(Part 3) Top products from r/learnmath

There would have been a time that I would have suggested getting a curriculum
text book and going through that, but if you're doing this for independent work
I wouldn't really suggest that as the odds are you're not going to be using a
very good source.

Going on the typical

Arithmetic > Algebra > Calculus

****

Arithmetic

Arithmetic refresher. Lots of stuff in here - not easy.


I think you'd be set after this really. It's a pretty terse text in general.

*****

Algebra

Algebra by Chrystal Part I

Algebra by Chrystal Part II

You can get both of these algebra texts online easily and freely from the search

chrystal algebra part I filetype:pdf

chrystal algebra part II filetype:pdf

I think that you could get the first (arithmetic) text as well, personally I
prefer having actual books for working. They're also valuable for future
reference. This filetype:pdf search should be remembered and used liberally
for finding things such as worksheets etc (eg trigonometry worksheet<br /> filetype:pdf for a search...).

Algebra by Gelfland

No where near as comprehensive as chrystals algebra, but interesting and well
written questions (search for 'correspondence series' by Gelfand).

Calculus

Calculus made easy - Thompson

This text is really good imo, there's little rigor in it but for getting a
handle on things and bashing through a few practical problems it's pretty
decent. It's all single variable. If you've done the algebra and stuff before
this then this book would be easy.

Pauls Online Notes (Calculus)

These are just a solid set of Calculus notes, there're lots of examples to work
through which is good. These go through calc I, II, III... So a bit further than
you've asked (I'm not sure why you state up to calc II but ok).

Spivak - Calculus

If you've gone through Chrystals algebra then you'll be used to a formal
approach. This text is only single variable calculus (so that might be calc I
and II in most places I think, ? ) but it's extremely well written and often
touted as one of the best Calculus books written. It's very pure, where as
something like Stewart has a more applied emphasis.

**

Geometry

I've got given any geometry sources, I'm not too sure of the best source for
this or (to be honest) if you really need it for the above. If someone has
good geometry then they're certainly better off, many proofs are given
gemetrically as well and having an intuition for these things is only going to
be good. But I think you can get through without a formal course on it.... I'm
not confident suggesting things on it though, so I'll leave it to others. Just
thought I'd mention it.

****

Oh, I'm terrible at calculus, haha. I teach discrete maths and logic, and never have to touch calculus at all, thank goodness :)

But a younger friend of mine is doing calculus just now, so I'll find out what he found useful and PM you. He did say that some of the books I'd recommended him were immensely useful for maths generally (not necessarily calculus in particular). In roughly ascending order of difficulty:

• Gowers, Mathematics: A Very Short Introduction (<http://amzn.to/1TQNEWY>)
  
• Devlin, The Language of Mathematics (<https://www.amazon.com/Language-Mathematics-Making-Invisible-Visible/dp/0805072543>)
  
• Stewart, Concepts of Modern Mathematics (<http://bit.ly/1L7rY4i>)
  
• Liebeck, A Concise Introduction to Pure Mathematics (<http://amzn.com/1439835985>)
  
  So you could pick one of those, and see if it meets the level you're after. There are also some standard books on "how to write proofs" which often get recommended: Velleman, How to Prove It: A Structured Approach <https://www.amazon.com/How-Prove-Structured-Daniel-Velleman-ebook/dp/B009XBOBL6>, and
  Solow, How to Read and Do Proofs: An Introduction to Mathematical Thought Processes <https://www.amazon.com/How-Read-Proofs-Introduction-Mathematical/dp/1118164024>, and they might be useful. I've never read either, shamefully, tho I probably should.
  
  Also: something that can be handy once you're past the basics is the amazing (but large and expensive) Princeton Companion to Mathematics. Which has something useful and interesting to say about pretty much every major area of modern mathematics.
  
  I hope that helps!

Yes it is definitely possible! I was in high school 12 years ago and I failed grade 9 math 3 times and now I have a degree with a minor in math.

I started university late and so not only did I do pretty poorly in high school math (I didn't have a conceptual understanding of anything), but it had been about 5 or 6 years since I'd used any of it, in other words I was in way worse shape than you are now.

If you are interested in science as opposed to math in itself, then you will be taking a calculus course, a linear algebra course and a stats course in university most likely, and fortunately it is surprisingly easy to get caught up!

Over the course of one month in the summer I worked through a book like http://www.amazon.com/Algebra-Trigonometry-Transcendentals-Calculus-Edition/dp/0321671031 and made sure I understood the concepts (not just memorized the answers). Anytime I didn't understand something I would look it up online, ask someone for help, or watch a khan video. The most important thing is working hard to practice and understand pre-calculus: doing a lot of exercises, and asking yourself "why does this work", whenever you can.

