to math language(several of several million existing books on the topic). You want to study several books because what's overlooked by one author will be covered by another:

Discrete Mathematics with Applications by Susanna Epp

Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand, Albert D. Polimeni, Ping Zhang

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

Numbers and Proofs by Allenby

Mathematics: A Discrete Introduction by Edward Scheinerman

How to Prove It: A Structured Approach by Daniel Velleman

Theorems, Corollaries, Lemmas, and Methods of Proof by Richard Rossi

Some special topics(elementary treatment):

Rings, Fields and Groups: An Introduction to Abstract Algebra by R. B. J. T. Allenby

A Friendly Introduction to Number Theory Joseph Silverman

Elements of Number Theory by John Stillwell

A Primer in Combinatorics by Kheyfits

Counting by Khee Meng Koh

Combinatorics: A Guided Tour by David Mazur


Just a nice bunch of related books great to have read:

generatingfunctionology by Herbert Wilf

The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates by by Manuel Kauers, Peter Paule

A = B by Marko Petkovsek, Herbert S Wilf, Doron Zeilberger

If you wanna do graphics stuff, you wanna do some applied Linear Algebra:

Linear Algebra by Allenby

Linear Algebra Through Geometry by Thomas Banchoff, John Wermer

Linear Algebra by Richard Bronson, Gabriel B. Costa, John T. Saccoman

Best of Luck.

I learned geometry with no book from a teacher (he taught me Euclid's Elements), and did many examples to learn. I have heard good things about this book, however --
https://www.amazon.com/dp/1888009187/?tag=stackoverfl08-20 --

A bit of advice concerning such proofs: Try and come up with a strategy before you begin the proof. Think about what you need to prove, and what information you need to make that true. It can be a challenge for sure, but once you've learned how to approach proving things, you'll find geometry rewarding.

You can feel free to message me with proofs if you need a hand and I'll do my best to assist!

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:

• math: grab a copy of one of the following (or some similar textbook) and go through the text as well as the problems
  
  • Thomas and Finney
    
  • Stewart (older editions of this are okay since they are cheaper. I have fourth edition which is good enough).
    
• physics:
  
  • for mostly conceptual discussion of physics, Feynman lectures
    
  • for beginner level problems sets in various branches of physics, any one of the following (older editions are okay):
    
    • Halliday and Resnick
      
    • Young and Freedman
      
    • Serway and Jewett
      
    • Giancoli
      
  • for intermediate level discussion (actually you can jump right into this if your calculus is good) on mechanics , the core branch of physics, Kleppner and Kolenkow
    
    
    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.

Intro Calculus, in American sense, could as well be renamed "Physics 101" or some such since it's not a very mathematical course. Since Intro Calculus won't teach you how to think you're gonna need a book like How to Solve Word Problems in Calculus by Eugene Don and Benay Don pretty soon.

Aside from that, try these:

Excursions In Calculus by Robert Young.

Calculus:A Liberal Art by William McGowen Priestley.

Calculus for the Ambitious by T. W. KORNER.

Calculus: Concepts and Methods by Ken Binmore and Joan Davies

You can also start with "Calculus proper" = Analysis. The Bible of not-quite-analysis is:

[Calculus by Michael Spivak] (http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&ie=UTF8&qid=1413311074&sr=1-1&keywords=spivak+calculus).

Also, Analysis is all about inequalities as opposed to Algebra(identities), so you want to be familiar with them:

Introduction to Inequalities by Edwin F. Beckenbach, R. Bellman.

Analytic Inequalities by Nicholas D. Kazarinoff.

As for Linear Algebra, this subject is all over the place. There is about a million books of all levels written every year on this subject, many of which is trash.

My plan would go like this:

1. Learn the geometry of LA and how to prove things in LA:

Linear Algebra Through Geometry by Thomas Banchoff and John Wermer.

Linear Algebra, Third Edition: Algorithms, Applications, and Techniques
by Richard Bronson and Gabriel B. Costa.

2. Getting a bit more sophisticated:

Linear Algebra Done Right by Sheldon Axler.

Linear Algebra: An Introduction to Abstract Mathematics by Robert J. Valenza.

Linear Algebra Done Wrong by Sergei Treil.

3. Turn into the LinAl's 1% :)

Advanced Linear Algebra by Steven Roman.

Good Luck.

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.

By the way (slightly related), I read a theorem in a big, thick book on Riemannian geometry about how (I'm reconstructing a real world example here), if you took two objects and rubbed them together for a long time, they will eventually have constant curvature (one object opposite of the other). If you start with edges that are macroscopically straight, but microscopically noisy, you could make pretty good straight edges with this method.... I think.

Sorry that I can't cite the exact statement from memory. I don't have the book within 2000 km. :-(

Edit: I'm talking about Marcel Berger's A Panoramic View of Riemannian Geometry, which is a really fascinating read by the way. Pretty sure it was somewhere in the first three chapters.

I agree- find some fun math books.

