That helps a little. I'm not too familiar with that world (I'm a physics major), but I took a look at a sample civil engineering course curriculum. If you like learning but the material in high school is boring, you could try self-teaching yourself basic physics, basic applied mathematics, or some chemistry, that way you could focus more on engineering in college. I don't know much about engineering literature, but this book is good for learning ODE methods (I own it) and this book is good for introductory classical mechanics (I bought and looked over it for a family member). The last one will definitely challenge you. Linear Algebra is also incredibly useful knowledge, in case you want to do virtually anything. Considering you like engineering, a book less focused on proofs and more focused on applications would be better for you. I looked around on Amazon, and I found this book that focuses on applications in computer science, and I found this book focusing on applications in general. I don't own any of those books, but they seem to be fine. You should do your own personal vetting though. Considering you are in high school, most of those books should be relatively affordable. I would personally go for the ODE or classical mechanics book first. They should both be very accessible to you. Reading through them and doing exercises that you find interesting would definitely give you an edge over other people in your class. I don't know if this applies to engineering, but using LaTeX is an essential skill for physicists and mathematicians. I don't feel confident in recommending any engineering texts, since I could easily send you down the wrong road due to my lack of knowledge. If you look at an engineering stack exchange, they could help you with that.

​

You may also want to invest some time into learning a computer language. Doing some casual googling, I arrived at the conclusion that programming is useful in civil engineering today. There are a multitude of ways to go about learning programming. You can try to teach yourself, or you can try and find a class outside of school. I learned to program in such a class that my parents thankfully paid for. If you are fortunate enough to be in a similar situation, that might be a fun use of your time as well. To save you the trouble, any of these languages would be suitable: Python, C#, or VB.NET. Learning C# first will give you a more rigorous understanding of programming as compared to learning Python, but Python might be easier. I chose these three candidates based off of quick application potential rather than furthering knowledge in programming. This is its own separate topic, but my personal two cents are you will spend more time deliberating between programming languages rather than programming if you don't choose one quickly.

​

What might be the best option is contacting a professor at the college you will be attending and asking for advice. You could email said professor with something along the lines of, "Hi Professor X! I'm a recently accepted student to Y college, and I'm really excited to study engineering. I want to do some rigorous learning about Z subject, but I don't know where to start. Could you help me?" Your message would be more formal than that, but I suspect you get the gist. Being known by your professors in college is especially good, and starting in high school is even better. These are the people who will write you recommendations for a job, write you recommendations for graduate school (if you plan on it), put you in contact with potential employers, help you in office hours, or end up as a friend. At my school at least, we are on a first name basis with professors, and I have had dinner with a few of mine. If your professors like you, that's excellent. Don't stress it though; it's not a game you have to psychopathically play. A lot of these relationships will develop naturally.

​

That more or less covers educational things. If your laziness stems from material boredom, everything related to engineering I can advise on should be covered up there. Your laziness may also just originate from general apathy due to high school not having much impact on your life anymore. You've submitted college applications, and provided you don't fail your classes, your second semester will probably not have much bearing on your life. This general line of thought is what develops classic second semester senioritis. The common response is to blow off school, hang out with your friends, go to parties, and in general waste your time. I'm not saying don't go to parties, hang out with friends, etc., but what I am saying is you will feel regret eventually about doing only frivolous and passing things. This could be material to guilt trip yourself back into caring.

​

For something more positive, try to think about some of your fun days at school before this semester. What made those days enjoyable? You could try to reproduce those underlying conditions. You could also go to school with the thought "today I'm going to accomplish X goal, and X goal will make me happy because of Y and Z." It always feels good to accomplish goals. If you think about it, second semester senioritis tends to make school boring because there are no more goals to accomplish. As an analogy, think about your favorite video game. If you have already completed the story, acquired the best items, played the interesting types of characters/party combinations, then why play the game? That's a deep question I won't fully unpack, but the simple answer is not playing the game because all of the goals have been completed. In a way, this is a lot like second semester of senior year. In the case of real life, you can think of second semester high school as the waiting period between the release of the first title and its sequel. Just because you are waiting doesn't mean you do nothing. You play another game, and in this case it's up to you to decide exactly what game you play.

​

Alternatively, you could just skip the more elegant analysis from the last few paragraphs and tell yourself, "If I am not studying, then someone else is." This type of thinking is very risky, and most likely, it will make you unhappy, but it is a possibility. Fair warning, you will be miserable in college and misuse your 4 years if the only thing you do is study. I guarantee that you will have excellent grades, but I don't think the price you pay is worth it.

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.

/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

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!

>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.

------------------------------------------------------

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.

--------------------------------------------------

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.

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.

LIST OF APPLICATIONS IN MY DIFF EQ PLAYLIST
Have you seen the first video in my series on differential equations?

I'm still working on the playlist, but the first video lists a bunch of applications that you might not have seen before. My goal was to provide a sample of the diversity of applications outside of mathematics, and I chose fairly concrete examples that include applications in engineering.

I don't go into any depth at all regarding any of the particular applications (it's just a short introductory video), but you might find the brief introduction to be helpful.

If you find any one of the applications interesting, then a Google search will reveal more detailed resources.

A COUPLE OF FREE OR INEXPENSIVE BOOKS
Also, off the top of my head, the books below have quite a few applications that you might not see in the more standard textbooks.

