(Part 2) Best calculus books according to redditors
We found 592 Reddit comments discussing the best calculus books. We ranked the 205 resulting products by number of redditors who mentioned them. Here are the products ranked 21-40. You can also go back to the previous section.
Maybe this will help you understand and feel better.
I would guess that career prospects are a little worse than CS for undergrad degrees, but since my main concern is where a phd in math will take me, you should get a second opinion on that.
Something to keep in mind is that "higher" math (the kind most students start to see around junior level) is in many ways very different from the stuff before. I hated calculus and doing calculations in general, and was pursuing a math minor because I thought it might help with job prospects, but when I got to the more abstract stuff, I loved it. It's easily possible that you'll enjoy both, I'm just pointing out that enjoying one doesn't necessarily imply enjoying the other. It's also worth noting that making the transition is not easy for most of us, and that if you struggle a lot when you first have to focus a lot of time on proving things, it shouldn't be taken as a signal to give up if you enjoy the material.
This wouldn't be necessary, but if you like, here are some books on abstract math topics that are aimed towards beginners you could look into to get a basic idea of what more abstract math is like:
Different mathematicians gravitate towards different subjects, so it's not easy to predict which you would enjoy more. I'm recommending these five because they were personally helpful to me a few years ago and I've read them in full, not because I don't think anyone can suggest better. And of course, you could just jump right into coursework like how most of us start. Best of luck!
(edit: can't count and thought five was four)
i have three categories of suggestions.
advanced calculus
these are essentially precursors to smooth manifold theory. you mention you have had calculus 3, but this is likely the modern multivariate calculus course.
out of these, if you were to choose one, i think the callahan book is probably your best bet to pull from. it is the most modern, in both approach and notation. it is a perfect setup for smooth manifolds (however, all of these books fit that bill). hubbard's book is very similar, but i don't particularly like its notation. however, it has some unique features and does attempt to unify the concepts, which is a nice approach. edwards book is just fantastic, albeit a bit nonstandard. at a minimum, i recommend reading the first three chapters and then the latter chapters and appendices, in particular chapter 8 on applications. the first three chapters cover the core material, where chapters 4-6 then go on to solidify the concepts presented in the first three chapters a bit more rigorously.
smooth manifolds
out of these books, i only have explicit experience with the first two. i learned the material in graduate school from john m. lee's book, which i later solidifed by reading tu's book. tu's book actually covers the same core material as lee's book, but what makes it more approachable is that it doesn't emphasize, and thus doesn't require a lot of background in, the topological aspects of manifolds. it also does a better job of showing examples and techniques, and is better written in general than john m. lee's book. although, john m. lee's book is rather good.
so out of these, i would no doubt choose tu's book. i mention the latter two only to mention them because i know about them. i don't have any experience with them.
conceptual books
these books should be helpful as side notes to this material.
i highly recommend all of these because they're all rather short and easy reads. the first two get at the visual concepts and intuition behind vectors, covectors, etc. they are actually the only two out of all of these books (if i remember right) that even talk about and mention twisted forms.
there are also a ton of books for physicists, applied differential geometry by william burke, gauge fields, knots and gravity by john baez and javier muniain (despite its title, it's very approachable), variational principles of mechanics by cornelius lanczos, etc. that would all help with understanding the intuition and applications of this material.
conclusion
if you're really wanting to get right to the smooth manifolds material, i would start with tu's book and then supplement as needed from the callahan and hubbard books to pick up things like the implicit and inverse function theorems. i highly recommend reading edwards' book regardless. if you're long-gaming it, then i'd probably start with callahan's book, then move to tu's book, all the while reading edwards' book. :)
i have been out of graduate school for a few years now, leaving before finishing my ph.d. i am actually going back through callahan's book (didn't know about it at the time and/or it wasn't released) for fun and its solid expositions and approach. edwards' book remains one of my favorite books (not just math) to just pick up and read.
