Reddit reviews Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics)
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McGraw-Hill Science Engineering Math
The answer is "virtually all of mathematics." :D
Although lots of math degrees are fairly linear, calculus is really the first big branch point for your learning. Broadly speaking, the three main pillars of contemporary mathematics are:
You might also think of these as the three main "mathematical mindsets" — mathematicians often talk about "thinking like an algebraist" and so on.
Calculus is the first tiny sliver of analysis and Spivak's Calculus is IMO the best introduction to calculus-as-analysis out there. If you thought Spivak's textbook was amazing, well, that's bread-n-butter analysis. I always thought of Spivak as "one-dimensional analysis" rather than calculus.
Spivak also introduces a bit of algebra, BTW. The first few chapters are really about abstract algebra and you might notice they feel very different from the latter chapters, especially after he introduces the least-upper-bound property. Spivak's "properties of numbers" (P1-P9) are actually the 9 axioms which define an algebraic object called a field. So if you thought those first few chapters were a lot of fun, well, that's algebra!
There isn't that much topology in Spivak, although I'm sure he hides some topology exercises throughout the book. Topology is sometimes called the study of "shape" and is where our most general notions of "continuous function" and "open set" live.
Here are my recommendations.
Analysis If you want to keep learning analysis, check out Introductory Real Analysis by Kolmogorov & Fomin, Principles of Mathematical Analysis by Rudin, and/or Advanced Calculus of Several Variables by Edwards.
Algebra If you want to check out abstract algebra, check out Dummit & Foote's Abstract Algebra and/or Pinter's A Book of Abstract Algebra.
Topology There's really only one thing to recommend here and that's Topology by Munkres.
If you're a high-school student who has read through Spivak in your own, you should be fine with any of these books. These are exactly the books you'd get in a more advanced undergraduate mathematics degree.
I might also check out the Chicago undergraduate mathematics bibliography, which contains all my recommendations above and more. I disagree with their elementary/intermediate/advanced categorization in many cases, e.g., Rudin's Principles of Mathematical Analysis is categorized as "elementary" but it's only "elementary" if your idea of doing math is pursuing a PhD. Baby Rudin (as it's called) is to first-year graduate analysis as Spivak is to first-year undergraduate calculus — Rudin says as much right in the introduction.
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
Calculus
Linear Algebra
Differential Equations
Number Theory
Proof-Writing
Analysis
Complex Analysis
Functional Analysis
Partial Differential Equations
Higher-dimensional Calculus and Differential Geometry
Abstract Algebra
Geometry
Topology
Set Theory and Logic
Combinatorics / Discrete Math
Graph Theory
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"
The problem you are having is that math education is shitty.
> What I want is to have a concrete understanding [...]
If you want to actually understand anything you learn in class, you'll have to seek it out yourself. Actual mathematics isn't taught until you get to college, and even then, only to students majoring in the subject.
"Why the fuck calculus works" typically goes under the name "analysis." You can look up a popular textbook, Baby Rudin, although I've never used it. I had this cheap-o Dover book. You can't beat it for $12. There's also this nice video series from Harvey Mudd.
The general pattern you see in actual, real mathematics isn't method-problem-problem-problem-problem, but rather definition-theorem-proof. The definitions tell you what you're working with. The theorems tell you what is true. The proofs give a strong technical reason to believe it.
> I know that to grasp mathematical concepts, it is advisable to do lots of problems from your textbook.
For some reason, schools are notorious for drilling exercises until you're just about to bleed from the fucking skull. Once you understand how an exercise is done, don't waste your time with another exercise of the same type. If you can correctly take the derivative of three different polynomials, then you probably understand it.
Just a heads up, analysis is built on the foundations of set theory and the real numbers. What you work with in high school are an intuitive notion of what a real number is. However, to do proper mathematics with them, it's better to have a proper understanding of how they are defined. Any good book on analysis will start off by giving a full, rigorous definition of what a real number is. This is typically done either in terms of cauchy sequences (sequences that seem like they deserve to converge), in terms of dedekind cuts (splitting the rational numbers up into two sets), or axiomatically (giving you a characterization involving least upper bounds of bounded sets). (No good mathematical book would ever talk about decimals. Decimals are a powerful tool, but pure mathematicians avoid them whenever possible).
