Best mathematical analysis books according to redditors
We found 510 Reddit comments discussing the best mathematical analysis books. We ranked the 148 resulting products by number of redditors who mentioned them. Here are the top 20.
2. An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements
25 mentions
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"
For real analysis I really enjoyed Understanding Analysis for how clear the material was presented for a first course. For abstract algebra I found A book of abstract algebra to be very concise and easy to read for a first course. Those two textbooks were a lifesaver for me since I had a hard time with those two courses using the notes and textbook for the class. We were taught out of rudin and dummit and foote as mainly a reference book and had to rely on notes primarily but those two texts were incredibly helpful to understand the material.
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If any undergrads are struggling with those two courses I would highly recommend you check out those two textbooks. They are by far the easiest introduction to those two fields I have found. I also like that you can find solutions to all the exercises so it makes them very valuable for self study also. Both books also have a reasonable amount of excises so that you can in theory do nearly every problem in the book which is also nice compared to standard texts with way too many exercises to realistically go through.
There are very few true textbooks - i.e. books designed to teach the material to those who don't already know the classical versions - written in this style.
While we're at it, a quick skim through the algebra chapter of Troelstra: Constructivism in Mathematics, vol. 2 should explain why there are no textbooks on abstract algebra written in the purely constructive tradition.
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 read this. I found it here a month ago on this subreddit, and it really stuck with me. I love those stories that help round out abstract concepts I've been thinking of.
More generally though... simple algebra used to be for the greatest thinkers alive. 'Ars Magna'. "The Great Art" written by Geromalo Cardano in the 1600s or whatever was the first European mathematical work that advanced beyond what was known by the Greeks... it gave a partial solution to how to find solutions to homogenous cubic polynomials (ax^3 + bx^2 + cx + d = 0)
he solved it with hilarious methods. Galileo used some incredibly painful notation where you're juggling ratios instead of... you know. Doing algebra the way we think of it. Fibonacci tried to encourage a switch to our standard number system, because arithmatic is RADICALLY easier when dealing with a simple base ten system instead of whatever crappy roman numeral type language they were using before. Took them 400 years to adopt our modern number system from the time the 'better' alternative was introduced.
All this is to say... you're absolutely right. The crystal core of the ideas we use can radically change our reach. What was impossible with one way of working becomes elementary when you can look at it right. But you've got a few layers of problems here. First... what's the right way of looking at it? I just read Judea Pearl's "Causality", and it's fascinating seeing a branch of math that's still so young, that there are arguments about what the definitions and axioms should even be. It's still a bubbling cauldron of ideas more so than an established branch. But even once you've gotten the 'right' way of looking at things (often there are many possible ways, you need to pick the right one for the job) now you're left with the arguably harder task of communication. How do you build a bridge to efficiently transmit a new way of thinking? I love 3blue1brown just because his whole shtick is finding new ways to graphically describe concepts that most people only vaguely understanding. The article I linked above (Ars Longa, Vita Brevis: 'long art, short life') breaks down the emergence of an art as being in 3 tiers... the inventors, the teachers, and the teacher teachers. The 'best' teachers I think are what you're asking about partly, but the right 'inventors' (what is the perfect framing that should be taught?) is part of the problem too.
Anyway, a related article you might also enjoy... thought as technology. A cool little exploration by Michael Nielson about the fact that 'how to think about things' is itself a technology, just one that's a pain in the ass to pass on compared to physical goods. He had some cool things to say on the topic you might also enjoy.
Also also... from a math perspective, I highly recommend you check out Alcock's how to think about analysis if you're looking for something fun to read. It's a very, very light introduction to real analysis, looking at the foundations of calculus, limits, series and convergence and so on. If you're interested in the 'heart' of what it means to learn math, I think you'll find that to be a pretty fun, approachable little book. You'll be able to blow through it in a couple weeks, but it'll give you some good framing for continuing the journey, if you're interested in doing so.
> Mathematical Logic
It's not exactly Math Logic, just a bunch of techniques mathematicians use. Math Logic is an actual area of study. Similarly, actual Set Theory and Proof Theory are different from the small set of techniques that most mathematicians use.
Also, looks like you have chosen mostly old, but very popular books. While studying out of these books, keep looking for other books. Just because the book was once popular at a school, doesn't mean it is appropriate for your situation. Every year there are new (and quite frankly) pedagogically better books published. Look through them.
Here's how you find newer books. Go to Amazon. In the search field, choose "Books" and enter whatever term that interests you. Say, "mathematical proofs". Amazon will come up with a bunch of books. First, sort by relevance. That will give you an idea of what's currently popular. Check every single one of them. You'll find hidden jewels no one talks about. Then sort by publication date. That way you'll find newer books - some that haven't even been published yet. If you change the search term even slightly Amazon will come up with completely different batch of books. Also, search for books on Springer, Cambridge Press, MIT Press, MAA and the like. They usually house really cool new titles. Here are a couple of upcoming titles that might be of interest to you: An Illustrative Introduction to Modern Analysis by Katzourakis/Varvarouka, Understanding Topology by Shaun Ault. I bet these books will be far more pedagogically sound as compared to the dry-ass, boring compendium of facts like the books by Rudin.
If you want to learn how to do routine proofs, there are about one million titles out there. Also, note books titled Discrete Math are the best for learning how to do proofs. You get to learn techniques that are not covered in, say, How to Prove It by Velleman. My favorites are the books by Susanna Epp, Edward Scheinerman and Ralph Grimaldi. Also, note a lot of intro to proofs books cover much more than the bare minimum of How to Prove It by Velleman. For example, Math Proofs by Chartrand et al has sections about doing Analysis, Group Theory, Topology, Number Theory proofs. A lot of proof books do not cover proofs from Analysis, so lately a glut of new books that cover that area hit the market. For example, Intro to Proof Through Real Analysis by Madden/Aubrey, Analysis Lifesaver by Grinberg(Some of the reviewers are complaining that this book doesn't have enough material which is ridiculous because this book tackles some ugly topological stuff like compactness in the most general way head-on as opposed to most into Real Analysis books that simply shy away from it), Writing Proofs in Analysis by Kane, How to Think About Analysis by Alcock etc.
Here is a list of extremely gentle titles: Discovering Group Theory by Barnard/Neil, A Friendly Introduction to Group Theory by Nash, Abstract Algebra: A Student-Friendly Approach by the Dos Reis, Elementary Number Theory by Koshy, Undergraduate Topology: A Working Textbook by McClusckey/McMaster, Linear Algebra: Step by Step by Singh (This one is every bit as good as Axler, just a bit less pretentious, contains more examples and much more accessible), Analysis: With an Introduction to Proof by Lay, Vector Calculus, Linear Algebra, and Differential Forms by Hubbard & Hubbard, etc
This only scratches the surface of what's out there. For example, there are books dedicated to doing proofs in Computer Science(for example, Fundamental Proof Methods in Computer Science by Arkoudas/Musser, Practical Analysis of Algorithms by Vrajitorou/Knight, Probability and Computing by Mizenmacher/Upfal), Category Theory etc. The point is to keep looking. There's always something better just around the corner. You don't have to confine yourself to books someone(some people) declared the "it" book at some point in time.
Last, but not least, if you are poor, peruse Libgen.
You are in a very special position right now where many interesing fields of mathematics are suddenly accessible to you. There are many directions you could head. If your experience is limited to calculus, some of these may look very strange indeed, and perhaps that is enticing. That was certainly the case for me.
Here are a few subject areas in which you may be interested. I'll link you to Dover books on the topics, which are always cheap and generally good.
Basically, don't limit yourself to the track you see before you. Explore and enjoy.
Found it.
https://www.amazon.com/Introduction-Error-Analysis-Uncertainties-Measurements/dp/093570275X
i have three categories of suggestions.
advanced calculus
these are essentially precursors to smooth manifold theory. you mention you have had calculus 3, but this is likely the modern multivariate calculus course.
out of these, if you were to choose one, i think the callahan book is probably your best bet to pull from. it is the most modern, in both approach and notation. it is a perfect setup for smooth manifolds (however, all of these books fit that bill). hubbard's book is very similar, but i don't particularly like its notation. however, it has some unique features and does attempt to unify the concepts, which is a nice approach. edwards book is just fantastic, albeit a bit nonstandard. at a minimum, i recommend reading the first three chapters and then the latter chapters and appendices, in particular chapter 8 on applications. the first three chapters cover the core material, where chapters 4-6 then go on to solidify the concepts presented in the first three chapters a bit more rigorously.
smooth manifolds
out of these books, i only have explicit experience with the first two. i learned the material in graduate school from john m. lee's book, which i later solidifed by reading tu's book. tu's book actually covers the same core material as lee's book, but what makes it more approachable is that it doesn't emphasize, and thus doesn't require a lot of background in, the topological aspects of manifolds. it also does a better job of showing examples and techniques, and is better written in general than john m. lee's book. although, john m. lee's book is rather good.
so out of these, i would no doubt choose tu's book. i mention the latter two only to mention them because i know about them. i don't have any experience with them.
conceptual books
these books should be helpful as side notes to this material.
i highly recommend all of these because they're all rather short and easy reads. the first two get at the visual concepts and intuition behind vectors, covectors, etc. they are actually the only two out of all of these books (if i remember right) that even talk about and mention twisted forms.
there are also a ton of books for physicists, applied differential geometry by william burke, gauge fields, knots and gravity by john baez and javier muniain (despite its title, it's very approachable), variational principles of mechanics by cornelius lanczos, etc. that would all help with understanding the intuition and applications of this material.
conclusion
if you're really wanting to get right to the smooth manifolds material, i would start with tu's book and then supplement as needed from the callahan and hubbard books to pick up things like the implicit and inverse function theorems. i highly recommend reading edwards' book regardless. if you're long-gaming it, then i'd probably start with callahan's book, then move to tu's book, all the while reading edwards' book. :)
i have been out of graduate school for a few years now, leaving before finishing my ph.d. i am actually going back through callahan's book (didn't know about it at the time and/or it wasn't released) for fun and its solid expositions and approach. edwards' book remains one of my favorite books (not just math) to just pick up and read.
Try picking up a book. I recommend this one. You can also use Rudin but it will be more difficult.
If you are using notes and online research, it may be that the exercises you've been working on are coming from many different areas and aren't really focused on one topic in particular. This may be the reason that every problem seems to require a new trick.
While it's certainly not the best or broadest advice, I've always found that, whenever a problem starts to get excessively complicated, the mean value theorem always seems to be the why-didn't-I-think-of-that trick that solves it.
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 resources such as large problem books (for instance: https://www.amazon.com/Schaums-Solved-Problems-Calculus-Outlines/dp/0071635343) where you can practice through different topics and figure out areas of weakness. Ultimately, it comes down to practice a lot for these foundational skills.
If what you are looking for is a more theoretical understanding, have a look into an introductory real analysis textbook. Hopefully this helps.
Please, simply disregard everything below if the info is old news to you.