This one drove me into insanity
I'm not going to say I'm great at math or anything, but I can say my power to actually proving things, solving many types of mathematical problems more effectively comes from reading this single fairly short book:
Elements of the Theory of Functions and Functional Analysis by Kolmogorov (the guy that formalized probability!)
It's difficult (especially if you try to solve everything he proposes -- which by the way is fairly essential to get the full experience), but I feel it really made me more mature (and some of the tools presented are actually quite useful in research).
It really shows how important proof is. Needless to say, thoroughly recommended, probably my favorite book ever.
(I heard it uses some uncommon terms due to Russian translation but I didn't have any problems e.g. discussing pure math at stackexchange)
If you've ever wondered "How the hell can someone come up with, and prove formally something such as the Universal approximation theorem", read this book.
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Other than that, I'd say one does not simply know enough linear algebra. Or in general one does not know enough about linear systems (including Markov chains), if you work with any kind of time-varying problems at all.
Online: http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx
If you can spare 14 bucks: http://www.amazon.com/Calculus-Early-Transcendentals-5th-included/dp/9812548831/ref=sr_1_8?ie=UTF8&s=books&qid=1250926453&sr=8-8
If not, check out your library. A calc textbook is going to be necessary to understand basic physics concepts.
For basic logic (first-order, classical) these are excellent textbooks...
I should also mention that there are some free, open source textbooks on standard logic (good books, written by scholars in the field) such as the following...
And if you want to go beyond classical logic, these are excellent textbooks...
> I think I need to read up on dealing with infinite sets.
Your confusion was that you didn't distinguish between an existential quantifier and a universal quantifier. I don't think it's infinite sets themselves that are your issue. You might be better served by reading up on basic logic and proof techniques before reading up on basic set theory. This text is the one I used in my freshman introduction to proofs class and I thought it was pretty good.
I'd like to give you my two cents as well on how to proceed here. If nothing else, this will be a second opinion. If I could redo my physics education, this is how I'd want it done.
If you are truly wanting to learn these fields in depth I cannot stress how important it is to actually work problems out of these books, not just read them. There is a certain understanding that comes from struggling with problems that you just can't get by reading the material. On that note, I would recommend getting the Schaum's outline to whatever subject you are studying if you can find one. They are great books with hundreds of solved problems and sample problems for you to try with the answers in the back. When you get to the point you can't find Schaums anymore, I would recommend getting as many solutions manuals as possible. The problems will get very tough, and it's nice to verify that you did the problem correctly or are on the right track, or even just look over solutions to problems you decide not to try.
Basics
I second Stewart's Calculus cover to cover (except the final chapter on differential equations) and Halliday, Resnick and Walker's Fundamentals of Physics. Not all sections from HRW are necessary, but be sure you have the fundamentals of mechanics, electromagnetism, optics, and thermal physics down at the level of HRW.
Once you're done with this move on to studying differential equations. Many physics theorems are stated in terms of differential equations so really getting the hang of these is key to moving on. Differential equations are often taught as two separate classes, one covering ordinary differential equations and one covering partial differential equations. In my opinion, a good introductory textbook to ODEs is one by Morris Tenenbaum and Harry Pollard. That said, there is another book by V. I. Arnold that I would recommend you get as well. The Arnold book may be a bit more mathematical than you are looking for, but it was written as an introductory text to ODEs and you will have a deeper understanding of ODEs after reading it than your typical introductory textbook. This deeper understanding will be useful if you delve into the nitty-gritty parts of classical mechanics. For partial differential equations I recommend the book by Haberman. It will give you a good understanding of different methods you can use to solve PDEs, and is very much geared towards problem-solving.
From there, I would get a decent book on Linear Algebra. I used the one by Leon. I can't guarantee that it's the best book out there, but I think it will get the job done.