Calculus and analysis can both be summed up shortly as "the cool things you can do with limits". Limits are the primary way we work with infinities in analysis. Their technical definition is often confusing the first time you see it, but the idea behind them is straightforward. Imagining a world where you can't measure things exactly, you have to rely on approximations. You want accuracy, though, and so you only have so much room for error. Suppose you want to make a measurement with a very small error. (We use ε for denoting the maximum allowable error). If the equipment you're using to make the measurement is calibrated well enough, then you can do this just fine. (The calibration of your machine is denoted δ, and so, these definitions commonly go by the name of "ε-δ definitions").
I was in the same position as you in high school (and am finishing my math major this semester). Calculus is not "math" in the sense you're referring to it, which is pure mathematics, without application, just theory and logic. Calculus, as it is taught in high school, is taught as a tool, not as a theory. It is boring, tedious, and has no aesthetic appeal because it is largely taught as rote memorization.
Don't let this bad experience kill your enthusiasm. I'm not sure what specifically to recommend to you to perk your enthusiasm, but what I did in high school was just click around Wikipedia entries. A lot of them are written in layman enough terms to give you a glimpse and you inspire your interest. For example, I remember being intrigued by the Fibonacci series and how, regardless of the starting terms, the ratio between the (n-1)th and nth terms approaches the golden ratio; maybe look at the proof of that to get an idea of what math is beyond high school calculus. I remember the Riemann hypothesis was something that intrigued me, as well as Fermat's Last Theorem, which was finally proved in the 90s by Andrew Wiles (~350 years after Fermat suggested the theorem). (Note: you won't be able to understand the math behind either, but, again, you can get a glimpse of what math is and find a direction you'd like to work in).
Another thing that I wish someone had told me when I was in your position is that there is a lot of legwork to do before you start reaching the level of mathematics that is truly aesthetically appealing. Mathematics, being purely based on logic, requires very stringent fundamental definitions and techniques to be developed first, and early. Take a look at axiomatic set theory as an example of this. Axiomatic set theory may bore you, or it may become one of your interests. The concept and definition of a set is the foundation for mathematics, but even something that seems as simple as this (at first glance) is difficult to do. Take a look at Russell's paradox. Incidentally, that is another subject that captured my interest before college. (Another is Godel's incompleteness theorem, again, beyond your or my understanding at the moment, but so interesting!)
In brief, accept that math is taught terribly in high school, grunt through the semester, and try to read farther ahead, on your own time, to kindle further interest.
As an undergrad, I don't believe I yet have the hindsight to recommend good books for an aspiring math major (there are plenty of more knowledgeable and experienced Redditors who could do that for you), but here is a list of topics that are required for my undergrad math degree, with links to the books that my school uses:
And a couple electives:
And a couple books I invested in that are more advanced than the undergrad level, which I am working through and enjoy:
Lastly, if you don't want to spend hundreds of dollars on books that you might not end up using in college, take a look at Dover publications (just search "Dover" on Amazon). They tend to publish good books in paperback for very cheap ($5-$20, sometimes up to $40 but not often) that I read on my own time while trying to bear high school calculus. They are still on my shelf and still get use.
My main hobby is reading textbooks, so I decided to go beyond the scope of the question posed. I took a look at what I have on my shelves in order to recommend particularly good or standard books that I think could characterize large portions of an undergraduate degree and perhaps the beginnings of a graduate degree in the main fields that interest me, plus some personal favorites.
Neuroscience: Theoretical Neuroscience is a good book for the field of that name, though it does require background knowledge in neuroscience (for which, as others mentioned, Kandel's text is excellent, not to mention that it alone can cover the majority of an undergraduate degree in neuroscience if corequisite classes such as biology and chemistry are momentarily ignored) and in differential equations. Neurobiology of Learning and Memory and Cognitive Neuroscience and Neuropsychology were used in my classes on cognition and learning/memory and I enjoyed both; though they tend to choose breadth over depth, all references are research papers and thus one can easily choose to go more in depth in any relevant topics by consulting these books' bibliographies.
General chemistry, organic chemistry/synthesis: I liked Linus Pauling's General Chemistry more than whatever my school gave us for general chemistry. I liked this undergraduate organic chemistry book, though I should say that I have little exposure to other organic chemistry books, and I found Protective Groups in Organic Synthesis to be very informative and useful. Unfortunately, I didn't have time to take instrumental/analytical/inorganic/physical chemistry and so have no idea what to recommend there.
Biochemistry: Lehninger is the standard text, though it's rather expensive. I have limited exposure here.
