This is a compilation of what I gathered from reading on the internet about self-learning higher maths, I haven't come close to reading all this books or watching all this lectures, still I hope it helps you.

General Stuff:
The books here deal with large parts of mathematics and are good to guide you through it all, but I recommend supplementing them with other books.

1. Mathematics: A very Short Introduction : A very good book, but also very short book about mathematics by Timothy Gowers, a Field medalist and overall awesome guy, gives you a feelling for what math is all about.
   
   
2. Concepts of Modern Mathematics: A really interesting book by Ian Stewart, it has more topics than the last book, it is also bigger though less formal than Gower's book. A gem.
   
   
3. What is Mathematics?: A classic that has aged well, it's more textbook like compared to the others, which is good because the best way to learn mathematics is by doing it. Read it.
   
   
4. An Infinitely Large Napkin: This is the most modern book in this list, it delves into a huge number of areas in mathematics and I don't think it should be read as a standalone, rather it should guide you through your studies.
   
   
5. The Princeton Companion to Mathematics: A humongous book detailing many areas of mathematics, its history and some interesting essays. Another book that should be read through your life.
   
   
6. Mathematical Discussions: Gowers taking a look at many interesting points along some mathematical fields.
   
   
7. Technion Linear Algebra Course - The first 14 lectures: Gets you wet in a few branches of maths.
   
   Linear Algebra: An extremelly versatile branch of Mathematics that can be applied to almost anything, also the first "real math" class in most universities.
   
   
8. Linear Algebra Done Right: A pretty nice book to learn from, not as computational heavy as other Linear Algebra texts.
   
   
9. Linear Algebra: A book with a rather different approach compared to LADR, if you have time it would be interesting to use both. Also it delves into more topics than LADR.
   
   
10. Calculus Vol II : Apostols' beautiful book, deals with a lot of lin algebra and complements the other 2 books by having many exercises. Also it doubles as a advanced calculus book.
    
    
11. Khan Academy: Has a nice beginning LinAlg course.
    
    
12. Technion Linear Algebra Course: A really good linear algebra course, teaches it in a marvelous mathy way, instead of the engineering-driven things you find online.
    
    
13. 3Blue1Brown's Essence of Linear Algebra: Extra material, useful to get more intuition, beautifully done.
    
    Calculus: The first mathematics course in most Colleges, deals with how functions change and has many applications, besides it's a doorway to Analysis.
    
    
14. Calculus: Tom Apostol's Calculus is a rigor-heavy book with an unorthodox order of topics and many exercises, so it is a baptism by fire. Really worth it if you have the time and energy to finish. It covers single variable and some multi-variable.
    
    
15. Calculus: Spivak's Calculus is also rigor-heavy by Calculus books standards, also worth it.
    
    
16. Calculus Vol II : Apostols' beautiful book, deals with many topics, finishing up the multivariable part, teaching a bunch of linalg and adding probability to the mix in the end.
    
    
17. MIT OCW: Many good lectures, including one course on single variable and another in multivariable calculus.
    
    Real Analysis: More formalized calculus and math in general, one of the building blocks of modern mathematics.
    
    
18. Principle of Mathematical Analysis: Rudin's classic, still used by many. Has pretty much everything you will need to dive in.
    
    
19. Analysis I and Analysis II: Two marvelous books by Terence Tao, more problem-solving oriented.
    
    
20. Harvey Mudd's Analysis lectures: Some of the few lectures on Real Analysis you can find online.
    
    Abstract Algebra: One of the most important, and in my opinion fun, subjects in mathematics. Deals with algebraic structures, which are roughly sets with operations and properties of this operations.
    
    
21. Abstract Algebra: Dummit and Foote's book, recommended by many and used in lots of courses, is pretty much an encyclopedia, containing many facts and theorems about structures.
    
    
22. Harvard's Abstract Algebra Course: A great course on Abstract Algebra that uses D&F as its textbook, really worth your time.
    
    
23. Algebra: Chapter 0: I haven't used this book yet, though from what I gathered it is both a category theory book and an Algebra book, or rather it is a very different way of teaching Algebra. Many say it's worth it, others (half-jokingly I guess?) accuse it of being abstract nonsense. Probably better used after learning from the D&F and Harvard's course.
    
    There are many other beautiful fields in math full of online resources, like Number Theory and Combinatorics, that I would like to put recommendations here, but it is quite late where I live and I learned those in weirder ways (through olympiad classes and problems), so I don't think I can help you with them, still you should do some research on this sub to get good recommendations on this topics and use the General books as guides.

I'd like to give you my two cents as well on how to proceed here. If nothing else, this will be a second opinion. If I could redo my physics education, this is how I'd want it done.

If you are truly wanting to learn these fields in depth I cannot stress how important it is to actually work problems out of these books, not just read them. There is a certain understanding that comes from struggling with problems that you just can't get by reading the material. On that note, I would recommend getting the Schaum's outline to whatever subject you are studying if you can find one. They are great books with hundreds of solved problems and sample problems for you to try with the answers in the back. When you get to the point you can't find Schaums anymore, I would recommend getting as many solutions manuals as possible. The problems will get very tough, and it's nice to verify that you did the problem correctly or are on the right track, or even just look over solutions to problems you decide not to try.

Basics

I second Stewart's Calculus cover to cover (except the final chapter on differential equations) and Halliday, Resnick and Walker's Fundamentals of Physics. Not all sections from HRW are necessary, but be sure you have the fundamentals of mechanics, electromagnetism, optics, and thermal physics down at the level of HRW.

Once you're done with this move on to studying differential equations. Many physics theorems are stated in terms of differential equations so really getting the hang of these is key to moving on. Differential equations are often taught as two separate classes, one covering ordinary differential equations and one covering partial differential equations. In my opinion, a good introductory textbook to ODEs is one by Morris Tenenbaum and Harry Pollard. That said, there is another book by V. I. Arnold that I would recommend you get as well. The Arnold book may be a bit more mathematical than you are looking for, but it was written as an introductory text to ODEs and you will have a deeper understanding of ODEs after reading it than your typical introductory textbook. This deeper understanding will be useful if you delve into the nitty-gritty parts of classical mechanics. For partial differential equations I recommend the book by Haberman. It will give you a good understanding of different methods you can use to solve PDEs, and is very much geared towards problem-solving.

From there, I would get a decent book on Linear Algebra. I used the one by Leon. I can't guarantee that it's the best book out there, but I think it will get the job done.

This should cover most of the mathematical training you need to move onto the intermediate level physics textbooks. There will be some things that are missing, but those are usually covered explicitly in the intermediate texts that use them (i.e. the Delta function). Still, if you're looking for a good mathematical reference, my recommendation is Lua. It may be a good idea to go over some basic complex analysis from this book, though it is not necessary to move on.

Intermediate

At this stage you need to do intermediate level classical mechanics, electromagnetism, quantum mechanics, and thermal physics at the very least. For electromagnetism, Griffiths hands down. In my opinion, the best pedagogical book for intermediate classical mechanics is Fowles and Cassidy. Once you've read these two books you will have a much deeper understanding of the stuff you learned in HRW. When you're going through the mechanics book pay particular attention to generalized coordinates and Lagrangians. Those become pretty central later on. There is also a very old book by Robert Becker that I think is great. It's problems are tough, and it goes into concepts that aren't typically covered much in depth in other intermediate mechanics books such as statics. I don't think you'll find a torrent for this, but it is 5 bucks on Amazon. That said, I don't think Becker is necessary. For quantum, I cannot recommend Zettili highly enough. Get this book. Tons of worked out examples. In my opinion, Zettili is the best quantum book out there at this level. Finally for thermal physics I would use Mandl. This book is merely sufficient, but I don't know of a book that I liked better.

This is the bare minimum. However, if you find a particular subject interesting, delve into it at this point. If you want to learn Solid State physics there's Kittel. Want to do more Optics? How about Hecht. General relativity? Even that should be accessible with Schutz. Play around here before moving on. A lot of very fascinating things should be accessible to you, at least to a degree, at this point.

Advanced

Before moving on to physics, it is once again time to take up the mathematics. Pick up Arfken and Weber. It covers a great many topics. However, at times it is not the best pedagogical book so you may need some supplemental material on whatever it is you are studying. I would at least read the sections on coordinate transformations, vector analysis, tensors, complex analysis, Green's functions, and the various special functions. Some of this may be a bit of a review, but there are some things Arfken and Weber go into that I didn't see during my undergraduate education even with the topics that I was reviewing. Hell, it may be a good idea to go through the differential equations material in there as well. Again, you may need some supplemental material while doing this. For special functions, a great little book to go along with this is Lebedev.

Beyond this, I think every physicist at the bare minimum needs to take graduate level quantum mechanics, classical mechanics, electromagnetism, and statistical mechanics. For quantum, I recommend Cohen-Tannoudji. This is a great book. It's easy to understand, has many supplemental sections to help further your understanding, is pretty comprehensive, and has more worked examples than a vast majority of graduate text-books. That said, the problems in this book are LONG. Not horrendously hard, mind you, but they do take a long time.

Unfortunately, Cohen-Tannoudji is the only great graduate-level text I can think of. The textbooks in other subjects just don't measure up in my opinion. When you take Classical mechanics I would get Goldstein as a reference but a better book in my opinion is Jose/Saletan as it takes a geometrical approach to the subject from the very beginning. At some point I also think it's worth going through Arnold's treatise on Classical. It's very mathematical and very difficult, but I think once you make it through you will have as deep an understanding as you could hope for in the subject.

Sure! I'll just assume knowledge of the more common stuff like OPS. I'll try to break it into learning resources v. interesting work to be read. Think my suggestions to OP might be structured a bit differently. I'll try to keep it moderately short.

​

Learning

The Book: Playing the Percentages in Baseball set the foundation for a lot of stuff seen today. Win expectancy, lineup optimization, "clutch" hitting, matchups, etc. A lot of it is common knowledge today, but probably because of this work. It's great to see them work through it.

This is a bit of a glossary to many of the more important stats, with links for further reading.

As well, not quite the same, but Analyzing Baseball Data With R is also a great introduction to learning R, which is probably preferable to Python for a lot of baseball-specific work (not to make a general statement on the two, at all).

​

Reading

A lot of good work is, somewhat annoyingly, scattered through the internet on blogs. I don't have time to dig up too much right now but I'll shamelessly plug some work a couple of friends did a few years ago that was rather successful. These are mostly just examples of the what these projects tend to look like.

• The Value of Draft Picks
  
• Projecting Prospects' Hitting Primes
  
• xxFIP p1 p2 p3
  
  Much of the more current work will probably be found on FanGraphs' community submissions section, which I honestly haven't up with recently. I imagine a lot of focus is on using all the new Statcast data.
  
  There's also the MIT Sloan Sports Analytics Conference, where a lot of really cool work comes from. The awesome part about Sloan is that there seems to be a strong emphasis on sharing; I looked for the data/code for two papers I was interested in and ended up getting it for three! My favourite work might be (batter|pitcher)2vec. This is more machine-learning oriented, which I think is a good direction.
  
  ​
  
  That's all I have time for rn, hope that helps!

I'm usually pretty optimistic for people when it comes to posts asking about "how do I get started in sabermetrics" because I was in that position once as well, and it's worked out okay for me, but I want to be a bit more realistic, because I think there is a big red flag that you should recognize in yourself in respect to this.

There are a couple ways to get jobs in fields that require sabermetrics, but you should be aware: there are very few, they are highly competitive, and they require a good amount of work.

The traditional progression for doing sabermetric work is usually something like:

Stage|Level of Sabermetric Experience|Work you're qualified to do|
--:|:--|:--|
1|You look up stats online to form arguments about baseball|Personal blogging, entry-level analytics writing (FanSided, SBN, other sites)|
2|You put stats into a spreadsheet to visualize data or calculate something new to form an argument about baseball|Personal blogging, entry-level analytics writing (FanSided, SBN, other sites), heavier stuff if you're very lucky and a good writer (bigger sites like FanGraphs, Baseball Prospectus), general baseball coverage that isn’t heavily analytical|
3|You use code with baseball stats to visualize data or calculate something new to form an argument about baseball|Heavier analytics writing (SBN, FanGraphs, Baseball Prospectus, The Athletic), entry-level baseball operations work|
4|You use code to create your own models, predictions, and projections about baseball.|Extremely heavy analytics writing, baseball operations/team analytics work|

From your post, it sounds like you're somewhere between #1 and #2 right now. However: "after trying [coding] out I did not like it." You have a very large barrier keeping you from making the jump to stage 3.

If you actually want to go into a sabermetric field as a career, you need to know how to code. Not with Javascript, mind you, but other languages (Python, R, SQL, etc.). I would advise that you try out Python or R (Analyzing Baseball Data with R is an excellent introduction and gives you a lot of practical skills) and see if those really suck you in - and believe me, they need to suck you in. If you really don't like it, don't force yourself to do it and find some other career path, because you won't be able to succeed if you can't enjoy the work that you do.

FanSided has very low barriers of entry and the compensation reflects that - you cannot make a career out of blogging for FanSided. Even if you get to where I am (stage 4), if you're lucky, you might land a contributing position at a site that pays decently for part-time work. There are extremely few people who are somewhere between #3 and #4 who can make a full-time living off of baseball work, and they do it because they like what they do - if you don't like coding and working with baseball data in that environment, you're not going to be able to beat out everybody else who's trying to get there.