This should cover most of the mathematical training you need to move onto the intermediate level physics textbooks. There will be some things that are missing, but those are usually covered explicitly in the intermediate texts that use them (i.e. the Delta function). Still, if you're looking for a good mathematical reference, my recommendation is Lua. It may be a good idea to go over some basic complex analysis from this book, though it is not necessary to move on.
Intermediate
At this stage you need to do intermediate level classical mechanics, electromagnetism, quantum mechanics, and thermal physics at the very least. For electromagnetism, Griffiths hands down. In my opinion, the best pedagogical book for intermediate classical mechanics is Fowles and Cassidy. Once you've read these two books you will have a much deeper understanding of the stuff you learned in HRW. When you're going through the mechanics book pay particular attention to generalized coordinates and Lagrangians. Those become pretty central later on. There is also a very old book by Robert Becker that I think is great. It's problems are tough, and it goes into concepts that aren't typically covered much in depth in other intermediate mechanics books such as statics. I don't think you'll find a torrent for this, but it is 5 bucks on Amazon. That said, I don't think Becker is necessary. For quantum, I cannot recommend Zettili highly enough. Get this book. Tons of worked out examples. In my opinion, Zettili is the best quantum book out there at this level. Finally for thermal physics I would use Mandl. This book is merely sufficient, but I don't know of a book that I liked better.
This is the bare minimum. However, if you find a particular subject interesting, delve into it at this point. If you want to learn Solid State physics there's Kittel. Want to do more Optics? How about Hecht. General relativity? Even that should be accessible with Schutz. Play around here before moving on. A lot of very fascinating things should be accessible to you, at least to a degree, at this point.
Advanced
Before moving on to physics, it is once again time to take up the mathematics. Pick up Arfken and Weber. It covers a great many topics. However, at times it is not the best pedagogical book so you may need some supplemental material on whatever it is you are studying. I would at least read the sections on coordinate transformations, vector analysis, tensors, complex analysis, Green's functions, and the various special functions. Some of this may be a bit of a review, but there are some things Arfken and Weber go into that I didn't see during my undergraduate education even with the topics that I was reviewing. Hell, it may be a good idea to go through the differential equations material in there as well. Again, you may need some supplemental material while doing this. For special functions, a great little book to go along with this is Lebedev.
Beyond this, I think every physicist at the bare minimum needs to take graduate level quantum mechanics, classical mechanics, electromagnetism, and statistical mechanics. For quantum, I recommend Cohen-Tannoudji. This is a great book. It's easy to understand, has many supplemental sections to help further your understanding, is pretty comprehensive, and has more worked examples than a vast majority of graduate text-books. That said, the problems in this book are LONG. Not horrendously hard, mind you, but they do take a long time.
Unfortunately, Cohen-Tannoudji is the only great graduate-level text I can think of. The textbooks in other subjects just don't measure up in my opinion. When you take Classical mechanics I would get Goldstein as a reference but a better book in my opinion is Jose/Saletan as it takes a geometrical approach to the subject from the very beginning. At some point I also think it's worth going through Arnold's treatise on Classical. It's very mathematical and very difficult, but I think once you make it through you will have as deep an understanding as you could hope for in the subject.
The best way to learn math is to read math text books.
I think this book prepared me well to explore math on my own:
http://www.amazon.com/gp/offer-listing/0534356389/ref=lp_g_1/103-3484664-0761418
Of course, it won't teach you any vector calculus. It's expensive, but supposedly, you can buy it used for 10 cents. (!?)
Finding a good maths book is harder than I thought. My favorite is a classic, Hamming's 'Methods of Mathematics Applied to Calculus, Probability, and Statistics'
https://www.amazon.com/gp/aw/d/0486439453/ref=mp_s_a_1_1?ie=UTF8&qid=1499896403&sr=8-1&pi=SL75_QL70&keywords=hamming+mathematics+book
It is the introductory part that I found the most exciting as it teaches mathematical thinking.
The most well known quote from Hamming is:
'The purpose of computation is insight, not numbers.'
