hey nerdinthearena,

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

helpful for intuition and basic understanding

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

  more advanced but still intuitive
  

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

  three general analysis favorites
  

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

> My lack of confidence, I think, is centered around the idea that complex math is a prerequisite to engineering. So I have a few questions.

The math in engineering is not hard, comparative to actual math. It's low tier within math departments (100/200 level), and most professors aren't expecting you to create some new theorem. You're there to apply theorems that have already been well established within the discipline.

> How hard is EE?

People say it's the hardest of the engineering disciplines. I've taken a few EE courses, and didn't have much trouble, granted, it's only circuits and the second physics course. Difficulty will probably depend on how much you catch on with the subject.

> What kind math will I need to learn?

Algebra, Trigonometry, Analytic Geometry, Calculus (Differential, Integral, Multivariable), and Differential Equations. Most universities merge Algebra, Trig, Analytic Geometry into the Calculus courses.

> Will the stuff from high school matter?

Of course. I am of the opinion that everything matters and is useful. All your science, math, english, and even history can come into play depending on your engineering discipline.

> At what point do you start?

Depends. I did Calculus in high school, but retook it since I added my engineering major in my junior year. Met some kids who came in with Calculus I and II credits and started in Multivariable. Some kids who started with Calc II and some who started with algebra or pre-calc. It really depends on you.

> Do professors assume you know close to nothing?

Depends on the professor. My statics professor was very hand holdy. A Physics professor was very hands off. My Calc III professor would teach you a concept and expect you to be able to apply it to contexts you were never taught. There is not a blanket statement to really apply to this question.

> Will I have to teach myself in order to catch up?

You should be teaching yourself because you want to improve, not to reach some arbitrary point.

If you want to be an engineer and you find deficiencies in your skills, then you need to take some time and think about what you want. I am, unfortunately, in the camp of, you should pursue what you're passionate in, as you are more likely to do that work without being told, hence, getting better. It took me several years to actually figure out that I wanted to be an engineer, and despite not having a math course in three years, I went in quite over prepared because I was already doing the work for fun.

There's no rush to make a decision, but in the event you do decide to work on being an engineer, here's some useful references.

Just-in-Time: http://www.amazon.com/Just--Algebra-Trigonometry-Transcendentals-Calculus/dp/0321671031

Great book to brush up and refine your algebra and trig skills

Schaum's Outlines: http://www.amazon.com/Schaums-Outline-Calculus-6th-Problems/dp/0071795537/ref=sr_1_1?s=books&ie=UTF8&qid=1465145307&sr=1-1&keywords=schaum+outline+calculus

I am in love with this series. I buy a book for every class I take (and they have one for every engineering/math/physics course). It's a great supplemental text, and this particular book covers Calculus I, II, III and a bit of Differential Equations!

Paul's Online Notes: http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx

Paul has great lecture notes and tons of practice problems. Invaluable resource.

Using Paul's site, you can also scan what kind of math is in engineering. Bit of warning, some of it will look scary at first. When I first started, the symbol for a partial derivative looked really intimidating. When you reach that point, you'll laugh at the fact you found it intimidating in the first place.

Best of luck OP!

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

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

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

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

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

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

I am not a math expert just a fellow student, so take this as peer discussion not expert opinion

> Is there a real benefit in dropping $150 for the course textbook, or is there cheaper books that help

Yes. Google "best calculus textbook", look at various reviews, and pick the one that seems the best fit for you. You can often find PDF extracts so you can preview them, or use Amazon to preview them. Then decrement the version number by a few points and check the Amazon price, its often in the $20-30 range, and nothing relevant has changed in calculus in the past 10 years anyway. Just get the matching solution guide for that same version and you are golden for a fraction of the cost. Best part is, you can buy more than one textbook to get multiple angles on the same topic. I used two textbooks when I took calculus, one was more rigorous and confusing and thus a bit over my head so the other one was easier to understand, but some examples and notation I preferred in the harder book and in those cases those were easier to understand. Etc. Then in calc 2 going back to refresh on some stuff from calc 1 I found I enjoyed the harder text more, it just took a bit of getting used to.

