You are in a very special position right now where many interesing fields of mathematics are suddenly accessible to you. There are many directions you could head. If your experience is limited to calculus, some of these may look very strange indeed, and perhaps that is enticing. That was certainly the case for me.

Here are a few subject areas in which you may be interested. I'll link you to Dover books on the topics, which are always cheap and generally good.

• The Nature and Power of Mathematics, Donald M. Davis. This book seems to be a survey of some history of mathematics and various modern topics. Check out the table of contents to get an idea. You'll notice a few of the subjects in the list below. It seems like this would be a good buy if you want to taste a few different subjects to see what pleases your palate.
  
  
• Introduction to Graph Theory, Richard J. Trudeau. Check out the Wikipedia entry on graph theory and the one defining graphs to get an idea what the field is about and some history. The reviews on Amazon for this book lead me to believe it would be a perfect match for an interested high school student.
  
  
• Game Theory: A Nontechnical Introduction, Morton D. Davis. Game theory is a very interesting field with broad applications--check out the wiki. This book seems to be written at a level where you would find it very accessible. The actual field uses some heavy math but this seems to give a good introduction.
  
  
• An Introduction to Information Theory, John R. Pierce. This is a light-on-the-maths introduction to a relatively young field of mathematics/computer science which concerns itself with the problems of storing and communicating data. Check out the wiki for some background.
  
  
• Lady Luck: The Theory of Probability, Warren Weaver. This book seems to be a good introduction to probability and covers a lot of important ideas, especially in the later chapters. Seems to be a good match to a high school level.
  
  
• Elementary Number Theory, Underwood Dudley. Number theory is a rich field concerned with properties of numbers. Check out its Wikipedia entry. I own this book and am reading through it like a novel--I love it! The exposition is so clear and thorough you'd think you were sitting in a lecture with a great professor, and the exercises are incredible. The author asks questions in such a way that, after answering them, you can't help but generalize your answers to larger problems. This book really teaches you to think mathematically.
  
  
• A Book of Abstract Algebra, Charles C. Pinter. Abstract algebra formalizes and generalizes the basic rules you know about algebra: commutativity, associativity, inverses of numbers, the distributive law, etc. It turns out that considering these concepts from an abstract standpoint leads to complex structures with very interesting properties. The field is HUGE and seems to bleed into every other field of mathematics in one way or another, revealing its power. I also own this book and it is similarly awesome. The exposition sets you up to expect the definitions before they are given, so the material really does proceed naturally.
  
  
• Introduction to Analysis, Maxwell Rosenlicht. Analysis is essentially the foundations and expansion of calculus. It is an amazing subject which no math student should ignore. Its study generally requires a great deal of time and effort; some students would benefit more from a guided class than from self-study.
  
  
• Principles of Statistics, M. G. Bulmer. In a few words, statistics is the marriage between probability and analysis (calculus). The wiki article explains the context and interpretation of the subject but doesn't seem to give much information on what the math involved is like. This book seems like it would be best read after you are familiar with probability, say from Weaver's book linked above.
  
  
• I have to second sellphone's recommendation of Naive Set Theory by Paul Halmos. It's one of my favorite math books and gives an amazing introduction to the field. It's short and to the point--almost a haiku on the subject.
  
  
• Continued Fractions, A. Ya. Khinchin. Take a look at the wiki for continued fractions. The book is definitely terse at times but it is rewarding; Khinchin is a master of the subject. One review states that, "although the book is rich with insight and information, Khinchin stays one nautical mile ahead of the reader at all times." Another review recommends Carl D. Olds' book on the subject as a better introduction.
  
  Basically, don't limit yourself to the track you see before you. Explore and enjoy.

Hello! I'm interested in trying to cultivate a better understanding/interest/mastery of mathematics for myself. For some context:

 




To be frank, Math has always been my least favorite subject. I do love learning, and my primary interests are Animation, Literature, History, Philosophy, Politics, Ecology & Biology. (I'm a Digital Media Major with an Evolutionary Biology minor) Throughout highschool I started off in the "honors" section with Algebra I, Geometry, and Algebra II. (Although, it was a small school, most of the really "excelling" students either doubled up with Geometry early on or qualified to skip Algebra I, meaning that most of the students I was around - as per Honors English, Bio, etc - were taking Math courses a grade ahead of me, taking Algebra II while I took Geometry, Pre-Calc while I took Algebra II, and AP/BC Calc/Calc I while I took Pre-Calc)

By my senior year though, I took a level down, and took Pre-Calculus in the "advanced" level. Not the lowest, that would be "College Prep," (man, Honors, Advanced, and College Prep - those are some really condescending names lol - of course in Junior & Senior year the APs open up, so all the kids who were in Honors went on to APs, and Honors became a bit lower in standard from that point on) but since I had never been doing great in Math I decided to take it a bit easier as I focused on other things.

So my point is, throughout High School I never really grappled with Math outside of necessity for completing courses, I never did all that well (I mean, grade-wise I was fine, Cs, Bs and occasional As) and pretty much forgot much of it after I needed to.

Currently I'm a sophmore in University. For my first year I kinda skirted around taking Math, since I had never done that well & hadn't enjoyed it much, so I wound up taking Statistics second semester of freshman year. I did okay, I got a C+ which is one of my worse grades, but considering my skills in the subject was acceptable. My professor was well-meaning and helpful outside of classes, but she had a very thick accent & I was very distracted for much of that semester.

Now this semester I'm taking Applied Finite Mathematics, and am doing alright. Much of the content so far has been a retread, but that's fine for me since I forgot most of the stuff & the presentation is far better this time, it's sinking in quite a bit easier. So far we've been going over the basics of Set Theory, Probability, Permutations, and some other stuff - kinda slowly tbh.

 




Well that was quite a bit of a preamble, tl;dr I was never all that good at or interested in math. However, I want to foster a healthier engagement with mathematics and so far have found entrance points of interest in discussions on the history and philosophy of mathematics. I think I could come to a better understanding and maybe even appreciation for math if I studied it on my own in some fashion.

So I've been looking into it, and I see that Dover publishes quite a range of affordable, slightly old math textbooks. Now, considering my background, (I am probably quite rusty but somewhat secure in Elementary Algebra, and to be honest I would not trust anything I could vaguely remember from 2 years ago in "Advanced" Pre-Calculus) what would be a good book to try and read/practice with/work through to make math 1) more approachable to me, 2) get a better and more rewarding understanding by attacking the stuff on my own, and/or 3) broaden my knowledge and ability in various math subjects?

Here are some interesting ones I've found via cursory search, I've so far just been looking at Dover's selections but feel free to recommend other stuff, just keep in mind I'd have to keep a rather small budget, especially since this is really on the side (considering my course of study, I really won't have to take any more math courses):
Prelude to Mathematics
A Book of Set Theory - More relevant to my current course & have heard good things about it
Linear Algebra
Number Theory
A Book of Abstract Algebra
Basic Algebra I
Calculus: An Intuitive and Physical Approach
Probability Theory: A Concise Course
A Course on Group Theory
Elementary Functional Analysis

if you're doing this to help prepare to switch careers, look at industries and companies you might be interested in. Every vertical has different tech stack choices that are common. Medicine has a lot of SAS, pharmaceutical researchers I've met all use R, main industry and research at this point is mostly Python. Python gets you the most bang for your buck. If you need to step outside ML and throw together a back end DB, a REST API and a front end to glue the whole thing together or whatever, Python's just as useful there as it will be with ML. I don't use R, but from what I hear it's much less versatile. The Stats libraries for R are a lot more mature though apparently, so if you want to get into doing some more intense statistical stuff, I've heard Python is a little less friendly. I haven't run into any of those limitations, but I've been more playing around with RL and stuff, and doing less intense statistical analysis with rigorous confidence bounds or whatever.

For forecasting from historical data, you're looking at time series. Unfortunately I don't know a ton about time series modeling yet. It's much more complicated than a situation where you're assuming N iid draws from a stationary distribution (the 'typical' entry point for classification and such that you see in supervised machine learning).