This applies particularly well to bioinformatics.
It's a rigorous upper-level math course focused on proving the fundamental theorems of calculus using basic logic and building up from set theory. Can confirm, was my lowest grade the semester I took it since unless you're Gauss you can't do the homework or tests without going to office hours.
Example question: "Prove this limit of this fucked up fractal looking puzzle piece bullshit function converges to 2" and you'd have to do a proof by analyzing the function's continuity, using some limit properties (which we defined using epsilon-delta, as well as a not so constructive definition using convergent sums), and then maybe using the modular properties of the fractal curve to simplify it into something that you can actually compute using basic algebra (which you also need to prove, obviously).
It's pretty cool though. Nice to see that math is super consistent and builds on its own concepts just starting with a few naive assumptions about what works (axioms). We literally went from set theory, and a single element "1" and another element "0" to the entire set of real numbers by defining "2" in terms of "1", defining "1/2" in terms of 1 and 2, and filling in the irrational numbers after proving the rationals are dense (which has a specific mathematical meaning) but not dense enough to be consistent with a few properties, so we need the irrational numbers to get the math we're used to.
Do I recommend taking it? Hell no! Read this reddit post a few times until it makes even less sense than the first time you read it and you have the gist of the class. Or read this unhelpful textbook: https://www.amazon.com/Advanced-Calculus-Applied-Undergraduate-Texts/dp/0821847910
As a rising senior, I'll be attending a prestigious research program for 7 weeks to do some materials research, most likely biochemistry or biophysics.
Also, my school only goes as high as BC Calc for math. I took AB this year since BC didn't fit into my schedule, and the Assistant Superintendent was nice enough to set up a teacher to teach me Multivariable one-on-one next year, so long as I teach myself BC over the summer. Should be easy, and I might even start on Multivariable if I finish early.
Very excited! Should be a productive summer!
Edit: I'll be teaching myself from this book. It was recommended because it goes very in depth on proving various theorems that are usually just introduced without regard to why they work. I was told learning the theory behind calculus will help for when I take an Analysis class.
If you're trying to learn calculus on your own you're better off buying a used version of either of these books for cheap (or going to a library)
http://www.amazon.com/Thomas-Calculus-11th-George-B/dp/0321185587
or Stewart: http://www.amazon.com/Calculus-Stewarts-James-Stewart/dp/0495011606/ref=sr_1_1?ie=UTF8&s=books&qid=1268447623&sr=1-1
Schaums provides basic insight, and several practice problems. If you want to understand the theory, go for Stewart or Thomas.
I'd say check with your professor first if it's for a class. You never know if you'll be missing a section. It helps to read what has changed in the newest edition. If it's minor cleanup or the addition of a single chapter you may be able to pass with the older version. However sometimes they change exercises and you'd be missing them for homework. Talk to the professor.
You can, however, check for Indian versions of books on Ebay or other places. These are usually paperbacks and are often in English, but they come at significantly reduced cost.
Otherwise, if this is for self-learning, I'd highly suggest looking at some Dover books. They pick up older classics or popular titles, often edit/update them a little, then publish in a cheap but nice looking and portable paperback.
E.g. Dover book on Infinitesimal Calculus for 4 bucks
There are hundreds of others. Many with good reviews, 4-5 star on Amazon. The presentation can be old-timey in some cases but the math is still relevant. I'm reading a book from the 1960's on "Information Theory" from Dover where you can see how this math motivated things like the internet and cell phones. It's based on Shannon's groundbreaking work in the 40s--much of it is still used to this day. They had the author (not Shannon) update it a bit for this new publication.
As a person who loves proofs and abstract math, here's why I support this book:
http://www.amazon.com/Calculus-Easy-Way-Barrons-Z/dp/0764129201/ref=pd_cp_b_1
You should spend some lovely evenings with my friend, Stewart. If you find my friend Stewart too hard on you, take some exercises from my little friend Thomas! And if you want even more fun, my friend Piskunov has some lovely exercises for you!