Texts I used for calc 1 and 2:

• Calculus: Early Transcendental Functions 4th Edition by Smith/Minton was significantly more confusing at first but more rewarding once I grasped the concepts. Looks like a "like new" used one is $50 right now and there are certainly older editions that are just as useful.
  
• Calculus 9th Edition by Anton is what I bought to help me understand Smith/Minton, based on reading reviews of both plus reviews of Stewart (which I never used), and the one I didn't use as much in calc 2
  
  Extra bonus: Smith/Minton is the text my school uses for calc 1-4, and I think Anton covers the same stuff. I didn't go past calc 2 so I can't attest to their usefulness at those levels.
  
  Then of course there's Spivak.

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.

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

I would assume that if you're a music major and "been good at math", you might be referring to the math of high school. In any case, it would help if you spend some time doing/reviewing calculus in parallel while you go through some introductory physics book. So here's what you could do:

• math: grab a copy of one of the following (or some similar textbook) and go through the text as well as the problems
  
  • Thomas and Finney
    
  • Stewart (older editions of this are okay since they are cheaper. I have fourth edition which is good enough).
    
• physics:
  
  • for mostly conceptual discussion of physics, Feynman lectures
    
  • for beginner level problems sets in various branches of physics, any one of the following (older editions are okay):
    
    • Halliday and Resnick
      
    • Young and Freedman
      
    • Serway and Jewett
      
    • Giancoli
      
  • for intermediate level discussion (actually you can jump right into this if your calculus is good) on mechanics , the core branch of physics, Kleppner and Kolenkow
    
    
    Other than that, feel free to google your question. You'll find good info on websites like physicsforums.com, physics.stackexchange.com, as well as past threads on this subreddit where others have asked similar questions.
    
    Once you're past the intro (i.e., solid grasp of calculus, and solid grasp of mechanics, which could take up to a year), you are ready to venture further into math and physics territory. In that regard, I recommend you look at posts by Gerard 't Hooft and John Baez.

Yeah, Frenkel is pretty solid and I haven't really worked through much of it, even though I own it. I'd really be interested in using it to learn some of the cool physics it contains.

I might start with learning about Riemannian holonomy in general, and then specializing to G2 later. A good starting point might be Dominic Joyce's 'Riemannian Holonomy Groups and Calibrated Geometry' https://www.amazon.com/Riemannian-Holonomy-Calibrated-Geometry-Mathematics/dp/0199215596 . (Calibrated geometry is also a beautiful subject that i'd be more than happy to learn more about. )

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

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

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

I think you need to list which books you didn't understand. I'm having a hard time understanding what you have trouble with. Studying general relativity, you should be familiar with metrics and curvature, but somethings you say indicate otherwise. It is also unclear to me what you want to learn. Do you want to learn differential geometry related to QFT, e.g. Yang-Mills connections?

Since you are familiar with GR, maybe you would appreciate O'Neill's book Semi-riemannian Geometry. Jurgen Jost also has a book Geometry and Physics that may help be a bridge between the language. The book is meant to be a bridge for mathematicians, but maybe it will also be helpful for going the opposite way.

Edit: Also, without more explanation to what you want, I think it would be useless to go back to stuff like undergraduate analysis. For example, you may be put off by geometry books giving a topological definition of manifold. This is a technical detail most geometers don't actually work with. The important thing to concentrate on is the smoothness and invertibility of the transition maps. For things involving groups, you could probably go very far just thinking of linear groups, e.g. special matrix groups.

The only possible issue I see is your selection of textbook: Principles of Mathematical Economics - I've honestly never heard of this book.

The graduate school go-to textbook is Mathematics for Economists by Simon and Blume. However, I think this book would be overkill for you, as it is geared towards pure, PhD level, economics. Also, I was in a similar place to you, with respect to mathematical training at one point, and Simon & Blume proved to be too large a leap.

My advice would be to use one of the following books (in order of my preference):
1. Essential Mathematics for Economic Analysis by Sydsaeter
2. Mathematics for Economics
by Hoy
3. Fundamental Methods of Mathematical Economics
by Chiang

They'll bring your basic command, of the basic required mathematics up to scratch AND these books cover linear algebra. You will also then be in a good place to tackle Simon & Blume if you ever need to in the future. Another piece of advice: PRACTISE PRACTISE PRACTISE. For what you are doing, you don't need to have a deep understanding of the mathematics you are using BUT, you do need to be very comfortable with applying the techniques.