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Algebraic geometry requires the knowledge of commutative algebra which requires the knowledge of some basic abstract algebra (consists of vector spaces, groups, rings, modules and the whole nine yards). There are many books written on abstract algebra like those of Dummit&Foote, Artin, Herstein, Aluffi, Lang, Jacobson, Hungerford, MacLane/Birkhoff etc. There are a million much more elementary intros out there, though. Some of them are:
Discovering Group Theory: A Transition to Advanced Mathematics by Barnard/Neil
A Friendly Introduction to Group Theory by Nash
Abstract Algebra: A Student-Friendly Approach by the Dos Reis
Numbers and Symmetry: An Introduction to Algebra by Johnston/Richman
Rings and Factorization by Sharpe
Linear Algebra: Step by Step by Singh
As far as DE go, you probably want to see them done rigorously first. I think the books you are looking for are titled something along the lines of "Analysis on Manifolds". There are famous books on the subject by Sternberg, Spivak, Munkres etc. If you don't know basic real analysis, these books will be brutal. Some elementary analysis and topology books are:
Understanding Analysis by Abbot
The Real Analysis Lifesaver by Grinberg
A Course in Real Analysis by Mcdonald/Weiss
Analysis by Its History by Hirer/Wanner
Introductory Topology: Exercises and Solutions by Mortad
Pick up mathematics. Now if you have never done math past the high school and are an "average person" you probably cringed.
Math (an "actual kind") is nothing like the kind of shit you've seen back in grade school. To break into this incredible world all you need is to know math at the level of, say, 6th grade.
Intro to Math:
These books only serve as samplers because they don't even begin to scratch the surface of math. After you familiarized yourself with the basics of writing proofs you can get started with intro to the largest subsets of math like:
Intro to Abstract Algebra:
There are tons more books on abstract/modern algebra. Just search them on Amazon. Some of the famous, but less accessible ones are
Intro to Real Analysis:
Again, there are tons of more famous and less accessible books on this subject. There are books by Rudin, Royden, Kolmogorov etc.
Ideally, after this you would follow it up with a nice course on rigorous multivariable calculus. Easiest and most approachable and totally doable one at this point is
At this point it's clear there are tons of more famous and less accessible books on this subject :) I won't list them because if you are at this point of math development you can definitely find them yourself :)
From here you can graduate to studying category theory, differential geometry, algebraic geometry, more advanced texts on combinatorics, graph theory, number theory, complex analysis, probability, topology, algorithms, functional analysis etc
Most listed books and more can be found on libgen if you can't afford to buy them. If you are stuck on homework, you'll find help on [MathStackexchange] (https://math.stackexchange.com/questions).
Good luck.
I used Principles of Mathematical Analysis by Walter Rudin. It's very thorough, and covers all the topics you mentioned.
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
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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.
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 most important thing you can do is memorize the definitions. I mean seriously have them down cold. The next thing I would recommend is to get another couple of analysis books (go cheap by getting old books, it isn't like the value of epsilon has changed over the past two hundred years) and look at their explanations, work those problems. Having a different set can be enlightening. Be prepared to spend a lot of time on it all.
Good luck!
EDIT: Back home now and able to put in some specific books. I used Rosenlicht and you wouldn't believe how happy I was to buy a textbook that, combined with a slice of pizza and a coke, was still less than $20. One of my books that I looked at for a different view point was Sprecher.
I also got a great deal of value out of Counterexamples in Analysis because after seeing things go wrong (a function that is continuous everywhere but nowhere differentiable? Huh?) I started to get a better feel for what the definitions really meant.
I hope you're also sensing a theme: Dover math books rock!
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.
> Pretending that dt is a variable
Infinitesimals are not variables, and you don't have to "pretend" they're variables to do the sort of manipulation the OP mentioned. There's an excellent book on this subject by John Bell, taking the nilpotent square approach to infinitesimals. What the OP did turns out to be valid.
Why not read an introductory text to numerical linear algebra like Trefethen and Bau?
This is the book I used. It's a solid read with lots of good problems and examples.
https://www.amazon.com/Numerical-Linear-Algebra-Lloyd-Trefethen/dp/0898713617
Analysis I by Terence Tao.
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!
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!
Your professors really aren't expecting you to reinvent groundbreaking proofs from scratch, given some basic axioms. It's much more likely that you're missing "hints" - exercises often build off previous proofs done in class, for example.
I appreciated Laura Alcock's writings on this, in helping me overcome my fear of studying math in general:
https://www.amazon.com/How-Study-as-Mathematics-Major/dp/0199661316/
https://www.amazon.com/dp/0198723539/ <-- even though you aren't in analysis, the way she writes about approaching math classes in general is helpful
If you really do struggle with the mechanics of proof, you should take some time to harden that skill on its own. I found this to be filled with helpful and gentle exercises, with answers: https://www.amazon.com/dp/0989472108/ref=rdr_ext_sb_ti_sims_2
And one more idea is that it can't hurt for you to supplement what you're learning in class with a more intuitive, chatty text. This book is filled with colorful examples that may help your leap into more abstract territory: https://www.amazon.com/Visual-Group-Theory-Problem-Book/dp/088385757X
You could try Abbott's Understanding Analysis. Quite a few students seem to like this book.
One concrete suggestion I can give you is when faced with a theorem or definition, try first to understand what it means in 'words' and then try to reason why it may be true, again in 'words'. I've noticed that often what trips students up is the symbolism -- often when I see incorrect answers from bright students, 10 to 1, its because they've got caught up in symbols and are now mentally running around in circles. This, I feel, is the unfortunate transition-pangs from school math to real math.
Remember math is not about symbols, formulas or equations, its about the concepts and ideas that hide behind those things.
It's hard to give an objective answer, because any sufficiently advanced book will be bound to not appeal to everyone.
You probably want Daddy Rudin for real analysis and Dummit & Foote for general abstract algebra.
Mac Lane for category theory, of course.
I think people would agree on Hartshorne as the algebraic geometry reference.
Spanier used to be the definitive algebraic topology reference. It's hard to actually use it as a reference because of the density and generality with which it's written.
Spivak for differential geometry.
Rotman is the group theory book for people who like group theory.
As a physics person, I must have a copy of Fulton & Harris.
Sorry, was a bad joke.
https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X
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.
Understanding Analysis by Abbott is a book that is more gentle than most.
Understanding Analysis is a very nice book I used to get a good grasp on the concepts behind real analysis. It goes at a very nice pace, perfect for the analysis novice.
By the end of reading the books you should be able to do all the problems, but I don't think that means doing literally every single problem before moving on. I'd google courses that use the books you're reading and try their example problems. If you try to read on and feel they weren't enough then you can always go back and do more.
Also, going to Rudin for self-study seems mildly sadistic. For future reference Pugh's book Real Mathematical Analysis also has a huge helping of fantastic exercises while also being written in a way that's trying to teach you and not impress you. You should be capable of Rudin by the end of a real analysis course, but Rudin itself is not necessarily the best way to get there.
One way of doing "non-standard" analysis that I think closely models what a physicist does it to posit the existence of "nil-square infinitesimal numbers" which have the following property: For any infinitesimal
dx, for any numbera, and any smooth functionf,f(a + dx) = f(a) + f'(a) * dx. Furthermore, infinitesimalsdxhave the property thatdx^2 = 0. You can derive most rules of calculus this way. E.g.,(f ° g)(a + dx) = f(g(a) + g'(a) * dx) = f(g(a)) + f'(g(a)) * g'(a) * dxso(f ° g)'(a) = f'(g(a)) * g'(a).If you want to work with higher order approximations you can use infinitesimals whose cube, fourth power, etc. is 0 instead of those whose square is 0. This is the approach of synthetic differential geometry. You can read a good intro here and check out this book if you want more.
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.
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
How to Prove It: A Structured Approach by Velleman is good for developing general proof writing skills.
How to Think About Analysis by Lara Alcock beautifully deconstructs all the major points of Analysis(proofs included).
The 1st Course In Math Analysis by Brannan
Analysis I by Terrence Tao
Yet Another Intro To Real Analysis by Bryant
Understanding Analysis Stephen Abbott
Elementary Analysis: The Theory of Calculus Kenneth A. Ross
Metric Spaces by Robert Reisel
A Problem Text in Advanced Calculus by John Erdman. PDF
Advanced Calculus by Shlomo Sternberg and Lynn Loomis.PDF
You are missing Abstract Algebra that usually comes before or after Real Analysis. As for that 4chan post, Rudin's book will hand anyone their ass if they havent seen proofs and dont have a proper foundation (Logic/Proofs/Sets/Functions). Transition to Higher Math courses usually cover such matters. Covering Rudin in 4 months is a stretch. It has to be the toughest intro to Real Analysis. There are tons of easier going alternatives:
Real Mathematical Analysis by Charles Pugh
Understanding Analysis by Stephen Abbot
A Primer of Real Functions by Ralph Boas
Yet Another Introduction to Analysis
Elementary Analysis: The Theory of Calculus
Real Analysis: A Constructive Approach
Introduction to Topology and Modern Analysis by George F. Simmons
...and tons more.
Really interested, actually! But I'm curious about a few things:
When exactly will it start in January? And when will it end? Will it be in the evenings? Which days of the week?
Will we need a text book? I have a Dover book on basic analysis already which I haven't cracked open.
Where will the class be held?
I had an incredibly hard time with calculus as a university student. I took it 5 times because I kept dropping it or withdrawing or not getting a passing grade. I almost got kicked out of my program because I pushed the limits of how many times I could repeat the course. There was a general disinterest on my part, but now, almost 10 years later, I am much more fascinated and genuinely interested in math, number theory, and also in many ways, analysis.
I started reading a book recently that finally explained what calculus actually was in simple terms. I feel like it's the first time that was ever done for me and I can say that helped my interest.
Anyway, I'd really hope to attend your class! The reason I'm curious about exact start date is that I'll be away from the HRM until mid-January. And it's a bummer to miss the first few classes of anything!
I'm on vacation, which means it's self study time. Definitely my favorite way to learn math.
I know enough algebra and modern algebraic geometry at this point that the best way to learn more alg geo should be to learn a bunch of differential geometry, so that's what I'm doing. A friend and I have been working through Global Calculus by Ramanan, and I'm looking to do directed reading from it once the school year starts.
We started with Lee's Introduction to Smooth Manifolds but it's dreadfully boring, index heavy, covers too many topics without indicating which ones are important, and doesn't block off important definitions and remarks. Also I don't really like einstein notation. Global Calculus strikes a really nice balance in which it uses categorical concepts to streamline and simplify the core ideas but doesn't randomly categorify everything for no reason (eg nLab on most things). However it is really, really dense, with my friend and I having spent hours discussing half a page at times, and lightly peppered with mistakes and little obscurisms.
The same friend and I, and our grad student mentor, are also polishing up a jointly written paper for submission, which will be my first. It's in combinatorics, and while it's very grounded, concrete stuff, a very non-concrete category of CW complexes popped up while we were trying to extend our ideas. That reminded me to reread Emily Riehl's excellent expositional paper A Leisurely Introduction to Simplicial Sets, so that's what I'm doing.