Keeping in mind that I have no business giving you advice where to start because I haven't made the trek yet myself, I've heard good things about Time Series Analysis and Its Applications. It's a grad level stats book though, so I hope you aren't joking about your math background, haha. The examples in that book are all in R too, as a head's up.

For a slightly easier (but still standard) introduction to the topic, I've also heard Wei's Time Series Analysis is decent. If you look around for a good introduction to multivariable time series analysis though, I'm sure you could find a lot of resources and judge for yourself what would most fit your needs. If you did pick one of those two books to pound out, I suspect you'll have a radically better idea how to go the rest of the way and get into practical application. As you're getting into the theory (whatever resource you use), I'd highly recommend picking a few datasets you're interested in (Kaggle might be a good source, to go with whatever you care to get into for your own reasons) and as you go, try applying the various methods you're learning on those few different datasets to get some sense of how it works and why. Pro-tip: one or two of your go-to toy datasets should be generated yourself with some simple to understand function to help give a really easily understandable case to play with, where your intuition can still hold up. y(t) = sin(t) +kt + N(0,b) maybe, or some simple dynamic process of the form y^t+1 = f(y^t ).

But either way, make sure you're rolling up your sleeves and cracking your assumptions against actual data in code to make sure you get the idea. All theory and no practical makes Jack a dull boy.

Edit: if you want a more broad introduction without necessarily having the rigorous focus on time series forecasting, 'applied predictive modeling' and 'introduction to statistical learning' are both good big picture intros. The new hands on machine learning book is good too, but more narrow and less comprehensive. Elements of Statistical Learning is kind of the defacto standard reference text going over all the common algorithms from a mathematical perspective. If you have the mathematical maturity to tackle ELS, that'd be a great way to start to get a deep foundation in the theoretical ideas across ML as a whole, though obviously none of that is going to be time series specific.

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.

> For those who prefer video lectures, Skiena generously provides his online. We also really like Tim Roughgarden’s course, available from Stanford’s MOOC platform Lagunita, or on Coursera. Whether you prefer Skiena’s or Roughgarden’s lecture style will be a matter of personal preference.
>
> For practice, our preferred approach is for students to solve problems on Leetcode. These tend to be interesting problems with decent accompanying solutions and discussions. They also help you test progress against questions that are commonly used in technical interviews at the more competitive software companies. We suggest solving around 100 random leetcode problems as part of your studies.
>
> Finally, we strongly recommend How to Solve It as an excellent and unique guide to general problem solving; it’s as applicable to computer science as it is to mathematics.
>
>
>
> [The Algorithm Design Manual](https://teachyourselfcs.com//skiena.jpg) [How to Solve It](https://teachyourselfcs.com//polya.jpg)> I have only one method that I recommend extensively—it’s called think before you write.
>
> — Richard Hamming
>
>
>
> ### Mathematics for Computer Science
>
> In some ways, computer science is an overgrown branch of applied mathematics. While many software engineers try—and to varying degrees succeed—at ignoring this, we encourage you to embrace it with direct study. Doing so successfully will give you an enormous competitive advantage over those who don’t.
>
> The most relevant area of math for CS is broadly called “discrete mathematics”, where “discrete” is the opposite of “continuous” and is loosely a collection of interesting applied math topics outside of calculus. Given the vague definition, it’s not meaningful to try to cover the entire breadth of “discrete mathematics”. A more realistic goal is to build a working understanding of logic, combinatorics and probability, set theory, graph theory, and a little of the number theory informing cryptography. Linear algebra is an additional worthwhile area of study, given its importance in computer graphics and machine learning.
>
> Our suggested starting point for discrete mathematics is the set of lecture notes by László Lovász. Professor Lovász did a good job of making the content approachable and intuitive, so this serves as a better starting point than more formal texts.
>
> For a more advanced treatment, we suggest Mathematics for Computer Science, the book-length lecture notes for the MIT course of the same name. That course’s video lectures are also freely available, and are our recommended video lectures for discrete math.
>
> For linear algebra, we suggest starting with the Essence of linear algebra video series, followed by Gilbert Strang’s book and video lectures.
>
>
>
> > If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.
>
> — John von Neumann
>
>
>
> ### Operating Systems
>
> Operating System Concepts (the “Dinosaur book”) and Modern Operating Systems are the “classic” books on operating systems. Both have attracted criticism for their writing styles, and for being the 1000-page-long type of textbook that gets bits bolted onto it every few years to encourage purchasing of the “latest edition”.
>
> Operating Systems: Three Easy Pieces is a good alternative that’s freely available online. We particularly like the structure of the book and feel that the exercises are well worth doing.
>
> After OSTEP, we encourage you to explore the design decisions of specific operating systems, through “{OS name} Internals” style books such as Lion's commentary on Unix, The Design and Implementation of the FreeBSD Operating System, and Mac OS X Internals.
>
> A great way to consolidate your understanding of operating systems is to read the code of a small kernel and add features. A great choice is xv6, a port of Unix V6 to ANSI C and x86 maintained for a course at MIT. OSTEP has an appendix of potential xv6 labs full of great ideas for potential projects.
>
>
>
> [Operating Systems: Three Easy Pieces](https://teachyourselfcs.com//ostep.jpeg)
>
>
>
> ### Computer Networking
>
> Given that so much of software engineering is on web servers and clients, one of the most immediately valuable areas of computer science is computer networking. Our self-taught students who methodically study networking find that they finally understand terms, concepts and protocols they’d been surrounded by for years.
>
> Our favorite book on the topic is Computer Networking: A Top-Down Approach. The small projects and exercises in the book are well worth doing, and we particularly like the “Wireshark labs”, which they have generously provided online.
>
> For those who prefer video lectures, we suggest Stanford’s Introduction to Computer Networking course available on their MOOC platform Lagunita.
>
> The study of networking benefits more from projects than it does from small exercises. Some possible projects are: an HTTP server, a UDP-based chat app, a mini TCP stack, a proxy or load balancer, and a distributed hash table.
>
>
>
> > You can’t gaze in the crystal ball and see the future. What the Internet is going to be in the future is what society makes it.
>
> — Bob Kahn
>
> [Computer Networking: A Top-Down Approach](https://teachyourselfcs.com//top-down.jpg)
>
>
>
> ### Databases
>
> It takes more work to self-learn about database systems than it does with most other topics. It’s a relatively new (i.e. post 1970s) field of study with strong commercial incentives for ideas to stay behind closed doors. Additionally, many potentially excellent textbook authors have preferred to join or start companies instead.
>
> Given the circumstances, we encourage self-learners to generally avoid textbooks and start with the Spring 2015 recording of CS 186, Joe Hellerstein’s databases course at Berkeley, and to progress to reading papers after.
>
> One paper particularly worth mentioning for new students is “Architecture of a Database System”, which uniquely provides a high-level view of how relational database management systems (RDBMS) work. This will serve as a useful skeleton for further study.
>
> Readings in Database Systems, better known as the databases “Red Book”, is a collection of papers compiled and edited by Peter Bailis, Joe Hellerstein and Michael Stonebreaker. For those who have progressed beyond the level of the CS 186 content, the Red Book should be your next stop.
>
> If you insist on using an introductory textbook, we suggest Database Management Systems by Ramakrishnan and Gehrke. For more advanced students, Jim Gray’s classic Transaction Processing: Concepts and Techniques is worthwhile, but we don’t encourage using this as a first resource.
>

> (continues in next comment)

Have you had a look at Carroll's general relativity notes? Chapters 2 and 3 are predominantly about developing the mathematics behind GR, and are very good introductions to this. I have a copy of Carroll's book and I can promise you that those chapters are almost unchanged in the book as compared to the lecture notes. This is my main suggestion really, as the notes are freely available, written by an absolute expert and a joy to read. I can't recommend them (and the book really) enough.

Most undergraduate books on general relativity start with a "physics first" type approach, where the underlying material about manifolds and curvature is developed as it is needed. The only problem with this is that it makes seeing the underlying picture for how the material works more difficult. I wouldn't neccessarily say avoid these sort of books (my favourite two of this kind would be Cheng's book and Hartle's.) but be aware that they are probably not what you are looking for if you want a consistent description of the mathematics.