These got me into math big time:

Euclid's Window (http://www.amazon.com/Euclids-Window-Geometry-Parallel-Hyperspace/dp/0684865246)

and

The Book of Numbers (http://www.amazon.com/gp/product/1554073618/ref=pd_lpo_k2_dp_sr_1?pf_rd_p=486539851&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=038797993X&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=0JCMX2MGQ5ZP3C8482D8)

the last one is the best- sort of like a beginner's math history book. :-) it even includes some basis proofs easy enough for a beginner.


and for the record- I got D's in algerbra in high school. And I HATED math.

I stumbled on Euclid's window and got interested in the historical side of math, and now I'm a mathematics major, with a 4.0. :-)

I plan on being a math history professor one day.

Good luck to you!

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.

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.

Oh, geez. Not Ana White. She doesn't know the first thing about joinery and her stuff is to woodworking as heating a can of spaghetti is to cooking. It is inadequate and doesn't teach you anything you need to know.

If you want to do some real woodworking, start with Tage Frid's books. Tage teaches you how to do real joinery both with hand and power tools. Traditional joinery is the difference between something left out for trash collection in a few years and a piece of furniture that gets used for 300 years. (Yes, well-made furniture will last that long or longer.) Screws and pocket jig stuff just doesn't hold up the way joinery techniques that have been used for hundreds of years will. Further, traditional joinery is not difficult and doesn't take much more time.

Second, learn about proportions and design. A good place to start is with the number Phi, also known as the Golden Section. There are other ways to proportion, but Phi always gives pleasing results. If you've never heard of 1.618 before, it's a little mind-blowing. You'll start seeing it everywhere.

Once you start sketching things out using Phi and using traditional joinery, you'll be making beautiful things that last for a very long time. Do not waste your time with crap. Make something wonderful that will still be in your family 200 years from now.

I'll be honest: most explanations of differential forms suck, and it makes you wonder if people really know what they're talking about. Of course, the answer, in general, is that they do; they just don't know how to communicate it. One issue is the absolute proliferation of different viewpoints regarding these things. Sometimes you get the impression that an author's intuition follows one viewpoint and his formal definitions follow another, resulting in a presentation that is, to say the least, a mess.

A fairly good treatment is the topology book Differential Forms in Algebraic Topology. Check the preview on there. You'll be able to see the basic introduction to forms in it and see if it makes sense to you (or potentially could make sense), that way you don't have to waste money on the book if it doesn't seem up your alley. That said, that book is pretty much as good as it's ever going to get when it comes to exposition on this subject.

Also, just in case it's not clear, when the authors there present the symbols dx_1, dx_2, ..., in the first paragraph, they mean those to be just formal symbols satisfying those two relations. They are not referencing some previous definition that they assume you know.

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!

If you divide a number in the Fibonacci Sequence by the previous number in the sequence, it will approach the golden ratio as you progress down the sequence. This ratio can be seen everywhere in nature. The golden ratio can be seen in the ratios of certain segments of the human body, petals on sunflowers, arms of pineapples, rabbit population growth, butterfly markings, sea shell spirals, and more! Here is a great book with more on the subject.

Edit: Grammar

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.

I think your background is definitely sufficient. The prerequisites for topology are very minimal, though analysis gives motivation to a lot concepts encountered.

The main idea of topology is generalizing the notion of neighbourhoods from metric spaces to more general spaces where there may not be a canonical choice of distance function, or even any notion of distance at all.

I'm a big fan of Ronald Brown's Topology book. It's under $10 for the ebook, and under $30 for the paperback on Amazon. Brown starts off with topology on the real line, and gives good motivation and intuition for the axioms. I personally prefer it to the often-recommended Munkres.

Loring Tu has a new book which discusses the basics of principal bundles and their characteristic classes towards the end. I have already read a significant chunk of it to get the forms perspective of characteristic classes on regular ol' vector bundles, and I can say that the book is excellent. To get a more detailed or advanced perspective though, Kobayashi & Nomizu seems to be the best place.

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

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

I don't know if it's ever used in research, but working with groupoids can be beneficial in basic algebraic topology because of the freedom allowed by not fixing a base point. Ronald Brown wrote a book that goes through a first course on algebraic topology using groupoids for homotopy theory, covering spaces, etc.

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"
  

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 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 Golden Ratio: The Story of Phi is a good read and gives you alot of examples of how it's used / why it's neat (nature, art, music, etc). Grab the eBook or head to your favorite bookstore and check it out.

Or maybe you just won't get it, which is fine.

I think I see what you mean by that, and if so, a change of basis could be considered a special case of what you're talking about, but change-of-basis is a linear transformation, so you'll only get specific cases of what you're looking for by change-of-basis.

(Actually, I just went back and looked at the video you linked, and he's dealing with explicitly invertible linear transformations, so yes he's basically doing change-of-basis.)

I brought up complex analysis because there are lots of ideas that are similar to what you're talking about in that subject. For instance, you might show that complex inversion transforms circles into circles (in this case, lines are considered circles of infinite radius). It might not be exactly what you're looking for, but it is about transformations and what they do to things like circles and lines.