Trust me, if I can go from failing grade 9 math three times to doing pretty well in courses like calculus and proof based classes like set theory, anyone can.

As others have mentioned, you can use various online resources such as Khan Academy. I'm assuming you know basic arithmetic (multiplication, exponents, working with fractions, percentages, etc...), if not you'll have to start with that. After that, you may need to learn what is usually called "Pre-Calculus". These are the topics you need to master before you can start learning Calculus. Usually they include study of functions (polynomials, logarithmic, etc...), trigonometry, analytic geometry.

Here's a list of stuff you should investigate:

• Arithmetic Refresher: Improve your working knowledge of arithmetic
  
  
• US Navy mathematics courses
  
  This is a collection of PDFs which are taken from some US Navy mathematics courses. The courses are:
  
  >US Navy course - Mathematics, Basic Math and Algebra NAVEDTRA 14139
  
  >US Navy course - Mathematics, Trigonometry NAVEDTRA 14140
  
  >US Navy course - Mathematics, Pre-Calculus and Introduction to Probability NAVEDTRA 14141
  
  >US Navy course - Mathematics, Introduction to Statistics, Number Systems and Boolean Algebra NAVEDTRA 14142
  
  The Basic Math and Algebra course covers arithmetic and could be used instead of the Arithmetic Refresher, if you'd like.
  
  You may also want to look into books on world problems, as that's where most people have difficulty with. World problems mirror our real-life problem solving process, where we have to translate some real problem into mathematical language in order to tackle it.
  
  
• http://www.artofproblemsolving.com
  
  > The Art of Problem Solving mathematics curriculum is specifically designed for outstanding math students in grades 6-12, and presents a much broader and deeper exploration of challenging mathematics than a typical math curriculum. The Art of Problem Solving texts have been used by tens of thousands of high-performing students, including many winners of major national contests such as MATHCOUNTS and the AMC.
  
  If you look at their bookstore, you will see that they have books for the various subjects your kid will encounter during school. They also have the Beast Academy which is an on-going project to release books for kids in grades 2-5.
  
  Note that they say that the books are for gifted math students since the exercises are taken from math competitions. What's nice about these books though is that they offer the full solutions (not just a final answer, but the full explanation). Also, for every book they have a Diagnostic Test (pre-test) to check and see if you are capable of starting the book.
  
  
• Pre-Calculus stuff...
  
  There are plenty of various Pre-Calculus books which contain all this material. I can't really recommend anything with certainty since I've never read any.
  
  But here's a book you could try by a well-known mathematician who also seems to write really well (it also appears to have solutions to the problems):
  http://www.amazon.com/Precalculus-Prelude-Calculus-Sheldon-Axler/dp/1118083768/
  
  
  ---------------------------
  Once you have some specific subject you're having difficulty with, you can always ask for help or look for a friendly book. The problem with math is that some authors/teachers teach subjects very dryly, so it makes it boring... the challenge for people who aren't naturally motivated for maths is to find teachers/books that excite the student -- and there are a few authors that can do it, so you just have to ask around.
  
  By the way, you should also look for various popular math books that could make studying the subject all the more interesting.

One thing I like to remind people, is that Linear Algebra is really cool and though it tends to come "after" calculus for some reason, it really has no explicit calc prerequisite.

I highly recommend Dr. Gilbert Strang's lectures on it, available on youtube and ocw.mit.edu (which has problems, solutions, etc, also)

I think it's a great topic for right around late HS, early college. And he stresses intuition and imo has the right balance of application and theory.

I'd also say that contrary to most peoples' perceptions, a student's understanding of a math topic will vary greatly depending on the teacher. And some teachers will be better for some students, others for others. That's just my opinion, but I firmly believe it. So if you find yourself struggling with a topic, find another teacher/resource and perhaps it will be more clear. Of course this shouldn't diminish the effort needed on your part, learning math isn't a passive activity, one really has to do problems and work with the material.

And finally, proofs are of course the backbone of mathematics. Here is an intro text I like on that.

Oh okay, one more thing, physics is a great companion to math. I highly recommend "Classical Mechanics" by Taylor, in that regard. It will be challenging right now, but it will provide some great accompaniment to what you'll learn in upcoming years.

Just as an add-on, Stewart's is definitely the best way to go for learning applied calculus as a beginner. It's EXHAUSTIVE, though I'd actually recommend the full-on "Calculus" textbook instead of "Early Transcendentals" or "Single (Multi) Variable" texts for this reason:

At the end of every chapter, there are "problems plus" that will really challenge the way you think about what you've just learned. You don't get these in the other books. They'll make you think like a mathematician or a scientist instead of a "plug-and-chugger."