I would also say avoid the harder end of the scale (Wald) till you've at least done your course. Wald is a tough book, and certainly not aimed at people seeing the material for the first time.

Another useful idea would be looking for lecture notes from other universities. As an example, there are some useful notes here from cambridge university. Generally I find doing searches like "general relativity site:.ac.uk filetype:pdf" in google is a good way to get started searching for decent lecture notes from other universities.

If you're willing to dive in a bit more to the mathematics, the riemannian geometry book by DoCarmo is supposed to be excellent, although I've only seen his differential geometry book (which was very good). As a word of warning, this book might assume knowledge of differential geometry from his earlier book. The book you linked by Bishop also looks fine, and there is also the book by Schutz which is supposed to be great and this book by Sternberg which looks pretty good, although quite tough.

Finally, if you would like I have a dropbox folder of collected together material for GR which I could share with you. It's not much, but I've got some decent stuff collected together which could be very helpful. As a qualifier, I had to teach myself GR for my undergrad project, so I know how it feels being on your own with it. Good luck!

Beyond intermediate texts, my classes ended up just reading papers from econ journals. You may want to pick up an econometrics text, get familiar with the methods, then read papers (here is a list of the 100 most cited).

I wrote my opinions on econometric textbooks I've used for another reddit comment, so I just pasted it in below. If you get into it, I'd recommend reading a less rigorous book straight through, then using a more rigorous text as reference or to do the practice stuff.

Less Mathematically Rigorous

• Kennedy - survey of modeling issues without the math. More about how to think about modeling rather than how do it. Easy to read, I liked it
  
  
• Angrist - similar to Kennedy, covers the why & how econometrics answers questions, very little math. Each chapter starts with a hitchhikers guide to the galaxy quote, which is fun. Just as good as Kennedy
  
  
• Long - this book is more about just "doing stuff" and presenting results, absolutely non-technical, but also dodges the heavy thinking in Angrist & Kennedy so I wasn't a big fan
  
  
• King - covers the thinking of Angrist & content of Maddala. It is more accessible but wordier, so give it a go if Kennedy or Angrist are too much. It is aimed at Poli Sci rather than econ.
  
  Middle of the Road
  
  
• Gujarati - I used this for a class. It wasn't hard to follow, but it mostly taught methodology and the how/why/when/what, and I didn't like that - a little too "push button" and slow moving.
  
  
• Woodlridge - a bit more rigorous than Gujarati, but it was more interesting and was clearer about motivations from the standpoint of interesting problems
  
  
• Cameron & Trivedi - I liked the few chapters I read, the math is there, but the methodology isn't driven by the math. I ddin't get too far into it
  
  More Mathematically Rigorous
  
  
• Greene - lots of math, so much it was distracting for me, but probably good for people who really want to learn the methodology
  
  
• Wooldridge - similar to Greene, you need a solid understanding before diving into this book. Some of the chapters are impenetrable
  
  
• Maddala - this book is best for probit/logit/tobit models and is somewhat technical but dated. My best econometrics teacher loved it

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.

I use [this book]http://www.amazon.com/Statistics-Manual-Edwin-L-Crow/dp/048660599X) as a reference. It's very small and inexpensive (you may have to buy it direct from Dover, though). It won't go through any derivations, but it covers most of the very important, basic, topics. I also have my old textbook on backup as well. Old editions of textbooks are cheaper, and all the information is the same.

Looking through amazon, this one looks pretty good as well, especially for an inexpensive text. A reviewer said it ends with what we just talked about! Any book that lays down the foundations well enough should be fine. See if you can find one that does correlation analysis, since you'll probably use that later.

Getting more advanced than this may depend on your field. If you're in biology, I'd recommend a book on designing and analyzing scientific experiments. I can't recommend a good title, though, because I'm only familiar with computer experiments (which tend to be easier).

I hope this helps!

Have you had a rigorous course on Analytical Mechanics? You will learn all about Noether's theorem there. How does Noether's theorem relate to charge conservation? For ANY continuous symmetry of the Lagrangian, we observe an associate Noether current made up of Noether charge. A continuous symmetry of the lagrangian is a symmetry that is generated by that lagrangian's Lie group. For example, the Lie group associated with electricity and magnetism is U(1). U(1) is the unitary group in 1 dimension and represents complex rotations in 1D. This is equivalent to SO(2), the 2 dimensional rotations in real space. If you apply this symmetry to the electromagnetic lagrangian using the proper covariant derivatives, you will obtain an associated four-current density that contains terms relating standard electrical current density. As for your question about special relativity and local gauge invariance. Strictly speaking, special relativity only has a continuous global symmetry, the poincare group, which is made up of the lorentz group (spacetime boosts, spacetime rotations) with the addition of spacetime translations. Jumping back to electricity and magnetism, enforcing local gauge invariance requires that the photon is massless. This is a definitely important for special relativity because the photon is assumed to be as such; only massless particles can keep pace with light. Neat info: because of the link to this gauge symmetry, you can actually experimentally verify charge conservation by measuring a zero mass photon. If the photon were massive, then the gauge symmetry is destroyed and you lose your conserved current. This is why you must have local charge conservation. No local charge conservation => massive photon => speed of light is not an invariant quantity.

Edit: Here is a link that has some information about the lorentz group. I wanted to mention the the four connected subgroups in my original post but didn't want to drone on. From them, you can derive the CPT symmetries and so forth.

http://en.wikipedia.org/wiki/Lorentz_group


Edit 2: Here is my favorite book on the topic of calculus of variations. This theoretical machinery is the foundation for mechanics, and really, your most important tool in theoretical physics. With it, you derive all of the fun contained in Noether's theorem. It is my opinion that no physics student should be without a copy of Weinstock's book.

http://www.amazon.com/Calculus-Variations-Applications-Physics-Engineering/dp/0486630692

Edit 3: Last one, I promise haha. Here is my other favorite, if you are interested in cutting your teeth in a more mathematically rigorous way. Also an excellent book on the topic, it contains a lot the the other book is missing. I want to say that Weinstock doesn't cover the calculation of the second variation(and beyond), which you use to prove that your extremized functional is a minimum or maximum.


http://www.amazon.com/Calculus-Variations-Dover-Books-Mathematics/dp/0486414485/ref=pd_bxgy_b_img_y

Had a very similar path. Decided to bite the bullet and take my school's remedial algebra course, as I cheated through middle and high school and thus knew nothing. Failed remedial algebra and retook it. Now I'm graduating with a math minor and am taking a calc-based probability theory course. Have hope!

Advice:

1. Find something to motivate you. I was inspired partially by a friend explaining couple of high-level concepts to me. What little I understood sounded fascinating, and I wanted to know more. Part of the reason math can get tough is that there might be no "light at the end of the tunnel" that will reward your hard work.
   
   
2. While immersing yourself in cool stuff can be good to keep you motivated, remember to do the "boring parts" too. Unfortunately, not everything can be awesome serendipity. There is no going around the fact that you're going to have to spend some time just going through practice problems. Way past the point when it stops being fun. You need to develop intuitions about certain things in order for the profundity of later things to really sink, and there's no way to do that aside from doing a bunch of problems.
   
   
3. Khan Academy's great. Right now they have tons of practice problems too.
   
   
4. I highly recommend this book: http://www.amazon.com/Humongous-Book-Algebra-Problems-Translated/dp/1592577229 Lots of problems broken down step-by-step. Skipped steps are one of the hardest things to deal with when you don't yet have much mathematical knowledge, especially during self-study. Look for other books by the author, W. Michael Kelly.
   
   
5. This blog has a lot of useful general study advice: http://calnewport.com/blog/
   
   
6. I also did uncharacteristically well in geometry. Try looking into discrete math which, among other things, is very directly related to computer science. There's a lot of "low-hanging fruit" - interesting stuff that doesn't require you to have as much experience under your belt. Any discrete math course will also have a section on symbolic logic, which you might be interested in if you liked the proofs some high school geometry.
   
   
7. An interesting take on math and math education, though a little bitter: http://www.maa.org/devlin/LockhartsLament.pdf
   
   Godspeed, sir!

Beginner Resources: These are fantastic places to start for true beginners.