Mathematics: When I was younger (i.e. before having learned calculus), I found the four-volume The World of Mathematics great for introducing me to a lot of new concepts and branches of mathematics and for inspiring interest; I would strongly recommend this collection to anyone interested in mathematics and especially to people considering choosing to major in math as an undergrad. I found the trio of Spivak's Calculus (which Amazon says is now unfortunately out of print), Stewart's Calculus (standard text), and Kline's Calculus: An Intuitive and Physical Approach to be a good combination of rigor, practical application, and physical intuition, respectively, for calculus. My school used Marsden and Hoffman's Elementary Classical Analysis for introductory analysis (which is the field that develops and proves the calculus taught in high school), but I liked Rudin's Principles of Mathematical Analysis (nicknamed "Baby Rudin") better. I haven't worked my way though Munkres' Topology yet, but it's great so far and is often recommended as a standard beginning toplogy text. I haven't found books on differential equations or on linear algebra that I've really liked. I randomly came across Quine's Set Theory and its Logic, which I thought was an excellent introduction to set theory. Russell and Whitehead's Principia Mathematica is a very famous text, but I haven't gotten hold of a copy yet. Lang's Algebra is an excellent abstract algebra textbook, though it's rather sophisticated and I've gotten through only a small portion of it as I don't plan on getting a PhD in that subject.
Computer Science: For artificial intelligence and related areas, Russell and Norvig's Artificial Intelligence: A Modern Approach's text is a standard and good text, and I also liked Introduction to Information Retrieval (which is available online by chapter and entirely). For processor design, I found Computer Organization and Design to be a good introduction. I don't have any recommendations for specific programming languages as I find self-teaching to be most important there, nor do I know of any data structures books that I found to be memorable (not that I've really looked, given the wealth of information online). Knuth's The Art of Computer Programming is considered to be a gold standard text for algorithms, but I haven't secured a copy yet.
Physics: For basic undergraduate physics (mechanics, e&m, and a smattering of other subjects), I liked Fundamentals of Physics. I liked Rindler's Essential Relativity and Messiah's Quantum Mechanics much better than whatever books my school used. I appreciated the exposition and style of Rindler's text. I understand that some of the later chapters of Messiah's text are now obsolete, but the rest of the book is good enough for you to not need to reference many other books. I have little exposure to books on other areas of physics and am sure that there are many others in this subreddit that can give excellent recommendations.
Other: I liked Early Theories of the Universe to be good light historical reading. I also think that everyone should read Kuhn's The Structure of Scientific Revolutions.
You should start with some gentler introduction to real analysis (e.g. the "baby" Rudin )that does the basic topology of the real line and Riemann integration rigorously.
There are essentially "two types" of math: that for mathematicians and everyone else. When you see the sequence Calculus(1, 2, 3) -> Linear Algebra -> DiffEq (in that order) thrown around, you can be sure they are talking about non-rigorous, non-proof based kind that's good for nothing, imo of course. Calculus in this sequence is Analysis with all its important bits chopped off, so that everyone not into math can get that outta way quick and concentrate on where their passion lies. The same goes for Linear Algebra. LA in the sequence above is absolutely butchered so that non-math majors can pass and move on. Besides, you don't take LA or Calculus or other math subjects just once as a math major and move on: you take a rigorous/proof-based intro as an undergrad, then more advanced kind as a grad student etc.
To illustrate my point:
Linear Algebra:
Linear Algebra Through Geometry by Banchoff and Wermer
3. Here's more rigorous/abstract Linear Algebra for undergrads:
Linear Algebra Done Right by Axler
4. Here's more advanced grad level Linear Algebra:
Advanced Linear Algebra by Steven Roman
-----------------------------------------------------------
Calculus:
Calulus by Spivak
3. Full-blown undergrad level Analysis(proof-based):
Analysis by Rudin
4. More advanced Calculus for advance undergrads and grad students:
Advanced Calculus by Sternberg and Loomis
The same holds true for just about any subject in math. Btw, I am not saying you should study these books. The point and truth is you can start learning math right now, right this moment instead of reading lame and useless books designed to extract money out of students. Besides, there are so many more math subjects that are so much more interesting than the tired old Calculus: combinatorics, number theory, probability etc. Each of those have intros you can get started with right this moment.
Here's how you start studying real math NOW:
Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers. Essentially, this book is about the language that you need to be able to understand mathematicians, read and write proofs. It's not terribly comprehensive, but the amount of info it packs beats the usual first two years of math undergrad 1000x over. Books like this should be taught in high school. For alternatives, look into
Discrete Math by Susanna Epp
How To prove It by Velleman
Intro To Category Theory by Lawvere and Schnauel
There are TONS great, quality books out there, you just need to get yourself a liitle familiar with what real math looks like, so that you can explore further on your own instead of reading garbage and never getting even one step closer to mathematics.