• Differential Equations and Their Applications: An Introduction to Applied Mathematics, Martin Braun (Amazon, PDF)
  
• Ordinary Differential Equations, Morris Tenenbaum and Harry Pollard (Amazon)
  
  I think you can find other legal PDFs of Braun's third edition, too. Pollard and Tenenbaum is an inexpensive paperback from Dover, and I actually found a copy at my local library.
  
  ENGINEERING BOOKS
  Of course, the books I listed are strictly devoted to differential equations, but you can find other applications if you look for books in engineering. For example, I used differential equations in a course on signals and systems that I tutored last semester (applications included electrical circuits and mass-spring-damper systems).
  
  NEAT VIDEO (SOFT BODY MODELING)
  By the way, here's a cool video of various soft body simulations based on mass-spring-damper systems modeled by differential equations.
  
  Here's a Wikipedia article on soft body dynamics. This belongs to the field of computer graphics, so I'm not sure if you're interested, but mass-spring-damper systems come up a fair amount in engineering courses, and this is an application of those ideas that might open your mind a bit to other possible applications.
  
  Edit: typo

I want to pay someone else to write the following book. And if someone else won't write it, maybe I will.

Applied Mathematics for Economists (max 700 pages)

aka Simon and Blume, but more mature, more compact, more focused.

Chapters:

1. Linear Algebra I (matricies, systems of equations, vector spaces, linear transformations, eigenthings; with applications to linear economic models)
   
2. Partial Differential Calculus (limits and continuity in R^(n); gradient, Jacobian, Hessian; maps from R^(n) to R^(m); implicit and inverse function theorems; unconstrained optimization, Lagrange multipliers, and equilibrium concepts; the focus is on systems of equations as seen throughout economics)
   
3. Multiple Integrals (standard topics with emphasis on applications to probability; the last section would cover Bayesian inference)
   
4. Linear Algebra II (least squares; singular value decomposition; LU, Cholesky, QR, Schur, and other funky matrix decompositions you occasionally use in econometrics; some extended attention is given to quadratic forms and symmetric matricies)
   
5. Theory of Optimization (Lagrangeans, Hamiltonians, Bellmans; the classical results regarding existence and uniqueness; the classical results regarding characterization, regularity, and sensitivity; Berge's theorem)
   
6. Complex Variables and Fourier Series (for the time-series folks)
   
   Prerequisite: Calculus I-II are assumed.
   
   Idea: Calculus III and linear algebra are developed in the first three chapters, with special emphasis on applications to economics. The fourth chapter covers advanced material in linear algebra and even touches on numerical issues. The fifth chapter develops optimization, both static and dynamic, in discrete and continuous time, with applications to economics throughout. The sixth chapter develops the complex variable theory that you need to do proper time-series analysis.
   
   Maybe sneak a chapter on analysis and topology in there. Maybe cut the optimization chapter into two or three pieces.
   
   The point of the book is to put all of the applied math that you'll actually use on a day-to-day basis in a single, relatively brief textbook. I think it can be done in less than 700 pages. Kaplan crammed all of advanced calculus for physics into 700 pages, so surely it can be done for economics.
   
   What isn't here: There is a market for books on advanced calculus. However, they are all geared towards engineering and physics students. They have a ton of material on differential equations and vector theory that are irrelevant for economists. So I cut all of those bits out and replace them with additional material on matrix analysis and optimization. Basically, I trade the theorems of Green, Gauss, and Stokes for the theorems of Lagrange, Hamilton, and Bellman. It's a reasonable tradeoff. I also throw away the (i,j,k) notation and spend more time on least squares, which would be brought back to its proper home in linear algebra.
   
   I have deliberately left out a detailed treatment of probability and statistics, as they could get their own book (of about 500 pages). I have left out a detailed treatment of numerical methods because they could get their own book as well.

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

There seems to often be this sort of tragedy of the commons with the elementary courses in mathematics. Basically the issue is that the subject has too much utility. Be assured that it is very rich in mathematical aesthetic, but courses, specifically those aimed at teaching tools to people who are not in the field, tend to lose that charm. It is quite a shame that it's not taught with all the beautiful geometric interpretations that underlie the theory.

As far as texts, if you like physics, I can not recommend highly enough this book by Lanczos. On the surface it's about classical mechanics(some physics background will be needed), but at its heart it's a course on dynamical systems, Diff EQs, and variational principles. The nice thing about the physics perspective is that you're almost always working with a physically interpretable picture in mind. That is, when you are trying to describe the motion of a physical system, you can always visualize that system in your mind's eye (at least in classical mechanics).

I've also read through some of this book and found it to be very well written. It's highly regarded, and from what I read it did a very good job touching on the stuff that's normally brushed over. But it is a long read for sure.

The iterative map x = cos(x) converges to the fixed point, the value of x for which x = cos(x). I don't think there's a nice analytic expression.

What's more interesting is why the map converges for all initial conditions. Firstly, the map x = cos(x) is what's called a dynamical system. An orbit for this dynamical system with initial condition x is the sequence {x, cos(x), cos(cos(x)), ...}. As you've found out by experimentation, all orbits converge to a single value. This value is called an attractor for the dynamical system, and an attracting fixed point in this case. To see why this is true, you can draw a cobweb plot. Wikipedia even has an illustration for the fixed point of x = cos(x).