Let's say that you work your rear end off, you get to stage three or stage four. What options are available to you? There's maybe a handful of people who work in the "public" sector - that is, writing for websites like FanGraphs, Baseball Prospectus, The Athletic - who make enough money to make sabermetrics their full-time job. It will take a hail fucking mary to land one of those jobs, regardless of how talented you are, and you'll basically need to work double-duty on both sabermetrics and whatever your main hustle is until one of those positions opens up, and even then, you're not guaranteed anything.

You could also work for a team! There are far more positions available, they pay better, you have more data to work with, better job security - this sounds great, right? Problem is, the market cap for analysts are at about 20 per team, so there's something like 600 analyst positions that could be available in the future (I can't promise that the MLB will ever have 600 analysts total at any given time, but that's an upper estimate). And almost half of those are already full! There's not a whole lot of brain drain from the industry, so it is still extremely hard to break in and you're still going to be competing with the absolute best people in the industry. You will have to love to code and do this work because everybody you're competing with already does, and everybody else is willing to work twice as hard for it.

My advice to you is this: try out R or Python with baseball data. See if it's enough to get you addicted. See if it starts to occupy every ounce of free time you have, and you feel comfortable with it, and you're willing to put yourself out there and advertise your own work. I'm a full time student and basically every ounce of my free time is put towards working with this stuff, like it's a second full-time job for the past three years, and I'm still a bit of a ways away from making a living off of this. If you can't learn to love it, your time and energy are best spent elsewhere.

i have three categories of suggestions.

advanced calculus

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

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

/u/another_user_name posted this list a while back. Actual aerospace textbooks are towards the bottom but you'll need a working knowledge of the prereqs first.

Non-core/Pre-reqs:

Mathematics:

Calculus.

1-4) Calculus, Stewart -- This is a very common book and I felt it was ok, but there's mixed opinions about it. Try to get a cheap, used copy.

1-4) Calculus, A New Horizon, Anton -- This is highly valued by many people, but I haven't read it.

1-4) Essential Calculus With Applications, Silverman -- Dover book.

More discussion in this reddit thread.

Linear Algebra

3) Linear Algebra and Its Applications,Lay -- I had this one in school. I think it was decent.

3) Linear Algebra, Shilov -- Dover book.

Differential Equations

4) An Introduction to Ordinary Differential Equations, Coddington -- Dover book, highly reviewed on Amazon.

G) Partial Differential Equations, Evans

G) Partial Differential Equations For Scientists and Engineers, Farlow

More discussion here.

Numerical Analysis

5) Numerical Analysis, Burden and Faires

Chemistry:

1. General Chemistry, Pauling is a good, low cost choice. I'm not sure what we used in school.
   
   

   Physics:
   

   
   2-4) Physics, Cutnel -- This was highly recommended, but I've not read it.
   
   

   Programming:
   

   
   

   Introductory Programming
   

   
   Programming is becoming unavoidable as an engineering skill. I think Python is a strong introductory language that's got a lot of uses in industry.
   
   

2. Learning Python, Lutz
   
   
3. Learn Python the Hard Way, Shaw -- Gaining popularity, also free online.
   
   

   Core Curriculum:
   

   
   

   Introduction:
   

   
   

4. Introduction to Flight, Anderson
   
   

   Aerodynamics:
   

   
   

5. Introduction to Fluid Mechanics, Fox, Pritchard McDonald
   
   
6. Fundamentals of Aerodynamics, Anderson
   
   
7. Theory of Wing Sections, Abbot and von Doenhoff -- Dover book, but very good for what it is.
   
   
8. Aerodynamics for Engineers, Bertin and Cummings -- Didn't use this as the text (used Anderson instead) but it's got more on stuff like Vortex Lattice Methods.
   
   
9. Modern Compressible Flow: With Historical Perspective, Anderson
   
   
10. Computational Fluid Dynamics, Anderson
    
    

    Thermodynamics, Heat transfer and Propulsion:
    

    
    

11. Introduction to Thermodynamics and Heat Transfer, Cengel
    
    
12. Mechanics and Thermodynamics of Propulsion, Hill and Peterson
    
    

    Flight Mechanics, Stability and Control
    

    
    5+) Flight Stability and Automatic Control, Nelson
    
    5+)[Performance, Stability, Dynamics, and Control of Airplanes, Second Edition](http://www.amazon.com/Performance-Stability-Dynamics-Airplanes-Education/dp/1563475839/ref=sr_1_1?ie=UTF8&qid=1315534435&sr=8-1, Pamadi) -- I gather this is better than Nelson
    
    

13. Airplane Aerodynamics and Performance, Roskam and Lan
    
    

    Engineering Mechanics and Structures:
    

    
    3-4) Engineering Mechanics: Statics and Dynamics, Hibbeler
    
    

14. Mechanics of Materials, Hibbeler
    
    
15. Mechanical Vibrations, Rao
    
    
16. Practical Stress Analysis for Design Engineers: Design & Analysis of Aerospace Vehicle Structures, Flabel
    
    6-8) Analysis and Design of Flight Vehicle Structures, Bruhn -- A good reference, never really used it as a text.
    
    
17. An Introduction to the Finite Element Method, Reddy
    
    G) Introduction to the Mechanics of a Continuous Medium, Malvern
    
    G) Fracture Mechanics, Anderson
    
    G) Mechanics of Composite Materials, Jones
    
    

    Electrical Engineering
    

    
    

18. Electrical Engineering Principles and Applications, Hambley
    
    

    Design and Optimization
    

    
    

19. Fundamentals of Aircraft and Airship Design, Nicolai and Carinchner
    
    
20. Aircraft Design: A Conceptual Approach, Raymer
    
    
21. Engineering Optimization: Theory and Practice, Rao
    
    

    Space Systems
    

    
    

22. Fundamentals of Astrodynamics and Applications, Vallado
    
    
23. Introduction to Space Dynamics, Thomson -- Dover book
    
    
24. Orbital Mechanics, Prussing and Conway
    
    
25. Fundamentals of Astrodynamics, Bate, Mueller and White
    
    
26. Space Mission Analysis and Design, Wertz and Larson

Entrepreneur Reading List

1. Disrupted: My Misadventure in the Start-Up Bubble
   
2. The Phoenix Project: A Novel about IT, DevOps, and Helping Your Business Win
   
3. The E-Myth Revisited: Why Most Small Businesses Don't Work and What to Do About It
   
4. The Art of the Start: The Time-Tested, Battle-Hardened Guide for Anyone Starting Anything
   
5. The Four Steps to the Epiphany: Successful Strategies for Products that Win
   
6. Permission Marketing: Turning Strangers into Friends and Friends into Customers
   
7. Ikigai
   
8. Reality Check: The Irreverent Guide to Outsmarting, Outmanaging, and Outmarketing Your Competition
   
9. Bootstrap: Lessons Learned Building a Successful Company from Scratch
   
10. The Marketing Gurus: Lessons from the Best Marketing Books of All Time
    
11. Content Rich: Writing Your Way to Wealth on the Web
    
12. The Web Startup Success Guide
    
13. The Best of Guerrilla Marketing: Guerrilla Marketing Remix
    
14. From Program to Product: Turning Your Code into a Saleable Product
    
15. This Little Program Went to Market: Create, Deploy, Distribute, Market, and Sell Software and More on the Internet at Little or No Cost to You
    
16. The Secrets of Consulting: A Guide to Giving and Getting Advice Successfully
    
17. The Innovator's Solution: Creating and Sustaining Successful Growth
    
18. Startups Open Sourced: Stories to Inspire and Educate
    
19. In Search of Stupidity: Over Twenty Years of High Tech Marketing Disasters
    
20. Do More Faster: TechStars Lessons to Accelerate Your Startup
    
21. Content Rules: How to Create Killer Blogs, Podcasts, Videos, Ebooks, Webinars (and More) That Engage Customers and Ignite Your Business
    
22. Maximum Achievement: Strategies and Skills That Will Unlock Your Hidden Powers to Succeed
    
23. Founders at Work: Stories of Startups' Early Days
    
24. Blue Ocean Strategy: How to Create Uncontested Market Space and Make Competition Irrelevant
    
25. Eric Sink on the Business of Software
    
26. Words that Sell: More than 6000 Entries to Help You Promote Your Products, Services, and Ideas
    
27. Anything You Want
    
28. Crossing the Chasm: Marketing and Selling High-Tech Products to Mainstream Customers
    
29. The Innovator's Dilemma: The Revolutionary Book that Will Change the Way You Do Business
    
30. Tao Te Ching
    
31. Philip & Alex's Guide to Web Publishing
    
32. The Tao of Programming
    
33. Zen and the Art of Motorcycle Maintenance: An Inquiry into Values
    
34. The Inmates Are Running the Asylum: Why High Tech Products Drive Us Crazy and How to Restore the Sanity
    
    

    Computer Science Grad School Reading List
    

    
    

35. All the Mathematics You Missed: But Need to Know for Graduate School
    
36. Introductory Linear Algebra: An Applied First Course
    
37. Introduction to Probability
    
38. The Structure of Scientific Revolutions
    
39. Science in Action: How to Follow Scientists and Engineers Through Society
    
40. Proofs and Refutations: The Logic of Mathematical Discovery
    
41. What Is This Thing Called Science?
    
42. The Art of Computer Programming
    
43. The Little Schemer
    
44. The Seasoned Schemer
    
45. Data Structures Using C and C++
    
46. Algorithms + Data Structures = Programs
    
47. Structure and Interpretation of Computer Programs
    
48. Concepts, Techniques, and Models of Computer Programming
    
49. How to Design Programs: An Introduction to Programming and Computing
    
50. A Science of Operations: Machines, Logic and the Invention of Programming
    
51. Algorithms on Strings, Trees, and Sequences: Computer Science and Computational Biology
    
52. The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation
    
53. The Annotated Turing: A Guided Tour Through Alan Turing's Historic Paper on Computability and the Turing Machine
    
54. Computability: An Introduction to Recursive Function Theory
    
55. How To Solve It: A New Aspect of Mathematical Method
    
56. Types and Programming Languages
    
57. Computer Algebra and Symbolic Computation: Elementary Algorithms
    
58. Computer Algebra and Symbolic Computation: Mathematical Methods
    
59. Commonsense Reasoning
    
60. Using Language
    
61. Computer Vision
    
62. Alice's Adventures in Wonderland
    
63. Gödel, Escher, Bach: An Eternal Golden Braid
    
    

    Video Game Development Reading List
    

    
    

64. Game Programming Gems - 1 2 3 4 5 6 7
    
65. AI Game Programming Wisdom - 1 2 3 4
    
66. Making Games with Python and Pygame
    
67. Invent Your Own Computer Games With Python
    
68. Bit by Bit

Personally I make a distinction between scripting and programming that doesn't really exist but highlights the differences I guess. I consider myself to be scripting if I am connecting programs together by manipulating input and output data. There is lots of regular expression pain and trial-and-error involved in this and I have hated it since my first day of research when I had to write a perl script to extract the energies from thousands of gaussian runs. I appreciate it, but I despise it in equal measure. Programming I love, and I consider this to be implementing a solution to a physical problem in a stricter language and trying to optimise the solution. I've done a lot of this in fortran and java (I much prefer java after a steep learning curve from procedural to OOP). I love the initial math and understanding, the planning, the implementing and seeing the results. Debugging is as much of a pain as scripting, but I've found the more code I write the less stupid mistakes I make and I know what to look for given certain error messages. If I could just do scientific programming I would, but sadly that's not realistic. When you get to do it it's great though.

The maths for comp chem is very similar to the maths used by all the physical sciences and engineering. My go to reference is Arfken but there are others out there. The table of contents at least will give you a good idea of appropriate topics. Your university library will definitely have a selection of lower-level books with more detail that you can build from. I find for learning maths it's best to get every book available and decide which one suits you best. It can be very personal and when you find a book by someone who thinks about the concepts similarly to you it is so much easier.
For learning programming, there are usually tutorials online that will suffice. I have used O'Reilly books with good results. I'd recommend that you follow the tutorials as if you need all of the functionality, even when you know you won't. Otherwise you get holes in your knowledge that can be hard to close later on. It is good supplementary exercise to find a method in a comp chem book, then try to implement it (using google when you get stuck). My favourite algorithms book is Numerical Recipes - there are older fortran versions out there too. It contains a huge amount of detailed practical information and is geared directly at computational science. It has good explanations of math concepts too.

For the actual chemistry, I learned a lot from Jensen's book and Leach's book. I have heard good things about this one too, but I think it's more advanced. For Quantum, there is always Szabo & Ostlund which has code you can refer to, as well as Levine. I am slightly divorced from the QM side of things so I don't have many other recommendations in that area. For statistical mechanics it starts and ends with McQuarrie for me. I have not had to understand much of it in my career so far though. I can also recommend the Oxford Primers series. They're cheap and make solid introductions/refreshers. I saw in another comment you are interested potentially in enzymology. If so, you could try Warshel's book which has more code and implementation exercises but is as difficult as the man himself.