If you want to consolidate your knowledge you get from books like those of Rodgers and Velleman and take it many, many steps further:
Basic Language of Math by Schaffer. It's a much more advanced book than those listed above, but contains all the basic tools of math you'll need.
I'd like to say soooooooooo much more, but I am sue you're bored by now, so I'll stop here.
Good Luck, buddyroo.
I used Principles of Mathematical Analysis by Walter Rudin. It's very thorough, and covers all the topics you mentioned.
Read Spivak's Calculus (and do the exercises) to learn the foundations of calculus rigorously. It's an excellent book, especially if you've only learned the computational aspect of calculus but haven't done much in the way of writing proofs.
Once you finish Spivak — or if you already know the material well enough — the logical next step is real analysis, to which Rudin's Principles of Mathematical Analysis is a solid and well-regarded introduction.
The pure mechanics component consists of multivariable differential calculus, a little bit of multivariable integral calculus, and a bit of linear algebra; plus substantial comfort what might be called "systems of equations differential calculus." The fastest way to cover this material is to work through the first five or so chapters of Kaplan's advanced calculus book or something similar. Do the exercises. Your basic Stewart Calculus doesn't adequately cover the systems-of-equations part and Kreyszig's Advanced Engineering Mathematics book is at the right technical level but has all the wrong emphasis and coverage for economists. Kaplan's book isn't ideal, but it's about as close as you're going to get. (This is a hole in the textbook market...)
The theoretical portion mainly consists of basic point-set topology and elementary real analysis. The fastest way to cover this material is to chop through the first eight chapters of Rudin's undergraduate book.
Yale has a lovely set of Math Camp notes that you should also work through side-by-side with Kaplan and Rudin.
To see economic applications, read those two books side-by-side with Simon and Blume's book.
The first chapter of Debreu's Theory of Value covers all the math you need to know and is super slick, but is also far too terse and technical to realistically serve as your only resource. Similarly you should peek at the mathematical appendices in MWG but they will likely not be sufficient on their own.
There are a few options. Firstly, if you are more familiar using infinity in the context of Calculus, then you might want to look into Real Analysis. These subjects view infinity in the context of limits on the real line and this is probably the treatment you are probably most familiar with. For an introductory book on the subject, check out Baby Rudin (Warning: Proofs! But who doesn't like proofs, that's what math is!)
Secondly, you might want to look at Projective Geometry. This is essentially the type of geometry you get when you add a single point "at infinity". Many things benefit from a projective treatment, the most obvious being Complex Analysis, one of its main objects of study is the Riemann Sphere, which is just the Projective Complex Plane. This treatment is related to the treatment given in Real Analysis, but with a different flavor. I don't have any particular introductory book to recommend, but searching "Introductory Projective Geometry" in Amazon will give you some books, but I have no idea if they're good. Also, look in your university library. Again: Many Proofs!
The previous two treatments of infinity give a geometric treatment of the thing, it's nothing but a point that seems far away when we are looking at things locally, but globally it changes the geometry of an object (it turns the real line into a circle, or a closed line depending on what you're doing, and the complex plane into a sphere, it gets more complicated after that). But you could also look at infinity as a quantitative thing, look at how many things it takes to get an infinite number of things. This is the treatment of it in Set Theory. Here things get really wild, so wild Set Theory is mostly just the study of infinite sets. For example, there is more than one type of infinity. Intuitively we have countable infinity (like the integers) and we have uncountable infinity (like the reals), but there are even more than that. In fact, there are more types of infinities than any of the infinities can count! The collection of all infinities is "too big" to even be a set! For an introduction into this treatment I recommend Suppes and Halmos. Set Theory, when you actually study it, is a very abstract subject, so there will be more proofs here than in the previous ones and it may be over your head if you haven't taken any proof-based courses (I don't know your background, so I'm just assuming you've taken Calc 1-3, Diff Eq and maybe some kind of Matrix Algebra course), so patience will be a major virtue if you wish to tackle Set Theory. Maybe ask some professors for help!
A relatively compact (excuse the pun) rundown of the basic definitions and theorems behind real analysis can be found in a book called "Baby Rudin"
https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X
But beware, this is definitely not ELIF. Math isn't really an ELIF type of thing, but I guess it depends on how deep you need to go to get where you're going.