I find the field of dynamical systems extremely interesting. In the Wikipedia article of the cobweb plot, there is an illustration of much more interesting behavior called chaos. A book that I can highly recommend, that starts with this kind of iterative maps and works its way up is Chaos: An Introduction to Dynamical Systems by Alligood, Sauer and Yorke, though it requires knowledge of Calculus.

15! Well then, you have plenty of time to figure this out. Well, a few years, in any case.

I think what you should do is learn some programming as soon as possible (assuming you don't already). It's easy, trust me. Start with C, C++, Python or Java. Personally, I started with C, so I'll give you the tutorials I learned from: http://www.cprogramming.com/tutorial/c/lesson1.html

You should also try out some electronics. There's too much theory for me to really explain here, but try and maybe get a starter's kit with a book of tutorials on basic electronics. Then, move onto some more complicated projects. It wouldn't hurt to look into some circuit theory.

For mechanical, well... that one is kind of hard to get practical experience for on a budget, but you can still try and learn some of the theory behind it. Start with learning some dynamics and then move onto statics. Once you've got that down, try learning about the structure and property of materials and then go to solid mechanics and machine design. There's a lot more to mechanical engineering than that, but that's a good starting point.

There's also, of course, chemical engineering, civil engineering, industrial engineering, aerospace engineering, etc, etc... but the main ones I know about are mechanical (what I'm currently studying), electrical and computer.

Hope this helped. I wasn't trying to dissuade you from pursuing engineering, but instead I'm just forewarning you that a lot of people go into it with almost no actual engineering skills and well, they tend to do poorly. If you start picking up some skills now, years before college, you'll do great.

EDIT: Also, try learning some math! It would help a lot to have some experience with linear algebra, calculus and differential equations. This book should help.

You are becoming a second year EE student, so Fourier transforms, vector calculus, and partial differential equations will be the primary math topics you should be familiar with. Keep in mind that this is based off of my own experience with EE; your university might differ in its approach.

In any case I would recommend that you get a text like this one and start learning the material early. Of course, check the sidebar for a link to Khan Academy for some good educational material on foundation topics like calculus and trig.

Best of luck and don't give up! Learning higher level math is extremely rewarding and will help you to solve problems in many other fields.

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!

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 :)

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

> don't think that there is a logical progression to approaching mathematics

Well, this might be true of the field as a whole, but def not true when it comes to learning basic undergrad level math after calc 1, as the OP asked about. There are optimized paths to gaining mathematical maturity and sufficient background knowledge to read papers and more advanced texts.

> Go to the mathematics section of a library, yank any book off the shelf, and go to town.

I would definitely NOT do this, unless you have a lot of time to kill. I would, based on recommendations, pick good texts on linear algebra and differential equations and focus on those. I mean focus because it is easy in mathematics to gloss over difficulties.

My recommendation, since you are self-studying, is to pick up Gil Strang's linear algebra book (go for an older edition) and look up his video lectures on linear algebra. That's a solid place to start. I'd say that course could be done, with hard work, in a summer. For a differential equations book, I'm not exactly sure. I would seek out something with some solid applications in it, like maybe this: http://amzn.com/0387978941

That is more than a summer's worth of work.

Sorry, agelobear, to be such a contrarian.

Oh sorry, I forgot to reply to you. There's two textbooks I typically reference for ODEs that will be more than sufficient by the sounds of it.

There is Braun - Differential Equations and their Applications. The author tries to motivate each problem by going to some interesting applications. I think he fails in this as the applications are quite standard at the end of the day, he just goes into more detail to give the context. But the book is overall great and quite well explained.

The other one is kind of tough to track down as it seems to be out of print now Differential Equations with applications and historical notes by Simmons. Again the book has a bit of a gimmick where it puts each of the big contributors, and methods, in historical context. It's an excellent resource I find, it gets pretty in depth on Laplace transforms.

It's really your call which one you go with, if you have some extra time once you make it through your course content I would recommend taking a look at the sections on qualitative theory, systems of ODEs, and numerical methods. Those topics are both fascinating and I think tend to have more active work associated with them.

I would check out Differential Equations, Dynamical Systems, and Linear Algebra by Hirsch and Smale (note this is different from Differential Equations, Dynamical Systems, and an Introduction To Chaos by Hirsch, Smale, and Devaney, which is a less self-contained, less rigourous, and more application-driven sequel).

The former book does rigourous proofs of all the results. It does applications as well, and is actually good at explaining things intuitively as well as rigourously. If you're okay with multivariable calculus, then I think you'd be okay with this book. While it's definitely easier if you already know linear algebra and analysis, this book doesn't assume those as prerequisites (the necessary linear algebra is mostly contained in the book, but the analysis results are usually stated without proof before being used to prove something else). That said, generally I would recommend that one knows linear algebra and real analysis, in addition to multivariable calculus, before reading this book or any other serious book on dynamical systems. They say in the introduction that a strong sophomore could handle this book, but that it's written more for an upper level undergrad or even graduate course.

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.).

I can highly recommend Paul's Online Math Notes for Differential equations - the range of topics he covers is comparable to just about any standard undergraduate course, and the exposition is clear with lots of worked examples.

If you're looking to invest in a physical textbook (probably a good idea if you're going into Aero engineering), then I've found Boyce & DiPrima to be useful and clearly written, with plenty of examples and interesting exercises.