Another option, if you want something a little more general, is you could look at a book like Garrity's Algebraic Geometry Book. Algebraic geometry is, in its full generality, a very difficult subject, but his book gives a very gentle introduction to the subject, and it starts with a description of affine transformations, which are sort of a generalization of these linear transformations. Affine transformations are linear transformations composed with translations. I bring up this subject because it fits the idea of "an equation defining a subset of the domain". Algebraic geometry is, at least in its simpler form, about studying curves and surfaces (and hypersurfaces) defined by polynomial equations.

"General Topology" by Stephen Willard, has always been my sentimental favorite. The sections and exercises I would recommend are

1. Chapter 6, Section 19, Compactification (Exercises 19H, 19J, and 19K; then if you are up for it, 19L and 19M.)
   
   
2. Chapter 7, Section 32, Metrization (Exercises 23B, 23C, 23D are well-wroth anybodies time.)
   
   The exercises from (1) anticipate the exercises form (2); at end of both sections and exercises I've suggested, you'll have seen some cool machinery from more than a couple of different perspectives.
   

"Topology and Groupoids" is a great one, think it would fit well in your list.

Oh, Stillwell probably also has some geometry books (I haven’t read anything of his other than the history book).

If you’re going to read Euclid, definitely buy this edition. Top notch book quality, way better at $25 than anything Springer sells for $100. I have no idea how the press makes any money on it. https://amzn.com/1888009187

Byrne’s illustrated version is also amazing: https://amzn.com/3836544717/

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 favorite two books in the whole world are Differential Forms in Algebraic Topology by R. Bott and L. Tu, and Characteristic Classes by J. Milnor and J. Stasheff. Every time I read them I learn something new.

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.

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.

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.

These were the most enlightening for me on their subjects:

• Differential and Integral Calculus, Vol. 1&2 by Richard Courant
  
• Linear Algebra by Georgi Shilov
  
• Differential Equations and Their Applications by Martin Braun
  
• Introduction to Geometry by H.S.M. Coxeter
  
• A Course in Mathematical Analysis, Vol. 1&2 by Edouard Goursat
  
• Introduction to Topology and Modern Analysis by George Simmons
  
• The Theory of Functions of a Complex Variable by A.I. Markushevich
  
• Basic Topology by M.A. Armstrong
  
• Topology by James Dugundji
  
• Ordinary Differential Equations by V.I. Arnold
  
• Partial Differential Equations by Fritz John
  
• Introduction to Probability Theory by Paul Hoel, Sidney Port and Charles Stone
  
• Linear Programming by Katta Murty
  
• Nonlinear Programming: Theory and Algorithms by M. Bazaraa, H. Sherali and C. Shetty
  
• A First Course in Numerical Analysis by Anthony Ralston and Philip Rabinowitz
  
• A First Course in Stochastic Processes by Samuel Karlin and Howard Taylor
  
• Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields by John Guckenheimer and Philip Holmes
  
• Homology Theory by James Vick

Almost anything Malcom Gladwell.

Disappearing Spoon
http://www.amazon.com/The-Disappearing-Spoon-Periodic-Elements/dp/B00401LQ2Q/ref=tmm_aud_title_0

Feynman's Rainbow: A Search for Beauty in Physics and in Life
http://www.amazon.com/Feynmans-Rainbow-Search-Beauty-Physics/dp/044653045X

Euclid's Window : The Story of Geometry from Parallel Lines to Hyperspace

http://www.amazon.com/Euclids-Window-Geometry-Parallel-Hyperspace/dp/0684865246/ref=la_B001IGP3W0_1_7?ie=UTF8&qid=1342497797&sr=1-7

The Drunkards Walk : How Randomness Rules Our Lives
http://www.amazon.com/The-Drunkards-Walk-Randomness-Rules/dp/0307275175/ref=la_B001IGP3W0_1_3?ie=UTF8&qid=1342497797&sr=1-3

Steve Jobs Walter Issacson Book
A Short History of Nearly Everything
Eat That Frog
In Defense of Food
Nullification

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 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.

I'd recommend something completely different, like projective geometry, or even Euclidean geometry, which can get quite involved.

A Panoramic View of Riemannian Geometry link

A massive book, thousands of references. Minimal proofs with the intent to get through and survey an amazing amount of material.

I really enjoyed Euclid's Window in high school

Wald does introduce the necessary DG, but if you're interested in a more mathematical perspective then O'Neill may suit you.

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

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.

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.

Yes that's the book I ended up doing catch up with when I took graduate differential topology with this book.

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.

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.

This book is what you're looked for. It's a rigorous calculus book. You'll learn why things are true.

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

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

That's why it's general topology and not just topology.
https://www.amazon.com/dp/0486434796/ref=rdr_ext_tmb

http://www.amazon.com/Geometry-Grades-9-11-Mcdougal-Littell/dp/0395977274/ref=sr_1_1?ie=UTF8&qid=1324530687&sr=8-1

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.

Do Carmo's Differential Geometry of Curves and Surfaces

Gallian's Contemporary Abstract 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.

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.

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.