I took a kind of mini-course in complex analysis that went through about the first three chapters in Stein and Shakarchi's Complex Analysis. I might be taking a Reimann surfaces course next quarter, so I've been working through the stuff I didn't see to make sure I'm up to speed with what somebody taking a standard intro course would know. I got to cut out most of the book because I have no interest in number theory and because no intro course is going to assume prior knowledge of fourier analysis, so at this point probably the only thing I have left to do is pick out and do the interesting exercises related to the Reimann mapping theorem. I say probably because I'm not sure whether I want to go through the material about conformal mappings into polygons. I have book format version of A Concise Course in Complex Analysis by Schlag arriving soon, so once I'm done with Stein that'll probably be my secondary work source after burning out on diff geo each day.
I have grad school applications this year, so I'm also working on getting my shit together for that. I have the math GRE in october and I barely remember a single trick for computing integrals so wish me luck.
I think these days it's really important to make it to the generalized stokes theorem, not just for an honors crowd but in general. This means covering differential forms. Hubbard and Hubbard has been mentioned.
Not a book but in my mind a very nice update on H&H is Ghrist's video lecture on multivariable calculus which covered traditional integral theorems (Green, Gauss and Stokes) while showing their full relationship to generalized stokes in a very natural way. I really think this is a kind of template how modern courses on multivariable/vector calculus should be taught these days. it's not just the content but also the order of presentation that is very neat and maximizes clarity.
There are a bunch of books that had treaded this path over the years. Loomis & Sternberg, and Harold Edwards are books worth considering, though H&H is in some sense most detailed while also having a nice pace.
I actually believe that there is a dearth of really good updated and polished books in the area, and that there are so few really good options calls for some effort to develop lecture notes into books on the topic.
Historically, mathematicians had a goal of obtaining all integrals of rational integrands as rational expressions, rational expressions that would be given explicitly (or in closed form) in terms of elementary expressions. However, it was realized eventually that such a goal is a hopeless goal, one that is not possible in the traditional sense, and that the traditional artificial restrictions imposed on elementary analysis are thus unjustified.
They were brought to this realization most popularly by the elliptic integrals, integrals of rational expressions (rational expressions with the square root of a polynomial of the 4th or 3rd degree as an argument) which does not resolve itself into an explicit elementary expression by the methods of substitution or integration by parts.
Instead, due to greater rigor as gifted to us by the field of mathematical analysis, we were thus able to justify processes of approximations with a level of confidence and certainty that was not offered before.
As an elementary example, from Richard Courant's and Fritz John's "Introduction to Calculus and Analysis I", page 410 - 411, an integral expression for the time period of an ordinary pendulum was obtained, it is an elliptic integral, which means, we cannot proceed by way of a simple transformation of the independent variable ("method of substitution") or by breaking the integral apart into smaller parts by way of integration by parts and still hope to obtain a simple explicit elementary expression.
So, instead, there is an expression in the integrand, being, 1/√[1 - u^2 sin^2 (𝜃/2)]. For sufficiently small values of 𝜃, we find that the expression is arbitrarily close to the value 1, and therefore, this entire expression in the integrand was reduced to the factor one, allowing us to approximate the elliptic integral in sufficiently small intervals of 𝜃.
Noting that, the margin of error must be calculated (as was done so in the book). At least the physicists now have an expression for the time period of an ordinary pendulum - an imperfect approximation.
.
Admitting defeat:
Over the decades and eventually, centuries, mathematicians decided to allow functions as integral expressions without requiring always that they must be solved explicitly in terms of elementary expressions due to the convenience offered, in fact, the famous Gamma function is exactly the example, a function that is usually expressed as an integral (and, of course, don't forget about the elliptic integrals).
In the end, not all integrals are meant to be solved the same way that an integral of an elementary polynomial is, this philosophy is not merely isolated to that of integral calculus, as its analog can be found in differential calculus as differential equations.
If you're interested in Fourier series in general, I'd recommend a couple of different books. They all contain these results (some contain more constructive versions than others).
[Stein and Shakarchi's Fourier Analysis: An Introduction] (http://www.amazon.com/Fourier-Analysis-Introduction-Princeton-Lectures/dp/069111384X) is probably the most accessible book I can think of. It doesn't assume much analysis background, and it's a pretty easy read. It contains all the classical goodies you should see on Fourier analysis and Fourier series without having to use any measure theory. It also springboards into the 3rd volume in this series, which is on measure theory.
Sticking with the classical camp but adding in a bit of measure theory and functional analysis, there's Katznelson's An Introduction to Harmonic Analysis and the infamous Zygmund Trigonometric Series. Zygmund is an exceedingly comprehensive introduction to Fourier series at the beginning graduate level. And I do mean comprehensive. It was published in 1935, and it's a fair bet that it captured close to everything that was known about convergence results concerning Fourier series at that time.
The last way I'd go (and I wouldn't really look at it until you have some background in the above) is Javier Duoandikoetxea's Fourier Analysis. The book makes very free use of measure theory and functional analysis. It also assumes a pretty good working familiarity with the theory of distributions (which it introduces at rapid speed).
You want to read http://www.amazon.com/Fourier-Analysis-Introduction-Princeton-Lectures/dp/069111384X, literally starts from the beginning talking about this- but I'm still reading it so I'll let someone who knows more actually answer you ;)
Stein and Shakarchi: http://www.amazon.com/Complex-Analysis-Princeton-Lectures-No/dp/0691113858
Basically all reading in mathematics will help with this. What kind of applied mathematician? What is your background?
If I had to pick one medium-sized book, I’d say read Trefethen & Bau’s Numerical Linear Algebra.
Well I'd recommend:
For a more basic stats refresh before you dive in, pretty much any introductory textbook will be sufficient. For a very basic but quick and dirty refresh on basic stats you can get: Statistics in Plain English
This image was used for the cover of a famous text on error analysis.
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.
Linear Algebra: An extremelly versatile branch of Mathematics that can be applied to almost anything, also the first "real math" class in most universities.
Calculus: The first mathematics course in most Colleges, deals with how functions change and has many applications, besides it's a doorway to Analysis.
Real Analysis: More formalized calculus and math in general, one of the building blocks of modern mathematics.
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.
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.
/u/LengthContracted this is a good book, as is Daphne Kollers book on PGMs as well as the associated course http://pgm.stanford.edu
A sample of what is on my reference shelf includes:
Real and Complex Analysis by Rudin
Functional Analysis by Rudin
A Book of Abstract Algebra by Pinter
General Topology by Willard
Machine Learning: A Probabilistic Perspective by Murphy
Bayesian Data Analysis Gelman
Probabilistic Graphical Models by Koller
Convex Optimization by Boyd
Combinatorial Optimization by Papadimitriou
An Introduction to Statistical Learning by James, Hastie, et al.
The Elements of Statistical Learning by Hastie, et al.
Statistical Decision Theory by Liese, et al.
Statistical Decision Theory and Bayesian Analysis by Berger
I will avoid listing off the entirety of my shelf, much of it is applications and algorithms for fast computation rather than theory anyway. Most of those books, though, are fairly well known and should provide a good background and reference for a good deal of the mathematics you should come across. Having a solid understanding of the measure theoretic underpinnings of probability and statistics will do you a great deal--as will a solid facility with linear algebra and matrix / tensor calculus. Oh, right, a book on that isn't a bad idea either... This one is short and extends from your vector classes
Tensor Calculus by Synge
Anyway, hope that helps.
Yet another lonely data scientist,
Tim.
Best:
Principles of Mathematical Analysis by Walter Rudin
Baby Rudin
If Rudin is too demanding for you, don't despair. A fairly good book that is not as challenging is Ross. Once you get through that, Rudin may be a bit more amenable.
I'm doing that, I guess, if you call 'advanced maths' anything proof-based (which is, generally, what people mean). I use the internet, my brain, and a lot of books. It was hard for sure. Only way to do it is to enjoy it and not burn yourself out working too hard.
This book is how I got started and probably the easiest way into anything proof based: http://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/0387950605.
Ofcourse you might not want to do analysis especially if you have't done any calc yet. At that level people (I think) do stuff like http://www.artofproblemsolving.com/. Also khan academy, MiT OCW, and competition-oriented books like https://www.google.com/webhp?sourceid=chrome-instant&amp;ion=1&amp;espv=2&amp;ie=UTF-8#q=complex%20numbers%20from%20a%20to%20z.
That said if you can work through that analysis book it'll open the doors to tons of undergrad level math like Abstract Algebra, for example.
Just keep at it?
It mentions Rosenlicht at the bottom. Lucky you, that book's only 8 bucks! It's a good book, too.
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.
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.
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.
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!
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).
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.
I would try http://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1375896302&amp;sr=1-1 and if that does not truly challenge you, then we can chat some more.
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&amp;qid=1346855198&amp;sr=8-5&amp;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&amp;qid=1346860077&amp;sr=8-1&amp;keywords=rudin
http://www.amazon.com/Invitation-Linear-Operators-Matrices-Bounded/dp/0415267994/ref=sr_1_4?ie=UTF8&amp;qid=1346860052&amp;sr=8-4&amp;keywords=from+matrix+to+bounded+linear+operators
http://www.amazon.com/Counterexamples-Analysis-Dover-Books-Mathematics/dp/0486428753/ref=sr_1_5?ie=UTF8&amp;qid=1346860077&amp;sr=8-5&amp;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&amp;ie=UTF8&amp;qid=1346860356&amp;sr=1-2&amp;keywords=chaos+and+dynamics
http://www.amazon.com/Numerical-Analysis-Richard-L-Burden/dp/0534392008/ref=sr_1_5?s=books&amp;ie=UTF8&amp;qid=1346860179&amp;sr=1-5&amp;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 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.
Schaum's 3000 Solved Calculus Problems saved my butt so many times throughout Calc 1-3 and now Differential Equations. You can find a PDF online if you're savvy enough.
I didn't struggle with Real Analysis mostly because it addresses all your "why?" questions from the get-go.
How to Think About Analysis by Lara Alcock is a nice book that walks you through the Analysis skeleton in a very short time especially if you have no problem with quantifiers.
I am struggling with Linear Algebra right now because of high school style "shut up and do these useless exercises" attitude of most LA books.
I found a book on LA:Real Linear Algebra by Fekete that seems to kick ass(deals with most of your "why" questions) if you are struggling with that as well. I found a free copy online, but it's shit quality, so I had to buy it for that exorbitant price. I think it's worth it since I am tired of crap Linear Algebra books.
I never understand the voting on this sub. Some unrelated posts are upvoted, but tangentially related posts downvoted. Hell, even two similar topics on the main page of this sub get different votes.
@OP, How To Think About Analysis by Lara Alcock.
I agree with all the suggestions to start with How to Prove It by Velleman. It's a great start for going deeper into mathematics, for which rigor is a sine qua non.
As you seem to enjoy calculus, might I also suggest doing some introductory real analysis? For the level you seem to be at, I recommend Understanding Analysis by Abbott. It helped me bridge the gap between my calculus courses and my first analysis course, together with Velleman. (Abbott here has the advantage of being more advanced and concise than Spivak, but more gentle and detailed than baby Rudin -- two eminent texts.)
Alternatively, you can start exploring some other fascinating areas of mathematics. The suggestion to study Topology by Munkres is sound. You can also get a friendly introduction to abstract algebra by way of A Book of Abstract Algebra by Pinter.
If you're more interested in going into a field of science or engineering than math, another popular approach for advanced high schoolers to start multivariable calculus (as you are), linear algebra, and ordinary differential equations.