So, as you are working through (for instance) Sydsaeter, I would be attempting the related practice questions you find in:

1. Schaum's Outline of Calculus
   
2. Schaum's Outline of Linear Algebra
   
3. Schaum's Outline of Introduction to Mathematical Economics
   
   Hope this helps.
   
   P.S. Almost all of these books are available for 'free' on Library Gensis

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.

These are listed in the order I'd recommend reading them. Also, I've purposely recommended older editions since they're much cheaper and still as good as newer ones. If you want the latest edition of some book, you can search for that and get it.

The Humongous Book of Basic Math and Pre-Algebra Problems https://www.amazon.com/dp/1615640835/ref=cm_sw_r_cp_api_pHZdzbHARBT0A


Intermediate Algebra https://www.amazon.com/dp/0072934735/ref=cm_sw_r_cp_api_UIZdzbVD73KC9


College Algebra https://www.amazon.com/dp/0618643109/ref=cm_sw_r_cp_api_hKZdzb3TPRPH9


Trigonometry (2nd Edition) https://www.amazon.com/dp/032135690X/ref=cm_sw_r_cp_api_eLZdzbXGVGY6P


Reading this whole book from beginning to end will cover calculus 1, 2, and 3.
Calculus: Early Transcendental Functions https://www.amazon.com/dp/0073229733/ref=cm_sw_r_cp_api_PLZdzbW28XVBW

You can do LinAlg concurrently with calculus.
Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) https://www.amazon.com/dp/0538735457/ref=cm_sw_r_cp_api_dNZdzb7TPVBJJ

You can do this after calculus. Or you can also get a book that's specific to statistics (be sure to get the one requiring calc, as some are made for non-science/eng students and are pretty basic) and then another book specific to probability. This one combines the two.
Probability and Statistics for Engineering and the Sciences https://www.amazon.com/dp/1305251806/ref=cm_sw_r_cp_api_QXZdzb1J095Y1


Differential Equations with Boundary-Value Problems, 8th Edition https://www.amazon.com/dp/1111827060/ref=cm_sw_r_cp_api_sSZdzbDKD0TQ9



After doing all of the above, you'd have the equivalent most engineering majors have to take. You can go further by exploring partial diff EQs, real analysis (which is usually required by math majors for more advanced topics), and an intro to higher math which usually includes logic, set theory, and abstract algebra.

If you want to get into higher math topics you can use this fantastic book on the topic:

This book is also available for free online, but since you won't have internet here's the hard copy.
Book of Proof https://www.amazon.com/dp/0989472108/ref=cm_sw_r_cp_api_MUZdzbP64AWEW

From there you can go on to number theory, combinatorics, graph theory, numerical analysis, higher geometries, algorithms, more in depth in modern algebra, topology and so on. Good luck!

I took calculus my senior year in high school, too, then took a calculus course in college in the 80s. (I graduated from high school in '65.) I took whatever the algebra class was the was the precursor for calculus, but I'm not sure you have time for that.

Ross Finney gets extremely high marks for his work, so his high school text may be of value to you to brush up on your skills.

Good luck and have fun. If I can make it with a 20-something year gap, I have no doubts about you.

>I might not be a ridiculous all-knowing physics major

I don't even know what that means.

>Trust me, it is brilliance

Well, I guess I can just forego my own critical thinking skills and desire to think my own thoughts now that you've arrived. As we know, trust in a self-proclaimed expert is the cornerstone of scientific inquiry.

I'm a grad student in chemistry. Granted, I'm not a physicist either, but his description ties a lot of poorly understood principles together with a little hand-waving and calls it an explanation of "where energy itself (which also means matter) comes from."

First, at most, this is a diagram illustrating concepts of mass-energy equivalency, or a diagram of a space-time grid. It's not an explanation or derivation of Einstein's mass-energy equivalency. Einstein already presented an explanation and derivation of the equation in his papers. Drawing a picture of a CO2 molecule doesn't explain the origin of CO2.

