Reddit mentions: The best mathematical analysis books
We found 510 Reddit comments discussing the best mathematical analysis books. We ran sentiment analysis on each of these comments to determine how redditors feel about different products. We found 148 products and ranked them based on the amount of positive reactions they received. Here are the top 20.
1. Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics)
- McGraw-Hill Science Engineering Math
Features:
Specs:
Height | 9.2 Inches |
Length | 6.4 Inches |
Number of items | 1 |
Weight | 1.39552611846 Pounds |
Width | 0.9 Inches |
2. Understanding Analysis (Undergraduate Texts in Mathematics)
- Springer
Features:
Specs:
Height | 9.21258 Inches |
Length | 6.14172 Inches |
Number of items | 1 |
Release date | August 2016 |
Weight | 14.72467447898 Pounds |
Width | 0.7499985 Inches |
3. Understanding Analysis (Undergraduate Texts in Mathematics)
- 18PC Total Solution Pyrex Glass Food
Features:
Specs:
Height | 9.21258 Inches |
Length | 6.14172 Inches |
Number of items | 1 |
Weight | 2.755778275 Pounds |
Width | 0.6874002 Inches |
4. Numerical Linear Algebra
Used Book in Good Condition
Specs:
Height | 10 Inches |
Length | 7 Inches |
Number of items | 1 |
Weight | 1.5 Pounds |
Width | 0.75 Inches |
5. How to Think About Analysis
Specs:
Height | 0.6 Inches |
Length | 7.6 Inches |
Number of items | 1 |
Release date | December 2014 |
Weight | 0.63493131456 Pounds |
Width | 5 Inches |
6. Introduction to Analysis (Dover Books on Mathematics)
- Heat preservation, heat insulation, EMI shielding
- Waterproof, cold and heat resistance, Ideal for outdoor application and durable
- Strong adhesion and easily peeled off
- UV resistance, flame retardant
- Moisture resistance, chemical and corrosion resistance
Features:
Specs:
Height | 8.5 Inches |
Length | 5.32 Inches |
Number of items | 1 |
Release date | February 1985 |
Weight | 0.65697754076 Pounds |
Width | 0.52 Inches |
7. Real and Complex Analysis (Higher Mathematics Series)
- McGraw-Hill Science Engineering Math
Features:
Specs:
Height | 9.5 Inches |
Length | 6.8 Inches |
Number of items | 1 |
Weight | 1.62921611618 Pounds |
Width | 0.8 Inches |
8. Elementary Analysis: The Theory of Calculus (Undergraduate Texts in Mathematics)
- Springer
Features:
Specs:
Height | 9.21258 Inches |
Length | 6.14172 Inches |
Number of items | 1 |
Weight | 16.68458398816 Pounds |
Width | 0.9373997 Inches |
9. Who Is Fourier?: A Mathematical Adventure
Used Book in Good Condition
Specs:
Height | 11.25 inches |
Length | 8.75 inches |
Number of items | 1 |
Weight | 2.3 Pounds |
Width | 1 inches |
10. Schaum's 3,000 Solved Problems in Calculus (Schaum's Outlines)
Specs:
Height | 10.8 Inches |
Length | 8.1 Inches |
Number of items | 1 |
Weight | 1.54984970186 Pounds |
Width | 0.73 Inches |
11. Introduction to Real Analysis
- Europe’s # 1 coffee container - Maintains aroma and rich, full coffee flavor
- Patented system creates a partial vacuum seal and keeps Coffee - Fresher for Longer
- Multiuse - Allows products to breathe and degas, without letting oxygen in
- NSF Certified, Food Grade - Airtight, Moisture free and Recyclable
- Coffee grounds or beans – 16 oz. / 500g / 1. 85 liter / Dimensions: 7-3/4" tall x 4-7/8" diameter (19. 5 cm x 12 cm)
Features:
Specs:
Height | 10.098405 Inches |
Length | 7.098411 Inches |
Number of items | 1 |
Weight | 1.72401488884 Pounds |
Width | 1.098423 Inches |
12. Advanced Calculus: A Differential Forms Approach
- Used Book in Good Condition
Features:
Specs:
Height | 9.99998 Inches |
Length | 7.00786 Inches |
Number of items | 1 |
Weight | 2.48240507012 Pounds |
Width | 1.0633837 Inches |
13. Fourier Analysis: An Introduction (Princeton Lectures in Analysis)
Specs:
Height | 9.64 Inches |
Length | 6.34 Inches |
Number of items | 1 |
Release date | April 2003 |
Weight | 1.3117504589 Pounds |
Width | 1 Inches |
14. Real Mathematical Analysis (Undergraduate Texts in Mathematics)
- Shutter mechanism inside the receptacle blocks access to the contacts unless a 2-prong plug is inserted, helping ensure hairpins, keys, etc., will be locked out