We used this one in my undergraduate analysis class, and I found it pretty straightforward to read and understand. And it's only $13.
You might like Rosenlicht's book, Introduction to Analysis. Google Books will show you the first 2 chapters for free. It's a Dover book, so it's good and also cheap. I believe that it is often used as the text for the first "serious" real analysis course.
No problem. For smooth infinitesimal analysis, there's an easy-to-read introduction by John Bell. For nonstandard analysis, on the other hand, there's Keisler.
The idea of significant figures is a simplification of error analysis. It doesn't produce perfect results, as you've found in your example. It's useful as a simple rule of thumb, especially for students, but any proper analysis would use real error analysis. Your approach of looking at the range of possible values is good, and is basically the next level of complexity after sig figs.
The problem with error analysis is that it's a bit of a bottomless rabbit-hole in terms of complexity: you can make things very complicated very quickly if you try to do things as accurately as possible (for example: the extreme values in your range of possible times are less likely than the central values, and since your using the inverse of the time, that produces a non-uniform distribution in the velocities. Computing the actual probability distribution is a proper pain in the ass).
My advice is this: if you're a highschool student or non-physics university student, stick to sig-figs: it's not perfect, but it's good enough for the sorts of problems you'll be working with. If you're a physics major, you should learn some basic error analysis from your lab courses. If you're really interested in learning to do it properly, I think the most common textbook is the 'Introduction to Error Analysis' by Taylor.
We used this in my undergrad Experimental Physics course: https://www.amazon.com/Introduction-Error-Analysis-Uncertainties-Measurements/dp/093570275X
Are you interested in systematic errors, random errors, or both? Ignoring systematic errors, with the information that you've given, here are the obvious things to consider:
For each of the items above, you can determine the uncertainties with a simple design-of-experiments. For validated instrumentation, the uncertainties will be specified as part of the IQ/OQ/PQ process, but even so, you should still verify them yourself.
Once you have these values, calculating how each of them contribute to the final error is relatively straightforward using principles of error propagation. There are many books and websites devoted to the topic of error propagation. I have a copy of John Taylor's book, which I like. It does have a significant number of errors in the book because it contains so many equations and works them out in detail. However, the principles of error propagation are taught very well in the book, and the minor math errors (I know it's ironic) are easy to spot.
In fact, it is possible to give error bars from one exact measurement. For example, let's say I count how many rain drops hit my hand in 5 seconds and the result is 25. The number of rain drops striking my hand in a given length of time will form a Poisson distribution. One can argue that based on my one measurement, the best estimate I can make of the true rate of rain striking my hand each 5 seconds is 25 +/- sqrt(25) = 25 +/- 5.
As you might intuit, the uncertainty of the mean number of drops striking my hand will decrease as more measurements are taken. This tends to drop like 1/Sqrt(N), where N is the number of 5-second raindrop measurements I make.
This style of problem is very standard in any introductory statistics textbook, but I can give you a particular book if you'd like to look into it further:
An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements by John R. Taylor
These plots are "distributions" in the sense that I meant the word distribution. Distributions are simply a collection of values placed side by side. When you arrange each month's datapoint side by side, that's a distribution.
It’s the cover of one of my favorite books I used in college. I still keep it on my desk. Error Analysis by John Taylor
An Introduction to Error Analysis by John R. Taylor is the text that undergrads at UC Berkeley use. It's pretty decent.
As an aside, I think that the undergraduate sequence at most schools does a terrible job of teaching about uncertainty and error analysis. I'm a PhD candidate at Berkeley (graduating in December!), and my dissertation involves high precision measurements that test the Standard Model. Thus, analyzing sources uncertainty is my bread and butter. I really appreciate how approximations, models, and measurement precision are interrelated.
I'm really curious to see what resources other people put here.
If you're serious about learning about Fourier Analysis, the book Who is Fourier?: A Mathematical Adventure is an excellent and very accessible introductory text on the subject.
I recommend thumbing through an introductory real analysis textbook like Abbot - and perhaps speaking to a professor - before declaring a second major. Mathematics beyond sophomore level are a lot different, even at the applied level.
FWIW, I quit a PChem PhD program to pursue applied math, it definitely gives you a lot more flexibility, but it's not for everyone.
Understanding Analysis by Stephen Abbott https://www.amazon.com/dp/1493927116/
Topics in Algebra by I.N. Herstein https://www.goodreads.com/book/show/1264762.Topics_in_Algebra
The Feynman lectures on physics http://www.feynmanlectures.caltech.edu/
I've got nothing for Economics, but the above would be my personal recommendations for self-study and just general reading.
Assuming you know analysis up to the Riemann integral and some basic stuff on uniform convergence of functions, then I think almost everything I mentioned is covered in chapters 2 and 3 of Fourier Analysis: An Introduction by Stein and Shakarchi. The only exception is Carleson's Theorem, which is very hard and if you really do need it then you'd be better off treating it as a black box.
Fair enough! That makes sense.
Since you did well in the topic I'd assume you know of the basics pretty well. If you'd still like to brush up on the topics, I really like Ahlfors' text. Its not everybody's cup of tea and its a bit terse but for someone looking for a second look at Complex Analysis it should be doable. If not then go for something less dense like Stein/Sarkachi or Gamelin.
If you are looking for topics then allow me to suggest you one: if you liked Geometry in university then I highly recommend looking into Complex Geometry, which is the study of complex manifolds. Holomorphic functions (or complex analytic, depending on what text you used) in [; \mathbb{C} ;] have really interesting/wacky properties as is (think analytic continuation, Louisville etc.). Now imagine the fun when you lift that up to manifolds! There are lots of tie ins with algebraic geometry as well(more so, imo than with differential geometry) so if that's something you liked, it is worth looking into.
I have to admit I don't know as much about this topic as I should but I think Complex Geometry is quite cool and if you found geometry in university at all interesting, I think it will be a fulfilling topic for you. Let me know if that sounds at all cool then we can talk literature.
Harold Edwards book takes this approach.
https://www.amazon.com/Advanced-Calculus-Differential-Forms-Approach/dp/0817637079
For me, a "good read" in mathematics should be 1) clear, 2) interestingly written, and 3) unique. I dislike recommending books that have, essentially, the same topics in pretty much the same order as 4-5 other books.
I guess I also just disagree with a lot of people about the
"best" way to learn topology. In my opinion, knowing all the point-set stuff isn't really that important when you're just starting out. Having said that, if you want to read one good book on topology, I'd recommend taking a look at Kinsey's excellent text Topology of Surfaces.
If you're interested in a sequence of books...keep reading.
If you are confident with calculus (I'm assuming through multivariable or vector calculus) and linear algebra, then I'd suggest picking up a copy of Edwards' Advanced Calculus: A Differential Forms Approach. Read that at about the same time as Spivak's Calculus on Manifolds. Next up is Milnor Topology from a Differentiable Viewpoint, Kinsey's book, and then Fulton's Algebraic Topology. At this point, you might have to supplement with some point-set topology nonsense, but there are decent Dover books that you can reference for that. You also might be needing some more algebra, maybe pick up a copy of Axler's already-mentioned-and-excellent Linear Algebra Done Right and, maybe, one of those big, dumb algebra books like Dummit and Foote.
Finally, the books I really want to recommend. Spivak's A Comprehensive Introduction to Differential Geometry, Guillemin and Pollack Differential Topology (which is a fucking steal at 30 bucks...the last printing cost at least $80) and Bott & Tu Differential Forms in Algebraic Topology. I like to think of Bott & Tu as "calculus for grown-ups". You will have to supplement these books with others of the cookie-cutter variety in order to really understand them. Oh, and it's going to take years to read and fully understand them, as well :) My advisor once claimed that she learned something new every time she re-read Bott & Tu...and I'm starting to agree with her. It's a deep book. But when you're done reading these three books, you'll have a real education in topology.
You might want to consider some kind of numerical linear algebra book like the very readable Trefethen and Bau.
While this topic isn't always included in an undergrad curriculum, it's hugely useful. It's critical for a bunch of more advanced areas like physical simulation, graphics optimization, and machine learning.
Elements of Statistical Learning covers KDE pretty well. (It does have a pretty heavy linear algebra prereq. If it is getting too hairy, you may want to look at a numerical linear algebra book, like Trefethen and Bau)
Also Computational Statistics covers it well from what I remember. These are both really good books.
But both are really great books.
The Nature of Computation
(I don't care for people who say this is computer science, not real math. It's math. And it's the greatest textbook ever written at that.)
Concrete Mathematics
Understanding Analysis
An Introduction to Statistical Learning
Numerical Linear Algebra
Introduction to Probability
I took a Stats for Sci & Eng class (it had this book). All I learned was that stats is really hard and you have to use way more calculus than I initially thought.
I'm potentially interested in picking up a textbook on error analysis. How do we feel about John R. Taylor's book?
This was on the cover of a college text book I had: An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements
>When's the last time someone flew a train into a building?
It's happened before.
This book about error analysis is really good
I think the rule about sig figs is that you want the sig figs on your error to be of the same place as your last sig fig in your calculation. So your numbers would be 5.77 ± 0.31.
This book, Who Is Fourier?: A Mathematical Adventure is the best starting-off resource I can think of. And then there's the stanford open course The Fourier Transform and its Applications. Good luck :)
I did a little research and it looks like if you're interested in the beauty then a good first read would be Who Is Fourier? Check out the reviews.
Now, if you actually want to learn fourier series, it would help a lot to be comfortable with some topics from linear algebra: vector spaces, spans, linear independence, bases, and orthogonality. You should ponder these in the context of a familiar vector space of functions (polynomials would be a good one since it is infinite-dimensional).
I always recommend Dover books if I can since they're so cheap and I happen to own a good one called Fourier Series and Orthogonal Functions by Davis. The first four chapters cover the information you're interested in. The first chapter reviews the above concepts from linear algebra and the second chapter starts exploring the concepts of orthogonality of functions and series of fuctions. The book starts getting juicy in the last section of the second chapter which applies the relatively general treatment of the five previous sections in the chapter to a specific problem of approximating a given function by a sum of sines and cosines. This motivates the material in the third chapter, the heart of the book, on Fourier series. The fourth chapter hints at the fact that Fourier series are merely a special case of a more fundamental idea and introduces series of Legendre polynomials and Bessel functions.
There are exercises at the end of every section.
You can solve this by inspection. The fundamental frequency is the lowest first order harmonic of all of your sin and cos terms.
100pi = 2pif0 -> f0 = 50Hz, or w0 = 100pi rads/s.
Period = 1/f0 = 20 ms.
Might I recommend this book: http://www.amazon.com/Who-Fourier-Mathematical-Adventure-2nd/dp/0964350432/ref=sr_1_fkmr0_3?ie=UTF8&amp;qid=1465228053&amp;sr=8-3-fkmr0&amp;keywords=who+is+joseph+fourier
I’m deeply in love with the scientist and engineers guide to dsp I’m doing audio dsp work and this was a big help, but it applies equally to shaders. It is not a how to guide but a core fundamentals book.
Along the same lines who is Fourier is a great but very quirky introduction to the fast Forier transform.
And Designing Audio Effect Plug-Ins in C++ is a good base intro to proper audio dsp development.