Introduction to Probability is an oldie but a goodie. This is a basic book about probability that is suited for the absolute beginner. Its written in an older style of english, but other than that it is a great place to start.

Bayes Rule is a really simple, really basic book that shows only the most basic ideas of bayesian stats. If you are completely unfamiliar with stats but have a basic understanding of probability, this book is pretty good.

A Modern Approach to Regression with R is a great first resource for someone who understands a little about probability but wants to learn more about the details of data analysis.

​

Advanced resources: These are comprehensive, quality, and what I used for a stats MS.

Statistical Inference by Casella and Berger (2nd ed) is a classic text on maximum likelihood, probability, sufficiency, large sample properties, etc. Its what I used for all of my graduate probability and inference classes. Its not really beginner friendly and sometimes goes into too much detail, but its a really high quality resource.

Bayesian Data Analysis (3rd ed) is a really nice resource/reference for bayesian analysis. It isn't a "cuddle up by a fire" type of book since it is really detailed, but almost any topic in bayesian analysis will be there. Although its not needed, a good grasp on topics in the first book will greatly enhance the reading experience.

John Fox's book is great. It's mostly linear regression models for continuous variables, but the GLM section is very helpful. If I remember correctly, the second edition is way more helpful with GLM than the first.

For categorical variables Scott Long's book is wonderfully helpful.

Unfortunately both are expensive. Hopefully your library has them.

Any more specificity in what types of variables you might be working with or what your data is like? Knowing what type of link function you're looking for my give you better results from some of the uber statisticians here.

I like Szekeres's A Course in Modern Mathematical Physics for referencing intro-grad-level material. It covers abstract linear algebra, differential geometry, measure theory, functional analysis, and Lie algebras, and teaches you some physics along the way.

More generally, the best "breadth" book on advanced mathematics is Princeton Companion to Mathematics by Gowers et al. and its slightly underachieving younger brother of a companion text, Princeton Companion to Applied Mathematics by Higham et al.. You won't properly learn advanced mathematics this way, but you'll get the bird's-eye view of modern research programs and the math underlying them.

If you want a more algebraic take on Szekeres's program to teach physicists all the math they need to know, check out Evan Chen's Napkin project, which is intended to introduce advanced undergrads (it's perfectly fine for grad students too) to a wide variety of advanced mathematics on the algebra side of things.

Since you're doing probability and statistics, check out Wasserman's All of Statistics and Knill's Probability Theory and Stochastic Processes for good, concise references for intro-grad-level material.

I will second what /u/Ovationification said, though. I didn't really learn anything with the above books, I just use them occasionally for reference or to think about pedagogy.

Mochizuki is kind of a special case.

Usually when there's some supposed big breakthrough in mathematics, various experts in the field will read though it to figure out how the proof works, Usually if there is an error, one will be found reasonably quickly. If it does work, there will likely be a number of expository articles and lectures from the original author and others in the field, and after maybe a few months to years, most experts in the field will at least have a working understanding of how the proof works, in a very big-picture high-level overview sense, even if they don't feel the need to learn the full proof. The ideas of the proof will then gradually be incorporated into other proofs in the field, until the breakthrough eventually just becomes a standard technique.

Wiles' proof of FLT roughly followed this pattern. He originally announced his proof in 1993, however a few months after that a hole in his proof was discovered. About a year later - with the help of Richard Taylor - he was able to correct the proof. The new proof was accepted and published less than a year later, in May 1995.

Even 2-3 years after this, other mathematicians (such as Fred Diamond) were already using Wiles' techniques to prove stronger results, and a graduate level book explaining the proof had been published. Six years after the proof, in 2001, Breuil, Conrad, Diamond and Taylor had already managed to extend Wiles' method to prove the full Taniyama–Shimura conjecture (Wiles had proved a special case of this to deduce FLT). Now, 20 years later, Taylor-Wiles patching has been used to prove a number of other important results, such as the Sato-Tate conjecture, and it's becoming a common technique in algebraic number theory.

Mochizuki's stuff is quite different from Wiles' in that it is far more complex, uses a bunch of new language that no one besides Mochizuki has ever used, and Mochizuki honestly hasn't done a great job of explaining it. To the best of my knowledge, neither Mochizuki or anyone else has given a good high-level overview of how the proof actually works, that would be accessible to people with a background in algebraic number theory, but who don't already know the specifics of Mochizuki's theory. I don't mean an actual explanation of the proof here, I literally mean something like: "This is the main strategy of the proof. This is why it is reasonable to expect that something like that could actually be used to prove something like the abc conjecture." To the best of my knowledge, no one's really been able to do that in a satisfying way. Giving the corresponding explanation for Wiles' proof of FLT can be done in a 1-3 hour lecture, and has been done countless times.

At this point it really isn't clear at all that Mochizuki's proof actually works. It's entirely possible that there's a very subtle error buried somewhere in the 500+ pages of technical gobbledygook, and that none of the 10-20 people who have actually read through it in any detail has actually noticed yet. The longer we go without getting a satisfying explanation as to why this proof even should work, the more skeptical I'm going to be.

At this point, most algebraic number theorists don't seem particularly inclined to try to learn the proof. It's the sort of thing that would take several years to fully learn, and there's still a decent chance that the whole thing will wind up being a giant waste of time

Sure! I have a lot of resources on this subject. Before I recommend it, let me very quickly explain why it is useful.

Bayes Rule basically means creating a new hypothesis or belief based on a novel event using prior hypothesis/data. So I am sure you can already see how useful it would be in medicine to think about. The Rule(or technically theorem) is in fact an entire field of statisitcs and basically is one of the core parts of probability theory.

Bayes Rule explains why you shouldn't trust sensitivity and specificity as much as you think. It would take too long to explain here but if you look up Bayes' Theorem on wikipedia one of the first examples is about how despite a drug having 99% sensitivity and specificity, even if a user tests positive for a drug, they are in fact more likely to not be taking the drug at all.

Ok, now book recommendations:

Basic: https://www.amazon.com/Bayes-Theorem-Examples-Introduction-Beginners-ebook/dp/B01LZ1T9IX/ref=sr_1_2?ie=UTF8&qid=1510402907&sr=8-2&keywords=bayesian+statistics

https://www.amazon.com/Bayes-Rule-Tutorial-Introduction-Bayesian/dp/0956372848/ref=sr_1_6?ie=UTF8&qid=1510402907&sr=8-6&keywords=bayesian+statistics

Intermediate/Advanced: Only read if you know calculus and linear algebra, otherwise not worth it. That said, these books are extremely good and are a thorough intro compared to the first ones.

https://www.amazon.com/Bayesian-Analysis-Chapman-Statistical-Science/dp/1439840954/ref=sr_1_1?ie=UTF8&qid=1510402907&sr=8-1&keywords=bayesian+statistics

https://www.amazon.com/Introduction-Probability-Chapman-Statistical-Science/dp/1466575573/ref=sr_1_12?s=books&ie=UTF8&qid=1510403749&sr=1-12&keywords=probability

I found Geometrical methods of mathematical physics by Bernard Schutz to be helpful, though it doesn't have many problems and it doesn't go into much depth on covariant differentiation. But it is good about discussing the modern view of tensors.

I would recommend watching the first half of the International Winter School on Gravity & Light (check the YouTube channel as well ) if you are interested in learning tensor calculus for use in differential geometry for GR.

I learned tensor calculus in bits from several different courses and texts, so I'm not sure what the best ones that are actually dedicated to the subject might be. In any case, I think you'll have a lot of fun learning the subject.

For anyone interested in a lay explanation of set theory in a challenging (for laymen) but tremendously well written and engaging book, I'd recommend Everything and More by David Foster Wallace. I'm sure it's beneath most mathematicians, but I really loved it.

Thank you! This is exactly what I was looking for!!!! I didn't think anyone was going to give me a sufficient reply because there are a lot of books (sorry), but this is what I wanted. Where would you place the two books I linked, Principles and Techniques in Combinatorics and Introduction to Combinatorial Mathematics, Liu, in that list or would you consider studying them a redundant exercise? I also did not include this book in the list, but where would you place Problems from the Book and its accompanying Straight from the Book?