Generally speaking, he does present facts that are indeed facts. However, he connects them in ways that make no sense. Alexander Pope said, "A little learning is a dangerous thing." The artist's description of his art illustrates this concept.

Anyway, I follow him on Planck lengths. Fair enough. All points in space-time are whole number multiples of the Planck length from all other points in space-time. So far, so good.

>Because there are no fractions the shape that arises is a two dimensional hexagon interacting with other hexagons creating larger and larger hexagons. If you view it three dimensionally you will see that it automatically creates three dimensional cubes that create larger and larger cubes. This is 'why', as Einstein stated, that space time has a grid like structure.

Think about it. Print his image on 2 pieces of paper. Stack one on top of the other and mentally connect the dots from the top layer to the bottom layer. You don't get cubes. When you render a 3d cube in 2d, one way you can do it is to draw a 2d regular hexagon. He's basically claiming that because 3d cubes can be represented by 2d hexagons, space-time must be represented by a grid. He's making an essential flaw in stating that his representation of reality is why reality is how it is. Maybe I'm misreading him, but there are some serious gaps between his explanation of Planck's length necessarily leading to a 3d cubic structure of spacetime.

Again, he accurately states the uncertainty principle. However, he unfortunately continues:

>Because of the Uncertainty principle each piece or slice of space time vibrates at the speed of light.

The uncertainty principle does not state that particles or points of space-time vibrate at the speed of light. Given that this speed of light vibration is the premise for the next paragraph and a half, and the premise is false, we can't really trust any conclusion he draws from this thought experiment. Bullshit.

>total Planck constants in the vibration (this equals mass)

Why would the sum of length measurements equal mass? Come on. You're a grad student in engineering. Measure your height. What can you conclude about your mass from this measurement? Nothing. For an object of macroscopic scale, we need the product of length, width, and height for a regular rectangular solid, combined with the density to determine the mass. The artist states that the total of 3 lengths equals the mass of a theoretical point in space-time. Bullshit.

>The equation at the bottome of the diagram describes energy in pure equation form.

Essentially, he claims that since A=B and B=C, then A=C. This is the property of transitivity, from basic algebra, and seems to think that this is an earth-shattering discovery.

>Combining these two equations and seeing the shape of space time vibrating at the speed of light is the key to understanding where energy comes from.

Again, there's no reason to think that space-time vibrates at the speed of light. Seeing his diagram with 2 equations and a mistaken understanding of the uncertainty principle doesn't help me explain the origin of energy and matter.

I think it's a stretch to call a regular polygon a fractal. That is, self-similarity at multiple scales is not the only criterion defining fractals. Yes, a regular hexagon can be divided infinitely into smaller regular hexagons (self-similarity). However, another defining feature of a fractal is that it is "too irregular to be easily described in traditional Euclidean geometric language." Source. A regular hexagon fails this criterion. So, in addition to misapplying his art to descriptions of the physical world, he mislabels his own geometric forms. Bullshit.

>every shape that exists is a fractal.

Bullshit. See previous paragraph.

>When this equation makes it into mainstream media it will be worth untold amount of money.

I hope he's talking about his diagram, not the hf=mc^2 equation. If so, I really hope he's trolling hard here. If not, he's painfully ignorant as to the progress of physics since Planck and Einstein. If he thinks he's the first to apply transitivity to these equations in the last 100+ years - again, a principle that could be applied by an intro algebra student without any knowledge of the principles of physics - well, he's wrong.

If you're going to try to explain where mass and energy come from, you're going to have to address the origin of the universe. All matter and energy currently in the universe has existed since the beginning of time, the moment of creation, the Big Bang, pick your label. If you're going to claim to explain where it all came from, you're going to have to explain what caused the Big Bang, the conditions that existed during it, etc., which elude physicists to this day, despite their best efforts and progress in the field.

Look, you don't have to be a physicist to see through his explanation. Now, it's a pretty picture. I'll give him credit for that. It would take a lot of time and patience with a ruler and a compass to recreate it. Let's not kid ourselves, though. It does not explain the origin of energy and matter. That is to say, it does not explain the origin of the universe.

Here's a little test. Show this to your research advisor and ask him/her if it's been published yet with a straight face. No cheating. Don't email it. Print it. Take it to his/her office. Hand over the diagram and read the description aloud. Tell me how it goes.