Who Is Fourier? A Mathematical Adventure
https://www.amazon.com/Who-Fourier-Mathematical-Adventure-2nd/dp/0964350432
Hello, I think you're spot on about it making your life easier after struggling, and by taking this class and putting in the time, it will make other math courses much easier for you. Because of what you gain from the struggle, I would really recommend you take this over 142, if you have the time. I took 140A last fall, and although I only got a C, it took an immense amount of effort to even get that. The class is set up so that if you put in the hard work to understand the concepts, the homework, the proofs and so forth, you're gonna do well, and If you truly understand how to solve the homework problems, then the tests will be familiar (doesn't mean it will be easy).
Expect to put a lot of work in. This statement needs to be taken seriously for this class, I've talk to some people in the class who say they put in 40 hours a week. This is usually because the concepts do not come immediately and you have to constantly repeat and approach at different angles to find a good understanding.
I recommend having a supplementary text while you are studying from the dreaded Rudin. For 140A, you should be looking at compactness and chapter 2 very early on as this is a big hurdle in that class. Other concepts will be more familiar but still challenging.
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Some recommended texts (definitely find your own that works for you)
https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Princeton/dp/0691172935 (If you prefer "casual" explanations of the concepts, this help me survive chapter 2 of Rudin. There are useful book recommendations in the very back)
https://www.amazon.com/Elementary-Analysis-Calculus-Undergraduate-Mathematics/dp/1461462703 (Ross is used for the 142 series, and I find it is very helpful if you are struggling. If you are having trouble, start with the easier version of a problem and build up from there. The book mainly stays within the R\^2 metric, which is what makes it simpler)
https://minds.wisconsin.edu/handle/1793/67009 (at some point, you're gonna get stuck and you will have to look at the solutions. This is ok, but don't become reliant on it, that really hurt me in the end when I did that. Some of the questions are fuccckkkiiinngg hard, so when you hit that wall, take a look here. They give solutions that skips over a ton of steps, or might not be that good of a way to solve the problem, but this is a great resource)
https://www.math.ucla.edu/~tao/preprints/compactness.pdf (Who doesn't know who Terence Tao is? This is very helpful for giving an answer to "what is compactness used for?". It gives some intuition about what it is, and you should read it a couple times during 140A.)
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So this is advice that I would give myself when entering the course, and maybe it won't apply to you. Since you got an A in 109 without too much trouble, you are definitely very ready for 140, and you have a very chance of succeeding. Stay curious, and don't stop at just the solution. Really question why it is true. You probably won't have this problem, but when it hits you (probably when you get to chapter 2) you have to keep at it and don't give up. Abuse office hours, ask lots of questions, study everyday etc. and you'll do well. If you want to get better at math then the pain is worth it.
I thought Elementary Analysis by Kenneth Ross was pretty accessible. As others have said, though, your goal seems somewhat unrealistic.
Here's my radical idea that might feel over-the-top and some here might disagree but I feel strongly about it:
In order to be a grad student in any 'mathematical science', it's highly recommended (by me) that you have the mathematical maturity of a graduated math major. That also means you have to think of yourself as two people, a mathematician, and a mathematical-scientist (machine-learner in your case).
AFAICT, your weekends, winter break and next summer are jam-packed if you prefer self-study. Or if you prefer classes then you get things done in fall, and spring.
Step 0 (prereqs): You should be comfortable with high-school math, plus calculus. Keep a calculus text handy (Stewart, old edition okay, or Thomas-Finney 9th edition) and read it, and solve some problem sets, if you need to review.
Step 0b: when you're doing this, forget about machine learning, and don't rush through this stuff. If you get stuck, seek help/discussion instead of moving on (I mean move on, attempt other problems, but don't forget to get unstuck). As a reminder, math is learnt by doing, not just reading. Resources:
math on irc.freenode.net
Here are two possible routes, one minimal, one less-minimal:
Minimal
Less-minimal:
NOTE: this is pure math. I'm not aware of what additional material you'd need for machine-learning/statistical math. Therefore I'd suggest to skip the less-minimal route.
Here are some great books that I believe you may find helpful :)
and last but definitely not least:
Later on:
Calculus by James Stewart is the best introductory Calculus book that I used in college - I definitely recommend it. It will get you through both single-variable calculus, as well as most of multi-variable calculus that you will need for for master's level probability and statistical theory. In particular, if you plan to use the book, you should focus on chapters 1-7 (for single variable calculus), chapter 11 (infinite sequences and series) and chapters 14 and 15 (partial derivatives and multiple integrals). These chapter numbers are based on the 7th edition.
If you have previously taken calculus, you might consider looking at Khan Academy for an overview instead.
If you have not previously taken linear algebra, or it has been awhile, you will definitely need to work through a linear algebra textbook (don't have any particular recommendations here) or visit Khan academy.
Finally, a book such as Stephen Abbott's Understanding Analysis is not necessary for master's level statistics, but could be helpful for getting into the mindset of calculus-based proofs.
I'm not sure what level of math you have previously completed, and what level of rigor the MS in Statistics program is, but you will likely need be very familiar with single- and multi-variable calculus as well as linear algebra to be successful in probability and statistical theory. It's certainly possible, just pointing out that there could be a lot of work! If you have any other questions, I'm happy to answer them.
>my first venture into proofs?
Have you had no prior experience with rigorous proofs, other than some elements of your linear algebra class? Not even something like a discrete math class? I'd worry that as an already-busy grad student, this might be biting off more than you can chew.
One additional question: is "grad analysis" a graduate-level class in analysis beyond an undergraduate-level class also offered at your school? I ask because typically, such a graduate-level class would assume considerable familiarity with undergrad-level analysis as a prerequisite. If you're in a situation where understanding the rigorous ε-δ definition of limit isn't something you've already internalized intuitively, then you'll likely find a grad-level introduction to something like measure theory to have a very steep learning curve.
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I second /u/Gwinbar's recommendation above of Stephen Abbott's Understanding Analysis as a textbook for self-directed learning. But even that might be premature if you don't first develop sufficient background in the basics of set theory and mathematical logic. In particular, lots of concepts in analysis involve logical quantifiers, meaning that you'll need to be comfortable with both the meaning of a statement like
and how you would take the logical negation of the above statement. If none of this is familiar or transparently clear to you, then you might be better served by taking an undergraduate class in real analysis. Another option, of course, would be to audit a class, though that would be less advantageous in the context of buttressing your CV.
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I think the best advice I can give you at this point would be to talk to someone at your school. Someone in the economics department would have the best sense of how valuable having a graduate-level analysis class could be for your pursuit of a doctorate—as well as how damaging flaming out from such a class might be. I'd recommend talking to someone at your school's math department, too, since the best way to evaluate your background would be through a conversation by someone who's familiar with your school's analysis curriculum. They're in the best position to make the recommendation that best fits your current background level in mathematics, given what your school's academic standards are for such analysis classes. They can also provide final exams from past iterations of the undergrad- and grad-level analysis courses, respectively. That might give you some additional data to illuminate what such classes entail.
I hope you can find more concrete information that's more custom-tailored to your specific circumstances. Good luck, whatever you decide!
In addition to Baby Rudin, I really liked this book when I first start learning analysis.
Square one...
You should have a solid base in math:
Introduction to Calculus and Analysis, Vol. 1 by Courant and John. Gotta have some basic knowledge of calculus.
Mathematical Methods in the Physical Sciences by Mary Boas. This is pretty high-level applied math, but it's the kind of stuff you deal with in serious physics/astrophysics.
You should have a solid base in physics:
They Feynman Lectures on Physics. Might be worth checking out. I think they're available free online.
You should have a solid base in astronomy/astrophysics:
The Physical Universe: An Introduction to Astronomy by Frank Shu. A bit outdated but a good textbook.
An Introduction to Modern Astrophysics by Carroll and Ostlie.
Astrophysics: A Very Short Introduction by James Binney. I haven't read this and there are no reviews, I think it was very recently published, but it looks promising.
It also might be worth checking out something like Coursera. They have free classes on math, physics, astrophysics, etc.
Rudin covers Hilbert spaces and Banach spaces in his Real and Complex Analysis, which is why he jumps straight into topological vector spaces in his book on functional analysis. So perhaps you could read those chapters from Real and Complex Analysis. Alternatively, check out the classic Functional Analysis by Reed and Simon or Conway's book. The reviews published by the MAA might also be interesting to you. And of course, there are many lecture notes available on the web. :-)
For functions of a single variable, Lebesgue integration is really just chopping up the y-axis, e.g. see Folland, Rudin, or almost any elementary treatment of real analysis. When extending to multiple dimensions, you must consider product spaces, and here the intuitive comparison to Riemann integration is not so clear.
Awesome! As mentioned, Rudin, Folland, and Royden are the gold standards of measure theory, at least from what I have heard from professors and the internet. I'm sure other people have found other good ones! Another few I somewhat enjoy are Capinski and Kopp and Dudley, as those are more based on developing probability theory. Two of my professors also suggested Billingsley, though I have not really had a good chance to look at it yet. They suggested that one to me after I specifically told them I want to learn measure theory for its own right as well as onto developing probability theory. What is your background in terms of analysis/topology? Also, I am teaching myself basic measure theory (measures, integration, L^p spaces), then I think that should be enough to look into advanced probability. Feel free to PM me if you need some help finding some of these books! I prefer approaching this from the pure math side, so mathematical statistics gets a bit too dense for me, but either way, I would look at probability then try to apply it to statistics, especially at a graduate level. But who am I to be doling out advice?!
*Edit: supplied a bit more context.
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.
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.
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.
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.
Can't go wrong with Schaum's
http://www.amazon.com/Schaums-000-Solved-Problems-Calculus/dp/0071635343/ref=sr_1_3?ie=UTF8&amp;qid=1335431258&amp;sr=8-3
In the grand scheme of math: jack shit. But who's to stop you after 2 months of studying?
What do you know so far? Are you comfortable with inequalities and math induction?
Check out the books below for a nice intro to Real Analysis:
How to Think About Analysis by Lara Alcock.
A First Course in Mathematical Analysis by D. A. Brannan.
Numbers and Functions: Steps to Analysis by R. P. Burn.
Inside Calculus by George R. Exner .
Discrete And Continuous Calculus: The Essentials by R. Scott McIntire.
Good Look.
Usual hierarchy of what comes after what is simply artificial. They like to teach Linear Algebra before Abstract Algebra, but it doesn't mean that it is all there's to Linear Algebra especially because Linear Algebra is a part of Abstract Algebra.
Example,
Linear Algebra for freshmen: some books that talk about manipulating matrices at length.
Linear Algebra for 2nd/3rd year undergrads: Linear Algebra Done Right by Axler
Linear Algebra for grad students(aka overkill): Advanced Linear Algebra by Roman
Basically, math is all interconnected and it doesn't matter where exactly you enter it.
Coming in cold might be a bit of a shocker, so studying up on foundational stuff before plunging into modern math is probably great.