I will likely end up replacing the Graph Theory book I have in the list, by Berge, with Modern Graph Theory by Bellobas, since Berge doesn't have exercises, but I will assume it stays in the same order of the sequence.


I apologize for not initially including them. I did not realize that I did not. Also, are there any other topics you would recommend I cover for establishing a solid foundation. I didn't buy Rudin's Complex Analysis because I didn't know if that kind of thing was necessary. I don't even know what other branches of mathematics Complex Analysis relates to. There could be other topics I'm not aware of as well. Please don't hesitate to make more recommendations. I appreciate it.

I used Carothers in my Real Analysis class, but if you're looking for a readable, introductory book which doesn't skip any steps, Bruckner is a great, free choice. I used this textbook in an Advanced Calculus class and really enjoyed reading it.

Did any of your calc classes include multivariate/vector calculus? E.g. things dealing with double and triple integrals.

If not, take another calc class or two; calculus is very important for statistics. It shouldn't be too hard to pick up the rest of the necessary calc since you've already got a good calc background.

If so, start taking probability and statistics courses in your school's math department if you can. The mathematical way (read: the right way) of understanding probability and statistics is based on probability distributions (like the normal distribution), defined by their probability functions. As such, you can use calculus to obtain a myriad of information from them! For instance, among many other things, within the first one or 2 courses, you'd likely be able to answer at least the Spearman's coefficient question, the Bernoulli process question, and the MLE question.

If you don't have room in your schedule to do the stats course, you could get a textbook and try learning on your own. There are tons of excellent resources. Hogg, Tanis, and Zimmerman is pretty good for an introduction, though I'm sure there's better out there.

Usual hierarchy of what comes after what is simply artificial. They like to teach Linear Algebra before Abstract Algebra, but it doesn't mean that it is all there's to Linear Algebra especially because Linear Algebra is a part of Abstract Algebra.

Example,

Linear Algebra for freshmen: some books that talk about manipulating matrices at length.

Linear Algebra for 2nd/3rd year undergrads: Linear Algebra Done Right by Axler

Linear Algebra for grad students(aka overkill): Advanced Linear Algebra by Roman

Basically, math is all interconnected and it doesn't matter where exactly you enter it.

Coming in cold might be a bit of a shocker, so studying up on foundational stuff before plunging into modern math is probably great.

Books you might like:

Discrete Mathematics with Applications by Susanna Epp

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

Building Proofs: A Practical Guide by Oliveira/Stewart

Book Of Proof by Hammack

Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al

How to Prove It: A Structured Approach by Velleman

The Nuts and Bolts of Proofs by Antonella Cupillary

How To Think About Analysis by Alcock

Principles and Techniques in Combinatorics by Khee-Meng Koh , Chuan Chong Chen

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!) by Carol Ash

Problems and Proofs in Numbers and Algebra by Millman et al

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

Mathematical Concepts by Jost - can't wait to start reading this

Proof Patterns by Joshi

...and about a billion other books like that I can't remember right now.

Good Luck.

Disclaimer: I'm an engineer, not a mathematician, so take my advice with a grain of salt.

Early in my grad degree I wanted to master probability and improve my understanding of statistics. The books I used, and loved, are

DeGroot, Probability and Statistics

Rozanov, Probability Theory: A Concise Course

The first is organized very well, with ever increasing difficulty and a good number of solved problems. I also appreciate that as things start to get complicated, he also always bridges everything back to earlier concepts. The books also basically does everything Bayesian and Frequentist side by side, so you get a really good idea of the comparison and arbitraryness.

The second is a good cheap short book basically full of examples. It has just enough math flavor to be mathier, without proofing me to death.

Also, if you're really just jumping into the subject, I would recommend some pop culture math books too, e.g.,

Paulos, Innumeracy

Mlodinow, The Drunkards Walk

Have fun!

You might find a book like Naked Statistics (https://www.amazon.com/Naked-Statistics-Stripping-Dread-Data/dp/1480590185) pretty helpful. The author uses a lot of common-place terminology and situations and helps the reader develop an intuition for the main ideas in statistics.

Imagine two buses... one is full of marathon runners and the other is full of participants in a festival of sausage. You stop each bus and weigh all the passengers. Since marathon runners tend to be lean and more uniform in size, more of them will be closer to the average weight. People attending a festival of sausage will be more diverse. Some will be thin others chunky, and others quite obese. Each individual's weight is more likely to be farther away from their average weight. In this case, the bus with marathon runners will have a lower variance in weight than the bus with the festival of sausage attendees.

The book does a better job than my paraphrased example.

Evolution of numbers is the "basic story". This is what Einstein said about this book
"This is beyond doubt the most interesting book on the evolution of mathematics which has ever fallen into my hands. If people know how to treasure the truly good, this book will attain a lasting place in the literature of the world. The evolution of mathematical thought from the earliest times to the latest constructions is presented here with admirable consistency and originality and in a wonderfully lively style."

Read Amazon reviews to get more info.

For a book that's less likely to defeat the reader (because even Baby Rudin is very tough), I'd like to recommend N. (Neal) L. Carothers - Real Analysis.

Metric spaces and Lebesgue measure on the real line only, but quite well written. Honestly, most of what's in there (on metric spaces) should probably be covered in your next analysis course, but owning more good books never hurt, right?

Dude, all you have to do is say, "Okay, I admit I don't understand how probability works" and then make an effort to learn about it. That's what I did. But I know intelligence is considered a liability in your precious little simian brain. Better to just stomp your feet and put your fingers in your ears when reality doesn't meet what Rush told you to think. Funny, seems like you're the one who wants a safe space.

https://www.amazon.com/Introduction-Probability-Statistics-Random-Processes/dp/0990637204/ref=sr_1_1?ie=UTF8&qid=1468191104&sr=8-1&keywords=introduction+to+probability+and+statistics

If you're comfortable with Python and some math notation Python Machine Learning, is a great resource for getting started. There's a great balance between explaining concepts and applying code.

In Machine Learning, knowledge of statistics is a huge help. This book explains basic concepts and this Pycon talk applies them in practice.

If you're looking to understand concepts and theories, Calculus and a bit of Linear Algebra will go a long way.

Do not enroll in a precalculus class until you have a solid grasp on the foundations of precalculus. Precalculus is generally considered to be the fundamentals required for calculus and beyond (obviously), and a strong understanding of precalculus will serve you well, but in order to do well in precalculus you still need a solid understanding of what comes before, and there is quite a bit.

I do not mean to sound discouraging, but I was tutoring a guy in an adult learning program from about December 2017-July 2018...I helped him with his homework and answered any questions that he had, but when he asked me to really get into the meat of algebra (he needed it for chemistry to become a nurse) I found a precalculus book at the library and asked him to go over the prerequisite chapter and it went completely over his head. Perhaps this is my fault as a tutor, but I do not believe so.

What I am saying is that you need a good foundation in the absolute basics before doing precalculus and I do not believe that you should enroll in a precalculus course ASAP because you may end up being let down and then give up completely. I would recommend pairing Basic Mathematics by Serge Lang with The Humongous Book of Algebra Problems (though any book with emphasis on practice will suffice) and using websites like khanacademy for additional practice problems and instructions. Once you have a good handle on this, start looking at what math courses are offered at your nearest CC and then use your best judgment to decide which course(s) to take.

I do not know how old you are, but if you are anything like me, you probably feel like you are running out of time and need to rush. Take your time and practice as much as possible. Do practice problems until it hurts to hold the pencil.

Razanov's Probability Theory was nice. It certainly won't cover everything you need to learn, but it requires the reader to put a great deal of effort into processing each proof, which helped me a lot with retention and application.

edit: Introductory Econometrics for Finance is great. I'd accidentally given you the wrong book earlier. Sorry!

edit #2: this is entirely unrelated but I'm reading it now and felt like sharing.

>I mean its to late now to enroll...

Why wait? Pick up a few books on math and use your Google Fu to get yourself started.

I really like this book.

And instead of studying geometry (which I doubt you'd be using in college), study Logic instead. The way problems are constructed is similar to geometry. In geometry you have theorem and postulates, in logic you make proofs. You start out with two or three opening statements, and by using different combinations of OR, AND & IF-THEN statements, you can prove the final statement.