There's also these free textbooks, though I haven't gone through them in enough detail to vouch for how they'd go for self-study

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

I agree that Arnold's ODE book is the best on the subject, but as you said it's not for beginners. For an intro to the subject, I think Martin Braun's Differential Equations and Their Applications is the best.

I've always thought that Fritz John's Partial Differential Equations is the best PDE book.

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.

Definitely split up the load and take classes over the summer. I often hear people say Calculus II is the hardest of the EPIC MATH TRILOGY. I certainly agree. If you've done well in Calc I and II and have a notion of what 3d vectors are (physics should of covered this well) then you'll have no problem with Calc III (though series' and summations can be tough).

Differential equations will be your first introduction to hard "pure"-style math concepts. The language will take some time to understand and digest. I highly recommend you purchase this book to supplement your textbook. If you take notes on each chapter and work through the derivations, problems, and solutions, you'll be golden.

In my experience, materials is not math heavy for ME's. All of my tests were multiple choice and more concept based. It's not too bad.

Thermodynamics and Engineering Dynamics will be in the top three as far as difficulty goes. Circuits or Fluids will also be in there somewhere. Make sure you allow plenty of time to study these topics.

Good luck!

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.

it might be very dated, but I remember being deeply impressed by Hirsch & Smale's "Differential Equations, Dynamical Systems, and Linear Algebra", a beautiful and seminal book: http://www.amazon.com/Differential-Equations-Dynamical-Mathematics-Academic/dp/0123495504/

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.

If you want to learn a modern (i.e., dynamical systems) approach, try Hirsch, Smale and Devaney for an intro-level book and Guckenheimer and Holmes for more advanced topics.

> a more Bourbaki-like approach

Unless you already have a lot of exposure to working with specific problems and examples in ODEs, it's much better to start with a well-motivated book with a lot of interesting examples instead of a dry, proof-theorem style book. I know it's tempting as a budding mathematician to have the "we are doing mathematics here after all" attitude and scoff at less-than-rigorous approaches, but you're really not doing yourself any favors. In light of that, I highly recommend starting with Strogatz which is my favorite math book of all time, and I'm not alone in that sentiment.

Algebra > Calculus > Linear algebra > ordinary differential equations is the logical progression. Technically you don't need calculus to do or understand linear algebra, but normally that will be the progression if you're a math major.

I no longer have any books on linear algebra or calculus, but I did keep my differential equations book. Check out "Differential Equations" by Blanchard, Devaney, Hall.. Now that I've looked it up and seen the price though... maybe only use it if you can find it in your library, which would be a good bet. It's a good book though, with a fair amount of techniques for solving things on paper and a discussion of how you can solve ODEs numerically.

Ordinary Differential Equations from the Dover Books on Mathematics series. I Just took my final for Diff Eq a few days ago and the book was miles better than the one my school suggested and is the best written math textbook I have encountered during my math minor. My Diff Eq course only covered about the first 40% of the book so there's still a TON of info that you can learn or reference later. It is currently $14 USD on amazon and my copy is almost 3" thick so it really is a great deal. A lot of the reviewers are engineering and science students that said the book helped them learn the subject and pass their classes no problem. Highly Highly recommend. ISBN-10: 9780486649405

​

https://www.amazon.com/gp/product/0486649407/ref=ppx_yo_dt_b_asin_title_o08_s00?ie=UTF8&psc=1

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.

All of the books I can see from top to bottom on Amazon:

1. http://www.amazon.com/Elements-Chemical-Reaction-Engineering-Edition/dp/0130473944 -- used price: $90.98.
   
2. http://www.amazon.com/Molecular-Thermodynamics-Donald-McQuarrie/dp/189138905X/ref=sr_1_1?s=books&ie=UTF8&qid=1407531821&sr=1-1&keywords=molecular+thermodynamics -- used price: $70.00 (paperback is $29.99)
   
3. http://www.amazon.com/Physical-Chemistry-Molecular-Donald-McQuarrie/dp/0935702997/ref=sr_1_1?s=books&ie=UTF8&qid=1407531925&sr=1-1&keywords=physical+chemistry+a+molecular+approach -- used price: $72.44 (paperback is $42.65)
   
4. http://www.amazon.com/Quantum-Physics-Molecules-Solids-Particles/dp/047187373X/ref=sr_1_1?s=books&ie=UTF8&qid=1407532022&sr=1-1&keywords=quantum+physics+of+atoms+molecules+solids+nuclei+and+particles -- used price: $52.66
   
5. http://www.amazon.com/Introduction-Chemical-Engineering-Thermodynamics-Mcgraw-Hill/dp/0073104450/ref=sr_1_1?s=books&ie=UTF8&qid=1407532094&sr=1-1&keywords=introduction+to+chemical+engineering+thermodynamics -- used price: $129.96 (paperback is $84.38)
   
6. http://www.amazon.com/Organic-Chemistry-8th-Eighth-BYMcMurry/dp/B004TSKJVE/ref=sr_1_5?s=books&ie=UTF8&qid=1407532227&sr=1-5&keywords=organic+chemistry+mcmurry+8th+edition -- used price: $169.33 (paperback is $79.86)
   
7. http://www.amazon.com/Elementary-Differential-Equations-William-Boyce/dp/047003940X/ref=sr_1_7?ie=UTF8&qid=1407532549&sr=8-7&keywords=Elementary+Differential+Equations+and+Boundary+Value+Problems%2C+9th+Edition+solutions -- used price: $8.00
   