- TR symbol on residential receptacles assures they meet the 2008 NEC requirement
- Ultrasonic heavy-duty construction offers long, trouble-free service life
- Heavy-gauge, rust-resistant steel mounting strap. Quickwire push-in and side wiring for easy installation
- Shallow design for maximum wiring room. Terminal screws accept up to No.12 AWG copper or copper-clad wire and Quickwire push-in terminals accept No.14 AWG solid copper wire only
- Wallplate sold separately
Features:
Specs:
Height | 9.21258 Inches |
Length | 6.14172 Inches |
Number of items | 1 |
Weight | 1.80338130316 Pounds |
Width | 0.999998 Inches |
15. A Mathematician's Apology (Canto)
- Used Book in Good Condition
Features:
Specs:
Height | 8.5 Inches |
Length | 5.5 Inches |
Number of items | 1 |
Weight | 0.4299014109 Pounds |
Width | 0.5 Inches |
16. Discovering Group Theory (Textbooks in Mathematics)
Specs:
Height | 9.25 Inches |
Length | 6.13 Inches |
Number of items | 1 |
Release date | December 2016 |
Weight | 0.7495716908 Pounds |
Width | 0.53 Inches |
17. The Elements of Integration and Lebesgue Measure
Specs:
Height | 9.09447 Inches |
Length | 6.003925 Inches |
Number of items | 1 |
Weight | 0.64374980504 Pounds |
Width | 0.401574 Inches |
18. Complex Analysis (Princeton Lectures in Analysis, No. 2)
Princeton University Press
Specs:
Height | 9.36 Inches |
Length | 6.5 Inches |
Number of items | 1 |
Release date | April 2003 |
Weight | 1.56307743758 Pounds |
Width | 1.18 Inches |
19. A Primer of Infinitesimal Analysis
Used Book in Good Condition
Specs:
Height | 9 Inches |
Length | 6 Inches |
Number of items | 1 |
Weight | 0.7275254646 Pounds |
Width | 0.38 Inches |
20. Numerical Analysis
Used Book in Good Condition
Specs:
Height | 10.25 Inches |
Length | 8.25 Inches |
Number of items | 1 |
Weight | 3.7699046802 Pounds |
Width | 1.25 Inches |
🎓 Reddit experts on mathematical analysis books
The comments and opinions expressed on this page are written exclusively by redditors. To provide you with the most relevant data, we sourced opinions from the most knowledgeable Reddit users based the total number of upvotes and downvotes received across comments on subreddits where mathematical analysis books are discussed. For your reference and for the sake of transparency, here are the specialists whose opinions mattered the most in our ranking.
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 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.
math is a funny thing... our culture gets so hung up on 'good at' and 'bad at', but the more I get into neurobiology and ML, the more amazing our general learning abilities seems to be. My partner and I are radically different, she's better at chess than I am in spite of having a poor ability with 'traditional' chess thinking, she relies almost entirely on pattern recognition, so she has to stand over the board looking down so her brain can feed up ideas from the books she's read (since chess layouts are always shown in those books from the top down).
All this is to say... there's a goddamn giant mountain in front of you, and it's easy to think that you're 'bad' at it because of where you're starting, or even because of base talents and interests that might not seem to line up with math at first glance. Just wanted to start out by saying that's horse shit. You're also 'bad' at judo and chinese (presumably), but given a few years of regular practice, you could get those reasonably under your belt as well. Math is a way of thinking and looking at problems, and it's incredibly helpful. It's kind of mind blowing the doors it can open... information theory, statistics, linear algebra, calculus, game theory, graph theory, group theory, representation theory, category theory... every branch opens up mind blowing new insights, tools, and models for looking at new problems. Don't look at it like this 'thing' you have to learn though. You can't learn all of math. You can just slowly learn new tools, get better at understanding what it even 'means' to learn one of those new fields, and how to organize your study to make real progress as you're slowly getting deeper.