Jensen comes closest to a detailed, general introduction from the books I've spent time with. Maybe focus on that first. I could go on for pages and pages about how I'd approach learning if I was back at undergrad so feel free to ask if you have any more questions.



Out of curiosity, is it DLPOLY that's irritating you so much?

For math you're going to need to know calculus, differential equations (partial and ordinary), and linear algebra.

For calculus, you're going to start with learning about differentiating and limits and whatnot. Then you're going to learn about integrating and series. Series is going to seem a little useless at first, but make sure you don't just skim it, because it becomes very important for physics. Once you learn integration, and integration techniques, you're going to want to go learn multi-variable calculus and vector calculus. Personally, this was the hardest thing for me to learn and I still have problems with it.

While you're learning calculus you can do some lower level physics. I personally liked Halliday, Resnik, and Walker, but I've also heard Giancoli is good. These will give you the basic, idealized world physics understandings, and not too much calculus is involved. You will go through mechanics, electromagnetism, thermodynamics, and "modern physics". You're going to go through these subjects again, but don't skip this part of the process, as you will need the grounding for later.

So, now you have the first two years of a physics degree done, it's time for the big boy stuff (that is the thing that separates the physicists from the engineers). You could get a differential equations and linear algebra books, and I highly suggest you do, but you could skip that and learn it from a physics reference book. Boaz will teach you the linear and the diffe q's you will need to know, along with almost every other post-calculus class math concept you will need for physics. I've also heard that Arfken, Weber, and Harris is a good reference book, but I have personally never used it, and I dont' know if it teaches linear and diffe q's. These are pretty much must-haves though, as they go through things like fourier series and calculus of variations (and a lot of other techniques), which are extremely important to know for what is about to come to you in the next paragraph.

Now that you have a solid mathematical basis, you can get deeper into what you learned in Halliday, Resnik, and Walker, or Giancoli, or whatever you used to get you basis down. You're going to do mechanics, E&M, Thermodynamis/Statistical Analysis, and quantum mechanics again! (yippee). These books will go way deeper into theses subjects, and need a lot more rigorous math. They take that you already know the lower-division stuff for granted, so they don't really teach those all that much. They're tough, very tough. Obvioulsy there are other texts you can go to, but these are the one I am most familiar with.

A few notes. These are just the core classes, anybody going through a physics program will also do labs, research, programming, astro, chemistry, biology, engineering, advanced math, and/or a variety of different things to supplement their degree. There a very few physicists that I know who took the exact same route/class.

These books all have practice problems. Do them. You don't learn physics by reading, you learn by doing. You don't have to do every problem, but you should do a fair amount. This means the theory questions and the math heavy questions. Your theory means nothing without the math to back it up.

Lastly, physics is very demanding. In my experience, most physics students have to pretty much dedicate almost all their time to the craft. This is with instructors, ta's, and tutors helping us along the way. When I say all their time, I mean up until at least midnight (often later) studying/doing work. I commend you on wanting to self-teach yourself, but if you want to learn physics, get into a classroom at your local junior college and start there (I think you'll need a half year of calculus though before you can start doing physics). Some of the concepts are hard (very hard) to understand properly, and the internet stops being very useful very quickly. Having an expert to guide you helps a lot.

Good luck on your journey!

I’m not sure precisely what you mean by “contemporary” or “geometric algebra” or “basic number elements and algebra”. What did you feel was missing from Lang’s book? (I’m not familiar with its contents.)

If you want something in line with the standard high school curriculum, but maybe a bit more rigorous than most, this book by Kiselev was the standard Russian school text for generations (review)

Or you could try the Art of Problem Solving geometry book (site).

There’s a lot of good stuff in Coxeter and Greitzer’s book Geometry Revisited, but I’d say it probably assumes a standard high school geometry course as a prerequisite.

Not really limited to plane geometry, but I really like Hilbert and Cohn-Vossen’s book Geometry and the Imagination (review). I’d recommend getting a used copy of the original printing; the recent ones are printed on demand and not as nice.

Also let me recommend Apostol and Mamikon’s lovely book New Horizons in Geometry (review), though it’s more about calculus than algebra per se.

If you want to study plane curves from a complex number perspective, you could try Zwikker’s 1963 The advanced geometry of plane curves and their applications

If by geometric algebra you mean Grassmann/Clifford/Hestenes style algebra, check out the stuff Jim Smith has been doing, or you could take a look at this thing (I haven’t read it), or try these papers.

They probably aren’t what you’re looking for, but I think Farouki’s Pythagorean Hodograph Curves are pretty neat (that book also has a lot of other interesting material in it). Also neat for formalistic theorizing about algebras for spline curves is Ramshaw’s monograph On Multiplying Points: The Paired Algebras of Forms and Sites (probably a bit abstract for what you want here).

What are your goals? Do you want to design lenses and mirrors for cameras? Model classical mechanics systems? Construct arbitrary shapes out of polynomial curves so you can draw fonts or animate characters on a computer screen? Design cut paths for CNC machines? Approximate transcendental functions by some type of function that you can more easily compute with? Find the prettiest proofs of thousand-year-old theorems about circles? Prepare yourself to study differential geometry or algebraic topology? ...

Mathematical Literature is a genre I don't think many people are aware of, I'm glad you're interested.

The Mathematical Experience is a great survey of mathematical ideas. This book toes the line perfectly - someone not knowledgable of advanced mathematics can follow easily yet the book does not dumb down complicated ideas. This is my top recommendation for anyone thinking about studying mathematics.

If you love geometry, then check out Geometry Revisited by H.S.M Coxeter. Coxeter is one of the greatest mathematicians of his time - he single handedly brought geometry back into vogue as a serious study.

Maybe for lighter reading, Ian Stewart has a bunch of good Mathematical survey books for the "layman" - I'd recommend if you have minimal mathematic knowledge.

There's a yearly collection of mathematical writings that you might like too. I've only bought and read the 2010 edition, but I assume the followups have been great. The essays collected vary from finance, game theory, geometry, social sciences, literature, etc. with connections to mathematics.

Hope you have a fun time with math, good luck!

>is there really no link to the role of this one-form dx and the role of the differential dx?

The differential dx is the one-form "dx", they're the same thing. The differential means nothing. In integration, there's really no need to have the "dx" and when you first do integration in Real Analysis it is usually omitted. If you do measure theory, then you may see d(mu), and this is just to represent the measure against which you're going the integration. It's a bookkeeping device. You can think of "Inta^(b) f(x) dx" as being analogous to "Sumi=a^(b) si". Limits of sums are analogous to limits of integrals, the summands are analogous to the integrand and "dx" is analogous to "i=", it's the same thing just in a different location.

In general, if M is an n-manifold, then it's space of n-forms is one dimensional. This means that it is equivalent to all things of the form w=f(x)dx1dx2...dxn (where these are wedge products). We can then view the integral as a linear function from n-forms to the real numbers. If we want to find Int(w), then we can cut up the manifold into flat pieces using Partitions of Unity, integrate the function f(x) over each of these patches using standard analysis, and then sum it all up.

If we have a line integral of a vector field on M, say the integral of (f(x,y),g(x,y)) along some curve C, then we usually write this as "IntC(f,g)·ds" and usually, we write ds=s'(t)dt so the integral is equal to "Int^(1)(f,g)·s'(t)dt". What we have a function s:[0,1]->M and a 1-form w=fdx+gdy and we're using Pullbacks to pull the 1-form w on M into a 1-form s^()(w)=(f,g)·s'(t)dt on the manifold [0,1]. We then use standard integration (since this is a 1-form on a 1-dimensional manifold) to integrate.

Something like a curve being embedded into a manifold, like above, is called a 1-Simplex and we can view the integral as pairing k-forms with k-simplexes and returning a real number, via integration of pullbacks. Stokes Theorem, which generalizes the divergence theorem, Green's Theorem, and the Fundamental Theorem of Calculus, is a specific statement about this kind of pairing. Generally, we can learn about a k-form (aka vector field) by how it integrates along these simplexes. Things like the Maxwell Equations are specific statements about what we can learn about these k-forms via integration. We can use Stokes Theorem to then, instead, treat them as statements about k-forms themselves rather than having to use integrals. The fact that if F is the electromagnetic force, then there is a 1-form A so that F=dA already takes care of half of Maxwell's equations.

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As for the d operator, if we have a 0-form f(x,y) (aka smooth function), then how are we going to get a 1-form? This is a 1-dimensional thing going into a 2-dimensional thing. What we do is see how f(x,y) interacts with both basis elements and see that we should get fxdx+fydy. This definition does not depend on the basis, so this means that for every 0-form f, we get a natural 1-form df. If we have a 1-form (now in 3D), w=Adx+Bdy+Cdz, where A,B,C are any three smooth functions (they don't have to be the respective partial derivatives of a single function), then how can I get a 2-form? The basis for the 2-forms is dxdy, dxdz and dydz (pretend these are wedges). I can play the same game, see how all the components compare to larger ones. This means I wedge w by each dx,dy,dz and reduce things, so wdx is Axdxdx+ Bxdydx+Cxdzdx = -Bxdxdy-Cxdxdz. Doing this kind of things for all the ones gives the standard formula for dw. We're essentially just combining all the possible wedges and seeing what we get. Following these, we'll always get zero after two successive applications. This is, essentially, because of combinatorics and the fact that partial derivatives commute. In the end, it doesn't depend on basis, so it's natural. The differential operator is just applying derivatives to differential forms in all possible combinations, adding them together and reducing the wedges.

Most importantly, the function d:Tk^() -> Tk+1^(*) so that d^(2)=0, df is the above function and d behaves well under the wedge product. These are the things that matter.

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How I see it, visualizations are a crutch. They're good for a little, but you can't run unless you give them up. Not being able to do Differential Geometry intuitively without having to visualize and interpret everything will eventually become taxing. If, however, the manipulation of the symbols becomes your intuition, then you'll be able to do much more. Visualization is good in Calc 3, but this should be seen as the time to get a feel for the symbols. Differential Geoemtry is glorified Calc 3, but everything is much more abstract and making it concrete will just give you Calc 3 in the end, rather than Differential Geometry. The physical interpretation of Maxwell's equations is elevated to statements in Differential Geometry. These are a lot more powerful, and the definitions are essentially a guidebook to recovering the physical interpretations when you actually need to compute things. I find the best way to gain an intuition for purely symbolic stuff is to use it, accept it for what it is and just go. Occasionally, take a step back, follow the definitions back to the familiar so that you can see how what you do abstractly actually is in line with what you already know.

Of course, I'm a number theorist, I'm pretty biased against physical interpretations. So maybe I'm not completely fair there.

As for references, I've heard that Lee and Spivak are good.

EDIT: As for your edit, I mean that for every smooth function there is an associated 1-form. If f is a smooth function, and D is an element of the tangent space, then D(f) is a real number. We can then view f as a map of tangent vectors D -> D(f). This means that f can be viewed as an element in the cotangent space. The associated cotangent vector is df.

> 1) I was in a school like that. I didn't join. No one hassled me. No one ever said anything to me. The really pro-union people kept to themselves and the vast majority did whatever and could actually care less.

That's good they didn't hassle you. Olof I decided to join a union I would lay back in the shadows and not be adamantly going on tangents why people should join. Glad you weren't harassed. During student teaching there was a teacher without fail that every Friday would wear her union shirt.

>2) Probably not, coming from a perspective of Power. Because it is so large and controls all of CPS, I doubt it would ever want to be split up -- even if those smaller unions are basically CPS lite.

Great point, probably.

>3) I know. Tell me about it. It did all across the state (WI). Most of the old teachers that were stuck in their ways were either asked not to come back by the district; felt like they had to retire or else they would lose all of their benefits (I'm still unclear where this hysteria came from); and, more district flexibility allowed districts to better craft budgets reflective of their priorities. It was a good 5-year window to get hired here.

I remember several years ago it had made news. Is hiring better now? I know you got a lot of flack as a state about the education stuff.

>4) There are many possible answers for this. One answer I've seen is that more conservative-minded people are in professions that typically pay more (accounting, business (management), etc.). Another answer is that that conservative ethos of conserving your wealth (being thrifty) is something harped on if you grow up in a conservative household and it is, therefore, something carried one through one's life. And there are other reasons but you should avoid blanket statements because, actually, if you (taking Republican and Democrat to be proxies for conservative-liberal, respectively) measure it, you'd see that Democrats have slightly, on average, a higher income. Believe it or not, wealth at the top quintile isn't a really good predictor of political ideology. It's actually pretty even split between R and D. In the lower quintile, you'd find a stronger correlation between income and D or R: the poorer one is, the more likely they are to vote D. Yet, a better way to examine that would be racial. There you'd see a clear split between black low income (D) and white low income (R). This whole idea of wealth impacting voting habits and ideology is something political scientists are trying to still better understand. One of the better books, written for the general public, on this subject is (still) (Red State, Blue State, Rich State, Poor State)[https://www.amazon.com/Red-State-Blue-Rich-Poor/dp/0691143935].

Thanks for that in depth answer, I'll be sure to look into that more.