Books you might like:
Discrete Mathematics with Applications by Susanna Epp
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Building Proofs: A Practical Guide by Oliveira/Stewart
Book Of Proof by Hammack
Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al
How to Prove It: A Structured Approach by Velleman
The Nuts and Bolts of Proofs by Antonella Cupillary
How To Think About Analysis by Alcock
Principles and Techniques in Combinatorics by Khee-Meng Koh , Chuan Chong Chen
The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!) by Carol Ash
Problems and Proofs in Numbers and Algebra by Millman et al
Theorems, Corollaries, Lemmas, and Methods of Proof by Rossi
Mathematical Concepts by Jost - can't wait to start reading this
Proof Patterns by Joshi
...and about a billion other books like that I can't remember right now.
Good Luck.
...here's a book I recommend
https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0198723539
I know someone else on /r/math has met the author
I've always liked Ross' Elementary Analysis
>My first goal is to understand the beauty that is calculus.
There are two "types" of Calculus. The one for engineers - the plug-and-chug type and the theory of Calculus called Real Analysis. If you want to see the actual beauty of the subject you might want to settle for the latter. It's rigorous and proof-based.
There are some great intros for RA:
Numbers and Functions: Steps to Analysis by Burn
A First Course in Mathematical Analysis by Brannan
Inside Calculus by Exner
Mathematical Analysis and Proof by Stirling
Yet Another Introduction to Analysis by Bryant
Mathematical Analysis: A Straightforward Approach by Binmore
Introduction to Calculus and Classical Analysis by Hijab
Analysis I by Tao
Real Analysis: A Constructive Approach by Bridger
Understanding Analysis by Abbot.
Seriously, there are just too many more of these great intros
But you need a good foundation. You need to learn the basics of math like logic, sets, relations, proofs etc.:
Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers
Discrete Mathematics with Applications by Epp
Mathematics: A Discrete Introduction by Scheinerman
I haven't heard of some of the lesser known books, but I just wanted to point out that Algebra Chapter 0 by Aluffi is a very advanced book (in comparison to other books on the list), and that you may want a more gentle introduction to Abstract Algebra before attempting that book. (Dummit and Foote is very standard, and there's plenty other good ones as well that are better motivated). Baby Rudin is also gonna be a tough one if you have no background in Analysis, even though it is concise and elegant I think it's best appreciated after knowing some analysis (something at the level of maybe Understanding Analysis by Abbott).
I know the symbols are scary! But you will be introduced to them gradually. Right now, everything probably looks like a different language to you.
Your university will either have an entire "Methods of Proof" course that proves basic results in number theory or some course (like real analysis) in which you learn methods of proof whilst immersed in a given course. In a course like this, you will learn what all those symbols you have been seeing mean, as well as some of the terminology.
Try reading an introductory analysis book (this one is a very easy read, as analysis books go). Or something like this. Or this
Anyways, don't be afraid! Everything looks scary right now but you really do get eased into it. Just enjoy the ride! Or you can always change your major to statistics! (I'm a double math/stat major, and I know tons of math majors who found the upper division stuff just wasn't for them and were very happy with stats).
I also really struggled with real analysis in the beginning. Stephen Abbot's Understanding Analysis saved my ass, I went from "reconsidering my career choice" to passing the course with a pretty good grade thanks to that book.
http://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/0387950605/ref=sr_1_1?ie=UTF8&amp;qid=1426932693&amp;sr=8-1&amp;keywords=understanding+analysis
A much, much more inexpensive copy with the same content is also available.
Rudin is definitely the classic, but for a more contemporary and "friendlier" (but no less rigorous) introduction to real analysis, some people prefer the book by Pugh.
Edit: The two books cover pretty much the same material in the same order. I've heard Pugh described as "Rudin, with pictures"
Sixty bucks!? Thirty bucks for Pugh and Rudin.
Hardy was a number theorist, but this book is straight analysis (I'm not sure how you approach analysis with number theory). From what I've seen, Hardy is very verbose and spends a lot of time on material you don't need to see the first time around. He also uses a lot of outdated terminology. Lastly, this book is calculus and analysis together. Presuming you've done calculus, you want to get straight to the analysis part. That's where the set theory and topology come in. "Modern" analysis (still pretty old) works in more general spaces and uses topological and set-theoretic ideas. It's actually very natural, and you'll wonder how you ever worked without them. You won't see this important modern presentation in Hardy, so you'd really be missing out. I'd buy Pugh/Rudin (or something easier) and use Hardy to supplement them, rather than the other way 'round.
Yes. However, you should probably read something that introduces you to proofs. My Intro to Higher Math classes (commonly called Intro to Proof-Writing or Intro to Analysis, the class or series of classes that introduce you to higher math and proofwriting skills) used this book alongside a prepackaged set of detailed lecture notes. I'd say that'd be a good place to start before reading about Abstract Algebra, plus the book is dirt cheap.
This is a pretty good book too. http://www.amazon.com/Introduction-Analysis-Dover-Books-Mathematics/dp/0486650383/ref=sr_1_1?ie=UTF8&amp;qid=1323212337&amp;sr=8-1
I don't know why more people on here don't recommend it, especially considering how cheap it is.
Introduction to Analysis (Dover Books on Mathematics) https://www.amazon.com/dp/0486650383/ref=cm_sw_r_cp_api_i_WTGPCbM6P2N4H
It’s actually a pretty decent book for a first look at Real Analysis.
Apologies for the serious comment on /r/funny.
https://www.amazon.com/Primer-Infinitesimal-Analysis-John-Bell/dp/0521887186
Of course! If you wanna read more, check out John Bell's Primer on Smooth Infinitessimal Analysis.
I have heard from professors that Rudin's Real and Complex Analysis is the go to book for analysis. I've also heard its a bit of a tough book to get through, but the understanding it provides is worth it.
If you end up using it let me know what you think! I'll be taking analysis next year.
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!
Well, there's here, of course. Hilbert spaces are a topic in analysis. I've heard good things about this book, which comes at it from a physics perspective.
If your background in analysis is up for it, they are covered in Rudin. This book is pretty intense.
Walter Rudin, principle of mathematical analysis
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).
I've found Rudin's Analysis useful. There's a lecture series on YouTube that roughly follows the book.
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.
THIS WILL PUT HAIR ON HIS CHEST:
http://www.amazon.com/Principles-Mathematical-Analysis-Third-Walter/dp/007054235X
Baby Rudin, The Inferno, DPV
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.
Is this the book you are recommending?
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.
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.
Rudin.
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?
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.
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.
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
> 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.
Is your objective to build a comprehensive understanding of the underlying topics of Calculus or is your objective to master quick problem solving, tricks, etc? If it is the latter I would suggest you pick this up as an auxiliary resource; Stuart is good but mastery of the mechanics of solving the problems will come only through ardent practice. You will need to see, and solve, a wider set of examples than is typically found in Stewart.
If your objective is the former I would grab this instead. Probably look for it on a used book seller's site like abebooks.com, though.
https://www.amazon.com/Schaums-Solved-Problems-Calculus-Outlines/dp/0071635343/ref=pd_sbs_14_img_0?_encoding=UTF8&amp;psc=1&amp;refRID=T66PCEAR6QPN9FAXE452
I'm going to go with a slightly different approach than starting from the very beginning.
How much do you know about calculus? If you know the basics of limits and derivatives, I would suggest to start learning at calculus. Go along with what you're being taught in class.
You can use Khanacademy/PatrickJMT to help you understand the concepts being taught in class. At the same time, as you're going through each concept, look up every term you're not familiar with. Don't take anything for granted. For instance, if you come across inverse functions in the explanation of something else, can you explain what inverse functions are? What's the difference between inverse and reciprocal? Or for the unit circle, do you know how the values came about? Question your understanding on every one of those concepts, and Google every single one of them. As you're going through the concept, make sure you commit it to memory. Try to build on your understanding. Even if you forget a little bit, the next time you come across the same concept, you'll have solidified your understanding a little more. The important thing is to be conscious of what you've just learned.
With this approach, it's going to take much more time and effort than your peers to get through some concepts, because you're using the opportunities in between to touch on previous concepts as well. So you really have to budget your time properly, but it'll be worth it in the end. If you don't have too much time, don't spend too much time rolling off the tangent looking up every single concept, just look up the thing that comes up and commit that to your memory.
Because you're going through a course, you don't have the luxury of being able to re-learn every single thing since grade one. The approach of learning as it comes up is much better suited for this situation imo. It's scary thinking there's a lot of things you don't know, but you can tackle those concepts as they come along. Don't panic.
Then at every available opportunity (winter break for example), practise what you've learned and drill yourself on the concepts.
I had a very similar problem of feeling like there are holes in my understanding and this was the approach I took. I'm in the middle of Calc 2 right now. As we're heading into winter break, I'm going to be reviewing everything that was taught this semester in Calc 2 and to review integration to prep for the second half of the course. I'll also be drilling myself with Shaum's 3000 problems book.
There are some good suggestions in this thread on Math Overflow as well.
Good luck!
I need either this or this. I'm taking Calculus II this semester for the second time. I'm aiming to be a math major, but I had difficulty last time. I'm already off to a better start this semester, but I want as much practice as possible. I'm aiming for a Masters in Math. I'm lucky that I have high grades and the F from last semester only dropped me down to a 3.2 GPA. I can't afford to have it drop any lower. I can't afford to spend any more time at this level. I have a Calculus workbook that my mom bought me, but it only covers Calc I and about two chapters of Calc II.
Actually.. Anything from my School Stuff WL is stuff I feel I need in order to do well at school. I really need to get organized with my school work and papers.. ._.
I have Abbott's and Charles Pugh's books. Both excellent and probably in your reserve library. There's another book I noticed on Amazon, I've never heard anybody on reddit or math.stackexchange mention, probably worth $20: https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0198723539
Also Spivak, Apostol, other books: https://www.reddit.com/r/math/comments/3drlya/what_mathematical_analysis_book_should_i_read/
There's lots of other threads here and math.SE that're helpful. Maybe looking thru Courant/Robbins What is Math witht he mindset that it's an enjoyable read
https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0198723539
Alcock is a Math Ed researcher with a huge focus on proofs in undergraduate mathematics.
I am sure this is the book you're referring to https://www.amazon.ca/Think-About-Analysis-Lara-Alcock/dp/0198723539
Yep, the stuff is quite hard and requires a lot of thinking about examples and counterexamples to understand what things mean. And you need time. You just can't learn this stuff in a cram session before an exam. A resource you might find helpful is
https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0198723539
I have a friend who swears by this book
https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0198723539
In case you also want some intro to Analysis(Calculus made a bit more rigorous), here's some:
How to Think About Analysis by Lara Alcock.
A First Course in Mathematical Analysis by Brannan.
There is actually a book called How to Think About Analysis which you might find useful. I have not read it myself, but I have read the author's other book and highly recommend her as an author.
Lara Alcock How to think about Analysis, How to study as a Mathematics Major
I hear that Rudin's book is pretty dense, so initially, I won't be using it, though I'm not entirely familiar with Spivak/Rudin beyond the comments on Amazon/Reddit.
Instead, I'm reading from Ross and [Bartle] (https://www.amazon.ca/Introduction-Real-Analysis-Robert-Bartle/dp/0471433314) right now, which I hear are good books for people starting out in Analysis. As I progress through the series, I might start teaching from Rudin and a variety of other sources.
This was true for me as well. A great introductory book is this one:
http://www.amazon.com/Elementary-Analysis-The-Theory-Calculus/dp/038790459X
Full of good pictures and lots of exposition.