And ask your questions here :-)
For your 14-month course:
A good book on probability theory and advanced statistics: I'm fond of Freund's Mathematical Statistics with Applications
Linear algebra: Lay's "Linear Algebra and its Applications" is all right for a first course.
The analysis course sounds a bit like what I had out of my Sally Series Advanced Calculus text.
One of my favorites is a Dover text by Flanigan.
Calculus 5th ed. by Stewart mostly for its table of integrals. I always find myself going back to Mechanical Behavior of Materials by Meyer and Thermodynamics of Materials by Gaskell, but I do not endorse these books. They are probably my least favorite textbooks on my shelf.
FYI, there's an answer book available for Spivak's Calculus that is very useful for self-study students.
https://www.amazon.com/Thomas-Calculus-13th-George-Jr/dp/0321878965
is that it? if it is, it is super expansive for me...
this is actually a good book - it lightly hits all the major points.
https://www.amazon.com/Calculus-Dummies-Math-Science/dp/1119293499/ref=sr_1_1
This book is cheap and it will teach you calculus. It's pretty corny but it really does work to explain the fundamentals.
Richard Hamming's calculus text has a focus on probability and statistics.
Might be worth checking out.
Fomin and Kolmogorov is a classic.
https://www.amazon.com/Elements-Functions-Functional-Analysis-Mathematics/dp/0486406830
Some of the terminology is out of date but it's a nice exposition.
> Second and third semester calculus
Is this vector calc? If so I enjoyed this book as it's very geometric, not at all rigorous and has lots of worked examples and exercises. Sorry it seems to be so expensive -- it wasn't when I bought it, and hopefully you can find it a lot cheaper if it's what you're looking for.
In general Stewart's big fat calculus book is a nice thing to have for autodidacts.
Obviously what you describe might include analysis, which these books won't help with.
>Formal logic theory (Think Kurt Godel)
I've heard Peter Smith's book on Godel is good, but haven't read it. Logic is a huge field and it depends a lot on what your background is and what you want to get out of it. You may need a primer on basic logic first; I like this one but again it's quite personal.
Take a look at nonstandard analysis. I believe some studies in the 90s showed that students better understood these methods.
As for books, I can recommend Henle or the free book by Keisler at the high school level.
Mathematics: A Discrete Introduction is really good. Very clear, good progression, assumes nothing, and has very good problem sets. Note that there is a new edition, but you can still order used copies of the older edition (the one I'm familiar with) very inexpensively.
Have you considered this?
https://www.amazon.com/Calculus-Dummies-Lifestyle-Mark-Ryan/dp/1119293499
I would grab the 410 textbook and start reading chapter 1 (first 2 sections) and chapter 2 (first 4 sections). Chapter 2, especially section 2.1, should be a pretty good indicator of your ability to succeed in 410. If you can follow the proofs and reproduce them on your own, you'll probably be fine without 310.
I would say that the best way to start is to pick a single book in Calculus, such as this or this or even this, and work all the way through it.
Then it is up to you; you could go straight towards Real Analysis; I recommend starting with a book that bears Intro in the name.
Or you could pursue a more collegiate curriculum and move onto Differential Equations and Linear Algebra, then Real Analysis.
I assume you are doing this all independently, so you should look at college sequences for math majors and the likes. You can mimic those, and look for online syllabi of the courses to make sure you are covering the appropriate material. This helps because it gives a nice structure to your learning.
Whatever the case, work through a calculus book, then decide what further direction you wish to take.
I just took Calc 3 this past quarter, and the new textbook is Thomas' Calculus 12th Edition.
Proofs: Hammack's Book of Proof. Free and contains solutions to odd-numbered problems. Covers basic logic, set theory, combinatorics, and proof techniques. I think the third edition is perfect for someone who is familiar with calculus because it covers proofs in calculus (and analysis).