Anyway, upvote for you for attempting a defense.

I really like Spivak's Classical Mechanics. It's a very well-written book, as you would expect from Spivak's other texts. For general relativity, I really like Barrett O'Neill. He tries to present as much of the theory as possible in a coordinate-free manner, which makes the underlying principles much more clear.

It's very brief, and in its 72 pages it has a grand total of 5 examples. It doesn't contain any exercises, either, and really doesn't discuss any actual physical applications of the subject. Neither does it have any graphics, but that (unfortunately) seems to be the norm with these kinds of "course notes" PDFs. For that kind of book on the subject, I prefer Semi-Riemannian Geometry and General Relativity by Shlomo Sternberg, which you can get here. It is 251 pages (but no graphics), has many exercises, and discusses physical applications.

I first learned the subject from the first edition of Riemannian Geometry by Gallot, Hulin and Lafontaine, and I appreciated all the illustrations it contains. The material on geodesics, in particular, really benefits from seeing some drawings.

The general problem that underlies the result is called Sylvester's Four Point problem. There's a great deal of literature on it.

My friend who first showed it to me got it from this book which he was able to obtain in electronic form somehow. I'm sure it's easy to find.

Perhaps we can get the special flair users in /r/math to setup some of this (the ones with the red background in their flair)?

I know nothing about any of these topics but we could use course notes from a school's Open Courseware.

Here are the relevant ones I've found. If a cell says "none" that just means I've left a placeholder for if people find something I can put in that spot. The ones with all nones means I either wasn't sure what to look for, or if what I found was the right thing (Lie Theory = Lie Groups? for example)



Subject | Source1 | Source2 | Source3| Source4
---|---|----|----|----
Algebraic Topology | MIT Seems to have all relevant readings as PDFs | Introductory Algebraic Topology I don not know the source for this one| Algebraic Topology by Hatcher is free | A Basic Course in Algebraic Topology by Massey - Not free
Algebraic Geometry | MIT Fall 2003 Has lecture notes| MIT Spring 2009 Also has lecture notes | Vakil's course notes| Eyal Goren Syllabus and course notes
Functional Analysis | MIT Lecture notes and assignments with solutions | Nottingham 2010 | Nottingham 2008 These ones not only have lecture notes, but audio of the lecture. | none
Lie Theory | MIT - Intro to Lie Groups | MIT - Topics in Lie Theory: Tensor Categories | none | none
General Relativity | Sean Carroll's Lecture Notes | Stanford video lectures on general relativity, Leonard Susskind | Lecture notes from Nobel Laureate Gerard Hooft on GR | Semi-Riemannian geometry with Applications to Relativity - Not free
Dynamical Systems | Very applied (Strogatz style) course notes for dynamical systems | More theoretical (Perko style) course notes for dynamical systems by the same author | none | none
Numerical Analysis | MIT Spring 2012 | MIT Spring 2004 | none | none

This is obviously not an exhaustive list. I thought Stanford and their own open courseware thing but it seems to just be a list of courses they have on Coursera.

Many people like Strang's book for a standard treatment. Then getting a bit more general is Axler's Linear Algebra Done Right. But for further understanding, especially for computer graphics I recommend learning about the exterior and geometric algebras.

https://www.amazon.com/Linear-Algebra-via-Exterior-Products/dp/140929496X/

https://www.amazon.com/Linear-Geometric-Algebra-Alan-Macdonald/dp/1453854932

A Panoramic View of Riemannian Geometry link

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

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

Remember the big thing is not to just watch lectures, but to spend a lot of time solving a lot of problems. Make sure you find text books with problems and solution manuals (easy to do if you look) and solve the problems. The biggest thing about learning to be an EE or subset of EE is that it takes a crap load of time and effort. It will be frustrating. Plow through it. There are big rewards at the end for those who truly want to know how and why electricity do what they do.

Schaums outlines are great:

http://www.amazon.com/Schaums-Outline-Calculus-6th-Edition/dp/0071795537

So many subjects with so many solved problems.