Poor Americans are more liberal than rich Americans in general. But there are distinct patterns of political preferences by income among racial groups and geography. Poor Blacks vote overwhelmingly Democratic, as do poor Whites, except in the South where poor White's preferences are a little murkier. Andrew Gelman does a good job of explaining this in Red State, Blue State, Rich State, Poor State.

The book's paradoxical conclusion is this: Rich states and poor people vote Democratic, while poor states and rich people vote Republican. The way to reconcile the contradiction is that in red states income is a much more robust predictor of your voting habits than in blue states. So, in Connecticut, a rich blue state, income does a less good job of predicting the voting habits of wealthy and middle class voters, many of whom vote Democratic despite their wealth. In Alabama, a poor red state, the votes of people who have above-average incomes are very well predicted by their incomes. And rich, white, Southerners are the most conservative people in the United States. In Mississippi in 2012, Obama only got 10% of whites in the state to vote for him. If you went to a polling station in a rich, white suburb of Atlanta, or Tallahassee, or Jackson, I'd guess that well over 95% of people would be voting Republican. Why? Gelman goes into the nuances in his book, but it has a lot to do with religion and values which are more important to rich Republicans than any other group.

Another interesting finding that comes out of people researching voting habits by demographic characteristics is the existence of two kinds of whites. White people's voting habits basically differ based on what side of the Mason-Dixon they live on. Southern Whites are extremely conservative, whereas Northerners basically split the vote between Democrats and Republicans. That's why Republicans handily win southern states with the country's largest minority populations.


TL;DR The poorest 10% of Americans are more liberal than the richest in general as measured by preference for the Democratic Party. This relationship is less strong outside the South

Source:
Red State, Blue State, Rich State, Poor State

Gelman's research

Crooked Timber

I'm in a similar boat as you. I'm a biologist by trade, but want to delve deeper into statistical analysis with R programming to add a new skill to my career. I'm also a huge baseball fan, especially love it for the stats.

A friend of mine gave me this book for a birthday gift and I've been working way my through it, albeit very slowly. So far (I'm only at Chapter 3), it's been easy to follow and a nice to guide through R. I'd suggest it.

The edx course, that /u/sin7 suggested sounds interesting as well.

We need to make a few definitions.

A group is a set G together with a pair of functions: composition GxG -> G and inverse G -> G, satisfying certain properties, as I'm sure you know.

A topological group is a group G which is also a topological space and such that the composition and inverse functions are continuous. It makes sense to ask if a topological group for example is connected. Every group is a topological group with the discrete topology, but in general there is no way to assign an interesting (whatever that means) topology to a group. The topology is extra information that comes with a topological group.

A Lie group is more than a topological group. A Lie group is a group G that is also a smooth manifold and such that the composition and inverse are smooth functions (between manifolds).

In the same way that O(n) is the set of matrices which fix the standard Euclidean metric on R^n, the Lorentz group O(3,1) is the set of invertible 4x4 matrices which fix the Minkowski metric on R^4. The Lorentz group inherits a natural topology from the set of all 4x4 matrices which is homeomorphic to R^16. It is some more work to show that the Lorentz group in fact smooth, that is, a Lie group.

It is easy to see the Lorentz group is not connected: it contains orientation preserving (det 1) matrices and orientation reversing (det -1) matrices. All elements are invertible (det nonzero), so the preimage of R+ and R- under the determinant are disjoint connected components of the Lorentz group.

There are lots of references. Munkres Topology has a section on topological groups. Stillwell's Naive Lie Theory seems like a great undergraduate introduction to basic Lie groups, although he restricts to matrix Lie groups and does not discuss manifolds. To really make sense of Lie theory, you also need to understand smooth manifolds. Lee's excellent Introduction to Smooth Manifolds is an outstanding introduction to both. There are lots of other good books out there, but this should be enough to get you started.

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.

When you say everyday calculations I'm assuming you're talking about arithmetic, and if that's the case you're probably just better off using you're phone if it's too complex to do in you're head, though you may be interested in this book by Arthur Benjamin.

I'm majoring in math and electrical engineering so the math classes I take do help with my "everyday" calculations, but have never really helped me with anything non-technical. That said, the more math you know the more you can find it just about everywhere. I mean, you don't have to work at NASA to see the technical results of math, speech recognition applications like Siri or Ok Google on you're phone are insanely complex and far from a "solved" problem.

Definitely a ton of math in the medical field. MRIs and CT scanners use a lot of physics in combination with computational algorithms to create images, both of which require some pretty high level math. There's actually an example in one of my probability books that shows how important statistics can be in testing patients. It turns out that even if a test has a really high accuracy, if the condition is extremely rare there is a very high probability that a positive result for the test is a false positive. The book states that ~80% of doctors who were presented this question answered incorrectly.

That's quite the opinion, my mathematician friend. I'm sure it's not all that off-putting from a physicist's perspective. The reason I called it insightful was strictly for its geometric description of contravariance and covariance (with respect to an orthonormal basis). The diagram it provides (on page six) is one of the most enlightening ones I've ever had presented to me, for it really clarified in my head why the metric tensor in the Euclidean plane can be taken as the Kronecker delta. Sometimes it's nice to just have something to hang your hat on so that you can move on with your own research.

Though if any physicists are looking for a nice introduction to differential geometry, which is the landscape for these concepts, I highly recommend John Lee's Intro to Smooth Manifolds. I agree that it's enlightening and serves one well to have a firm understanding of geometry.

edit: justified some comments

First, please make sure everyone understands they are capable of teaching the entire subject without a textbook. "What am I to teach?" is answered by the Common Core standards. I think it's best to free teachers from the tyranny of textbooks and the entire educational system from the tyranny of textbook publishers. If teachers never address this, it'll likely never change.

Here are a few I think are capable to being used but are not part of a larger series to adopt beyond one course:
Most any book by Serge Lang, books written by mathematicians and without a host of co-writers and editors are more interesting, cover the same topics, more in depth, less bells, whistles, fluff, and unneeded pictures and other distracting things, and most of all, tell a coherent story and argument:

Geometry and solutions

Basic Mathematics is a precalculus book, but might work with some supplementary work for other classes.

A First Course in Calculus

For advanced students, and possibly just a good teacher with all students, the Art of Problem Solving series are very good books:
Middle & high school:
and elementary linked from their main page. I have seen the latter myself.

Some more very good books that should be used more, by Gelfand:

The Method of Coordinates

Functions and Graphs

Algebra

Trigonometry

Lines and Curves: A Practical Geometry Handbook

Honestly, if you're wanting an understanding of statistics, I'd recommend Statistics for Dummies. Don't be deceived by the title, you'll still have to do some real thinking on your own to grasp the ideas discussed. You might consider using textbooks or other online resources as secondary supports to your study.

I can also give you a basic breakdown of the topics you'd want to develop an understanding of in beginning to study statistics.

Descriptive Statistics

Descriptive statistics is all about just describing your sample. Major ideas in being able to describe the sample are measures of center (e.g., mean, median, mode), measures of variation (e.g., standard deviation, variance, range, interquartile range), and distributions (e.g., uniform, bell-curve/normally distributed, skewed left/right).

Inferential Statistics

There is a TON of stuff related to this. However, I would first recommend beginning with making sure you have some basic understanding of probability (e.g., events, independence, mutual exclusivity) and then study sampling distributions. Because anything you make an inference about will depending upon the measures in your sample, you need to have a sense of what kinds of samples are possible (and most likely) when you gather data to form one. One of the most fundamental ideas of inferential statistics is based upon these ideas, The Central Limit Theorem. You'll want to make sure you understand what it means before progressing to making inferences.

With that background, you'll be ready to start studying different inferences (e.g., independent/dependent sample t-tests). Again, there are a lot of different kinds of inference tests out there, but I think the most important thing to emphasize with them is the importance of their assumptions. Various technologies will do all of the number crunching for you, but you have to be the one to determine if you're violating any assumptions of the test, as well as interpret what the results mean.

As a whole, I would encourage you to focus on understanding the big ideas. There is a lot of computation involved with statistics, but thanks to modern technology, you don't have to get bogged down in it. As a whole, keep pushing towards understanding the ideas and not getting bogged down in the fine-grained details and processes first, and it will help you develop a firm grasp of much of the statistics out there.

Yes. This is a classic question and the typical answer is

f(x) = x^2 sin(1/x) if x != 0

f(x) = 0 if x = 0

The proof that f is continuous, and f' exists but is not continuous is left as an exercise for the reader. :-)

The book Counterexamples in Analysis has this and more. Having this book handy will do wonders for you and your class and I highly recommend it. Thank god Dover got hold of the copyright and re-printed it, it is a great book and the original is hard to find.

Here are some books I'd recommend.

General Books

These are general books that are more focused on proving things per se. They'll use examples from basic set theory, geometry, and so on.

1. How to Prove It: A Structured Approach by Daniel Velleman
   
2. How to Solve It: A New Aspect of Mathematical Method by George Pólya
   
   Topical Books
   
   For learning topically, I'd suggest starting with a topic you're already familiar with or can become easily familiar with, and try to develop more rigor around it. For example, discrete math is a nice playground to learn about proving things because the topic is both deep and approachable by a beginning math student. Similarly, if you've taken AP or IB-level calculus then you'll get a lot of out a more rigorous treatment of calculus.
   
   

• An Invitation to Discrete Mathematics by Jiří Matoušek and Jaroslav Nešetřil
  
• Discrete Mathematics: Elementary and Beyond by László Lovász and Jaroslav Pelikan
  
• Proofs from THE BOOK by Martin Aigner and Günter Ziegler
  
• Calculus by Michael Spivak
  
  I have a special place in my hear for Spivak's Calculus, which I think is probably the best introduction out there to math-as-she-is-spoke. I used it for my first-year undergraduate calculus course and realized within the first week that the "math" I learned in high school — which I found tedious and rote — was not really math at all. The folks over at /r/calculusstudygroup are slowly working their way through it if you want to work alongside similarly motivated people.
  
  General Advice
  
  One way to get accustomed to "proof" is to go back to, say, your Algebra II course in high school. Let's take something I'm sure you've memorized inside and out like the quadratic formula. Can you prove it?
  
  I don't even mean derive it, necessarily. It's easy to check that the quadratic formula gives you two roots for the polynomial, but how do you know there aren't other roots? You're told that a quadratic polynomial has at most two distinct roots, a cubic polynomial has a most three, a quartic as most four, and perhaps even told that in general an n^(th) degree polynomial has at most n distinct roots.
  
  But how do you know? How do you know there's not a third root lurking out there somewhere?
  
  To answer this you'll have to develop a deeper understanding of what polynomials really are, how you can manipulate them, how different properties of polynomials are affected by those manipulations, and so on.
  
  Anyways, you can revisit pretty much any topic you want from high school and ask yourself, "But how do I really know?" That way rigor (and proofs) lie. :)

(Note: I wrote this elsewhere)

Discrete Mathematics. It teaches the basics of the following 5 key concepts in theoretical computer science:

• function
  
• set
  
• recursion
  
• proposition
  
• proof
  
  When you master these concepts, you will see that all hairy, formal definitions boil down to functions, sets, and propositions (with quantifiers). Recursion appears in many interesting structures in computer science, as well as in proofs of theorems (which are themselves propositions).
  
  I don't know of any good online introduction to discrete math, but here are a few book ideas.
  
  
• For a good textbook on discrete mathematics, check Schaum's Outline of Discrete Mathematics. It's also cheap.
  
• For a less textbook-ish overview written in an entertaining manner, Discrete Mathematics: Elementary and Beyond is my personal favourite.
  
• If you want to learn-by-doing, then The Haskell Road to Logic, Maths and Programming teaches you discrete math by having you manipulate discrete math concepts in the programming language Haskell. Here is a good free online Haskell tutorial.

Ordinary Differential Equations and Dynamical Systems by Gerald Teschl is a really good intro to ODE theory on the first-year graduate level. It also has the benefit of being freely available online. At the undergrad level, I haven't used this book personally but Differential Equations, Dynamical Systems, & and Introduction to Chaos by Hirsch, Smale, and Devaney seems to be a common choice.

For PDE, there are lots of standard texts that don't take the "toolbox" approach: at the undergrad level you have Walter Strauss, and at the begininning graduate level you've got Evans and Folland. For a slightly more advanced treatment, I like John Hunter's PDE notes, also free online.

Prerequisites: you should have a firm grasp of introductory analysis, say at the level of Baby Rudin, before diving into either of these subjects. You should also know your undergraduate linear algebra well.

I'm preparing to go from a pure maths/stats background to an applied maths graduate program in the fall, and I bought both of these books:

1. Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, by H.M. Schey, and
   
2. A Student's Guide to Maxwell's Equations, by Daniel Fleisch.
   
   Hoping they'll help me when I get home to read them, maybe they'll help you too? The Amazon feedback seems pretty positive. Good luck!

1. I was in a school like that. I didn't join. No one hassled me. No one ever said anything to me. The really pro-union people kept to themselves and the vast majority did whatever and could actually care less.
   
   
2. Probably not, coming from a perspective of Power. Because it is so large and controls all of CPS, I doubt it would ever want to be split up -- even if those smaller unions are basically CPS lite.
   