Calculus: Spivak's Calculus. A difficult but rewarding book on calculus that also introduces analysis. Good problems, and a solution manual is available. Another option is Apostol's Calculus which also covers linear algebra. Knowledge of proofs is recommended.
Number Theory: Hardy and Wright's An Introduction to the Theory of Numbers. As he explains in a foreword to the sixth edition, Andrew Wiles received this book from his teacher in high school and was a starting point for him. It also covers the zeta function. However, it may be too difficult for absolute beginners as it doesn't contain any problems. Another book is Stark's An Introduction to Number Theory which has a great section on continued fractions. You should have familiarity with proof before learning number theory.
I looked it up, and considering its three dollars on amazon then why not. Here is a link to the book i used for calc 1, 2 and 3 in college.
https://www.amazon.com/gp/aw/d/0321878965/ref=pd_aw_sbs_14_2?ie=UTF8&psc=1&refRID=0HC5R5JDCF2Q1GTVJ2P9
I would look and see if you can figure out what class you'll be taking next semester, and what book they use. The guide that guy posted above looked really good too.
Nope. https://www.amazon.com/gp/aw/d/1464125260/ref=mp_s_a_1_1?ie=UTF8&qid=1469827583&sr=8-1&pi=AC_SX236_SY340_FMwebp_QL65&keywords=calculus+rogawski+3rd+edition
Its for Math 1920
Try Kolmogorov and Fomin's Elements of the Theory of Functions and Functional Analysis.
PROS:
CONS:
SELLING Atoms First Except it's paperback
SELLING Calculus: Early Transcendentals
I did my schooling in India and we have a super hard on for old school Russian texts. One of the books I referred to for Calculus was by N. Piskunov.
I kept on returning to the book well until my undergrad math courses. It has 2 volumes, with a fairly good number of problems, most of them pretty good; and some solution manuals floating around somewhere.
EDIT:
Another undergraduate book you may want to check out would be Hall & Knight's Higher Algebra.
The book was first published in 1887 and has stood the test of time. It just contains pure, brutal, sadistic shit tonnes of algebra problems.
I’m finishing up my stats degree this summer. For math, I took 5 courses: single variable calculus , multi variable calculus, and linear algebra.
My stat courses are divided into three blocks.
First block, intro to probability, mathematical stats, and linear models.
Second block, computational stats with R, computation & optimization with R, and Monte Carlo Methods.
Third block, intro to regression analysis, design and analysis of experiments, and regression and data mining.
And two electives of my choice: survey sampling & statistical models in finance.
Here’s a book for intro to probability. There’s also lectures available on YouTube: search MIT intro to probability.
For a first course in calculus search on YouTube: UCLA Math 31A. You should also search for Berkeley’s calculus lectures; the professor is so good. Here’s the calc book I used.
For linear algebra, search MIT linear algebra. Here’s the book.
The probability book I listed covers two courses in probability. You’ll also want to check out this book.
If you want to go deeper into stats, for example, measure theory, you’re going to have to take real analysis & a more advanced course on linear algebra.
Logic, Number theory, Graph Theory and Algebra are all too much for you to handle on your own without first learning the basics. In fact, most of those books will probably expect you to have some mathematical maturity (that is, reading and writing proofs).
I don't know how theoretical your CS program is going to be, but I would recommend working on your discrete math, basic set theory and logic.
This book will teach you how to write proofs, basic logic and set theory that you will need: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
I can't really recommend a good Discrete Math textbook as most of them are "meh", and "How to Prove It" does contain a lot of the material usually taught in a Discrete Math course. The extra topics you will find in discrete maths books is: basic probability, some graph theory, some number theory and combinatorics, and in some books even some basic algebra and algorithm analysis. If I were you I would focus mostly on the combinatorics and probability.
Anyway, here's a list of discrete math books. Pick the one you like the most judging from the reviews:
Don't bother trying to learn too much too soon, as you really do need to let time for the math to sink in.