8. http://www.amazon.com/Numerical-Methods-Engineers-Sixth-Edition/dp/0073401064/ref=sr_1_1?ie=UTF8&qid=1407532859&sr=8-1&keywords=numerical+methods+for+engineers+6th+edition -- used price: $47.99 (paperback is $22.48)
   
9. http://www.amazon.com/Applied-Partial-Differential-Equations-Mathematics/dp/0486419762/ref=sr_1_5?s=books&ie=UTF8&qid=1407532927&sr=1-5&keywords=applied+partial+differential+equations -- used price: $8.32 (paperback is $1.96)
   
10. http://www.amazon.com/Transport-Phenomena-2nd-Byron-Bird/dp/0471410772/ref=sr_1_1?s=books&ie=UTF8&qid=1407533036&sr=1-1&keywords=transport+phenomena+bird+stewart+lightfoot+2nd+edition -- used price: $28.00
    
11. http://www.amazon.com/Basic-Engineering-Data-Collection-Analysis/dp/053436957X/ref=sr_1_2?s=books&ie=UTF8&qid=1407533106&sr=1-2&keywords=data+collection+and+analysis -- used price: $80.00
    
12. http://www.amazon.com/Calculus-9th-Dale-Varberg/dp/0131429248/ref=sr_1_1?s=books&ie=UTF8&qid=1407533219&sr=1-1&keywords=calculus+varberg+purcell+rigdon+9th+edition+pearson -- used price: $11.97 (paperback is $2.94)
    
13. http://www.amazon.com/Elementary-Principles-Chemical-Processes-Integrated/dp/0471720631/ref=sr_1_1?s=books&ie=UTF8&qid=1407533286&sr=1-1&keywords=elementary+principles+of+chemical+processes -- used price: $161.72
    
14. http://www.amazon.com/Inorganic-Chemistry-4th-Gary-Miessler/dp/0136128661/ref=sr_1_1?s=books&ie=UTF8&qid=1407533412&sr=1-1&keywords=inorganic+chemistry+messler -- used price: $75.00
    
15. http://www.amazon.com/Fundamentals-Heat-Transfer-Theodore-Bergman/dp/0470501979/ref=sr_1_1?s=books&ie=UTF8&qid=1407533484&sr=1-1&keywords=fundamental+of+heat+and+mass+transfer -- used price: $154.99 (loose leaf is $118.23)
    
16. http://www.amazon.com/Biochemistry-Course-John-L-Tymoczko/dp/1429283602/ref=sr_1_1?s=books&ie=UTF8&qid=1407533588&sr=1-1&keywords=biochemistry+a+short+course -- used price: $139.00 (loose leaf is $115)
    
17. http://www.amazon.com/Separation-Process-Principles-Biochemical-Operations/dp/0470481838 -- used price: $93.50 (international edition is $49.80)
    
18. http://www.amazon.com/University-Physics-Modern-13th/dp/0321696867/ref=sr_1_1?s=books&ie=UTF8&qid=1407545099&sr=1-1&keywords=university+physics+young+and+freedman -- used price: $83.00
    
    Books & Speakers | Price (New)
    ---|---
    Elements of Chemical Reaction Engineering (4th Edition) | $122.84
    Molecular Thermodynamics | $80.17
    Physical Chemistry: A Molecular Approach | $89.59
    Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles | $128.32
    Introduction to Chemical Engineering Thermodynamics (The Mcgraw-Hill Chemical Engineering Series) | $226.58
    Organic Chemistry 8th Edition | $186.00
    Elementary Differential Equations | $217.67
    Numerical Methods for Engineers, Sixth Edition | $200.67
    Applied Partial Differential Equations | $20.46
    Transport Phenomena, 2nd Edition | $85.00
    Basic Engineering Data Collection and Analysis | $239.49
    Calculus (9th Edition) | $146.36
    Elementary Principles of Chemical Processes, 3rd Edition | $206.11
    Inorganic Chemistry (4th Edition) | $100.00
    Fundamentals of Heat and Mass Transfer | $197.11
    Biochemistry: A Short Course, 2nd Edition | $161.45
    Separation Process Principles: Chemical and Biochemical Operations | $156.71
    University Physics with Modern Physics (13th Edition) | $217.58
    Speakers | $50.00
    
    Most you can get is $1476.86 (selling all of the books (used and hard cover) in person), and if you sell it on Amazon, they take around 15% in fees, so you'll still get $1255.33. But wait...if you sell it to your university's book store, best they can do is $.01.
    
    Total cost: $2832.11 (including speakers)
    
    Net loss: -$1355.25 (books only). If sold on Amazon, net loss: -$1576.78 (books only). Speakers look nice; I wouldn't sell them.
    
    Edit: Added the two books and the table. /u/The_King_of_Pants gave the price of speakers. ¡Muchas gracias para el oro! Reminder: Never buy your books at the bookstore.
    
    Edit 2: Here are most of the books on Library Genesis
    Thanks to /u/WhereToGoTomorrow

The Cauchy-Schwarz Master Class is a great book on inequalities that will really improve your understanding of how and when to apply specific techniques. Highly recommended, and the paperback version isn't too overpriced.