Ah yeah you're at a more advanced stage than I thought. In that case an analysis text might appeal -- I like Abbot's Understanding Analysis but, again, it's quite pricey.
I suspect you'd love Galois theory, but I can't recommend a good text for self-study offhand.
This book.
I would just dive into it to see if it makes more sense! Here is a guide about delta epsilon proofs, which is one of the most common basic proofs you learn about in pure mathematics. Real Mathematical Analysis is a great textbook about real analysis. Also, if you're worried about the math, I would look into philosophical logic—Logic by Hodges is a good text for that and it won't involve any necessary background in math.
My school does a one semester intro using Understand Analysis and then a year long sequence using Rudin. I've been reading Real Mathematical Analysis and Pugh and I have to say that I am really enjoying it. Chapter two goes into more depth on topology that Rudin does in his book. There is also a lot of pictures and I am a visual learner.
If you really want to understand probability then you'll need to learn measure theory, which will require some background knowledge in real analysis. This is the book I used, which I highly recommend (and it's cheap!): http://www.amazon.com/Introduction-Analysis-Dover-Books-Mathematics/dp/0486650383/ref=sr_1_1?ie=UTF8&amp;qid=1414974523&amp;sr=8-1&amp;keywords=introduction+to+analysis
As for an actual book on probability, I'm not too sure since my probability course was based on lecture notes provided by the professor, although I just ordered this book because it looked decent: http://www.amazon.com/Graduate-Course-Probability-Dover-Mathematics-ebook/dp/B00I17XTXY/ref=sr_1_1?ie=UTF8&amp;qid=1414974533&amp;sr=8-1&amp;keywords=graduate+book+on+probability
https://www.amazon.com/Introduction-Analysis-Dover-Books-Mathematics/dp/0486650383
Here is the mobile version of your link
To piggy back off of danielsmw's answer...
> Fourier analysis is used in pretty much every single branch of physics ever, seriously.
I would phrase this as, "partial differential equations (PDE) are used in pretty much every single branch of physics," and Fourier analysis helps solve and analyze PDEs. For instance, it explains how the heat equation works by damping higher frequencies more quickly than the lower frequencies in the temperature profile. In fact Fourier invented his techniques for exactly this reason. It also explains the uncertainty principle in quantum mechanics. I would say that the subject is most developed in this area (but maybe that's because I know most about this area). Any basic PDE book will describe how to use Fourier analysis to solve linear constant coefficient problems on the real line or an interval. In fact many calculus textbooks have a chapter on this topic. Or you could Google "fourier analysis PDE". An undergraduate level PDE course may use Strauss' textbook whereas for an introductory graduate course I used Folland's book which covers Sobolev spaces.
If you wanted to study Fourier analysis without applying it to PDEs, I would suggest Stein and Shakarchi or Grafakos' two volume set. Stein's book is approachable, though you may want to read his real analysis text simultaneously. The second book is more heavy-duty. Stein shows a lot of the connections to complex analysis, i.e. the Paley-Wiener theorems.
A field not covered by danielsmw is that of electrical engineering/signal processing. Whereas in PDEs we're attempting to solve an equation using Fourier analysis, here the focus is on modifying a signal. Think about the equalizer on a stereo. How does your computer take the stream of numbers representing the sound and remove or dampen high frequencies? Digital signal processing tells us how to decompose the sound using Fourier analysis, modify the frequencies and re-synthesize the result. These techniques can be applied to images or, with a change of perspective, can be used in data analysis. We're on a computer so we want to do things quickly which leads to the Fast Fourier Transform. You can understand this topic without knowing any calculus/analysis but simply through linear algebra. You can find an approachable treatment in Strang's textbook.
If you know some abstract algebra, topology and analysis, you can study Pontryagin duality as danielsmw notes. Sometimes this field is called abstract harmonic analysis, where the word abstract means we're no longer discussing the real line or an interval but any locally compact abelian group. An introductory reference here would be Katznelson. If you drop the word abelian, this leads to representation theory. To understand this, you really need to learn your abstract/linear algebra.
Random links which may spark your interest:
At what level? I'm really enjoying Stein & Shakarchi but that's closer to graduate level.
This is a good option: http://www.amazon.com/Complex-Analysis-Princeton-Lectures/dp/0691113858
I learned from Differential Forms: A Complement to Vector Calculus and Advanced Calculus: A Differential Forms Approach. In first book many of the exercises seem tedious, but you should do them anyway.
Depends what you're interested in, but since we're in the ML subreddit it's probably about computation.
Numerical/computational linear algebra studies how to implement the ideas introduced in a 1st LA course on a finite-precision computer.
Linear programming, integer programming, non-linear optimization, and differential equations all heavily rely on linear algebra. The latter two mainly because of Taylor expansions which allow us to approximate functions in terms of linear and quadratic forms.
For ML you're probably best off skimming through the high level ideas in numerical linear algebra, and then diving into linear programming and non-linear optimization.
Not sure if it covers the same topics as Math 110, but this textbook is extremely friendly: https://www.amazon.com/Numerical-Linear-Algebra-Lloyd-Trefethen/dp/0898713617
Thanks for sharing I'll look into that one! Thanks:)
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Edit: They actually write in that course "The book Numerical Linear Algebra by Trefethen and Bau is recommended." so It might be some further applications!
Earlier this year I finished my PhD in aero (researching computational fluid dynamics). I'll go ahead and reiterate a couple of the other recommendations in this thread, I think they've given you pretty good advice so far.
Numerical Recipes is great, and you can even read their older editions for free online. Don't worry about them being older, their content really hasn't changed much over the years beyond switching around the programming language. A word of warning, though. The code itself in these books come with rather restrictive licenses, and what it ends up meaning for you is you can copy their code and use it yourself, but you aren't allowed to share it (although I don't think this is carefully enforced). If you want to share code, you'll either have to pay for their license, or use their code only as inspiration for writing your own. If you pay close attention to their licensing, they don't even let you store on your computer more than one copy of any of their functions (again, I can't imagine they actually have a way of enforcing this, but it makes me disappointed they do things this way nevertheless), so it can get problematic fast.
If you want more reading material, I've only paged through it myself but Chapra and Canale's book seems like a nice intro text (if it wasn't your textbook already), and uses MATLAB. Reddy has a well-liked intro to finite element methods. Some more graduate level texts are Moin, LeVeque (he has a bunch of good ones), and Trefethen.
Project Euler is indeed great.
I would also recommend you learn some other (any other, really) programming language. MATLAB is a fine tool, but learning something else as well will make you a better programmer and help you be versatile. I don't really recommend you go and learn half a dozen other languages, or even learn every feature available one language--just getting reasonably comfortable with one will do. I'd say pick any of: C, C++, Fortran 90 (or higher), or Python, but there are others as well. Python is probably the easiest to get into and there are lots of packages that will give it a similar "feel" to Matlab, if you like. One nice way of learning (I think) is going through Project Euler in your language of choice.
Slightly more long term, take other numerical/computational courses. As you take them, think about what you like to use computation for (if you don't have a good idea already). If you like to analyze data, develop more or less "simple" simulations to direct design decisions, and don't care so much for heavy simulations, you'll get a better idea of what to look for in industry. If you like physics simulations and solving PDEs, you may lean toward the research end of things and possibly dumping Matlab altogether in favor of more portable and high performance tools.
Try buying a new hardcover of this linear algebra book!
Hi OP,
I found myself in a similar situation to you. To add a bit of context, I wanted to learn optimization for the sake of application to DSP/machine learning and related domains in ECE. However, I also wanted sufficient intuition and awareness to understand and appreciate optimization it for it's own sake. Further, I wanted to know how to numerically implement methods in real-time (embedded platforms) to solve the formulated problems (Since my job involves firmware development). I am assuming from your question that you are interested in some practical implementation/simulations too.
&#x200B;
< A SAMPLE PIPELINE >
Optimization problem formulation -> Enumerating solution methods to formulated problem -> Algorithm development (on MATLAB for instance) -> Numerical analysis and fixed-point modelling -> Software implementation -> Optimized software implementation.
&#x200B;
So, building from my coursework during my Masters (Involving the standard LinAlg, S&P, Optimization, Statistical Signal Processing, Pattern Recognition, <some> Real Analysis and Numerical methods), I mapped out a curriculum for myself to achieve the goals I explained in paragraph 1. The Optimization/Numerical sections of the same is as below:
&#x200B;
OPTIMIZATION MODELS:
NUMERICAL METHODS:
&#x200B;
Personally I think this might be a good starting point, and as other posters have mentioned, you will need to tailor it to your use-case. Remember that learning is always iterative and you can re-discover/go deeper once you've finished a first pass. Front-loading all the knowledge at once usually is impractical.
&#x200B;
All the best and hope this helped!
Numerical Linear Algebraby Nick Trefethen is a pretty friendly intro to graduate linear algebra/matrix theory from a numerical analysis angle:http://www.amazon.com/Numerical-Linear-Algebra-Lloyd-Trefethen/dp/0898713617
Introduction to Numerical Analysisis very comprehensive, more advanced, but reads like an encyclopedia in a way. A good reference, though not very good as a lone textbook.http://www.amazon.com/Introduction-Numerical-Analysis-J-Stoer/dp/038795452X
Coincidentally, this is my favorite trainwreck of all time.
I would agree that people can get lost in the illusion of what science can and cannot reasonably do. My course of study was very careful to make sure that people were not indoctrinated. The undergraduate courses were of course devoted to learning basic terminology and principles that have been around for decades if not centuries, but the upper division courses never presented you with "this is the answer spit it back" types of courses. It was all about teaching us how to design experiments and how to think critically. For example, one of my favorite courses was Advanced Molecular Genetics where our professor (who had a Nobel Prize and was just teaching for the hell of it- and because he loved tormenting students) would present us with papers that had been published and point us to the "further questions" section and say "design an experiment that would determine what is actually going on." We were judged based on the experiments we designed, and we actually had the equipment to run the experiments, which we did. It would have been a very good reality TV show called "So You Think You're a Scientist." He was brutal. Imagine Gordon Ramsey as a scientist. He would tear you a new one if your experiment was shit and he had nothing to lose. That class was awesome. You had to have balls to show up every day because he'd shit all over everything you did unless you had solid facts to back you up.
Come to think of it- I'd watch the shit out of that show.
But yeah- this book was required reading for all of us. It explicitly lays out what science can do and (more importantly) lays out what science can't do.
Relating to our conversation- people severely overestimate what science can and cannot do. GMOs (or any other technology for that matter) can potentially help or potentially harm. What we have to weigh is the potential harm of the new technology versus the actual harm of the current technology.
Here's an example for a though experiment: Horses vs automobiles. Automobiles emit greenhouse gases and require mining of minerals to make, among other things, catalytic converters. There are risks of using automobiles, but compare them to the hazards of using horses. Piles of manure attracting rats and spreading disease. Millions of acres of cropland being grown to provide fuel for the horses, etc.
Old vs new. Neither is perfect. If we wait for something perfect we'll never do anything and become stagnant.
But thanks for the conversation. And just so you know I have rather thick skin so your insults didn't phase me at all. Glad we could get to the point where we're having civil discourse.