I wish you luck!
tl;dr: you need to learn proofs to read most math books, but if nothing else there's a book at the bottom of this post that you can probably dive into with nothing beyond basic calculus skills.
Are you proficient in reading and writing proofs?
If you aren't, this is the single biggest skill that you need to learn (and, strangely, a skill that gets almost no attention in school unless you seek it out as an undergraduate). There are books devoted to developing this skill—How to Prove It is one.
After you've learned about proof (or while you're still learning about it), you can cut your teeth on some basic real analysis. Basic Elements of Real Analysis by Protter is a book that I'm familiar with, but there are tons of others. Ask around.
You don't have to start with analysis; you could start with algebra (Algebra and Geometry by Beardon is a nice little book I stumbled upon) or discrete (sorry, don't know any books to recommend), or something else. Topology probably requires at least a little familiarity with analysis, though.
The other thing to realize is that math books at upper-level undergraduate and beyond are usually terse and leave a lot to the reader (Rudin is famous for this). You should expect to have to sit down with pencil and paper and fill in gaps in explanations and proofs in order to keep up. This is in contrast to high-school/freshman/sophomore-style books like Stewart's Calculus where everything is spelled out on glossy pages with color pictures (and where proofs are mostly absent).
And just because: Visual Complex Analysis is a really great book. Complex numbers, functions and calculus with complex numbers, connections to geometry, non-Euclidean geometry, and more. Lots of explanation, and you don't really need to know how to do proofs.
Sorry, was a bad joke.
https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X
This book is full of proofs you can work through. It could keep you busy for quite a while and it's considered a standard for analysis.
https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X
You've taken some sort of analysis course already? A lot of real analysis textbooks will cover Lebesgue integration to an extent.
Some good introductions to analysis that include content on Lebesgue integration:
Walter Rudin, principle of mathematical analysis, I think it is heavily focused on the real numbers, but a fantastic book to go through regardless. Introduces Lebesgue integration as of at least the 2nd edition (the Lebesgue theory seems to be for a more general space, not just real functions).
Rudin also has a more advanced book, Real and Complex Analysis, which I believe will cover Lebesgue integration, Fourier series and (obviously) covers complex analysis.
Carothers Real Analysis is the book I did my introductory real analysis course with. It does the typical content (metric spaces, compactness, connectedness, continuity, function spaces), it has a chapter on Fourier series, and a section (5 chapters) on Lebesgue integration.
Royden's real analysis I believe covers very similar topics and again has a long and detailed section on Lebesgue integration. No experience with it, recommended for my upcoming graduate analysis course.
Bartle, Elements of Integration is a full book on Lebesgue integration. Again, haven't read it yet, recommended for my upcoming course. It is supposed to be a classic on the topic from what I've heard.
Baby Rudin
Best:
Principles of Mathematical Analysis by Walter Rudin
I know that in the long run competition math won't be relevant to graduate school, but I don't think it would hurt to acquire a broader background.
That said, are there any particular texts you would recommend? For Algebra, I've heard that Dummit and Foote and Artin are standard texts. For analysis, I've heard that Baby Rudin and also apparently the Stein-Shakarchi Princeton Lectures in Analysis series are standard texts.
Here are some suggestions :
https://www.coursera.org/course/maththink
https://www.coursera.org/course/intrologic
Also, this is a great book :
http://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X/ref=sr_1_5?ie=UTF8&qid=1346855198&sr=8-5&keywords=history+of+mathematics
It covers everything from number theory to calculus in sort of brief sections, and not just the history. Its pretty accessible from what I've read of it so far.
EDIT : I read what you are taking and my recommendations are a bit lower level for you probably. The history of math book is still pretty good, as it gives you an idea what people were thinking when they discovered/invented certain things.