   
3. I know. Tell me about it. It did all across the state (WI). Most of the old teachers that were stuck in their ways were either asked not to come back by the district; felt like they had to retire or else they would lose all of their benefits (I'm still unclear where this hysteria came from); and, more district flexibility allowed districts to better craft budgets reflective of their priorities. It was a good 5-year window to get hired here.
   
   
4. There are many possible answers for this. One answer I've seen is that more conservative-minded people are in professions that typically pay more (accounting, business (management), etc.). Another answer is that that conservative ethos of conserving your wealth (being thrifty) is something harped on if you grow up in a conservative household and it is, therefore, something carried one through one's life. And there are other reasons but you should avoid blanket statements because, actually, if you (taking Republican and Democrat to be proxies for conservative-liberal, respectively) measure it, you'd see that Democrats have slightly, on average, a higher income. Believe it or not, wealth at the top quintile isn't a really good predictor of political ideology. It's actually pretty even split between R and D. In the lower quintile, you'd find a stronger correlation between income and D or R: the poorer one is, the more likely they are to vote D. Yet, a better way to examine that would be racial. There you'd see a clear split between black low income (D) and white low income (R). This whole idea of wealth impacting voting habits and ideology is something political scientists are trying to still better understand. One of the better books, written for the general public, on this subject is (still) (Red State, Blue State, Rich State, Poor State)[https://www.amazon.com/Red-State-Blue-Rich-Poor/dp/0691143935].

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

It would be hard to include all those variables, but you could probably make a simpler model using differential equations from the perspective of dynamical systems. Or maybe it would be more natural to think of it as a discrete dynamical system. The starting of a mosh pit could be thought of as a bifurcation in the system.

A great book on this subject is the one by Hirsch, Smale, and Devaney. Necessary prerequisites are calculus and linear algebra.

As far as growth of the mosh pit I have a feeling that statistical mechanics might be the place to look, but I don't really know much about it.

And sorry I didn't respond sooner, the reddits prevented me :)

It's pretty basic stuff, but the first three chapters of this book was a game-changer for me

https://www.amazon.com/Introduction-Probability-2nd-Dimitri-Bertsekas/dp/188652923X

My mind was blown when I finally understood the connection between random variables and the "basic" probability theory with events and sample spaces. For me they had always been two seperate things.

The notation is also really nice.

Having solid fundamentals makes it much easier to study advanced topics, so I would start here.

There's also a great EDX course which is based on the book, but it's a complement and not a substitute. Get the book.

That is a pretty big red flag. Most departments offer a statistics course for non-math majors, I've TA'd that course before, its not good for people who enjoy math. Make sure there is at the very least a calculus pre-req for the course, and you should take a probability course first anyways.

Probability and Statistics in general is such a great field, it would be really unfortunate if a bad class designed for psych majors turned you off from the subject. I would wait until you can take the right classes so you can at least see the material presented in the right way, if your curious what a course sequence should look like:

1. Calculus based probability (undergrad, math major prob)
   
   
2. Calculus based statistics (undergrad, math major stat)
   
   
3. Measure-theoretic probability (grad prob)
   
   
4. Statistical Theory (grad stat - here is where analysis and some really neat math merges with the stats, but undergrad stat does this to some degree as well.
   
   Here are links to textbooks in the same order:
   
   
5. http://www.amazon.com/First-Course-Probability-9th-Edition/dp/032179477X
   
   
6. http://www.amazon.com/Mathematical-Statistics-Analysis-Available-Enhanced/dp/0534399428
   
   
7. http://www.amazon.com/Probability-Measure-Patrick-Billingsley/dp/1118122372
   
   
8. http://www.amazon.com/Mathematical-Statistics-Selected-Topics-Edition/dp/0132306379
   
   You can find most of those in pdf format somewhere online. I'm not saying those are the best textbooks to use, but they should at least provide a guide so that you can be sure you taking the right courses.
   
   Edit: If your school offers a combined prob/stat course (usually offered for engineers) that has a calc 3 pre-req it would probably serve well as a compact introduction to the subject.
   

Start With "Foundations Of Analysis" By Edmund Landau

http://www.amazon.com/Foundations-Analysis-AMS-Chelsea-Publishing/dp/082182693X

It's a tiny book, but is very good at explaining basic abstract algebra.

Here is the description from Amazon:

"Why does $2 \times 2 = 4$? What are fractions? Imaginary numbers? Why do the laws of algebra hold? And how do we prove these laws? What are the properties of the numbers on which the Differential and Integral Calculus is based? In other words, What are numbers? And why do they have the properties we attribute to them? Thanks to the genius of Dedekind, Cantor, Peano, Frege and Russell, such questions can now be given a satisfactory answer. This English translation of Landau's famous Grundlagen der Analysis-also available from the AMS-answers these important questions."

With the above book you should then have enough knowledge to move on to calculus.

I recommend the two volume series called "Calculus" by Tom M. Apostol.

The first volume is single variable calculus and the second is multivariate calculus

http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051/ref=sr_1_4?ie=UTF8&s=books&qid=1239384587&sr=1-4

http://www.amazon.com/Calculus-Vol-Multi-Variable-Algebra-Applications/dp/0471000078/ref=sr_1_3?ie=UTF8&s=books&qid=1239384587&sr=1-3

There are some really good books that you can use to give yourself a solid foundation for further self-study in mathematics. I've used them myself. The great thing about this type of book is that you can just do the exercises from one side of the book to the other and then be confident in the knowledge that you understand the material. It's nice! Here are my recommendations:

First off, three books on the basics of algebra, trigonometry, and functions and graphs. They're all by a guy called Israel Gelfand, and they're good: Algebra, Trigonometry, and Functions and Graphs.

Next, one of two books (they occupy the same niche, material-wise) on general proof and problem-solving methods. These get you in the headspace of constructing proofs, which is really good. As someone with a bachelors in math, it's disheartening to see that proofs are misunderstood and often disliked by students. The whole point of learning and understanding proofs (and reproducing them yourself) is so that you gain an understanding of the why of the problem under consideration, not just the how... Anyways, I'm rambling! Here they are: How To Prove It: A Structured Approach and How To Solve It.

And finally a book which is a little bit more terse than the others, but which serves to reinforce the key concepts: Basic Mathematics.

After that you have the basics needed to take on any math textbook you like really - beginning from the foundational subjects and working your way upwards, of course. For example, if you wanted to improve your linear algebra skills (e.g. suppose you wanted to learn a bit of machine learning) you could just study a textbook like Linear Algebra Done Right.

The hard part about this method is that it takes a lot of practice to get used to learning from a book. But that's also the upside of it because whenever you're studying it, you're really studying it. It's a pretty straightforward process (bar the moments of frustration, of course).

If you have any other questions about learning math, shoot me a PM. :)

Wow, ambitious! I'd highly recommend V.I. Arnold's book on ODEs: http://www.amazon.com/Ordinary-Differential-Equations-V-Arnold/dp/0262510189 ... not only is it a great book in itself, but it should give you an excellent foundation for differential geometry and more advanced geometric mechanics (e.g., Lagrangian/Hamiltonian mechanics, dynamical systems, etc.).

What level of dynamical systems are we talking here? Graduate or undergraduate. In the former case I would recommend: http://www.amazon.com/Introduction-Dynamical-Encyclopedia-Mathematics-Applications/dp/0521575575
and for an undergraduate approach I would recommend:
http://www.amazon.com/Differential-Equations-Dynamical-Introduction-Mathematics/dp/0123497035
Both are pretty fun introductions to the subject. Good luck in your search

The pymc3 documentation is a good place to start if you enjoy reading through mini-tutorials: pymc3 docs

Also these books are pretty good, the first is a nice soft introduction to programming with pymc & bayesian methods, and the second is quite nice too, albeit targeted at R/STAN.

• Bayesian Methods for Hackers
  
• Doing Bayesian Data Analysis (Kruschke)

Mathematical Statistics and Data Analysis.

I learned from this textbook and have found it quite good. It's pretty expensive, but may be what you're looking for. I really don't know how much statistics your classes covered, but the table of contents should give you a good idea on what to expect.

I also had success with cheap supplemental books from Dover, which can cover quite a lot of undergraduate statistics at an affordable price. I found good use in Statistical Inference by Rohatgi.

Study what you find the most interesting!

Does your linear algebra include the spectral theorem or Jordan canonical form? IMHO, a pure math subject that is relatively the easiest to learn and is useful no matter what you do is linear algebra.

Group theory (representation theory) has also served me well so far.

If you want to learn GR and Hamiltonian mechanics in-depth, learning smooth manifolds would be a must. Smooth manifolds are basically spaces that locally look like Euclidean spaces and we can do calculus on. GR is on a pseudo-Riemannian manifold with changing metric (because of massive stuffs). Hamiltonian mechanics is on a cotangent bundle, which is a symplectic manifold (whereas Lagrangian mechanics is on a tangent bundle.) John Lee's book is a gentle starting point.

Edit: If you feel like the review of topology in the appendix is not enough, Lee also wrote a book on topological manifolds.

The second book that gerschgorin listed is very good, though a little old fashioned.

Since you are finishing up your math major, I'd recommend Hirsch & Smale & Devaney, an excellent book if you have a little bit of mathematical background.

There is also a video series I'm making meant to be a quick overview of many of the key topics. Maybe useful, maybe not. Also, the MIT lectures are excellent.

I'm going to suggest what I suggest to most people your age with promise. Work through Gelfand's set of high school textbooks:

Algebra

Trigonometry

Functions and Graphs

They're written for self-study, so they're structured to help the reader figure things out for themselves. They're challenging at times, but with hard work I'm sure you'll do fine.

You might also pick up a book about proof, like the free Book of Proof and teach yourself how to prove things which is, after all, the basis for math.

I agree with brushing up on your math. EM requires good mathematical intuition as you need to visualize both electric and magnetic field lines. I strong understanding of vector calculus helps with this immensely.

That said: I've heard great things about A Student's Guide to Maxwell's Equations. Note you'll probably be learning things other than just Maxwell's Equations however, such as transmission lines.

I would try Mathematical Statistics and Data Analysis by Rice. The standard intro text for Mathematical Statistics (this is where you get the proofs) is Wackerly, Mendenhall, and Schaeffer but I find this book to be a bit too dry and theoretical (and I'm in math). Calculus is less important than a thorough understanding of how random variables work. Rice has a couple of pretty good chapters on this, but it will require some mathematical maturity to read this book. Good luck!

Here are some great books that I believe you may find helpful :)

• Elements of Algebra by Leonhard Euler
  
• Foundations of Mathematics by Ian Stewart
  
• Geometry Revisited by H. S. M. Coxeter
  
• Excursions in Calculus by Robert M. Young
  
• New Horizons in Geometry by Tom M. Apostol and Mamikon A. Mnatsakanian
  
• The Calculus Gallery: Masterpieces From Newton to Lebesgue by William Dunham
  
• A Primer of Real Functions by Ralph P. Boas
  
• Understanding Analysis by Stephen Abbot
  
• Chaos: Making a New Science by James Gleick
  
• Logic: An Introduction to Elementary Logic by Wilfred Hodges
  
  and last but definitely not least:
  
  
• A Mathematical Gift I, II, III by Kenji Ueno et al.
  
  Later on:
  
  
• Conceptual Mathematics: A First Introduction to Categories by F. William Lawvere
  
• Twelve Landmarks of Twentieth-Century Analysis by D. Choimet & H. Queffélec
  
• Theory of Complex Functions by Reinhold Remmert
  
• Algebraic Combinatorics. Walks, Trees, Tableaux & More by Richard P. Stanley

Depending on how strong your math/stats background is you might consider Statistical Inference by Casella and Berger. It's what we use for our first year PhD Mathematical Statistics course.

That might be a little too difficult if you're not very comfortable with probability theory and basic statistics. If you look at the first few chapters on Amazon and it seems like too much I recommend Mathematical Statistics and Data Analysis by Rice which I guess I would consider a "prequel" to the Casella text. I worked through this in an advanced statistics undergrad course (along with Mostly Harmless Econometrics and the Goldberger's course in Econometrics).

Let's see, if you're interested in Stochastic Models (Random Walks, Markov Chains, Poisson Processes etc), I recommend Introduction to Stochastic Modeling by Taylor and Karlin. Also something I worked through as an undergrad.

If you're looking for the "bible" of PDE - Evans is typically considered the standard at the graduate level. For an undergraduate exposition of differential equations (ODE), then my professor liked to use Zill for ODE and Haberman for PDE.

​

If you're a little more specific I might be able to direct you to better sources - hope you enjoy the above, I have them all and really like them.

I dove into this stuff almost two years ago with very little preparation or background. Now I'm in an MS program for Applied Statistics, and doing quite well. Here are some tips that worked for me:

• If you don't have time to back up and regroup, check out Khan Academy, and this guy's YouTube videos. These can help with specific concepts.
  
  
• If you have time to back up and regroup, check out Coursera, Udacity, EdX, and the other MOOCs. Coursera in particular has some very good courses dealing with statistics.
  
  
• Take a look at Statistics for Dummies and Naked Statistics.
  
  
• Use Reddit and StackOverflow. But use them wisely, and only after you've exhausted other means.
  
  Good luck.