I learnt everything in Do Carmo . A caveat about Riemannian geometry is that there is a high barrier of entry in terms of abstract notations (tensors all over the place, connections, endless equations with Christoffel symbols, etc.) which makes any reference material look very obstruse, and arguably not very geometric. But it turns out that once this vocabulary has been assimilated, it is often not really needed to read modern research topics in Riemannian geometry: things à la Gromov, min-max stuff, optimal transport, Ricci flow. In a way it's a bit like the baby version of spending an inordinate amount of energy to learn schemes and then never needing that level of generality in your own work.

Cool! Here are a couple of resources that I know about:

• Fantasy Art & RPG Maps Book
  
• A Very Short Introduction to Fractals Book
  
• YouTube Channel which goes into WAY too much technical detail
  
• YouTube video discussing Plate Techtonics
  
• Fractal World Generator (www)

Right, and technology based calc is based on this book and is not a full calculus class.

Like I said earlier, Ok State offers the ability to get a full FPE degree (part of which is taking regular calc), but I don't think a single person has actually done that yet.

Maybe this book?

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

I don't know exactly what math class you're in, but the "Schaum's Outline" series contains a ton of solved problems. They're also MUCH cheaper than buying a textbook.

Like I said I'm not sure what your skill level is, but here are a few I found on Amazon.

Precalculus

Trigonometry

Calculus

See if your parents/guardian would let you find a tutor. If this isn't practical, get a big book of calculus exercises with solutions, there's loads on Amazon (e.g. this one).

If you're looking for a book that will give you lots of insights, then look no further than the MIT Calculus book. It's free online, I highly suggest printing it out (if you have access to some sort of binding machine, do that, I have it comb bound in 3 volumes), and I swear to god every time I sit down to read it, whether to learn more about something I'm currently learning in school or just to see what it has to say about an older topic, I end up having some sort of earth-shattering revelation within the first few paragraphs, without fail, no hyperbole.

CANNOT recommend it enough, it is what gave me a huge edge over everyone in my Calc 2 class when it came to infinite series. It's my calculus bible. And it's free!

EDIT: My school had that calc book you mentioned as the assigned text. It's worthless. What you want to invest in for extra practice problems is a Schaum's Outline. This combined with the MIT calc book is all you'll need, and then use Wolfram Alpha to give you a step-by-step solution for problems you're struggling with/stuck on.

This was basic math for Civil engineers, these two books to be specific but equally vague:

http://www.amazon.ca/Calculus-Analytic-Geometry-Henry-Edwards/dp/0137363311

And then also Erwin Kreyzig Advanced engineering mathematics.

"Analytic geometry and calculus: A unified treatment" by Frederic H. Miller, (c) 1949 by John Wiley & Sons, Inc. (Note: No ISBN number is available because the ISBN book numbering system had not been invented yet)

Instructor solution manual of Calculus and Analytic Geometry by Thomas and Finney

This is the book https://www.amazon.com/Calculus-Analytic-Geometry-George-Thomas/dp/0201531747

I want a file which contains solutions to problems of all 14 chapters.

$5

Riemannian Geometry by Manfredo do Carmo.

https://www.amazon.com/Riemannian-Geometry-Manfredo-Perdigao-Carmo/dp/0817634908/ref=sr_1_4?s=books&ie=UTF8&qid=1492966206&sr=1-4&keywords=differential+geometry+manfredo

This one is highly regarded. You can get it for $5.00, maybe less if you search around.

https://www.amazon.com/Calculus-Analytic-Geometry-Alternate-George/dp/B000K1UMJS?ie=UTF8&*Version*=1&*entries*=0

https://smile.amazon.com/Schaums-Outline-Calculus-6th-Problems/dp/0071795537?sa-no-redirect=1

Same series, has a book on just about every undergrad physics/math subject.

I learned DG from Riemannian Geometry by do Carmo.

I am a mathematician though, not a physicist.

Haven't read it, but I'd start here. Can anyone comment on Fractals: a very short introduction?

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

For the fractals part, I've been planning on picking up Fractal Geometry, by Falconer.

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

How about a Shaums? Their cheap and paperback and have lots of solved problems.

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/

TL;DR Start here

_

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

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

try and find a pdf of shaum's outlines. they are broken up by class and the higher level stuff is broken up by subject.

here is calc on amazon