And once again, I'm gonna plug Smith's "Transition to Advanced Mathematics" for an introduction to proof-writing and set theory and the most basic of analysis. http://www.amazon.com/Transition-Advanced-Mathematics-Douglas-Smith/dp/0495562025/ref=sr_1_1?ie=UTF8&qid=1371247275&sr=8-1&keywords=douglas+smith+transition

Though definitely get an older edition to save more money. And I realize you can't get books delivered--you can find pdf versions of older editions.......

As for the lower, pre-calculus stuff, just look to the right on this reddit for Khan Academy and just browse through the topics there. If you're as good a student as you say you are, you just need the few holes filled in and a quick refresher, and Khan is perfect for it.

What you're probably looking for is a book on foundations, like logic or set theory. The thing is though that such books often do assume a certain level of mathematical maturity and experience with proofs. In addition, while the main results should be self-contained, the examples and exercises may make use of things that are supposed to be familiar but which you haven't learned due to your background (or lack thereof).

My advice is to be patient and focus on really learning the elementary topics, up to and including the standard calculus sequence. After that things really start to open up and you can usually find quality undergraduate-level books on just about anything that interests you.

But it probably isn't too soon to pick up a book on writing proofs and proof techniques. The one I used when I was younger was Peter Eccles' An Introduction to Mathematical Reasoning. I often hear Polya's How to Solve It recommended as well. I'm not sure how similar these are to the AoPS series though.

Anyway, good luck! Stick with it, mathematics is very worth it.

Also, in another comment you asked for a topic where you could do some work on your own. I highly recommend Dudley's Elementary Number Theory for this purpose. Right in the middle of the exposition he will give you an exercise to do, so they actually mix in with what you're learning rather than only being at the end of the section. I absolutely love this book and I think it would be perfect for what you need since it really is elementary (but not necessarily easy) and doesn't require any previous knowledge.

By the way, your English is amazing.

I'm currently working through Munkres' book on Topology, and I am using the video lectures found here. I know these are in an annoying form factor, but, trust me, these are the only videos that go into any depth you will find on the internet. They use Munkres, too, which is a plus.

On the same site are video lectures for Algebraic Topology. For this, I definitely recommend buying Artin's "Algebra" (1st edition can be found cheaply, and I don't think there's really any significant difference from 2nd), and watch these video lectures by Harvard. Then, you can finally move on to the Algebraic Topology video lectures which uses the free textbook "Algebraic Topology" by Allen Hatcher.

Hope this helps.

I'm 2 years into a part time physics degree, I'm in my 40s, dropped out of schooling earlier in life.

As I'm doing this for fun whilst I also have a full time job, I thought I would list what I'm did to supplement my study preparation.

I started working through these videos - Essence of Calculus as a start over the summer study whilst I had some down time. https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr

Ive bought the following books in preparation for my journey and to start working through some of these during the summer prior to start

Elements of Style - A nice small cheap reference to improve my writing skills
https://www.amazon.co.uk/gp/product/020530902X/ref=oh_aui_detailpage_o02_s00?ie=UTF8&psc=1

The Humongous Book of Trigonometry Problems https://www.amazon.co.uk/gp/product/1615641823/ref=oh_aui_detailpage_o08_s00?ie=UTF8&psc=1

Calculus: An Intuitive and Physical Approach
https://www.amazon.co.uk/gp/product/0486404536/ref=oh_aui_detailpage_o09_s00?ie=UTF8&psc=1

Trigonometry Essentials Practice Workbook
https://www.amazon.co.uk/gp/product/1477497781/ref=oh_aui_detailpage_o05_s00?ie=UTF8&psc=1

Systems of Equations: Substitution, Simultaneous, Cramer's Rule
https://www.amazon.co.uk/gp/product/1941691048/ref=oh_aui_detailpage_o05_s00?ie=UTF8&psc=1

Feynman's Tips on Physics
https://www.amazon.co.uk/gp/product/0465027970/ref=oh_aui_detailpage_o07_s00?ie=UTF8&psc=1

Exercises for the Feynman Lectures on Physics
https://www.amazon.co.uk/gp/product/0465060714/ref=oh_aui_detailpage_o08_s00?ie=UTF8&psc=1

Calculus for the Practical Man
https://www.amazon.co.uk/gp/product/1406756725/ref=oh_aui_detailpage_o09_s00?ie=UTF8&psc=1

The Feynman Lectures on Physics (all volumes)
https://www.amazon.co.uk/gp/product/0465024939/ref=oh_aui_detailpage_o09_s00?ie=UTF8&psc=1

I found PatrickJMT helpful, more so than Khan academy, not saying is better, just that you have to find the person and resource that best suits the way your brain works.