So... my recommendation for where to start? Start with the meta learning. What is math? How can you learn it? How should you study? The best glimpse into those questions I've found is how to think about analysis. It takes a complete beginner's perspective (explaining how to read the standard math notation even... the summation symbol, epsilon, etc) slowly builds up an introduction to the guts of what calculus is, basically. You can read it in a week or two, so it's not a huge time investment, and it'll do a lot I think to arm you for the road ahead.
I'm personally a fan of bottom up learning as much as possible, but that's just because I hat trying to play with half a deck. There's plenty of people though that just treat pieces they're working with like 'black boxes'. You can use a decision tree without any fucking clue about information theory, or even what the decision surface actually looks like for the resulting tree. Finding good visualizations when wrapping your head around that stuff can be really helpful... so if you're struggling with one resource, don't be afraid to look for another. Sometimes a git article with some good graphs can make all the difference.
I don't know what road is best for you, but the only barrier in front of you is your patience, and your willingness to spend time every single week, and turn this into a practice instead of just a hobby. I started a year and a half ago after ten years in an unrelated industry, and while I still have a long way to go, I've also covered a ton of ground too. I'd never even had stats before at all, even in high school... now I'm comfortably following some pretty gnarly multivariate derivations in Bishop's pattern recognition and machine learning. You just keep putting one foot in front of the other, pay attention to your goal, follow your curiosity, and before you know it... people start looking at you funny, because you know things most people don't know, and you can build things most people don't even understand. I can't imagine a more exciting thing to be learning, especially at this time in history. If you have the patience and interest for it, whichever road you take I think you'll find it well worth your time.
My own personal suggestion by the way... take a little time for fundamentals on the regular (starting with linear algebra, a proper textbook with a lot of exercises if possible) and practical (actually implementing stuff, doing Kaggle competitions, whatever). Eventually in the distant future, you'll meet in the middle, and find you have the insight to start pursuing your own questions... possibly even questions no one has ever solved before, and you'll have an enormous amount of practical good to bring to whatever field you've been working in, if you choose to continue there. Good luck!
Machine learning is largely based on the following chain of mathematical topics
Calculus (through Vector, could perhaps leave out a subsequent integration techniques course)
Linear Algebra (You are going to be using this all, a lot)
Abstract Algebra (This isn't always directly applicable but it is good to know for computer science and the terms of groups, rings, algebras etc will show up quite a bit)
General Topology (Any time we are going to deal with construction of a probability space on some non trivial manifold, we will need this. While most situations are based on just Borel sets in R^n or C^n things like computer vision, genomics, etc are going to care about Random Elements rather than Random Variables and those are constructed in topological spaces rather than metric ones. This is also helpful for understanding definitions in well known algorithms like Manifold Training)
Real Analysis (This is where you learn proper constructive formulations and a bit of measure theory as well as bounding theorems etc)
Complex Analysis (This is where you will get a proper treatment of Hilbert Spaces, Holomorphic functions etc, honestly unless you care about QM / QFT, P-chem stuff in general like molecular dynamics, you are likely not going to need a full course in this for most ML work, but I typically just tell people to read the full Rudin: Real and Complex Analysis. You'll get the full treatment fairly briefly that way)
Probability Theory (Now that you have your Measure theory out of the way from Real Analysis, you can take up a proper course on Measure Theoretic Probability Theory. Random Variables should be defined here as measurable functions etc, if they aren't then your book isn't rigorous enough imho.)
Ah, Statistics. Statistics sits atop all of that foundational mathematics, it is divided into two main philosophical camps. The Frequentists, and the Bayesians. Any self respecting statistician learns both.
After that, there are lots, and lots, and lots, of subfields and disciplines when it comes to statistical learning.
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.
hey nerdinthearena,
i too find this area to be fascinating and wish i knew more on the upper end myself. i'm just going to list off a few resources. in my opinion, graduate school will concentrate a lot on progressing your technical knowledge, but will likely not give you a lot of time to hone your intuition (at least in the first few years). so, the more time you spend in undergraduate school doing so, the better.
helpful for intuition and basic understanding
more advanced but still intuitive
hopefully this helps. if i were to revisit geometric analysis, i would basically use the above books to help bone up my understanding, intuition, and technical skill before moving on. these are also mainly geometry books, so learning analysis (like functional analysis) would be good as well. i mainly have three suggestions there.
three general analysis favorites
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.
> 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.
/u/another_user_name posted this list a while back. Actual aerospace textbooks are towards the bottom but you'll need a working knowledge of the prereqs first.