I'll give you this link about it but I'm hesitant to because it has lots of scary symbols and letters. Here. But save that for later. If you want to get started, take a look at truth tables .

Logic is so much more interesting than geometry because it'll help your Google Fu get even better. You can make Boolean statements when you enter a Google query. It also gets you on the path to learning SQL (which your brother may also be able to help you with). SQL is all about sets - sets of records, and how you can join them and select those that have certain values, etc.

You may even find this book a nice, gentle introduction to logic that doesn't require much math.

Basically, what I'm saying to you is this: you live in the most incredible time to be alive ever. The Internet is a super-powerful tool you can use to educate yourself and you should make full use of it.

I also want you to know that if you don't have a specialized skill, you're going to be treated like a virtual slave for the rest of your life. Working at WalMart is not a good career choice. That's just choosing a life of victimhood. Make full use of the Internet, and your lack of a car will seem less problematic.

When I was that age, I used to really enjoy reading books by Clifford Pickover. He has many books, and there are many reviews on Amazon that can aid you in choosing one that you think would be good for your brother-in-law.

Side note: I had not read a Pickover in a long time, but I recently stumbled across The Math Book, which I purchased and quite enjoy. While it may not be the best option for inspiring a 13-year-old, members of Mathit may find it interesting, so I recommend checking it out!

Here's mine

To understand life, I'd highly recommend this textbook that we used at university http://www.amazon.com/Campbell-Biology-Edition-Jane-Reece/dp/0321558235/ That covers cell biology and basic biology, you'll understand how the cells in your body work, how nutrition works, how medicine works, how viruses work, where biotech is today, and every page will confront you with what we "don't yet" understand too with neat little excerpts of current science every chapter. It'll give you the foundation to start seeing how life is nothing special and just machinery (maybe you should do some basic chemistry/biology stuff on KhanAcademy first though to fully appreciate what you'll read).

For math I'd recommend doing KhanAcademy aswell https://www.khanacademy.org/ and maybe a good Algebra workbook like http://www.amazon.com/The-Humongous-Book-Algebra-Problems/dp/1592577229/ and after you're comfortable with Algebra/Trig then go for calc, I like this book http://www.amazon.com/Calculus-Ron-Larson/dp/0547167024/ Don't forget the 2 workbooks so you can dig yourself out when you get stuck http://www.amazon.com/Student-Solutions-Chapters-Edwards-Calculus/dp/0547213093/ http://www.amazon.com/Student-Solutions-Chapters-Edwards-Calculus/dp/0547213107/ That covers calc1 calc2 and calc3.

Once you're getting into calc Physics is a must of course, Math can describe an infinite amount of universes but when you use it to describe our universe now you have Physics, http://www.amazon.com/University-Physics-Modern-12th/dp/0321501217/ has workbooks too that you'll definitely need since you're learning on your own.

At this point you'll have your answers and a foundation to go into advanced topics in all technical fields, this is why every university student who does a technical degree must take courses in all those 3 disciplines.

If anything at least read that biology textbook, you really won't ever have a true appreciation for the living world and you can't believe how often you'll start noticing people around you spouting terrible science. If you could actually get through all the work I mentioned above, college would be a breeze for you.

You've taken some sort of analysis course already? A lot of real analysis textbooks will cover Lebesgue integration to an extent.

Some good introductions to analysis that include content on Lebesgue integration:

Walter Rudin, principle of mathematical analysis, I think it is heavily focused on the real numbers, but a fantastic book to go through regardless. Introduces Lebesgue integration as of at least the 2nd edition (the Lebesgue theory seems to be for a more general space, not just real functions).

Rudin also has a more advanced book, Real and Complex Analysis, which I believe will cover Lebesgue integration, Fourier series and (obviously) covers complex analysis.

Carothers Real Analysis is the book I did my introductory real analysis course with. It does the typical content (metric spaces, compactness, connectedness, continuity, function spaces), it has a chapter on Fourier series, and a section (5 chapters) on Lebesgue integration.

Royden's real analysis I believe covers very similar topics and again has a long and detailed section on Lebesgue integration. No experience with it, recommended for my upcoming graduate analysis course.

Bartle, Elements of Integration is a full book on Lebesgue integration. Again, haven't read it yet, recommended for my upcoming course. It is supposed to be a classic on the topic from what I've heard.

I've wasted too much time trying to find the so-called "right" statistics book. I'm still early in my journey, going through calculus using Prof. Leonards videos while working through a Linear Algebra book all in prep for tackling a stats book. Here's a list of books that I've had a look at so far.

​

• Probability and Statistical Inference by Hogg, Tanis and Zimmerman
  
• Mathematical Statistics with Applications by Wackerly
  
  These seem to be of a similar level with regards to rigour, as they aren't that rigourous. That's not to say you can get by without the calculus prereq and even linear algebra
  
  ​
  
  The other two I've been looking at which seem to be a lot more complex are
  
  
• Introduction to Mathematical Statistics by Hogg as well. I'd think it's the more rigorous version of the book mentioned above by the same author
  
• All of Statistics by Wasserman which seems to require a lot of prior knowledge in statistics, but I think tackles just the perfect topics for machine learning
  
  And then there's Casella and Berger's Statistical inference, which I looked at once and decided not to look at again until I can manage at least one of the aforementioned books. I think I'm leaning most to the first book listed. Whichever one you decide to use, good luck with your journey.
  
  ​

I'm a fan of Hogg, Mckean &Craig. This is a graduate level text so don't feel like you need to understand everything in it, but it could be a good way to get a better understanding of the topics you've already covered but don't quite grock. Also, don't be intimidated just because it's a graduate level textbook: it's fairly accessible, certainly more so than Casella & Berger, which someone else probably would have already suggested if I'd gotten to this later.

For probability I'd recommend Introduction to Probability Theory by Hoel, Port & Stone. It has the best explanations of any probability book I've seen, great examples, and answers to most of the problems are in the back (making it well-suited for self-study). I think it's still the best introductory book on the subject, despite its age. Amazon has used copies for cheap.

For statistics, you have to be more precise as to what you mean by an "average undergraduate statistics" course. There's a difference between the typical "elementary statistics" course and the typical "mathematical statistics" course. The former requires no calculus, but goes into more detail about various statistical procedures and tests for practical uses, while the latter requires calculus and deals more with theory than practice. Learning both wouldn't be a bad idea. For elementary stats there are lots of badly written books, but there is one jewel: Statistics by Freedman, Pisani & Purves. For mathematical statistics, Introduction to Mathematical Statistics by Hogg & Craig is decent, though a bit dry. I don't think that Statistical Inference by Casella & Berger is really any better. Those are the two most-used textbooks on the subject.

This imo is a good book for basic probability and mathematical statistics. Super easy read with a lot of examples. [You also mentioned pdf's for books and someone told you library gensis. I can promise this one is on there :)]

Dear OP, I would recommend going through a GR book like Carroll first. Then when you feel like you have a decent enough grasp turn to a mathematical physics book like this one http://www.amazon.com/Course-Modern-Mathematical-Physics-Differential/dp/0521829607/ref=sr_1_1?ie=UTF8&qid=1345340408&sr=8-1&keywords=szekeres

This should give you all you need.

http://www.amazon.com/Applied-Regression-Analysis-Generalized-Linear/dp/0761930426

http://www.amazon.com/Regression-Categorical-Dependent-Quantitative-Techniques/dp/0803973748

http://www-stat.stanford.edu/~tibs/ElemStatLearn/

http://www-bcf.usc.edu/~gareth/ISL/

http://www.amazon.com/Extending-Linear-Model-Generalized-Nonparametric/dp/158488424X/ref=sr_1_2?s=books&ie=UTF8&qid=1380716057&sr=1-2

http://www.amazon.com/Generalized-Edition-Monographs-Statistics-Probability/dp/0412317605

hope it helps man. good luck. im learning this stuff for my project too

My math stats textbook is Hogg McKean Craig. I don't think the math would be too much for a computation statistics major, but it would give you a great overview if you're interested in that direction.

http://www.amazon.com/Introduction-Mathematical-Statistics-7th-Edition/dp/0321795431

Are you looking for a more introductory book or something more graduate level? If your note dealing with measure theory and you want something elementary this is one of the best books, i've found and it comes with a solution manual as well: Intro to Prob. If you need something more advanced let me know.