The absolute best book I've found for someone with a frequentist background and undergraduate-level math skills is Doing Bayesian Data Analysis by John Kruschke. It's a fantastic book that goes into mathematical depth only when it needs to while also building your intuition.

The second edition is new and I'd recommend it over the first because of its improved code. It uses JAGS and STAN instead of Bugs, which is Windows-only now.

Discrete mathematics and any proof based math in general is what college based math should be like- if you continue to take upper level math and CS courses, you will undoubtedly face this style of math again. Plug and chug (which is what a lot of calculus is) will no longer be the norm.

There is often a very large learning curve for students who are not used to seeing this type of math- so don't stress out too much about it. Eventually, you'll break a point where everything will make (sort of) sense. I went through the exact same thing when I took discrete for the first time, and I felt like I was getting destroyed on everything (I still suck at some topics) until I suddenly hit a point of clarity where I could see how most topics were tied in together. Mathematics, and especially an introductory discrete course, is cruel in that way- that every topic you learn is inherently related to each other, so if you already fall behind just a little, the mountain to catch up just becomes incredibly massive incredibly fast- and it's hard to even pinpoint a place to even start to catch up.

You may be lost in learning elementary proof techniques, or number theory, and then the next topic (say it's graph theory) utilizes a bunch concepts and previous proofs from number theory, and then the next topic might use something proved in graph theory and number theory, and so on. All of a sudden, nothing makes sense, and to learn topic ___, you need to know graph theory, but to know graph theory, you need to know number theory, but you don't know number theory that well, and some topics in number theory can perhaps be explained by another topic in graph theory (or any topic for that matter) The chain is all interlinked and it may difficult to even see where to start- but it is for this reason that once you cross this steep barrier, most things will suddenly become clear to you.

So I'd advise you to just continue visiting professor office hours, asking more questions, asking for other students' help, doing more and more practice. It may seem like you're getting nowhere, but you're essentially learning a new language right now, so it'll obviously take sometime until you feel as if you know what you're doing. Figuring out where people get the intuition to suggest seemingly random functions or a set of numbers or some assumption will come to you slowly, and slowly you'll break more and more of this chain.

https://www.amazon.com/Discrete-Mathematics-Laszlo-Lovasz/dp/0387955852 is another book my professor enjoyed using as a supplmenet.

These are, in my opinion, some of the best books for learning high school level math:

• I.M Gelfand Algebra {[.pdf] (http://www.cimat.mx/ciencia_para_jovenes/bachillerato/libros/algebra_gelfand.pdf) | Amazon}
  
• I.M. Gelfand The Method of Coordinates {Amazon}
  
• I.M. Gelfand Functions and Graphs {.pdf | Amazon}
  
  These are all 1900's Russian math text books (probably the type that /u/oneorangehat was thinking of) edited by I.M. Galfand, who was something like the head of the Russian School for Correspondence. I basically lived off them during my first years of high school. They are pretty much exactly what you said you wanted; they have no pictures (except for graphs and diagrams), no useless information, and lots of great problems and explanations :) There is also I.M Gelfand Trigonometry {[.pdf] (http://users.auth.gr/~siskakis/GelfandSaul-Trigonometry.pdf) | Amazon} (which may be what you mean when you say precal, I'm not sure), but I do not own this myself and thus cannot say if it is as good as the others :)
  
  
  I should mention that these books start off with problems and ideas that are pretty easy, but quickly become increasingly complicated as you progress. There are also a lot of problems that require very little actual math knowledge, but a lot of ingenuity.
  
  Sorry for bad Englando, It is my native language but I haven't had time to learn it yet.

I wouldn't call it a "branch" exactly, but pathological functions are pretty much the definition of "weird." Things like Weierstrass functions, the Cantor function, the Conway base 13 function. There's a good book with a lot of this stuff in it called Counterexamples in Analysis. There's another one on topology I haven't read yet.

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

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

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

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

> Furthemore, why did the South shift from being a Democratic stronghold to a Republican one?

There have been plenty of threads here about the Southern Strategy and the partisan realignment. The tl;dr is that the Republican Party appealed to racism against blacks and opposition to civil rights among southern white voters. Those voters, previously strongly Democratic voters, switched to supporting the Republican Party, where they remain to this day. (For an academic look, you can see here.)

> Why is it that after '92 the Northeast and West coast became consistently Democratic, and the South and midwest become consistently Republican?

It's largely a function of population demographics. Another tl;dr is that the coasts are far more urbanized than the South and midwest. Highly urban areas tend to be more Democratic-leaning. Essentially, blue states are blue because they're disproportionately urban, while red states are red because they're disproportionately rural. Even in states like California, you see large swaths of Republican counties because they're heavily rural areas.

As for the central thrust of your question, Andrew Gelman would likely argue that, even though rich people tend to vote Republican quite overwhelmingly, 1) there are far more poor people in those blue states, and poor people tend to vote Democratic, and 2) rich people on the coasts care more about social issues that Democrats favor. In general, I think it's just a function of population demographics.

This is a really popular theoretical differential equations book http://www.amazon.com/Ordinary-Differential-Equations-V-I-Arnold/dp/0262510189

It's Ordinary Differential Equations by V.I. Arnold, it's highly regarded and I see people recommend it over on math.stackexchange all the time.

However I'm not sure if it's the kind of book you're looking for because I don't believe it's an introductory book at all. From what I've heard it's pretty advanced.

Hopefully someone more knowledgeable than I can explain whether this book is appropriate for you or not.

I have very few universal recommendations. Think the only one that actually comes to mind is "Introduction to Probability" by Blitzstein and Hwang. It is probably the best book on probability that I've found for a broad audience. It also has a corresponding video lecture series.

If you want any more, please answer this:

• What is your interest?
  
• What is your background?
  
• What do you want to learn to do?
  
  Maybe I can see what I have laying around that meets your criteria.
  

For proofs in general, I like D'Angelo and West's Mathematical Thinking. http://www.amazon.com/Mathematical-Thinking-Problem-Solving-Proofs-Edition/dp/0130144126

For discrete math, especially combinatorics, I loved Miklos Bona's A Walk Through Combinatorics. http://www.amazon.com/Walk-Through-Combinatorics-Introduction-Enumeration/dp/9814335231/

For induction proofs, you check your base case, assume the induction hypothesis (true for k), and then check k+1.

You should be able to manipulate the k+1 term into something involving the k term, and that will then lead to the k+1 conclusion.

Example For all n >= 4, 2^(n) < n!

Base case: n = 4. 2^(4) = 16 < 24 = 4!

IH: Assume true for some k >= 4.

Then 2^(k+1) = 2*2^(k)

2*2^(k) < 2*k! (Induction Hypothesis used here)

2*k! < (k+1)k! (k > 3, so k+1 > 2)

(k+1)k! = (k+1)! (definition of factorial)

The short version is that in a bayesian model your likelihood is how you're choosing to model the data, aka P(x|\theta) encodes how you think your data was generated. If you think your data comes from a binomial, e.g. you have something representing a series of success/failure trials like coin flips, you'd model your data with a binomial likelihood. There's no right or wrong way to choose the likelihood, it's entirely based on how you, the statistician, thinks the data should be modeled. The prior, P(\theta), is just a way to specify what you think \theta might be beforehand, e.g. if you have no clue in the binomial example what your rate of success might be you put a uniform prior over the unit interval. Then, assuming you understand bayes theorem, we find that we can estimate the parameter \theta given the data by calculating P(\theta|x)=P(x|\theta)P(\theta)/P(x) . That is the entire bayesian model in a nutshell. The problem, and where mcmc comes in, is that given real data, the way to calculate P(x) is usually intractable, as it amounts to integrating or summing over P(x|\theta)P(\theta), which isn't easy when you have multiple data points (since P(x|\theta) becomes \prod_{i} P(x_i|\theta) ). You use mcmc (and other approximate inference methods) to get around calculating P(x) exactly. I'm not sure where you've learned bayesian stats from before, but I've heard good things , for gaining intuition (which it seems is what you need), about Statistical Rethinking (https://www.amazon.com/Statistical-Rethinking-Bayesian-Examples-Chapman/dp/1482253445), the authors website includes more resources including his lectures. Doing Bayesian data analysis (https://www.amazon.com/Doing-Bayesian-Data-Analysis-Second/dp/0124058884/ref=pd_lpo_sbs_14_t_1?_encoding=UTF8&psc=1&refRID=58357AYY9N1EZRG0WAMY) also seems to be another beginner friendly book.

Differential forms as they build up to the general Stokes theorem are extremely satisfying because they give you the full picture of multivariable integration generalized to arbitrary k-dimensional objects in n-dimensional spaces. They basically relieve you of that feeling you (maybe) had in calc 3 that there's got to be more to the story than greene's theorem and stokes theorem.

However, I don't know that they give you better intuition for vector calculus and maxwell's equations, eg stuff in R^3. The way I got intuition for those was by doing problems and going through the proofs of curl and divergence from their definitions as limits of integrals. Work through the proof that this is equivalent to the usual definition of curl, and you'll understand curl and stokes theorem. Do the same for divergence

For maxwell's equations, this is an excellent book for intuition.

We are using Mathematical Thinking: Problem Solving & Proofs 2nd Edition. We get lectures notes because the text book is difficult to understand, but they dont really help..

EDIT: I realize now its the same book! Great! Any help?

IQ tests are calibrated to return an average of 100. Absent any evidence to the contrary, we assume the null hypothesis and place all subgroups at the global average. Here's a good starting point if you're interested in learning more.

Educational barriers for African Americans are well documented and muddy the relationship between intelligence and education. You'll note that the educational data I provided earlier was solely for whites, where the relationship is clear cut. I'm unaware of any reliable data for blacks.

Now, since you're presumably white, and we do have that data, would you mind telling the audience a little about your education, and we'll see what inferences we can draw?

Here are some suggestions :

https://www.coursera.org/course/maththink

https://www.coursera.org/course/intrologic

Also, this is a great book :

http://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X/ref=sr_1_5?ie=UTF8&qid=1346855198&sr=8-5&keywords=history+of+mathematics

It covers everything from number theory to calculus in sort of brief sections, and not just the history. Its pretty accessible from what I've read of it so far.


EDIT : I read what you are taking and my recommendations are a bit lower level for you probably. The history of math book is still pretty good, as it gives you an idea what people were thinking when they discovered/invented certain things.

For you, I would suggest :

http://www.amazon.com/Principles-Mathematical-Analysis-Third-Edition/dp/007054235X/ref=sr_1_1?ie=UTF8&qid=1346860077&sr=8-1&keywords=rudin

http://www.amazon.com/Invitation-Linear-Operators-Matrices-Bounded/dp/0415267994/ref=sr_1_4?ie=UTF8&qid=1346860052&sr=8-4&keywords=from+matrix+to+bounded+linear+operators

http://www.amazon.com/Counterexamples-Analysis-Dover-Books-Mathematics/dp/0486428753/ref=sr_1_5?ie=UTF8&qid=1346860077&sr=8-5&keywords=rudin

http://www.amazon.com/DIV-Grad-Curl-All-That/dp/0393969975

http://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0738204536/ref=sr_1_2?s=books&ie=UTF8&qid=1346860356&sr=1-2&keywords=chaos+and+dynamics

http://www.amazon.com/Numerical-Analysis-Richard-L-Burden/dp/0534392008/ref=sr_1_5?s=books&ie=UTF8&qid=1346860179&sr=1-5&keywords=numerical+analysis

This is from my background. I don't have a strong grasp of topology and haven't done much with abstract algebra (or algebraic _____) so I would probably recommend listening to someone else there. My background is mostly in graduate numerical analysis / functional analysis. The Furata book is expensive, but a worthy read to bridge the link between linear algebra and functional analysis. You may want to read a real analysis book first however.

One thing to note is that topology is used in some real analysis proofs. After going through a real analysis book you may also want to read some measure theory, but I don't have an excellent recommendation there as the books I've used were all hard to understand for me.

Interdisciplinary connections spring up from generality. You'd be hard pressed to find a spontaneous connection between something like particle phenomenology and an unrelated field.

To illustrate this idea of generality, consider the methods of statistical mechanics, which are so general that they can be used to describe everything from black holes to ferromagnets. However, the methods have also been used to model neural networks and social dynamics (the latter being accurate enough to successfully recreate historical events.)

What makes statistical mechanics more general than other branches? Probably the fact that it's almost more mathematics than physics, specifically a branch of probability theory regarding highly correlated random variables.

With this in mind, perhaps you'd benefit from focusing your attention on the mathematical ideas that drive physics rather than physics itself. Take the calculus of variations which, whilst developed for problems in classical mechanics, has found applications in mathematical optimisation. Another example being brownian motion, the mathematics of which have been generalised to higher dimensions and applied to finance. The mathematics behind relativity is differential geometry, which has been applied to too many fields to list.

I'd recommend having a look at Mathematical Methods for Physicists by Arfken, Weber and Harris for a broad overview of the methods.

I'd recommend Discrete Mathematics, Elementary and Beyond By Lovász, Pelikán, and Vesztergombi. It's the book I'm using in my undergraduate discrete math course, and I think it's a great introductory book that explores many areas of discrete math, and should allow you to see which field interests you most.