Non-core/Pre-reqs:
Mathematics:
Calculus.
1-4) Calculus, Stewart -- This is a very common book and I felt it was ok, but there's mixed opinions about it. Try to get a cheap, used copy.
1-4) Calculus, A New Horizon, Anton -- This is highly valued by many people, but I haven't read it.
1-4) Essential Calculus With Applications, Silverman -- Dover book.
More discussion in this reddit thread.
Linear Algebra
3) Linear Algebra and Its Applications,Lay -- I had this one in school. I think it was decent.
3) Linear Algebra, Shilov -- Dover book.
Differential Equations
4) An Introduction to Ordinary Differential Equations, Coddington -- Dover book, highly reviewed on Amazon.
G) Partial Differential Equations, Evans
G) Partial Differential Equations For Scientists and Engineers, Farlow
More discussion here.
Numerical Analysis
5) Numerical Analysis, Burden and Faires
Chemistry:
Physics:
2-4) Physics, Cutnel -- This was highly recommended, but I've not read it.
Programming:
Introductory Programming
Programming is becoming unavoidable as an engineering skill. I think Python is a strong introductory language that's got a lot of uses in industry.
Core Curriculum:
Introduction:
Aerodynamics:
Thermodynamics, Heat transfer and Propulsion:
Flight Mechanics, Stability and Control
5+) Flight Stability and Automatic Control, Nelson
5+)[Performance, Stability, Dynamics, and Control of Airplanes, Second Edition](http://www.amazon.com/Performance-Stability-Dynamics-Airplanes-Education/dp/1563475839/ref=sr_1_1?ie=UTF8&qid=1315534435&sr=8-1, Pamadi) -- I gather this is better than Nelson
Engineering Mechanics and Structures:
3-4) Engineering Mechanics: Statics and Dynamics, Hibbeler
6-8) Analysis and Design of Flight Vehicle Structures, Bruhn -- A good reference, never really used it as a text.
G) Introduction to the Mechanics of a Continuous Medium, Malvern
G) Fracture Mechanics, Anderson
G) Mechanics of Composite Materials, Jones
Electrical Engineering
Design and Optimization
Space Systems
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.
​
< 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!
>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.
---
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.
---
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!
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 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.
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.
&#x200B;
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.
&#x200B;
I'd recommend these books, which are for the most part short and easy to read, on mathematical thinking:
&#x200B;
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.
&#x200B;
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.
&#x200B;
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:
&#x200B;
A Book of Abstract Algebra, Pinter ( https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178 )
&#x200B;
Understanding Analysis, Abbott ( https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 ).
&#x200B;
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.
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.
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.
&#x200B;
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.)
&#x200B;
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.
Generally the algorithmic approach is done purposely. The majority of students first encountering calculus simply don't care about the why, instead they want to know how. It seems you are not the average student, and this is great, but I heavily recommend you get comfortable with the algorithmic side. You should be very comfortable with all the integration techniques and have a decent understanding of infinite series by the time you leave calc 2.
As far as a book recommendation goes, I used Thomas' Calculus for Calculus 1,2, and 3 at my school. This book does have the algorithm approach rather than the conceptual approach, but it was definitely a nice read (to me). You can probably find a free copy somewhere online in pdf form to let you see what the book is like.
If you are majoring in Mathematics you will certainly want to be as comfortable with this material as possible. Jumping right into a more theoretical book for calc 2 will also likely be very difficult. Generally a first Analysis course (rigorous calculus) will cover calc 1, and 2 as well as some other topics.
The treatment of rigorous calculus will have you proving general things about integrals and series' rather than simply evaluating things. If you want to see what the future holds for you (and maybe these will be helpful for you) check out
Long story short, Thomas' is a good easy read with a more algorithmic approach, the other two are very rigorous, and so they will be much more challenging and the Calc 2 stuff will be built up from the rigorous exposition of the calc 1 stuff.
The Book
The Review of The Book:
~
~OK... Deep breaths everybody...
It is not possible to overstate how good this book is. I tried to give it uncountably many stars but they only have five. Five is an insult. I'm sorry Dr. Rudin...
This book is a good reference but let me tell you what its really good for. You have taken all the lower division courses. You have taken that "transition to proof writing" class in number theory, or linear algebra, or logic, or discrete math, or whatever they do at your institution of higher learning. You can tell a contrapositive from a proof by contradiction. You can explain to your grandma why there are more real numbers than rationals. Now its time to get serious.