If you want theoretical / mathematical I would suggest reading a few math, stats or engineering books.

Dover is a great place to find some cheaper reading material. They republish old scientific and math texts that were popular in their time in a smaller sized paperback. They're a nice size to bring around with you and they don't cost much.

Math and stats findings of today build on this knowledge, and much of it is still used in state-of-the-art applications. Or, that math/stats is used as part of some state-of-the-art algorithm. Lots of the newest ML algorithms are blending math from a variety of areas.

Statistical analysis of experimental data

Principals of Statistics

Information Theory

Statistics Manual

Some theory of sampling

Numerical Methods for Scientists and Engineers (Hamming)

Mathematical Handbook for Scientists Engineers

Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables

==

There is also the Data-Science Humble Bundle for more technical / practical skill building.

Everything and More: A Compact History of Infinity

It inspired me to become a mathematician. Maybe not the most profound, but as far as mathematical books, it's still, to this day, the liveliest and most easily accessible book I've ever read.

The Math Book by Clifford Pickover is kind of a coffeetable style book, I love coming home from work and flipping to a random page as a starting point to exploring whatever concept is discussed on it. The book traces chronologically through 250 great discoveries in mathematics.

What do you mean by the "Whole Proof"? How much do you want to assume? If you assume everything that is learned in a standard phd program that is loosely related to number theory (so including things like Class Field Theory, basic theory of Elliptic curves and Modular forms), you would need the Langlands-Tunnell Theorem, which is a whole book on its own, Ribet's Theorem and the analysis of Frey Curves, Deformation Theory, Hecke Algebras, a boatload of advanced Commutative Algebra and many computational results on particular elliptic/modular curves. Then you can begin to talk about Wiles' contributions. It wouldn't be just one book.

If you want something that contains the general knowledge of the proof, but is brief when it needs to be, then Modular Forms and Fermat's Last Theorem is pretty solid.

Buy this book http://www.amazon.com/exec/obidos/ASIN/1402757964/cliffordpickover

Try look at numbers from an aesthetic point of view besides just logical numbers.

Number theory and such will help because it opens a world of mystery and you begin to realize just how important numbers really are.
http://en.wikipedia.org/wiki/Number_theory

Geometry is also really cool http://mathworld.wolfram.com/topics/Geometry.html

With computers math has become very visual which allows for a greater experience, when you come across a mathematical concept try and Google an image for it, for instance Google "prime number visualization" hit "image search" and your off to the races with incredible stuff.

William W.S. Wei's Time Series Analysis is an amazing book. It's really clear and well structured.

Is it Number: The Language of Science by Dantzig? It covers the early history of math pretty well. I strongly recommend it. So does Albert Einstein.

Thanks! I'll use those books. I'm currently studying stats using Introduction to Probability, Statistics, and Random Processes. Is that a good book for probability or would you recommend something else?

I have this one:
http://www.amazon.com/gp/product/0956372848/ref=pd_luc_rh_bxgy_01_02_t_lh?ie=UTF8&psc=1

Haven't read it yet, but thumbed through it quickly. It looked solid.

I also have Efron's book, also have not read it yet, but given how awesome the Bootstrap is, anything penned by Efron/Tibshirani is going to get in my collection.

The following are all really good and all very different. Check out the reviews and decide what fits you best. If I had to pick only one, I would pick one of the first three listed.

http://www.amazon.com/gp/product/0471530123

http://www.amazon.com/gp/product/1592577229

http://www.amazon.com/The-Humongous-Book-Calculus-Problems/dp/1592575129

http://www.amazon.com/gp/product/0817636773

http://www.amazon.com/gp/product/0760706603

http://www.amazon.com/gp/product/B000ZEC19Y

http://www.amazon.com/gp/product/0070293317

For Variational Calculus, the best references are Landau and Lifchitz and Gelfand and Fomin. The former is really a mechanics book that incorporates variational calculus in a very rigorous manner that one would expect from a theoretical physicist. The latter is a straight-up variational calculus book. Both are relatively cheap (you can find landau for cheaper than the amazon price).

For non-commutative geometry, there is this classic paper. /u/hopffiber gave the classic references for the rest of the topics, although you should think about learning quantum field theory since all the applications of Lie algebras come from QFT and String Theory. There are some excellent notes by David Tong that you can find with google-fu.

Author's name: http://www.amazon.com/Principles-Techniques-Combinatori-Chen-Chuan-Chong/dp/9810211392

I accept your apology.

Awesome book by the way.

If you would like to become an expert in probability theory, you need to have a solid ground in measure theory. I would suggest to study analysis out of Carothers. This covers most of what Rudin covers but I find it easier to read, and it goes into more detail about measure and Lebesgue integration on the real line. If you work through this, you'll have a solid background for heavier measure theory books and for upper level probability theory.

Principles of Statistics (Bulmer) - this is a very nice introduction to probability and statistics. It takes you through the important distributions (binomial, normal, poisson, etc), laws of probability, central limit theorem, etc. And it's like $10 as an eBook or $15 in paperback.

http://www.amazon.com/Principles-Statistics-Dover-Books-Mathematics/dp/0486637603/ref=sr_1_5?ie=UTF8&qid=1463424228&sr=8-5&keywords=statistics

Found Naked Statistics to be a great casual read.

https://www.amazon.com/dp/1480590185

I really enjoyed Dantzig's, Number. It more explores the development of the numerical system, but I think that is tied into what you are interested in. The book doesnt really get too far into modern notation though.

Ah, this is great! Thank you :)

I didn't manage to find the book I need for MATH 3215 though. Is there any way you could get Probability and Statistical Inference, Ninth Edition by Hogg, Tanis and Zimmerman?

Ah yes, if you ever get stuck there's a lovely book to consult along the way!

Introduction to Linear Algebra by Gilbert Strang (https://www.amazon.com/Introduction-Linear-Algebra-Gilbert-Strang/dp/0980232775/)

(Quite good when taken along with the online course: https://www.youtube.com/playlist?list=PLE7DDD91010BC51F8)

I like the format in which they are written.

For people who want a good grasp of Cantor's idea in prose. I heartily recommend the book Everything and More by the incomparable David Foster Wallace.

This is really good. But you should be aware of this might be outdated. The basics are the same. I browsed someone has recommended Andrew Ng's course. I've been doing this for my master's degree. You definitely need to know math and statistics. For statistics, I recommend you check this out :https://www.openintro.org/stat/textbook.php
they have high school, university, maybe middle school. Anyway, so many people recommend you lots of things, but I don't think most of them consider you are 13 years old (No offence ). Some knowledge in math are a bit difficult for you to understand at the moment, but don't worry; you can remember it first and try to find some introductionary book. such as https://www.amazon.com/Naked-Statistics-Stripping-Dread-Data/dp/1480590185 it's a good read anyway.

I really believe that Michael Kelly's "Humongous Book of" series are the best resources for getting through all math classes up to Calculus II. These books contain every single type of problem you will ever encounter in Algebra I & II, Geometry, Trig, and Calc I & II, all solved in great detail. They are like Schaums Outlines but much more reliable.

https://www.amazon.com/Humongous-Basic-Pre-Algebra-Problems-Books/dp/1615640835

https://www.amazon.com/Humongous-Book-Algebra-Problems-Books/dp/1592577229

https://www.amazon.com/Humongous-Book-Geometry-Problems-Books/dp/1592578640

https://www.amazon.com/Humongous-Book-Trigonometry-Problems-Comprehensive/dp/1615641823

https://www.amazon.com/Humongous-Book-Calculus-Problems-Books/dp/1592575129

Since you're an applied math PhD, maybe the following are good. They are not applied though.

This is the book for first year statistics grad students at OSU.
http://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126/ref=sr_1_1?ie=UTF8&qid=1368662972&sr=8-1&keywords=casella+berger

But, I like Hogg/Craig much more.
http://www.amazon.com/Introduction-Mathematical-Statistics-7th-Edition/dp/0321795431/ref=pd_sim_b_2

I believe each can be found in international editions, and for download on the interwebs.