For you, I would suggest :
http://www.amazon.com/Principles-Mathematical-Analysis-Third-Edition/dp/007054235X/ref=sr_1_1?ie=UTF8&qid=1346860077&sr=8-1&keywords=rudin
http://www.amazon.com/Invitation-Linear-Operators-Matrices-Bounded/dp/0415267994/ref=sr_1_4?ie=UTF8&qid=1346860052&sr=8-4&keywords=from+matrix+to+bounded+linear+operators
http://www.amazon.com/Counterexamples-Analysis-Dover-Books-Mathematics/dp/0486428753/ref=sr_1_5?ie=UTF8&qid=1346860077&sr=8-5&keywords=rudin
http://www.amazon.com/DIV-Grad-Curl-All-That/dp/0393969975
http://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0738204536/ref=sr_1_2?s=books&ie=UTF8&qid=1346860356&sr=1-2&keywords=chaos+and+dynamics
http://www.amazon.com/Numerical-Analysis-Richard-L-Burden/dp/0534392008/ref=sr_1_5?s=books&ie=UTF8&qid=1346860179&sr=1-5&keywords=numerical+analysis
This is from my background. I don't have a strong grasp of topology and haven't done much with abstract algebra (or algebraic _____) so I would probably recommend listening to someone else there. My background is mostly in graduate numerical analysis / functional analysis. The Furata book is expensive, but a worthy read to bridge the link between linear algebra and functional analysis. You may want to read a real analysis book first however.
One thing to note is that topology is used in some real analysis proofs. After going through a real analysis book you may also want to read some measure theory, but I don't have an excellent recommendation there as the books I've used were all hard to understand for me.
I would try http://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X/ref=sr_1_1?s=books&ie=UTF8&qid=1375896302&sr=1-1 and if that does not truly challenge you, then we can chat some more.
My buddy (phd student) told me that if I were to do a reading course, or just want to do self study that I should use Munkres. I think you can find international editions for much cheaper than that. We were using Rudin for our analysis class and spent a lot of time on ch.2. These are my only suggestions because I haven't done much with topology or analysis.
The reason for this is that you haven't studied enough math. You simply haven't seen a mathematically precise treatment of algebra and arithmetic yet. You'll feel the same way about calculus before having studied real analysis, and functional analysis (specifically L^p spaces) before having studied measure theory. Let's stick to algebra and arithmetic for now. In analysis, the real numbers are constructed as a complete ordered field. The "rules" of real arithmetic follow from the construction of the reals: addition and multiplication are defined. The reals with these operations can be shown to satisfy some properties, namely those that make it a field. Most of the "rules" of algebra in fact hold in every field, either by the definition of a field or as something that can be proven from the definition of a field. See Rudin's Principles of Mathematical Analysis for a construction of the reals and derivations of many of its properties. Definitions and theorems (true statements that can be proven based on definitions and previous theorems) may still be a challenge to understand, but at least they provide you with something more than just rules to follow. Being content with definitions and theorems is called mathematical maturity.
I think learning proofs-based calculus and linear algebra are solid places to start. To complete the trifecta, look into Arnold for a more proofy differential equations course.
After that, my suggestions are Rudin and, to build on your CS background, Sipser. These are very standard references, though Rudin's a slightly controversial suggestion because he's notorious for being terse. I say, go ahead and try it, you might find you like it.
As for names of fields to look into: Real Analysis, Complex Analysis, Abstract Algebra, Topology, and Differential Geometry mostly partition the field of mathematics with corresponding undergraduate courses. As for computer science, look into Algorithmic Analysis and Computational Complexity (sometimes sold as a single course called Theory of Computation).
If you are interested enough in machine learning that you are going to work through ESL, you may benefit from reading up on some math first. For example:
Without developing some mathematical maturity, some of ESL may be lost on you. Good luck!
This book is not a calculus book, but a real analysis book at the level of baby Rudin.
It's also essentially designed to be used as a book for a Moore method style course, so it is not a textbook in any regular sense. Erdman teaches his classes by having students present the solutions to lots of problems, with only minimal lecturing.
I learned a lot from getting a copy of Rudin (however, this book is very challenging and probably not the best to self study from. I was able to get to about continuity before taking my analysis course and it was challenging, but worth while). You can probably find it online somewhere for free.
A teacher lent Introduction to Analysis to me and suggested I use it instead of the book by Rudin. It was a well written book and had exercises which were much more approachable (although it included very difficult ones as well). The layout of this book (and I'd bet many others) is quite similar to that of Rudin. It was nice to be able to read them together.
For linear algebra, I can't speak to the quality of many books, but there are plenty which can fairly easily be found online. You will likely be recommended Linear Algebra Done Right however I found it a bit challenging as a first introduction to linear algebra and never got quite far.
My university course used Larson, Falvo Linear Algebra and it was enjoyable and helps you learn the computations very well and gives a decent understanding of proofs.
If it helps, here are some free books to go through:
Linear Algebra Done Wrong
Paul's Online Math Notes (fantastic for Calc 1, 2, and 3)
Basic Analysis
Basic Analysis is pretty basic, so I'd recommend going through Rudin's book afterwards, as it's generally considered to be among the best analysis books ever written. If the price tag is too high, you can get the same book much cheaper, although with crappier paper and softcover via methods of questionable legality. Also because Rudin is so popular, you can find solutions online.