Get this book. Start at page one. Read until you come to the word Theorem. Do not read the proof. Prove it yourself. Or at least try. If you get stuck read a line or two until you see what to do.
Thrust, repeat.
If you make it through the first six or seven chaptors like this then there shall be no power in the verse that can stop you. Enjoy graduate school. You half way there.
Now some people complain about this book being too hard. Don't listen to them. They are just trying to pull you down and keep you from your true destiny. They are the same people who try to sell you TV's and lobodemies.
"The material is not motivated." Not motivated? Judas just stick a dagger in my heart. This material needs no motivation. Just do it. Faith will come. He's teaching you analysis. Not selling you a used car. By the time you are ready to read this book you should not need motivation from the author as to why you need to know analysis. You should just feel a burning in you chest that can only be quenched by arguments involving an arbitrary sequence {x_n} that converges to x in X.
Finally, some people complain about the level of abstraction, which let me just say is not that high. If you want to see abstraction grab a copy of Spanier's 'Algebraic Topology' and stare at it for about an hour. Then open 'Baby Rudin' up again. I promise you the feeling you get when you sit in a hottub for like twenty minutes and then jump back in the pool. Invigorating.
No but really. Anyone who passes you an analysis book that does not say the words metric space, and have the chaptor on topology before the chapter on limits is doing you no favors. You need to know what compactness is when you get out of an analysis course. And it's lunacy to start talking about differentiation without it. It's possible, sure, but it's a waste of time and energy. To say a continuous function is one where the inverse image of open sets is open is way cooler than that epsilon delta stuff. Then you prove the epsilon delta thing as a theorem. Hows that for motivation?
Anyway, if this review comes off a combative that's because it is. It's unethical to use another text for an undergraduate real analysis class. It insults and short changes the students. Sure it was OK before Rudin wrote the thing, but now? Why spit on your luck? And if you'r a student and find the book too hard? Try harder. That's the point. If you did not crave intellectual work why are you sitting in an analysis course? Dig in. It will make you a better person. Trust me.
Or you could just change your major back to engineering. It's more money and the books always have lots of nice pictures.
In conclusion: Thank you Dr. Rudin for your wonderfull book on analysis. You made a man of me.
There are essentially "two types" of math: that for mathematicians and everyone else. When you see the sequence Calculus(1, 2, 3) -> Linear Algebra -> DiffEq (in that order) thrown around, you can be sure they are talking about non-rigorous, non-proof based kind that's good for nothing, imo of course. Calculus in this sequence is Analysis with all its important bits chopped off, so that everyone not into math can get that outta way quick and concentrate on where their passion lies. The same goes for Linear Algebra. LA in the sequence above is absolutely butchered so that non-math majors can pass and move on. Besides, you don't take LA or Calculus or other math subjects just once as a math major and move on: you take a rigorous/proof-based intro as an undergrad, then more advanced kind as a grad student etc.
To illustrate my point:
Linear Algebra:
Linear Algebra Through Geometry by Banchoff and Wermer
3. Here's more rigorous/abstract Linear Algebra for undergrads:
Linear Algebra Done Right by Axler
4. Here's more advanced grad level Linear Algebra:
Advanced Linear Algebra by Steven Roman
-----------------------------------------------------------
Calculus:
Calulus by Spivak
3. Full-blown undergrad level Analysis(proof-based):
Analysis by Rudin
4. More advanced Calculus for advance undergrads and grad students:
Advanced Calculus by Sternberg and Loomis
The same holds true for just about any subject in math. Btw, I am not saying you should study these books. The point and truth is you can start learning math right now, right this moment instead of reading lame and useless books designed to extract money out of students. Besides, there are so many more math subjects that are so much more interesting than the tired old Calculus: combinatorics, number theory, probability etc. Each of those have intros you can get started with right this moment.
Here's how you start studying real math NOW:
Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers. Essentially, this book is about the language that you need to be able to understand mathematicians, read and write proofs. It's not terribly comprehensive, but the amount of info it packs beats the usual first two years of math undergrad 1000x over. Books like this should be taught in high school. For alternatives, look into
Discrete Math by Susanna Epp
How To prove It by Velleman
Intro To Category Theory by Lawvere and Schnauel
There are TONS great, quality books out there, you just need to get yourself a liitle familiar with what real math looks like, so that you can explore further on your own instead of reading garbage and never getting even one step closer to mathematics.