You can easily simplify the URL - it works just as well.
Not really. I have no clue what "honor's algebra" is or what sort of math it deals with.
I was able to breeze through calculus 1, 2, and most of 3 without studying (most of it from this book, and most of this book http://www.amazon.com/Thomas-Calculus-11th-George-B/dp/0321185587 ) Yet I still needed to study for this 200 level class.
My point being, you can't make sweeping generalizations for who should and shouldn't be in college based off of whether they need to study for a class or not. Yes, there are some people that shouldn't be there, but that has little to do with how much they study.
And even then, people that struggle on the simple stuff can often surprise you. I know one student that went from "couldn't pass basic algebra" to "top of the class in a Differential equations and Linear algebra class". All from a change in study habits and extra effort.
You could try "Precalculus" by Stitz & Zeager. Chapters 10 and onwards is their Trigonometry book. This should be a very smooth book to work through.
Have you already picked out a Calculus textbook? Also, what are her plans as an MIT student? If she's going into engineering and the like, I would say Larson's "Calculus" (solutions manuals vol 1, 2) would be good enough.
If she plans on being a math student, though, I would say give her a a few months with Velleman's "How To Prove It". Afterwards, I can't recommend Spivak's Calculus (Answer Book) and Jim Hefferon's Linear Algebra (solutions manual on same page) enough. This is a good time to introduce mathematical rigor as a normal thing in mathematics because, really, this is what math is about.
I second Horowitz and Hill, its one of those rare books that is almost universally suggested. The third edition just came out so second editions are a bit easier to find cheap.
The book does a good job of pointing out which mathematical areas can be skipped, but anyone wanting to design filters will need their calculus up to scratch. Thomas is the best intro text that I know of.
I personally found this $7.83 book to be a lifesaver during my complex analysis course. Furthermore, I enjoyed the prose quite a bit.
Something like this might be what you're looking for:
http://www.amazon.com/Special-Functions-Their-Applications-Mathematics/dp/0486606244/ref=sr_1_1?s=books&ie=UTF8&qid=1335655173&sr=1-1
Perhaps non-standard calculus/analysis. There are some free texts here otherwise this one has good reviews.
http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?ie=UTF8&qid=1342068971&sr=8-1&keywords=spivak%27s+calculus
This book starts with basic properties of numbers (associativity, commutativity, etc), then moves onto some proof concepts followed by a very good foundation (functions, vectors, polar coordinate). Be forewarned that the content is VERY challenging in this book, and will definitely require a determined effort, but it will certainly be good if you can get through it.
A more gentle introduction to Calculus is http://www.amazon.com/Thomas-Calculus-12th-George-B/dp/0321587995/ref=sr_1_1?s=books&ie=UTF8&qid=1342069166&sr=1-1&keywords=thomas%27+calculus and it is a much easier book, but you don't prove much in this one. Both of these can likely be found online for free. Also, if you want to get a decent understanding I recommend, http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521675995/ref=sr_1_1?s=books&ie=UTF8&qid=1342069253&sr=1-1&keywords=how+to+prove+it or http://www.people.vcu.edu/~rhammack/BookOfProof/index.html the latter is definitely free.
You may also need a more introductory text for trig and functions. I can't find the book my school used for precalc, hopefully someone else can offer a good recommendation.
Also, getting a dummies book to read alongside was pretty helpful for me, and Paul's online notes(website) is very nice.
I strongly suggest you take your time learning calculus, because anything you don't grasp completely will come back to haunt you.
But the good news is that there are lots of great resources you can use. MIT OCW has a full course with lectures, notes, and exams. Here are three free online books. If you're looking to buy a textbook, some good choices are Thomas, Stewart, and Spivak. (You can find dirt-cheap copies of older editions at abebooks.com.)
If you want more guidance, another great place to find it is at /r/learnmath.