If you want something better than online notes for univariate Calculus, get Spivak's Calculus, as it'll walk you through single-variable Calculus using more theory than a standard math class. If you're able to get through that and Rudin, you should be good to go once you get good at linear algebra.
If you want to learn serious mathematics, start with a theoretical approach to calculus, then go into some analysis. Introductory Real Analysis by Kolmogorov is pretty good.
As far as how to think about these things, group theory is a strong start. "The real numbers are the unique linearly-ordered field with least upper bound property." Once you understand that sentence and can explain it in the context of group theory and the order topology, then you are in a good place to think about infinity, limits, etc.
Edit: For calc, Spivak is one of the textbooks I have heard is more common, but I have never used it so I can't comment on it. I've heard good things, though.
A harder analysis book for self-study would be Principles of Mathematical Analysis by Rudin. He is very terse in his proofs, so they can be hard to get through.
I guess I don't have a clear idea what an "elementary math degree" entails, so let me put it this way:
I learned about space-filling curves in my second semester of Real Analysis. First-semester Real Analysis was the first upper-division math class people take at my college; the second-semester is typically taken Junior or Senior year by those who are particularly passionate about the subject. It is not, as a general rule, a subject I recommend learning without the benefit of an instructor - at least, not from the book I used. To be clear: its a good reference book, and I developed a healthy respect for its approach to the subject in time, but its not the most user-friendly book as you're getting going.
To briefly paraphrase the argument: you basically construct a fractal via a sequence of functions, and then argue based on the convergence and continuity properties of the function family that a) the function they converge to is continuous and b) it passes through every point in the area to be covered.
Have you ever seen how much technical books cost?
For example, here's the standard text for mathematical analysis: Principles of Mathematical Analysis. That's $87 for a 325 page book.
Nobody's pretending that printing/binding/distributing is a significant fraction of that cost so an ebook would likely be similarly priced, maybe slightly less, possibly slightly more.
Manning, in particular, focuses on texts in computer science and programming for which such prices are pretty standard. The price difference between the ebook and print+ebook varies (I think it's proportional for most of their texts) but if the ebook is $35 then the physical+ebook is usually around $45. Again, this is very reasonable for a quality text in the field.
Good advice, but I'd add that if you do revisit calc get an intro to analysis textbook to understand how we derived the rules that calc uses. For instance, an integral is not defined as an antiderivative, that had to be proven.
Edit: My class used Principles of Mathematical Analysis by Rudin. It requires little to no initial knowledge and essentially builds multivariable calculus from the ground up.
> You're not wrong, you're just an asshole. Anything else you'd like to say about how great you are? Tell me about me your thesis. I'll bet it's extremely groundbreaking stuff.
My thesis is on chaotic behavior of swarm traffic, swarm traffic analysis and using spectral graph theory to predict traffic patterns. Very fun, but something you really need schooling for.
You're right, I'm an asshole. And that may be so. Maybe you should put down the drugs and try to learn something that takes actual mental capacity like Real Analysis, and maybe I won't be such an asshole.
Edit: If you want to learn it on your own Rudin is the best.
Oh. I'm sorry. I thought your name was in reference to the mathematician walter rudin. He wrote some popular upper undergraduate and graduate math books on analysis (baby rudin and papa rudin respectively). There are many math definitions and proofs in these books with very little background into what purpose they may serve in an applied mathematical field.
baby rudin
papa rudin
Try Baby Rudin. I think the first chapter covers what you are looking for very thoroughly.
You might also find Analysis: With an Introduction to Proof to be rather helpful.
A course I took previously used this book; it has a chapter on introductory real analysis, which is what you want to get at. I would not suggest going directly to a book like Rudin, as he (in my opinion) tends to amplify the "general route" problem that you mention.
This is super helpful, thank you!
And nothing against simulation, I know it's a powerful tool. I just don't want my foundations built on sand (I'm familiar with intro stats already).
Would Rubin's book on Real Analysis suffice: http://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X
Or are there even more advanced texts to pursue for Real Analysis?
Rudin.
If you want to do more math in the same flavor as Apostol, you could move up to analysis with Tao's book or Rudin. Topology's slightly similar and you could use Munkres, the classic book for the subject. There's also abstract algebra, which is not at all like analysis. For that, Dummit and Foote is the standard. Pinter's book is a more gentle alternative. I can't really recommend more books since I'm not that far into math myself, but the Chicago math bibliography is a good resource for finding math books.