If you want to consolidate your knowledge you get from books like those of Rodgers and Velleman and take it many, many steps further:
Basic Language of Math by Schaffer. It's a much more advanced book than those listed above, but contains all the basic tools of math you'll need.
I'd like to say soooooooooo much more, but I am sue you're bored by now, so I'll stop here.
Good Luck, buddyroo.
I 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.
Not much, the nice thing for upper math courses is they do a good job of building up from bare bones. If you have some linear algebra and a multivariable calc course you should be good. The big requirement is however mathematical maturity. You should be able to read, understand, and write proof.
A very basic intro to proofs course is usually taught to first year math students, this covers set notations, logic, and some basic proof techniques. A common reference is "How to prove it: a structured approach", I learned from Intro to mathematical thinking. The latter isn't as liked, it does seem to cover some material that I think should be taught early. A lot of classical number theory and algebra, for example fundamental theorem of arithmetic, and Fermat's little (not last) theorem are proven. Try to find a reference for that stuff if you can.
It's really important to do a proof based linear algebra class. It helps build the maturity I mentioned and will make life easier with topology. But even more importantly teaching linear algebra in a more abstract way is important for a physics undergrad as it can serve as a foundation for functional analysis, the theory upon which quantum mechanics is built. And in general it is good to stop thinking of vectors as arrows in R^n as soon as possible. A great reference is Axler's LADR.
Again not strictly required, but it helps build maturity and it serves as a good motivation for many of the concepts introduced in a topology class. You will see the practical side of compact sets (namely they are closed and bounded sets in R^(n)), and prove that using the abstract definition (which is the preferred one in topology). You will also prove some facts about continuous functions which will motivate the definition of continuity used in topology, and generally seeing proofs about open sets will show you why open sets are important and why you may wish to look at spaces described only by their open sets (as you will in topology). The reference for real analysis is typically Rudin, but that can be a little tough (I'm sorry, I can't remember the easier book at the moment)
Edit: I will remove this as it doesn't meet the requirements for an /r/askscience question, we usually answer questions about the science rather than learning references. If you feel my answer wasn't comprehensive enough feel free to ask on /r/math or /r/learnmath
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.
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).
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).
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!
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.
Like justrasputin says, there usually is quite a lot of work to be done before you start to really see the beauty everyone refers to. I'd like to suggest a few book about mathematics, written by mathematicians that explicitly try to capture the beauty -
By Marcus Du Sautoy (A group theorist at oxford)
By G.H. Hardy,
Also, a good collection of seminal works -
God Created the Integers
And a nice starter -
What is Mathematics
Good luck and don't give up!
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.
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 heard good things about it, but honestly as an applied mathematician I found its table of contents too lackluster. Its coverage appears to be in a weird spot between "for physicists" and "for mathematicians" and I don't know who its target audience is. I think the standard recommendation for classical mechanics from the physics side is Goldstein, which is a perfectly good book with plenty of math!
For an actual mathematicians' take on classical mechanics, you'll have to wait until you take more advanced math, namely real analysis and differential geometry. Common references are Spivak and Tu. When you have that background, I think Arnold has the best mathematical treatment of classical mechanics.
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.
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.
&#x200B;
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.
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!
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.
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!
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.
>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
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.
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.
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.
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.
Yes, they're awesome. Brought up pretty frequently on /r/math, too. I'm pretty sure I have at least 10 Dover books. Two excellent titles that come to mind are Pinter's A Book of Abstract Algebra and Rosenlicht's Introduction to Analysis.
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).
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.
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!
An Introduction to Manifolds by Tu is a very approachable book that will get you up to Stokes. Might as well get the full version of Stokes on manifolds not just in analysis. From here you can go on to books by Ramanan, Michor, or Sharpe.
A Guide to Distribution Theory and Fourier Transforms by Strichartz was my introduction to Fourier analysis in undergrad. Probably helps to have some prior Fourier experience in a complex analysis or PDE course.
Bartle's Elements of Integration and Legesgue Measure is great for measure theory. Pretty short too.
Intro to Functional Analysis by Kreysig is an amazing introduction to functional analysis. Don't know why you'd learn it from any other book. Afterwards you can go on to functional books by Brezis, Lax, or Helemskii.
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'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?
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.
Here are some great books that I believe you may find helpful :)
and last but definitely not least:
Later on:
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.