That being said, competition mathematics also requires that you be able to recognize which technique to use (and then do it) quickly. Improving your speed is really accomplished by doing problem sets, and I don't have a good collection to suggest here.

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!

If you're interested in doing mathematical biology later on I'd recommend keeping dynamical systems stuff fresh in your memory. Maybe read and do some exercises from Tenenbaum and Pollard once in a while? Also, looks like you haven't taken Linear Algebra yet, so maybe self-study from Linear Algebra Done Right by Axler.

Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.

Amazon search for Dover Books on mathematics

There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm

Pre-Calculus / Problem-Solving

• The Art of Problem Solving - Lehoczky/Ruczyk - in two volumes, both have full solutions available separately
  
  Calculus
  
  
• Calculus - Stewart - ignore the bad reviews, it's very clear and covers all of basic calculus well - it's just the book that most intro students use and as a consequence gets a bad rap
  
  
• Calculus - Spivak - this will take your understanding of calculus to the next level
  
  
• Calculus - Apostol - there are two volumes and they are very comprehensive, albeit difficult
  
  Linear Algebra
  
  
• Linear Algebra - Lay
  
  
• Linear Algebra Done Right - Axler
  
  Differential Equations
  
  
• Elementary Differential Equations - Boyce/DePrima
  
  
• Ordinary Differential Equations - Tenenbaum
  
  Number Theory
  
  
• Number Theory - Pommersheim for a change of pace and proof skills
  
  Proof-Writing
  
  
• Transition to Higher Mathematics - Dumas/McCarthy - helps to bridge the gap between "lower" and higher math
  
  Analysis
  
  
• The Way of Analysis - Strichartz
  
• Principles of Mathematical Analysis - Rudin
  
• Real and Complex Analysis - Rudin
  
  Complex Analysis
  
  
• Complex Analysis - Ahlfors
  
  Functional Analysis
  
  
• Functional Analysis - Rudin
  
  
• Introductory Functional Analysis with Applications - Kreyszig
  
  Partial Differential Equations
  
  
• Partial Differential Equations for Scientists and Engineers - Farlow
  
  Higher-dimensional Calculus and Differential Geometry
  
  
• Calculus on Manifolds - Spivak - for a modern take on multivariable calculus
  
  
• Differential Geometry - Kreyszig
  
  Abstract Algebra
  
  
• A First Course in Abstract Algebra - Fraleigh
  
• Abtract Algebra - Dummit and Foote
  
  Geometry
  
  
• The Elements - Euclid - for culture and for understanding where math has come from, plus it's still an awesome book after all these years; don't need to go through all of it
  
• Euclidean and Non-Euclidean Geometries - Greenberg
  
  Topology
  
  
• Introduction to Topology - Mendelson
  
  
• Topology - Munkres
  
  Set Theory and Logic
  
  
• Introduction to Set Theory - Hrbacek/Jech
  
  
• Elements of Set Theory - Enderton
  
  
• Mathematical Logic - Kleene
  
  
• Set Theory and the Continuum Hypothesis - Cohen
  
  Combinatorics / Discrete Math
  
  
• Combinatorics - Cameron
  
  
• Concrete Math - Graham/Knuth/Patashnik
  
  
• Discrete Math - Biggs
  
  Graph Theory
  
  
• A First Course in Graph Theory - Chartrand/Zhang
  
  P. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"
  

It's hard to say, in my day, Advanced Calculus was a class that was taught out of a book like this which basically covered what might be termed "Vector Calculus" i.e. the study of functions f:R^n -->R^m . There would be little, if any, talk of the material in a book like Apostol. That would be reserved for a class called "Real Analysis".

For grad school in physics I'd say that the material in Kaplan (above) would be absolutely essential. I'm not sure how useful the Apostol book is going to be for a physicist, but I'm not a physicist... That said, Spivak, which was mentioned elsewhere, strikes me a being very useful for a physicist and it would be a lot easier to digest after a book like Apostol.

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.

Depends on how in depth you are ready to go. Maths is the starting point to grow your knowledge of the universe and you would eventually want to get to the point of being able to read something like this.

Also found this which seems like it might be a bit easier.

Probably not what you are looking for and my advice would be to get an engineering degree.

These are my personal favourites for introductory books on ODEs - [Simmons & Krantz's Differential equations: theory, techniques and practice](https://www.amazon.com/Differential-Equations-Steven-Krantz-Simmons/dp/0070616094) is a great book with examples from physics and engineering along with lots of historic notes.

[Braun's differential equations and their applications](https://www.amazon.com/Differential-Equations-Their-Applications-Introduction/dp/0387978941/ref=sr\_1\_1?crid=35EOUTZZ32HDA&keywords=braun+differential+equations&qid=1556968795&s=books&sprefix=braun+Differenz%2Cstripbooks-intl-ship%2C215&sr=1-1) is another applications oriented differential equations book that is a bit more involved than Simmon's but has a much broader perspective with introductions to bifurcation theory and applications in mathematical biology.

​

If you're not planning to do research in ODE theory, but want to learn the basic theory more rigorously, then [Hurewicz's Lectures](https://www.amazon.com/Lectures-Ordinary-Differential-Equations-Hurewicz/dp/1258814889/ref=sr\_1\_1?crid=341Z3D48AUTBU&keywords=hurewicz+differential&qid=1556969136&s=gateway&sprefix=hurewicz+%2Cstripbooks-intl-ship%2C216&sr=8-1) is a perfect short book that covers the basic theorems for existence and uniqueness of solutions of ODEs.