To supplement and expand, this book is a great read, with very clearly presented data, that describes and explores this phenomenon.

One way is picking a distribution with a mode and a "concentration" around that mode that reflects what you have. John Kruschke does an amazing job at explaining how to pick Beta priors based off of that in Doing Bayesian Data Analysis (which, may I note, has the best cover of any statistics book I've ever read).

this book really helped me in undergrad. Has a lot of really good concepts. It went along with a course but it does a great job on its own explaining some of the most relevant concepts to computer science.

I didn't full understand the material that well when I was in school but I wanted to learn it better after school. I, like you, tried to find something to supplement my existing texting books. I came across the A student's guide to maxwell equations and I began to understand more. It's a small book and what the author does is break down what the equation means. One chapter might be just on what does the surface integral mean.. Or another chapter might be on just the E vector. I found breaking it down to be more understandable than trying to take the entire equation(s) in together.

Depending on your level, i have used PDEs by Evans which is very well written, and the most recommended book i know of on the subject. It is pretty advanced though.

The book for my undergrad diff eqs class. I highly recommend it if you have an introductory background in ODEs, but even if you don't (I didn't going in), it's a great book.

Hey I'm a physics BSc turned mathematician.

I would suggest starting with topology and functional analysis. Functional analysis is the foundation of quantum mechanics, and topology is necessary to properly understand manifolds, which are the foundation of relativity.

I would suggest Kreyszig for functional analysis. It's probably the most gentle functional analysis book out there.

For topology, I would suggest John Lee. This topology text is unique because it teaches general topology with a view towards manifolds. This makes it ideal for a physicist. If you want to know about Lie algebras and Lie groups, the sequel to this text discusses them.

Counterexamples in Analysis is a wonderful menagerie of mathematical oddities—it's full of pathological examples. It's the most fun math book I know of.

To the nay-sayers, I'll offer a contrary opinion: It is doable. Especially if you do linear algebra and multivariable calculus at the same time, since a lot of the underlying ideas and techniques are the same. It will, however, take focus.

I am by no means a mathematical genius, but with consistent, daily studying, I was able to take calc III and linear algebra in the same 5 weeks, and differential equations in the regular semester following that. By prepared to work hard, do lots of problems, and carefully dissect new ideas as they are presented, but it can be done.

EDIT:

In fact, I'd like to recommend a superb textbook that covers all three of these topics:
http://www.amazon.com/Calculus-Vol-Multi-Variable-Applications-Differential/dp/0471000078
If you're interested in self-study, it's often difficult when different textbook authors use different notation, or different but practically equivalent definitions and methods. Not only does this avoid that problem, but it's an extremely lucid and thorough book, with lots of exercises, and you can keep it for the rest of your career for reference.

My favourite used to be Calculus on Manifolds until I started reading Munkres' Analysis on Manifolds. It covers the same material and then some and does a better job at explaining it. Spivak's purpose was a graduate reference book, and I think it does a good job at that. But in terms of learning Multivariable Analysis from it, it is very dense, and leaves out some stuff which I feel hinders it.

In terms of DE you could look at this one by Hirsh. It has some humour like Spivak, and is very theoretical, it has some applications in it but we skipped them when we took DE at my uni. There's also the dover book Advanced Ordinary Differential Equations (I think) which was used for the same course. However DE/Dynamical systems/chaos isn't a really concrete subject as opposed to analysis, so there are many ways of approaching it.

I can recommend a very good book, I am using it and it is beautiful.

Only if you are too dumb to know how to use it. Knowing a median is actually quite useful when making future predictions.

I would encourage you to read up on statistics, so you can focus on things that matter, rather than on the odds that Urban Meyer wins a game when there is snow within 100 miles and he is wearing khakis.

http://www.amazon.com/Statistics-Dummies-Deborah-J-Rumsey/dp/0470911085

I’d recommend Blitzstein’s Into to Probability book- it’s the book used for Harvard’s Stat110 which has free lectures online as well.

https://www.amazon.com/Introduction-Probability-Chapman-Statistical-Science/dp/1466575573

I looked at similar (WA resident also) but there's only a few community college classes that are interesting (linear algebra, probability, ODE) so then you're looking at UW/WSU tuition. There's a couple applied tracks you could consider: machine learning and financial math:

https://metacademy.org/roadmaps/

http://www.deeplearningweekly.com/pages/open_source_deep_learning_curriculum

https://www.quantstart.com/articles/Quantitative-Finance-Reading-List

-----------

Self study: math for physics texts like Arfken/Harris/Weber, Boas, Riley/Hobson, Thomas Garrity

http://www.goldbart.gatech.edu/PostScript/MS_PG_book/bookmaster.pdf

https://www.amazon.com/Mathematical-Methods-Physicists-Seventh-Comprehensive/dp/0123846544

In all seriousness, the applications of analysis to geometry can be really interesting and insightful, but to get to them you would have to first have background in differential topology, which it seems you lack. That might be a good subject to start with. A good book would be John Lee's An Introduction to Smooth Manifolds.

I'd start with a discrete math course (often offered for intro computer-science, but make sure the curriculum doesn't consist of any coding). Then move on to real analysis.


I really like this book as an intro.

Lee is still the easiest and best for self study. https://www.amazon.com/Introduction-Topological-Manifolds-Graduate-Mathematics/dp/1441979395/ref=sr_1_3?keywords=lee+manifolds&qid=1558265795&s=gateway&sr=8-3

followed by:

https://www.amazon.com/Introduction-Smooth-Manifolds-Graduate-Mathematics/dp/1441999817/ref=sr_1_2?keywords=lee+manifolds&qid=1558265795&s=gateway&sr=8-2

​

It's long, taking almost two volumes to get to Stokes theorem. If conciseness is important, you can just read Warner:

https://www.amazon.com/Foundations-Differentiable-Manifolds-Graduate-Mathematics/dp/0387908943/ref=sr_1_1?keywords=Warner+manifolds&qid=1558265966&s=gateway&sr=8-1

​

Tu looks good, but I haven't read it carefully.

Optics takes a fair amount of math. If you want to read something useful, I recommend:

• All the Mathematics You Missed But Need to Know for Graduate School
  
  
• A Student's Guide to Maxwell's Equations

I personally think you should brush up on frequentist statistics as well as linear models before heading to Bayesian Statistics. A list of recommendations directed at your background:

• For Frequentist Statistics and Probability: Introduction to Probability, Mathematical Statistics With Applications or since you are more interested in applications of Stats to Machine Learning, All of Statistics.
  
  
• For Bayesian Statistics, I would start with Statistical Rethinking and Introduction to Bayesian Statistics for an intro.
  
  

The Nature of Computation

(I don't care for people who say this is computer science, not real math. It's math. And it's the greatest textbook ever written at that.)

Concrete Mathematics

Understanding Analysis

An Introduction to Statistical Learning

Numerical Linear Algebra

Introduction to Probability

Ah yes, traditionally math learning is a fairly linear progression and is bottlenecked up until you take your first proof/analysis class, after which your path can branch out. Seeing as how you already have a link there, a textbook is listed for that class and that one happens to be popular so maybe you can buy that one. Me personally, I used this one when I went through the fundamentals

Analyzing Baseball with R is the best book, I believe:

https://www.amazon.com/Analyzing-Baseball-Data-Chapman-Hall/dp/1466570229

I also would download PitchRX and Baseball on a Stick to round out your toolkit!

-Kyle

I love Jack Lee's series on manifolds:

Introduction to Topological Manifolds

Introduction to Smooth Manifolds

I've heard Munkres' Topology is fantastic as an introduction to general topology, but never read it myself.

There is an excellent series of Counterexamples in ... books which might be relevant to this thread:

counterexamples in...

• calculus
  
• analysis
  
• topology
  
• probability & statistics
  
• several complex variables
  
  Or just run a search on amazon for "Counterexamples in".

Available free as well:

https://www.amazon.com/Students-Guide-Maxwells-Equations/dp/0521701473

http://www.amazon.co.uk/A-Students-Guide-Maxwells-Equations/dp/0521701473

This book is very short and explains it all from the bottom up. I'd definitely recommend if you're new to Electromagnetism and/or haven't really studied vector calculus.

Mostly because I wanted to analyze baseball stats, and at the time (4-5 years ago) that was mostly done in R. If the last industry conference I went to is any indication, it still is, many of the presentations features plots that were clearly ggplot2. There are also books like this one floating around: https://www.amazon.com/Analyzing-Baseball-Data-Chapman-Hall/dp/1466570229/ref=nodl_.

I like "Mathematical Thinking." You can get the PDF online quite easily

http://www.amazon.com/gp/aw/d/0130144126

I have never argued that draw is good for the game. If you read my posts around this subreddit, I have critized mojang for not putting in the proper way to the hand limit and have argued that it makes the game less tactical on several occasions. As for the rest, try this:
http://www.amazon.co.uk/Statistics-For-Dummies-Deborah-Rumsey/dp/0470911085

This depends on your current level of knowledge and experience, generally you would start with multiple choice problems and then move on to International Maths Olympiad (IMO) type problems that require written solutions.

Most competitors at the IMO go through training and selection programs to make their national team. Many of the countries running these programs publish their material, for example South Africa : http://www.mth.uct.ac.za/imo/imopub.html.

Another great resource is http://www.artofproblemsolving.com/.

There are a lot of books as well, a small sample :
http://books.google.co.za/books/about/In_P%C3%B3lya_s_Footsteps.html?id=Z3p_MToD32MC&redir_esc=y
http://www.amazon.com/Mathematical-Olympiad-Handbook-Introduction-Publications/dp/0198501056/ref=sr_1_1?s=books&ie=UTF8&qid=1375034070&sr=1-1&keywords=lets+solve+some+math+problems
http://www.amazon.com/Erd%25f6s-Kiev-Problems-Mathematical-Expositions/dp/0883853248/ref=sr_1_2?s=books&ie=UTF8&qid=1375033538&sr=1-2
http://www.amazon.com/dp/0883856190

Our site www.examify.net will email you multiple choice 'math competition' papers that the site will mark and send you worked solutions, most of the content is at a very introductory level at the moment.

> My problem is that I have never really been introduced to sets or other things,

How did that happen? I know that at my alma mater, you're supposed to have some kind of proofs-oriented course before you take intro abstract algebra (either "abstract linear algebra", which is a proofs-heavy intro to linear algebra, or "fundamental mathematics" or "theory of computation"). Does this course not have appropriate prereqs, or did you disregard them?

Edit: the text that the fundamental mathematics class there uses is Mathematical Thinking: Problem-Solving and Proofs. It's written by a couple of the professors from the university. I don't know much about West, but I had D'Angelo for real analysis, and he was both meticulous and clear in lecture; I'd be surprised if any book that he put his name on was not.

Is this the book you are looking for https://www.amazon.com/Functions-Graphs-Dover-Books-Mathematics/dp/0486425649/ref=cm_cr_arp_d_pdt_img_top?ie=UTF8 ?

There are more freely available books from erstwhile USSR published by Mir Publishers https://mirtitles.org/

http://www.amazon.com/Introduction-Probability-Edition-Dimitri-Bertsekas/dp/188652923X/ref=sr_1_1?ie=UTF8&qid=1394424420&sr=8-1&keywords=bertsekas+probability

You can find the video lectures from http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/ or taking the course on edX https://www.edx.org/course/mitx/mitx-6-041x-introduction-probability-1296

*Solutions to the book exercises can be found on the book's website. Perfect for self-taught learner.

Sorry, I went on vacation and totally blanked about posting these for you!

Anyway, here are some books

Linear Algebra Done Right (Undergraduate Texts in Mathematics) https://www.amazon.com/dp/3319110799/ref=cm_sw_r_cp_api_1L8Byb5M5W9D3

This one is actually for analysis but depending on your appetite, it might help greatly with the proof side of your class. You can buy it here: Counterexamples in Analysis (Dover Books on Mathematics) https://www.amazon.com/dp/0486428753/ref=cm_sw_r_cp_api_GS8BybQWYBFXX

But there's also a PDF hosted here: http://www.kryakin.org/am2/_Olmsted.pdf

Here you go! It's very helpful and has a wide range of topics so you can learn whatever you want. It uses Retrosheet, Lahman and Pitch Fx

https://www.amazon.com/Analyzing-Baseball-Data-Chapman-Hall/dp/1466570229/ref=sr_1_1?ie=UTF8&qid=1494296330&sr=8-1&keywords=analyzing+baseball+data+with+r

https://www.amazon.com/Analyzing-Baseball-Data-Chapman-Hall/dp/1466570229

This book covers everything related to how to get the data (Retrosheet, Lahman's, pitchf/x IIRC) and then how to do a lot of different stuff with R. It's a good place to start. You could probably find it cheaper than that Amazon link though.

Incorporating expert opinion into a Bayesian model is usually done through prior distributions instead of an additional feature. (As an aside, doing so is considered subjective Bayesian inference versus objective Bayesian inference).

As a quick overview, Bayesians usually make inference on the posterior distribution - a combination of the prior distribution (in your case, expert opinion), and the likelihood. As a really basic example, consider a setting where you have data on MI outcomes (no covariates at this point) - a series of 1's and 0's. A frequentist would likely take the mean of the data. As a Bayesian, you would consider this binomial likelihood and likely combine it with a beta prior. The default (non informative) prior would be to use a beta(1, 1) distribution. However, if in a prior dataset, you had observed four patients, three with an MI and one without, you could use a prior of beta(1+3, 1+1). See here for more details on beta-binomial.