I also had about a 12 year break between HS and college, and like you got through Trig just fine and then found myself drowning in Calc 1. Here's what helped me:
-attended another section of the class with another professor
-books that translated the mathy language into intuition
(http://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606 and http://www.amazon.com/Calculus-Easy-Way-Douglas-Downing/dp/0764129201/ref=sr_1_1?s=books&ie=UTF8&qid=1415864089&sr=1-1&keywords=calculus+the+easy+way)
-MIT OCW videos
-Khan Academy
Good luck. If you make it through this.. well, I'm not going to say it's easy going after, but you will know how to be confused and work through that confusion, and that is a priceless skill in the rest of the curriculum.
For multivariable calc and linear algebra, maybe this one:
http://www.amazon.com/Vector-Calculus-Linear-Algebra-Differential/dp/B008VRPQV2/ref=pd_sim_14_1?ie=UTF8&dpID=31PLLAqcnhL&dpSrc=sims&preST=_AC_UL160_SR160%2C160_&refRID=1FH9KJ7N0PQ9EM0BJMWA
For probability and stats, I like Wasserman's All of Statistics
and for Optimization, Boyd's Convex Optimization.
Looking for the full solutions manual for the textbook below. ($5)
https://www.amazon.com/Calculus-Standalone-book-Jon-Rogawski/dp/1464125260
Edit: still need.
You can try Piskunov's Calulus series Vol 1 and 2. Amazon links below:
Vol 1:
http://www.amazon.com/Differential-Integral-Calculus-Vol-I/dp/8123904924/ref=pd_sim_sbs_b_1?ie=UTF8&refRID=0SXQVY4M0QXKTR7JTVH6
Vol2:
http://www.amazon.com/Differential-Integral-Calculus-Vol-II/dp/8123904932/ref=pd_sim_b_1?ie=UTF8&refRID=0C13CBBF0587M907R2P1
It has the rigor that you are looking for. I was looking for the same thing and this book has helped me a lot to understand calculus, the insights, the geometrical view point. Hope it helps.
Complex Variables by Francis J. Flanigan is a good book.
For compsci you need to study tons and tons and tons of discrete math. That means you don't need much of analysis business(too continuous). Instead you want to study combinatorics, graph theory, number theory, abstract algebra and the like.
Intro to math language(several of several million existing books on the topic). You want to study several books because what's overlooked by one author will be covered by another:
Discrete Mathematics with Applications by Susanna Epp
Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand, Albert D. Polimeni, Ping Zhang
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Numbers and Proofs by Allenby
Mathematics: A Discrete Introduction by Edward Scheinerman
How to Prove It: A Structured Approach by Daniel Velleman
Theorems, Corollaries, Lemmas, and Methods of Proof by Richard Rossi
Some special topics(elementary treatment):
Rings, Fields and Groups: An Introduction to Abstract Algebra by R. B. J. T. Allenby
A Friendly Introduction to Number Theory Joseph Silverman
Elements of Number Theory by John Stillwell
A Primer in Combinatorics by Kheyfits
Counting by Khee Meng Koh
Combinatorics: A Guided Tour by David Mazur
Just a nice bunch of related books great to have read:
generatingfunctionology by Herbert Wilf
The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates by by Manuel Kauers, Peter Paule
A = B by Marko Petkovsek, Herbert S Wilf, Doron Zeilberger
If you wanna do graphics stuff, you wanna do some applied Linear Algebra:
Linear Algebra by Allenby
Linear Algebra Through Geometry by Thomas Banchoff, John Wermer
Linear Algebra by Richard Bronson, Gabriel B. Costa, John T. Saccoman
Best of Luck.
Projective Geometry by H.S.M. Coxeter.
Euclidean and Non-Euclidean Geometries by Marvin Greenberg.
Linear Programming by Katta Murty
Geometry of Complex Numbers by Hans Schwerdtfeger
Special Functions & Their Applications by N. Lebedev.