Boyce and DiPrima is a very popular and IMO very good introduction to differential equations textbook. It discusses Existence & Uniqueness, Linear Systems, Dynamical Systems, and Stability Theory among other things. It's worth a look.

An Introduction to Ordinary Differential Equations - $7.62

Ordinary Differential Equations - $14.74

Partial Differential Equations for Scientists and Engineers - $11.01

Dover books on mathematics have great books for very cheap. I personally own the second and third book on this list and I thought they were a great resource, especially for the price.

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.

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.

As far as textbooks go, Alligood's Choas is a great introduction to Dynamical Systems. It might be a little advanced (it's designed for late undergrad or early grad courses), but I find it to be very accessible, and it includes a lot of supplementary material on the topic (plus pretty pictures).

Also, one of the authors (James Yorke) is a pretty big player in Chaos Theory; he coined many terms such as "chaos", "crisis", etc. in the landmark paper "Period 3 Implies Chaos" (which is actually a document included in the first chapter of the book, and is pretty accessible).

I really like this one and this one. For what it's worth, I am PhD student in an applied math department and PDEs are all we do. I found these books helpful for the basic stuff.

Hello guys. I'm looking for Differential Equations book. When I search on amazon I got this Nagle and [Zill] (https://www.amazon.com/Differential-Equations-Boundary-Value-Problems-8th/dp/1111827060/ref=sr_1_5?ie=UTF8&qid=1505387954&sr=8-5&keywords=zill+differential+equation) Which one do you think is best for self study?

The best differential equations book that I have used was my first ODE course in Undergrad, by Blanchard, Hall, and Delaney. It was a textbook that didn't take a lot of time going into the solutions of ODEs, but the modeling and interpretation aspect much more. The book treats you like a human too and it reads so well. Highly suggest it.

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

For DEs try:
Ordinary Differential Equations by Tennenbaum

Its a great book with a TON of worked examples and solutions to all the exercises. This text was my holy book during my undergrad engineering courses.

These were the most enlightening for me on their subjects:

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

Kaplan's Advanced Calculus might be for you, then. Link. I've found this through more nefarious sources :) PM me if you can't find it then. Just be forewarned that this is a very proof-heavy book!

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.

Available free as well:

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

This is the best one I have used.

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.

When I took the class 3 years ago we used Boyce and DiPrima's Elementary Differential Equations. I thought it was a great book since I ended up skipping most of the lectures and teaching the material to myself.

Not sure what your background is, but I quite liked Steele's book.

this one

i used this book, the one that was required for the class sucked, this one is much better and it's super cheap. Also, answers and steps are included in the sections, so you can actually check if you're doing it correctly or not.

We used Elementary Differential Equations by Boyce and DiPrima and I thought it was fantastic, as I had to learn the majority from the book.

I've never really used MIT OCW however I've used Paul's OMN a lot back when I was studying multivar calc. I do recommend books, though. I have books both on multivar calc and differential equations and they're both well, however, I've moved on from calculus (that is, I don't actively study it anymore) so I can't really say much more.


The books I have:

> https://www.amazon.com/Multivariable-Calculus-Clark-Bray/dp/1482550741/ref=sr_1_3?s=books&ie=UTF8&qid=1500976188&sr=1-3&keywords=multivariable+calculus

> https://www.amazon.com/Ordinary-Differential-Equations-Dover-Mathematics/dp/0486649407/ref=sr_1_1?s=books&ie=UTF8&qid=1500976233&sr=1-1&keywords=differential+equations

I had a hard time getting through dif eq also, because the book was unreadable (to me). I also hate reading anything by Hibbler. The Munson fluid mechanics book is... barely tolerable. When that happens, I tend to look, with more vigor than usual, for other sources. Dif eq: I was lucky, and our tutoring center has dif eq tutors. Fluids: I found a wonderful lecture series done by UC Irvine OpenCourseWare. Hibbler... well, I've been S.O.L. on that so far. Generally, I also try to find a solutions manual. If I'm having a terrible time with a problem, I work through it and check myself each step of the way. I often try to find a different book, too. The only reason you need the required book is so you know what to look for in your chosen book.

I recently discovered there is a very highly-rated dif eq book available used on Amazon for about $13, so I ordered it in the hopes that it will be readable, as I now need to brush up on dif eq and can't stand the book we used in class.

Looking to get deeper into both as my eventual goal is to learn stochastic DE's.

Thanks for the recommendations. I am looking into Arnold right now. Do you also have an opinion on this book by Blanchard? I've been eyeing that one for some time.

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.

This Book is really cheap and really good.

General Advice

• Try to put in a consistent amount of work daily.
  
• Make sure you're doing exercises (not just reading watching videos)
  
  Specific Advice
  
  
• Videos/Course: MIT Calculus Course. Watch the videos, supplement with notes if you need to. Do the assignments and check your solutions. Work towards getting passing grades in the exams. It's not important to get this done before college, just work on it and you will be more prepared.
  
• ODE Textbook: Love this book, working my way through it now, not sure if a better ODE book exists. It's also fairly simple but you might want to do some work on the MIT course first. It's not legally free, but...
  