In the above example, it's easy to incorporate prior information because we used a conjugate prior. While probably not exactly what you are doing for your dissertation, here's an overview of a conjugate prior with a linear regression from wikipedia. There are many more resources online for this that you can find by searching for something along the lines of "bayesian linear regression subjective conjugate prior". For a more detailed (introductory) overview of bayesian statistics, check out this book.

To be honest, as much as I'm a Bayesian, I think that creating an automatic model that incorporates expert opinion will be really difficult. Usually, subjective priors are chosen carefully, and there not always as interpretable as the beta-binomial posterior presented above. I think this goal is possible, but it would require a lot of though about how the prior is automatically constructed from a data set of surgeon's predictions. If you have any followup questions/would like more resources, let me know!

Edit: I guess I never really addressed the issue of predictive models. However, the difficult part will be constructing the prior automatically. If you can do this, predicting outcomes will be a simple change to make, especially in the case of linear model.

I am no expert (undergrad applied maths), but from what I have heard, Evans is the go to text. I have also heard good things about Salsa as a general overview/ course on PDEs.

I found Prof. Joseph Blitzstein's course, at Harvard, on statistics engaging. First I watched his lectures and worked through the problem sets. This was extemely rewarding, so I went on to work through his book on probability. According to me, what separates him from other Profs is that he takes a lot of effort to build intuition about statistical concepts.
Stat110 is the course website. You can find his book here.

My favorite introductory discrete math textbook is https://www.amzn.com/0387955852. (It also appears to be available for less unreasonable prices.)

Entire books have been written on this subject--here is a good place to start

This book might be a helpful start: https://www.amazon.com/Analyzing-Baseball-Data-Chapman-Hall/dp/1466570229

This is a fan favorite if you can read proofs, but i can't personally testify to it: https://www.amazon.com/Introduction-Probability-2nd-Dimitri-Bertsekas/dp/188652923X

This seems to be lecture notes corresponding to the book: https://www.google.com/url?sa=t&source=web&rct=j&url=http://vfu.bg/en/e-Learning/Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf&ved=0ahUKEwi0tPaYhfXVAhVJro8KHQVuB9sQFghnMAo&usg=AFQjCNHg2bvy0qIa4qilsIT9qVtC3xX8VQ

Also, this StackOverflow answer to your question is highly-rated: https://math.stackexchange.com/q/31838/200344

I've had a few colleagues recommend A Student's Guide to Maxwell's Equations.

As an undergrad physics major, I would recommend this as well. If you're going to continue and do graduate PDE work, I would just jump into Evans after that.

You might like Geometry Revisited by Coxeter and Greitzer.

http://www.amazon.com/Geometry-Revisited-Mathematical-Association-Textbooks/dp/0883856190

At least re: random variables, events, PDF, and CDF, I like the diagrams from Prof. Joe Blitzstein's textbook:

http://i.imgur.com/aBkgHGC.jpg

> The set/subset relation could be considered an inverse relation as well.

Let A = {1, 2} be a set. Then B = {1} is a subset of A. Let's define a relation between them, f: A -> B given by f(1) = 1 and f(2) = 1. f is, actually, a function. But this function f doesn't have an inverse. Why? Find out from Mathematical Thinking: Problem-Solving and Proofs by D'Angelo and West.

Therefore

> Then that would also make the infinite/finite relation an inverse relation.

doesn't follow.

They're not free, but Doing Bayesian Data Analysis and Statistical Rethinking are worth their weight in gold.

Shankar's book teaches almost everything you need: calculus, vectors, series, complex variables, ODE, linear algebra in only ~300pag.
http://www.amazon.com/Basic-Training-Mathematics-Fitness-Students/dp/0306450364


For more advanced topics check out Arfken.

In order to understand the modern approach to PDEs in full generality you must have a minimum background of ODEs, basic topology, complex analysis, and basic differential geometry.
Many of the foundational theorems for these fields are directly applicable to the study of PDEs and it would be fruitless to try to study PDEs in full generality without that basic understanding. That being said, Evans ( http://www.amazon.com/Partial-Differential-Equations-Graduate-Mathematics/dp/0821849743 ) is an excellent well-rounded introduction to the general theory.
If this is too difficult for you to tackle at the moment, you will need to work your way through the above topics first. PDEs, studied in full generality instead of in particular cases, is not a light topic.

This is the book I used when I was studying statistics and probability. https://www.amazon.com/Introduction-Probability-2nd-Dimitri-Bertsekas/dp/188652923X

"Math isn't a spectator sport", but you shouldn't make yourself hate math by doing hundreds of problems. Study what you find interesting.

That's nothing. At least you are comparing different books, so maybe the new, expensive one benefits from something that has changed since 1960.

Look at this: Apostol, "Calculus", Volume 2. A brand new copy of the current edition in hardback is $270. That's the 2nd edition.

That book was about $20 when I bought a hardback copy in 1976 at Caltech. Guess what edition we were using? The 2nd edition, from 1969.

Same story with volume I. The nearly $300 edition they sell new today is the 1967 2nd edition. (Some sites list it as 1991, but it's still just the 1967 2nd edition text).

Asking every single person is definitely not the only way to get accurate numbers. For starters you could give this read:

http://www.amazon.com/Statistics-For-Dummies-Deborah-Rumsey/dp/0470911085/ref=sr_1_5?ie=UTF8&qid=1412872846&sr=8-5&keywords=intro+to+statistics

But you're right about this little piece of click bait. I'm not sure why more people aren't commenting on the NBC/Survey Monkey Ad they were just tricked into reading.

If you have problems with probability take the MITx probability class on edX. That is as good as it can get as a EECS probability class. It teaches you tons of stuff but assumes nothing but multivariable calculus from you. If you have time, read Introduction to Probability by the class instructors.

Note the class alone is a huge time sink.

A very user-friendly treatment that hits every criterion you mention is John Kruschke's Doing Bayesian Data Analysis, Second Edition.

An intermediate resource between the Downey book and the Gelman book is Doing Bayesian Analysis. It's a bit more grounded in mathematics and theory than the Downey, but a little less mathy than the Gelman.

• topology: Introduction to Topology and Modern Analysis by G. Simmons
  
• real analysis: A First Course in Real Analysis by Protter and Morrey
  
• complex analysis: Theory of Functions of a Complex Variable by A.I. Markushevich
  
• differential equations: Ordinary Differential Equations by V.I. Arnold
  
• linear algebra: Linear Algebra by G. Shilov
  
• calculus: Calculus with Applications and Computing (vol.1) by P. Lax
  
• probability: Introduction to Probability Theory by Hoel, Port and Stone

what I meant is this one

A lot of people on this thread could do with

1. Reading the article and not just the headline
   
2. Reading this book
   Statistics For Dummies, 2E https://www.amazon.co.uk/dp/0470911085/ref=cm_sw_r_cp_apa_w4xUAbNGYEP37

>what can I show her so she can be properly informed?

this

EDIT: or a bunch of leftist professors discussing it

http://freakonomics.com/2016/01/07/the-true-story-of-the-gender-pay-gap-a-new-freakonomics-radio-podcast/

Calculus Volume 2 - Apostol.

If you need to brush up on some of the more basic topics, there's a series of books by IM Gelfand:

Algebra

Trigonometry

Functions and Graphs

The Method of Coordinates

http://www.amazon.com/Counterexamples-Analysis-Dover-Books-Mathematics/dp/0486428753 click to look inside.

Counterexamples in topology

Counterexamples in analysis

Ah but did your tour have a guide?

a student's guide to maxwell's equations

http://www.amazon.com/Students-Guide-Maxwells-Equations/dp/0521701473

For when you get into Electricity and Magnetism - This

As of today, these books sell for:

• Measure and Integral by Weeden and Zygmund: $91
  
• Partial Differential Equations by Evans: $88
  
• Linear and Nonlinear Functional Analysis with Applications by Ciarlet: $95
  
  At $274, this is probably the most expensive tofu presser I've ever seen.

Graduate or undergraduate level?


If graduate, this is THE book to get.

This is much more applied.

Geometry Revisited by Coxeter and Greitzer

http://www.amazon.com/Geometry-Revisited-New-Mathematical-Library/dp/0883856190/

Coxeter, maybe?

I suggest either Tu or (easy) Lee.

https://www.amazon.com/Analyzing-Baseball-Data-Chapman-Hall/dp/1466570229

https://www.amazon.com/Analyzing-Baseball-Data-Chapman-Hall/dp/1466570229/

It's an introduction to baseball data, statistical analysis, and the R programming language.

Analyzing Baseball Data with R

https://www.amazon.com/Analyzing-Baseball-Data-Chapman-Hall/dp/1466570229

​

Walks you through learning the program using baseball stats as the foundation.

Blitzstein and one of his students published a probability textbook

I'm going to guess this one based on high reviews and a description that mentions R.

[citation needed]

http://www.amazon.ca/Statistics-For-Dummies-Deborah-Rumsey/dp/0470911085

Here is the lowest end dumbed down version.

Should be perfect.

I think learning proofs-based calculus and linear algebra are solid places to start. To complete the trifecta, look into Arnold for a more proofy differential equations course.

After that, my suggestions are Rudin and, to build on your CS background, Sipser. These are very standard references, though Rudin's a slightly controversial suggestion because he's notorious for being terse. I say, go ahead and try it, you might find you like it.

As for names of fields to look into: Real Analysis, Complex Analysis, Abstract Algebra, Topology, and Differential Geometry mostly partition the field of mathematics with corresponding undergraduate courses. As for computer science, look into Algorithmic Analysis and Computational Complexity (sometimes sold as a single course called Theory of Computation).

If you weren't satisfied with geometry in your school, then I can suggest this wonderful text: http://www.amazon.com/gp/product/0883856190/

No, I am not familiar with vector calculus. Do I need a lot of background before I can try to learn that or is it okay to jump right in? I know there are a lot of gradients and that is something I hadn't seen before.

I was also looking at getting this.

Read this: https://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773 and you're more than set for algebraic manipulation.

And if you're looking to get super fancy, then some of that: https://www.amazon.com/Method-Coordinates-Dover-Books-Mathematics/dp/0486425657/

And some of this for graphing practice: https://www.amazon.com/Functions-Graphs-Dover-Books-Mathematics/dp/0486425649/

And if you're looking to be a sage, these: https://www.amazon.com/Kiselevs-Geometry-Book-I-Planimetry/dp/0977985202/ + https://www.amazon.com/Kiselevs-Geometry-Book-II-Stereometry/dp/0977985210/

If you're uncomfortable with mental manipulation of geometric objects, then, before anything else, have a crack at this: https://www.amazon.com/Introduction-Graph-Theory-Dover-Mathematics/dp/0486678709/

https://www.amazon.com/Introduction-Probability-2nd-Dimitri-Bertsekas/dp/188652923X/ref=sr_1_1?ie=UTF8&qid=1523289228&sr=8-1&keywords=bertsekas

No busques mas porque explica todo de 10. Si todos los textos fuesen asi...

For math there isn't much better undergraduate/beginning graduate review than Arfken, Weber, Harris. This will cover most mathematics you'll encounter in your first and maybe second years of graduate studies. Personally I'm not a huge fan of the complex contour integration sections you'll encounter in that book - I much prefer Ahlfors or Rudin for something more on the pure side or Churchill for something more on the applied side of complex analysis. The other sections are, in my opinion, stellar - although I have only the third edition in my possession.

Unfortunately for you, 251 learning is mostly from lecture and recitation lessons, for which there is not an official textbook (student informally use the Concepts of Mathematics Textbook, which is quite decent).

This is the public course website: https://colormygraph.ugrad.cs.cmu.edu/15251-s12/

Course materials are located in the calendar tab and many of them are public.

You will have a tough time learning anything of consequence without something like videos of the lectures, etc. (Which even students don't have access to)

I’m finishing up my stats degree this summer. For math, I took 5 courses: single variable calculus , multi variable calculus, and linear algebra.

My stat courses are divided into three blocks.

First block, intro to probability, mathematical stats, and linear models.

Second block, computational stats with R, computation & optimization with R, and Monte Carlo Methods.

Third block, intro to regression analysis, design and analysis of experiments, and regression and data mining.

And two electives of my choice: survey sampling & statistical models in finance.

Here’s a book for intro to probability. There’s also lectures available on YouTube: search MIT intro to probability.

For a first course in calculus search on YouTube: UCLA Math 31A. You should also search for Berkeley’s calculus lectures; the professor is so good. Here’s the calc book I used.

For linear algebra, search MIT linear algebra. Here’s the book.


The probability book I listed covers two courses in probability. You’ll also want to check out this book.

If you want to go deeper into stats, for example, measure theory, you’re going to have to take real analysis & a more advanced course on linear algebra.

I used Mathematical Thinking: Problem-Solving and Proofs by D’Angelo and West, and I remember it being quite a good book as an introduction to proofs. We didn’t use the book extensively in that course, but when we did need it I had no complaints.