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.
If you are interested enough in machine learning that you are going to work through ESL, you may benefit from reading up on some math first. For example:
Without developing some mathematical maturity, some of ESL may be lost on you. Good luck!
This 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?
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 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.
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").
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.
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.
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.
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
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.
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.
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"
If you want to do more math in the same flavor as Apostol, you could move up to analysis with Tao's book or Rudin. Topology's slightly similar and you could use Munkres, the classic book for the subject. There's also abstract algebra, which is not at all like analysis. For that, Dummit and Foote is the standard. Pinter's book is a more gentle alternative. I can't really recommend more books since I'm not that far into math myself, but the Chicago math bibliography is a good resource for finding math books.
Edit: I should also mention Evan Chen's Infinite Napkin. It's a very condensed, free book that includes a lot of the topics I've mentioned above.
If 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.
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.
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:
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.
Definitely check out Chebfun for interpolation, quadrature, root finding, solving ODEs and PDEs, convolutions, etc. of smooth functions. The team there at Oxford has done a big pile of great work sifting through past ideas, inventing a bunch of new ideas, and implementing everything concretely in a friendly-to-use software library.
If you want to learn numerical linear algebra, check out Trefethen & Bau’s 1997 book.
If you’re looking for a general numerical methods book, this one by Corless/Fillion seems decent.
Please, simply disregard everything below if the info is old news to you.
------------
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
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.
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.
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!
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.
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.
Numerical Linear Algebra
by 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 Analysis
is 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
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.
> 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.
I used Intro to Real Analysis by Bartle and Sherbert for my first analysis course. I just checked, and it has quite a few examples for most sections. The book was sufficient without lecture, as far as learning proofs and applying theorem goes. It was also relatively easy to read; there are a lot of analysis books which are hard to read due to terseness. You can probably find it for cheaper too. Good luck.
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
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.
I also have a lot of love for the Bartle real analysis text. It's light on prerequisites other than calculus and the text spends time explaining the logical reasoning. A lot of upper-level math texts are simply a collection of theorems. The good ones present a coherent narrative that give context as each theorem is presented.
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).
It depends on where they are and what the purpose is. If you are trying to discourage them (and there might be valid reasons to do that), I'd say try measure theory.
Maybe use the Bartle book.
That would give them a taste for how abstract things can get and also drive home the point tiny books can require a lot o work.
On the other hand, if you want to do something that will help them, they An Introduction to Mathematical Reasoning.
It won't break the bank and, despite a few small typos, covers a lot material fairly gently.
http://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/0387950605
You can thank me later- it's really good. Also, a full solutions manual can be found with some googlefu.
Thanks for sharing I'll look into that one! Thanks:)
&#x200B;
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!
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. :-)
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.
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.
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.
It mentions Rosenlicht at the bottom. Lucky you, that book's only 8 bucks! It's a good book, too.
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.
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.
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.
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.
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
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.
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
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
Who Is Fourier?: A Mathematical Adventure is a great book, I suggest you look into it.
Best:
Principles of Mathematical Analysis by Walter Rudin
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.
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.
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.
For real analysis, I would avoid Rudin. I think it's overrated as a good book to learn from, especially for people who aren't math majors. I'd go with Introduction to Analysis by Rosenlicht. It's basically a friendlier version of Rudin, and a heck of a lot cheaper.
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
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.
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 a friend who swears by this book
https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0198723539
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"
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
> 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.
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
I thought Elementary Analysis by Kenneth Ross was pretty accessible. As others have said, though, your goal seems somewhat unrealistic.
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.
We used this one in my undergraduate analysis class, and I found it pretty straightforward to read and understand. And it's only $13.
Thanks! I assume it's this text, right?
In addition to Baby Rudin, I really liked this book when I first start learning analysis.
I've found Rudin's Analysis useful. There's a lecture series on YouTube that roughly follows the book.
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.
This is a good option: http://www.amazon.com/Complex-Analysis-Princeton-Lectures/dp/0691113858
I used Burden and Faires for three courses in numerical methods. I really enjoyed the book and it comes with free code online in a variety of languages. It is a little pricey but if you search hard enough (cough cough, first link) you will find it.
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
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.
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.
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.
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
...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
For the physics perspective you need something like A Primer of Infinitesimal 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.
At what level? I'm really enjoying Stein & Shakarchi but that's closer to graduate level.
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
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.
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).