  
  

I'm in engineering, and back when I took DE I mainly used Braun for applications I think. I remember liking it quite a bit. http://www.amazon.com/Differential-Equations-Their-Applications-Introduction/dp/0387978941/

Edit: also IIRC, Churchill (Complex Variables and Applications) had a section about applications of DE in Laplace transforms.

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.

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

http://www.amazon.com/Ordinary-Differential-Equations-Dover-Mathematics/dp/0486649407

Ordinary Differential Equations by Pollard and Tenenbaum (Dover)

Someone else recommended the book by Zill, don't waste your time, it's a piece of garbage, just another Pearson money grab.

I'm a particular fan of Tenenbaum and Pollard. It's both really well-explained and cheap.

I just finished my differential equations 2 final (B overall) we covered 2 chapters during the semester from ( http://www.amazon.com/Elementary-Differential-Equations-William-Boyce/dp/0470458321) chapter 5 used series solution techniques and chapter 7 used linear systems of differential eguations.

I've been reviewing topics in Hirsch & Smale's Dynamical Systems text.

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.

I would also recommend Cauchy-Schwarz Master Class. It has a lot of interesting problems without requiring a whole lot of prerequisite equipment. It also does an incredible job explaining how to go about solving problems.
http://www.amazon.com/The-Cauchy-Schwarz-Master-Class-Introduction/dp/052154677X

I just bought this for $10. Not all textbook companies are jokes. Just most.

I used Tenenbaum. One of my favorite undergrad books. Only downside that it doesn't use any Linear Algebra

Tenenbaum and Pollard's book is fine. It is cheap too (published by Dover methinks)
https://www.amazon.com/Ordinary-Differential-Equations-Dover-Mathematics/dp/0486649407

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).

Tenenbaum and Pollard's ODE book made the subject come quite easily when all my $150 textbook did was confuse me.

This is my favourite linear algebra book. This covers all of Calculus, Linear Algebra, and introduces you to ODEs.

Now that might sound like a little bit much, but when learning Linear Algebra you should learn it at the same time you are taking 3 dimensional calculus(Calc 3).

Smale, Hirsh

Like the other comments say, polynomial division is closely related to Jordan form for linear maps (see pages 12-13 of this lecture for instance https://see.stanford.edu/materials/lsoeldsee263/12-jcf.pdf), and although this has no immediately obvious connection with statistical data, this description of what linear transformations can be like, and has generalizations to non-linear phenomena as in this book https://www.amazon.co.uk/Differential-Equations-Dynamical-Systems-Mathematics/dp/0123495504/ref=pd_sbs_14_t_0/258-1780192-4312467?_encoding=UTF8&pd_rd_i=0123495504&pd_rd_r=30e40ed0-89f8-4fbc-9b41-d4d065c63d73&pd_rd_w=aukE3&pd_rd_wg=W7tNp&pf_rd_p=e44592b5-e56d-44c2-a4f9-dbdc09b29395&pf_rd_r=7CC41Q1EX6Z0J2AS7P43&psc=1&refRID=7CC41Q1EX6Z0J2AS7P43 or others.

 

Also, polynomial division is the main element of the polynomial-time primality test of AKS.

 

The tacit assumption that you can understand phenomena by exclusively applying least-squares analysis (regression, correlation, ANOVA, t-test, f test, etc etc) occurs even in parts of "Freakonomics" and damages and weakens that otherwise wonderful text. Polynomials, complex numbers, Laplace transforms etc etc are indeed weirdly and unfortunately specific, but the aim should not be to discard particular conceptual tools and make things more specific, rather to try to widen and connect together the conceptual tools we do have. And, crucially, to learn not to over-depend on any particular one.

For the most part, a complex system is just a dynamical system that isn't well understood. Choas theory is a small part of dynamical systems.

You only need up to an undergraduate differential equations class to understand chaos theory. A good book is Alligood and Yorke

Tenenbaum.

I use this one

Dover's ODE textbook.

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.

Calculus Volume 2 - Apostol.

Ah but did your tour have a guide?

what I meant is this one

Ordinary Differential Equations By: Morris Tenenbaum, Harry Pollard. Under $20, got me an A in my ODE class.

Graduate or undergraduate level?


If graduate, this is THE book to get.

This is much more applied.

a student's guide to maxwell's equations

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

The one and only , if you're willing to dedicate the time

Ordinary Differential Equations (Dover Books on Mathematics)
https://www.amazon.com/dp/0486649407/

For when you get into Electricity and Magnetism - This

Ordinary Differential Equations by Tenenbaum and Pollard is a classic. I thought it explained things well and was more rigorous than some other treatments of subject that I've come across.

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).

For ODEs, I'd seriously suggest buying this. Lots and lots of exercises, and full solutions. Plus, at $15, it hopefully won't break the bank too badly.

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.

With regards to your edit, if your friend is still incarcerated after reading his calculus text, send him Ordinary Differential Equations by Tenenbaum and Pollard. It contains zillions of worked problems showing how ODEs can be applied to physical problems.

What about the second example that triggers the same domain problem for 0. Do you get the same result?

https://www.amazon.ca/Ordinary-Differential-Equations-Morris-Tenenbaum/dp/0486649407/ref=mp_s_a_1_1?keywords=ordinary+differential+equations&qid=1567534646&s=gateway&sprefix=ordinary+differ&sr=8-1

This is the link for the textbook.