We used this book in my intro level physics lab for error analysis.
You need this book.
Until then - the general formula for error propagation in a function q(x, y, z, ....) with uncertainties &delta;x, &delta;y, &delta;z .... is equal to sqrt( (&delta;x&part;q/&part;x)^2 + (&delta;y&part;q/&part;y)^2 .....)
For your simple case where q = log10(x), &delta;q = &delta;x/(x*ln(10)).
Hope this helps.
A lack of this xD: http://www.amazon.com/Introduction-Error-Analysis-Uncertainties-Measurements/dp/093570275X
If you're looking to apply basic error analysis, I recommend Taylor's book:
http://www.amazon.com/Introduction-Error-Analysis-Uncertainties-Measurements/dp/093570275X
It's pretty common to find this book on physics grad students' shelves. You may have already seen it, though, and you may be asking for something deeper.
This image is used in one of my all time favorite textbook covers of all time: Introduction to Error Analysis
https://www.amazon.com/Introduction-Error-Analysis-Uncertainties-Measurements/dp/093570275X
Refer to this book, it will most definitely have the answer for you. Refer pages 60-62, I think it is what you are looking for, if not that chapter should have the answer for you.
This reminds me of a textbook I used for a physics class
This book does a good job:
http://www.amazon.com/Who-Fourier-Mathematical-Transnational-College/dp/0964350408
Though what was perhaps even better for me was realizing that a two dimensional representation for a sine wave is actually an unnatural representation.
The most natural representation is actually more like a 3-dimensional spiral staircase or stretched out slinky,
http://www.theoryofmind.org/misc-info/Physics/spring.jpg
Forget about the cos(x) + isin(x) part for now. Just figure out why the e^(ix) part looks like the slinky.
Then from there, the rest is pretty easy to see.
For example,
cos(x) = e^(ix) - isin(x)
is pretty intuitive. cos(x) is a 2d function. We get that by starting with the 3D spiral and then subtracting off the imaginary part to get a nice 2d graph.
Obviously, I have leaving out a lot of information. But the key, IMO, is to really understand why the graph of e^(ix) looks like it does, and then the rest will fall in place.
Who Is Fourier?: A Mathematical Adventure is a great book, I suggest you look into it.
I like to think of it as using different "lenses" to look at the data. Sometimes you want to use a microscope. Other times you want to use a telescope. Not to be taken literally of course, but you need the right tools.
Btw if you want to indulge that inner quant on this topic, check this book out. What I found amazing is that this is actually a kid's book in Japan.
I'm asking someone to show me how to do the problem. So I know how to do it. I learned a lot from
ss0317 2 points 3 hours ago
You can solve this by inspection. The fundamental frequency is the lowest first order harmonic of all of your sin and cos terms.
100pi = 2pif0 -> f0 = 50Hz, or w0 = 100pi rads/s.
Period = 1/f0 = 20 ms.
Might I recommend this book: http://www.amazon.com/Who-Fourier-Mathematical-Adventure-2nd/dp/0964350432/ref=sr_1_fkmr0_3?ie=UTF8&amp;qid=1465228053&amp;sr=8-3-fkmr0&amp;keywords=who+is+joseph+fourier
True, I do seem to be asking for a very specific plan. Is it this book?
https://www.amazon.com/Who-Fourier-Mathematical-Adventure-2nd/dp/0964350432
Oh also forgot to mention, I have taken a RA class that covered through roughly chapter ~35 of Elementary Analysis - Ross
I liked the one by Ross
http://www.amazon.com/Elementary-Analysis-Calculus-Undergraduate-Mathematics/dp/1461462703/ref=cm_cr_arp_d_product_top?ie=UTF8
I believe abstract algebra will be more useful. It'll teach you useful skills regardless of the field of physics. Analysis on the other hand will just make you a wizard with limits. You shouldn't need analysis for things like differential geometry. I would recommend this textbook for analysis though. While a deep understanding of calculus is nice to have, it's not often useful. Abstract algebra allows you to explore a whole new world of math.
I wish I was only taking those two. I've also got Abstract Algebra II (Ring Theory), and teaching the one class on top of that. This is my "tough" semester. The next two I'll probably only be taking 2 classes each semester, plus teaching.
What book are you using for Topo? We're using Munkres.
And what are you using for Real Analysis? I know Baby Rudin is sort of the standard, but we're using Ross.
Question about Spivak's Calculus and Ross' Elementary Classical Analysis:
Are they books treating mathematics on the same level? Do they treat the rigorous theoretical foundation and computational techniques equally well? Can each one be an alternative to the other? Could someone please give brief comparative reviews/comments on them?
This question is also on r/learnmath: HERE.
I found 'Understanding Analysis' by Stephen Abbott ( https://www.amazon.co.uk/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 ) to be super helpful/enlightening post Real Analysis insofar that it helped me build an intuition and understanding for some of the key ideas. Earlier today someone highly recommended this book as well: 'A Story of Real Analysis'
http://textbooks.opensuny.org/how-we-got-from-there-to-here-a-story-of-real-analysis/ (download link on the right). I had a quick glance through it and it seems pretty good.
This free pdf book should help you: Proof, Logic, and Conjecture - The Mathematician's Toolbox
It's really well written (I like it better than Velleman's How to Prove It.) After this you should go through something easier than Rudin, like Spivak Calculus. Then you can try a real analysis book, but try using Abbott or Pugh instead; I hear those books are much better than Rudin.
Your question is pretty vague because studying "mathematics" could mean a lot of things. And yes, your observation is correct: "There are a lot of Mathematical problems which are extremely difficult". In fact, that's true for a lot of people as well. So I suggest that you choose a certain field and delve into that.
For proof based subjects, the most basic to start with is Real Analysis. I recommend Stephen Abbott's Understanding Analysis as it is a pretty well-explained book.
This is just my perspective, but . . .
I think there are two separate concerns here: 1) the "process" of mathematics, or mathematical thinking; and 2) specific mathematical systems which are fundamental and help frame much of the world of mathematics.
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Abstract algebra is one of those specific mathematical systems, and is very important to understand in order to really understand things like analysis (e.g. the real numbers are a field), linear algebra (e.g. vector spaces), topology (e.g. the fundamental group), etc.
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I'd recommend these books, which are for the most part short and easy to read, on mathematical thinking:
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How to Solve It, Polya ( https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X ) covers basic strategies for problem solving in mathematics
Mathematics and Plausible Reasoning Vol 1 & 2, Polya ( https://www.amazon.com/Mathematics-Plausible-Reasoning-Induction-Analogy/dp/0691025096 ) does a great job of teaching you how to find/frame good mathematical conjectures that you can then attempt to prove or disprove.
Mathematical Proof, Chartrand ( https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094 ) does a good job of teaching how to prove mathematical conjectures.
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As for really understanding the foundations of modern mathematics, I would start with Concepts of Modern Mathematics by Ian Steward ( https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Books/dp/0486284247 ) . It will help conceptually relate the major branches of modern mathematics and build the motivation and intuition of the ideas behind these branches.
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Abstract algebra and analysis are very fundamental to mathematics. There are books on each that I found gave a good conceptual introduction as well as still provided rigor (sometimes at the expense of full coverage of the topics). They are:
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A Book of Abstract Algebra, Pinter ( https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178 )
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Understanding Analysis, Abbott ( https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 ).
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If you read through these books in the order listed here, it might provide you with that level of understanding of mathematics you talked about.
introduction to calculus and analysis (3 book set) by courant and john:
volume 1
volume 2 book 1
volume 2 book 2
You may be thinking of Courant's original Differential and Integral Calculus in two volumes. What I have is the solutions to problems from the updated and expanded to about twice the size, Introduction to Calculus and Analysis, in three volumes.
Terry Tao has two books on undergrad analysis. The first volume should be sufficient for your course. A lot of people use Rudin's Principles of Mathematical Analysis as well.
Analysis I and II by Terence Tao https://www.amazon.com/Analysis-Third-Texts-Readings-Mathematics/dp/9380250649
John Taylor did a great job at writing this book, I suggest it as a good read. It is still used in physics classes to this day: https://www.amazon.com/Introduction-Error-Analysis-Uncertainties-Measurements/dp/093570275X
It boils down to this: Uncertainty in percent is effectively an error.
So if your left leg does 200 Watts (As measured by some mythical leg powermeter device that is 100% accurate) you will get a measurement of +/- 2% from the Stages unit meaning the reading drifts 196 to 204 (left leg only, remember that). Now if you double that (as Stages does) you get a reading of 392-408. This gives you a variance of 4% (assuming left leg power = right leg power).
As for your other question: The claim of accuracy needs to be made about their measurement, not their calculated value. The calculated value (as you have pointed out) is based onan assumption of both legs putting out the same power. You can't account for that in marketing.
I think OkCupid data is all fair and good. It represents a certain set of people, those that are comfortable with doing online dating on one particular website. Now if we look at the whole population, including those that do not feel the need to use a dating site, we might see a different result.
If you take in account a proper bell curve you'll see that the population of men that are taller than women 5'10 or greater is actually about half the population. so that majority, if one at all, is a small majority. There are two separate bell curves: one for men and one for women. So is it really about a scarcity of suitors; if not, what could it be?
It could be something like men on a certain dating website are intimidated because of previous social interactions skewing the data or a smaller population of tall women on said site. All in all it is more often than not a person making the decision to rule out a specific group.
So, why don't we blame shallow people instead people with a certain physical trait?
Here's a nice book on errors
TL;DR: you don't have data, you have a graph. Shallow people are to blame.
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.
I prefer the third review of this classic math book!
http://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X
Why shouldn't everyone just begin with Baby Rudin?
https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X
Here you go OP.
> Calculus has a huge foundation in mathematical analysis that at most universities takes roughly half a year to a year of graduate/upper-undergrad study to develop (at least this is how it is at my university).
Graduate/upper undergrad? At Copenhagen University (KU) material corresponding roughly to Abbott's Understanding Analysis is covered in the first year. Plus some linear algebra and other stuff.
KU does have the advantage that it doesn't have to teach any engineers. They are all over at DTU in Lyngby learning to use maths to compute things leaving the mathematics department at KU to focus on teaching maths students to prove things.
> This is objectively untrue. How in the world did you come away with that conclusion?
You literally just google scholar'd "economics quantifying uncertainty" and assumed things would be there. There isn't, did you look? It's all about the psychological concept of uncertainty like "ohhh I don't know what the Fed does next" instead of measurement uncertainty. Do you know what that is? John Taylor wrote a good book on it.
> The statistic only applies to experimental economics
What? "Economics is only non-replicable when we actually try it out in real life." I'm actually starting to feel bad for stressing you out with this argument.
> I assume you wouldn't tell me that psychology is all fake. Then again, maybe you would.
Nearly all of psychology is fake.
> 5) Let's summarize. One study of 18 papers in experimental economics found that 60% were replicated, 40% were not, and you're ready to throw out the entire field?
Let me be very clear. Experimental economics is the shining star of economics. It is the upper bound. 1000x as credible as macro-econmics. And it still fails to replicate 40% of the time.
> Even studies in medicine often fail replication
Medicine has a bad problem with replication as well due to low sample sizes necessary due to high experiment cost and p-hacking due to pressure to publish.