Edit: I should also mention Evan Chen's Infinite Napkin. It's a very condensed, free book that includes a lot of the topics I've mentioned above.
If desired, it is possible to make an elementary argument that (1+x/n)^n converges, for each x, to a function e(x) satisfying e(x)e(y) = e(x+y), using just inequalities to show convergence of the needed limits. This is outlined, for example, in the chapter on the AM-GM inequality in this book: https://www.amazon.com/Inequalities-Journey-into-Linear-Analysis/dp/0521876249
There's also an exercise in the first chapter of Baby Rudin outlining how to define exponentials using least upper bounds and monotonicity properties:
https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X
Honestly though, while in general I support showing students the details, this is a case where I think that, pedagogically, it's right to pull the wool over students' eyes until the time is right. It's so much more elegant to define the exponential function as the solution of a differential equation, or as the sum of a power series, or as the inverse of the logarithm (defined as an integral), that one should simply put off a fully rigorous definition until it can be given in one of these forms.
The reasoning in doing so is not circular: The basic properties of integrals, power series, and solutions of differential equations are established through abstract theorems, and then one can use these tools to define the exponential and logarithmic functions and derive their properties. (See https://proofwiki.org/wiki/Definition:Exponential and https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Complex_Exponential)
Until then about all that needs to be mentioned is that a^m is a product of m copies of a, a^1/n is the nth root, a^m/n = (a^(1/n))^(m), and that this extends in a natural way to irrational exponents; as well as compound interest and the fact that (1+x/n)^n converges to a power of a special number e approx 2.718281827459, which is the "natural base" of the logarithm for reasons to be explained later.
Is this the book you are recommending?
RPCV checking in. This is a good idea... you're going to have a lot of downtime and it's a great opportunity to read all the things you've wanted to but haven't yet found the time for. That could mean math, or languages, or just old novels.
When I was learning functional analysis, if found this book by Bollobas to be incredibly helpful. Of course, the only real analysis reference you need is Baby Rudin, but if you want to learn measure theory you may want his Real & Complex Analysis instead.
For texts on the other subjects, take a look at this list. You should be able to find anything you need there.
If you have any questions about Peace Corps, feel free to PM me. Good luck!
For the whys and hows, you're gonna need a full-blown analysis textbook like baby Rudin. Calc I and II at most universities don't even scratch the surface when it comes to understanding the whys of anything. Anyways, yeah. Engineering is cool.
Baby Rudin, The Inferno, DPV
THIS WILL PUT HAIR ON HIS CHEST:
http://www.amazon.com/Principles-Mathematical-Analysis-Third-Walter/dp/007054235X
You could try Principles of mathematical analysis by Rudin. This is too much for me, so be warned.
I find Spivak's Calculus to be a lot more palatable, but I've read less of it than Rudin.
I've found Rudin's Analysis useful. There's a lecture series on YouTube that roughly follows the book.
Hmm. I kept almost all my textbooks. I just looked through them and the most expensive one I could find cost $47.97 in 1987. That calculator says it would be $100.60 in 2014 dollars. I just checked Amazon, and it's now $109.15. Pretty close.
I seem to recall one book costing $80 or more, but I didn't write the prices on all my books. My books were math or statistics, and cost more than nonmathematical texts, but I always figured that was the cost of typesetting (which I'd guess is not as much a consideration as it once was).
Walter Rudin, principle of mathematical analysis
Why shouldn't everyone just begin with Baby Rudin?
https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X
Here you go OP.
I prefer the third review of this classic math book!
http://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X
Hawking is a theoretical physicist. His craft is closer to math than it is to classical physics.
You made a lot of erroneous and hot-headed statements, but that's understandable. Since you seem to be very, very ignorant of math, I don't even know where to even begin to show you the differences - I am at a disadvantage here :) How about we talk about levels, then?
Most math an engineer knows is barely a first year material for a math undergrad. Math is so vast that even the grad students of math are at the very base of a huge mountain.
Here's Basic Algebra for a math major(flip through the first pages and checkout the contents).
Here's Algebra for engineers.
Notice how the algebra for engineers is a very small part of general algebra and non-rigorous at that.
Here's Calculus for engineers.
Here's Calculus for math majors.
This is not to say engineers are mentally inferior to mathematicians, it's just these two professions are concerned with fundamentally different things.