This may be good for example:
http://www.amazon.com/The-Humongous-Book-Algebra-Problems/dp/1592577229/ref=pd_sim_b_5?ie=UTF8&refRID=0H1GD8HDQZB58PWTY0F5
You could take a look and see if it suits you.

But don't trust me on this. Others on /r/learnmath or /r/matheducation may be more knowledgeable than me about good algebra workbooks.

PDF WARN: Introduction to Math Stat by Hogg

Not to be confused with Probability and Math Stat by Tannis and Hogg which is a "first semester" course.

Good blend of theory and "talky-ness", good exercises that test your understanding, most should be do-able from just applying the basics.

The Math Book. Beautiful pictures on every page, and MOST of the pages are simple enough that a literate six-year-old could understand them. I've never met a person who didn't enjoy flipping through that book.

...so even if your kid doesn't like it, you will ;)

What about Number: The Language of Science by Tobias Dantzig? I came across it at a Border's closing and rather enjoyed it.

http://www.amazon.com/Number-Language-Science-Tobias-Dantzig/dp/0452288118/ref=sr_1_1?ie=UTF8&qid=1347469608&sr=8-1&keywords=Number

That's pretty cool. Unfortunately finding geodesics is a pain because you end up trying to solve evil nonlinear systems of differential equations. This is a great book if you're interested in learning some more about calculus of variations. If you have any questions I can try to answer them.

There is a very interesting book called Number which explores this a bit. It goes through the history of the development of the concept of what a number is, including the shift to writing math with symbols instead of words.

I can't recommend this book highly enough.

I quite like Schutz’s book:

Geometrical Methods of Mathematical Physics

https://www.amazon.co.uk/dp/0521298873/

I think it's this: https://www.amazon.com/Introduction-Mathematical-Statistics-Robert-Hogg/dp/0321795431/ref=mt_hardcover?_encoding=UTF8&me=

But really, if I remember right, they "use" it the same way we "used" the textbook in 400. (I do like both books though.)

I learned from Baez & Muniain; Gauge Fields, Knots, and Gravity.

Toward the end of the course, I met Brian Greene at a public talk, and he recommended Schutz; Geometrical Methods of Mathematical Physics.

Try the two:

https://www.amazon.com/Introduction-Mathematical-Statistics-Robert-Hogg/dp/0321795431

https://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126

introduction to mathematical statistics by craig and statistical inference by george casella.

Number: The Language of Science

Changed my thinking about math/science to be more compatible with my artistic/philosophical leanings.

http://www.amazon.com/The-Humongous-Book-Algebra-Problems/dp/1592577229

Maybe? I'm thinking about picking this up when I finish Khan academy algebra.

found it! Apparently they've gone through several editions and added a coauthor since I bought my copy.

My father is a statistician and he is the one who recommended Hogg and Craig when I complaining about Casella and Berger. I spent a summer working my way through Hogg and Craig and then reviewed everything from my classes that previous year as my way for studying for the written quals. I passed so it worked. And then I promptly forgot everything.

This book supposedly covers the proof and much of the background material: Modular Forms and Fermat's Last Theorem (Springer). Of course, you'll probably have to consult many, many other books along the way, but this looks good as a point of reference.

Everything and More by David Foster Wallace. Just got back into calculus, mostly so I can have at least a literate understanding of it. Not sure how math majors feel about it.

Take a 'probability theory' class that covers or read something like this Dover book:

http://www.amazon.com/Probability-Theory-Concise-Course-Mathematics/dp/0486635449/

The statistics survey courses are generally horrible.

My intro/grad class used Bulmer's book. Its an enjoyable read, easy to follow and answers to odd exercises in the back... and a hell of alot cheaper than current textbooks.

It's a lot of work but with this book I lost my math anxiety and actually started to enjoy math. The author's philosophy is the only way to get better at algebra is to just do a lot of algebra, it starts out with the most basic fundamentals you need to know too, like if you have trouble with negative numbers or fractions (as I did). It's possible you just need a recap on the foundational stuff you forgot in grade school + more practice. By the end of the book you'll be working with functions and logarithms and you'll understand it.

Absolutely. The stress tensor is a (2, 0) tensor (called contravariant in the physicists definition), which means that it takes two vector inputs to produce a real number.

If you input a vector, say e1 (this may be x-hat, the unit vector in the x direction in Cartesian coordinates), it will return a vector which represents the force per unit area in that direction. It actually returns a 1-form (covariant vector), but in the case of the stress tensor, which is a Cartesian tensor, covariant vectors are the same as contravariant vectors, their duals.

This operation is called tensor contraction, where the tensor only acts on one input and returns another tensor of rank (n-1, m-1), or in the case of the stress tensor, it returns a (1, 0) tensor which is just a covariant vector, or in the case of cartesian tensors, it is just a vector (contravariant).

I encourage anybody who is interested in this stuff to read Schutz's Geometrical Methods of Mathematical Physics, as this book describes tensors fully in the newer language (my definition number 2), and does so with applications to physics. Most tensors in physics are taught in the old indices/transformation law language, and can be quite confusing for first timers.

this is one of the best for self teaching. the examples are very clear so you don't get tripped up on them jumping steps. You will need to get more problems from somewhere like a more formal textbook, but this will help you get the idea of what to do instead of fuming at an impasse.

http://www.amazon.com/The-Humongous-Book-Calculus-Problems/dp/1592575129

there's also trig and precalc versions if he needs the review.

http://www.amazon.com/The-Humongous-Book-Algebra-Problems/dp/1592577229/ref=pd_bxgy_b_text_z

http://www.amazon.com/The-Humongous-Book-Trigonometry-Problems/dp/1615641823/ref=pd_bxgy_b_text_y

I'm late to this party, but as a lot of other people have said, missing a negative sign somewhere is not an indication that you're bad at math. What is important in math is understanding why things are the way that they are. If you can look at the spot where you missed a negative sign and understand exactly why there should have been a negative sign there then you're doing fine. Being good at math isn't so much about performing the calculations—I mean, computers can find the roots of a quadratic function polynomial pretty reliably, so probably no one's going to hire you to do that by hand—but it's following the chain of reasoning that takes you from problem to solution and understanding it completely.

That said, there are things you can do to make yourself better at performing the calculations. Go back to basics, and I mean wayyyy back to like grade 5. A lot of students are seriously lacking skills that they should have mastered in around grade 5, and that will really screw up your ability to do algebra well. For instance, know your times tables. Know, and I mean really know and understand, how arithmetic involving fractions works: how and why and when do we put two fractions over a common denominator, what does it mean to multiply and divide by a fraction, and so on. It's elementary stuff but if you can't do it with numbers then you'll have an even harder time doing it with x's and y's. Make sure you understand the rules of exponents: Do you know how to simplify (a^(2)b^(3))^(2)? How about (a^(3)b)/(ab^(5)) How about √(3^(4))? What does it mean to raise a number to a negative power? What about a fractional power? These things need to be drilled into you so that you don't even think twice about them, and the only way to make it that way is to go through some examples really carefully and then do as many problems as you can. Try to prove the things to yourself: why do exponents behave the way that they do? Go out and get yourself something like this and just work through it and make sure you understand exactly why everything is the way that it is.

Feel free to PM me if you are stuck on specific stuff.

No worries.

There's also a book called Humongous Book of Algebra Problems

I can't remember, honestly. That's how bad it was. I'll dig out my notes tomorrow.

Prob Theory and Math Stats together was basically this book.

I've been working my way through the Humongous Book of Algebra Problems. It's about a thousand math problems with complete (and very good explanations). The only way to get good is to get out the paper and plow through problems. I supplement that with videos from Khan Academy (which has it's own math quiz system that is also excellent). I try to do every problem, even if I hate it (looking at you matrices!). And if I get it wrong, no matter the mistake, I re-do the whole problem.

After I do that one, it's on to the Humongous book on Trig. Then calculus. All for the randy hell of it (I grew up with an interest in science and bad math teachers).