Now I'm deep in calculus and quantum mechanics, I would say the important things are:

Algebra - practice practice practice, get good, make it smooth.

Trig - again, practice practice practice.

Try not to learn by rote, try understand the why, play with things, draw triangles and get to know the unit circle well.

Good luck, it's going to cause frustrating moments, times of doubt, long nights and early mornings, confusion, sweat and tears, but power through, keep on trucking, and you will start to see that calculus and trig are some of the most beautiful things in the world.

Mathematical Proofs: A Transition to Advanced Mathematics was the book for my college proofs class. I found it to be a good resource and easy to follow. It covers some introductory set theory as well. Just be prepared to work through the proof exercises if you really want a good intuition on the topic.

I'd pick up Stillwell's
https://www.amazon.com/Pillars-Geometry-Undergraduate-Texts-Mathematics/dp/0387255303

or, if on a slightly lighter budget, I'd get Pedoe's book
https://www.amazon.com/Geometry-Comprehensive-Course-Dover-Mathematics/dp/0486658120

Take your time with the material and try to understand what's going on rather than memorize formulas. Geometry has lots of proofs. Treat the material with the respect and time it deserves and you'll understand more geometry than even some math undergrads do. You can go through Euclid's Elements, if still interested, afterwards.

Happy learning!

Parametric and vector functions will be very useful to have a basic grasp of. If you're in engineering, you'll be doing a lot of work with vector calculus in that class, and any familiarity you can get with it will help you immensely.

Get a copy of Div, Grad, Curl and All That. http://www.amazon.com/Div-Grad-Curl-All-That/dp/0393925161

It's a great book to read along with your textbook. Wouldn't hurt to start reading it now, actually, while you have the time to go back and refresh yourself on any topics the book assumes knowledge on that are rusty for you.

I got my bachelors in Spanish and I have one more semester left to finish my masters in mathematics. As someone coming from a liberal arts degree, proofs were foreign to me, and handling anything with more than one variable was just asking too much. When I took Linear Algebra (my first proofs class), I had peers like the ones you mentioned, who just "got it" without taking notes while missing a third of the class lectures. And here I was slaving away, lost in a web of confusion. That class almost broke me. But over time, I learned a few things which were catalysts for my math competency:

• I learned the framework of proofs and logic from this awesome book so that whenever I saw a theorem, I already had an idea of how to tackle it
  
• I gave no shits about people thinking I'm stupid and thus asked A LOT of questions in class
  
• I showed up to office hours religiously; it's insane not to utilize this one-on-one time with a PhD with tuition rates being as they are
  
• I drew everything I could to help me understand concepts; if you can see the forest from the trees, the details will fall out
  
• Related to the above, I tried building a visual intuition of things I was learning which helped me see past the slew of variables and greek letters
  
• I memorized all definitions and stuck them in Anki; you're screwed trying to do proofs without definitions
  
• Those other smart guys had to do everything I was doing at
  some point in time, they just got a head start; so I put in the hours and caught up (even surpassed in some areas)
  
• I immersed myself in math: gave talks at conferences, got a job related to math, talked to other students about it, blogged about it, etc...
  
  Anyway, the struggle is real, but after all those focused hours of engaged studying, the intuition will finally be there and your brain will then "compress" that information so you have room to learn more. Ad infinitum.
  
  Hope that helps.

Also pick a book on Number Theory, pretty useful in computer science. I would recommend: https://www.amazon.com/Elementary-Number-Theory-Second-Mathematics/dp/048646931X/ref=sr_1_1?ie=UTF8&qid=1500074415&sr=8-1&keywords=elementary+number+theory+dudley

I know a few people who highly recommend How to Prove It by Velleman. I've never read it so I can't say for sure. The first book I used to learn mathematical logic was Lay's Analysis with an Intro to Proof. I can't recommend that book enough. The first quarter of the book or so is a pretty gentle introduction to mathematical logic, sets, functions, and proof techniques. I imagine it will get you where you need to be pretty quickly.

I haven't read it personally, but some agree on Quick Calculus being an approachable book for covering both the techniques and the concepts applied in calculus. The "why" behind the techniques often gets hidden away from non-maths majors, so this book supposedly works as a good self-supplement.

http://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics/dp/0486404536/ref=sr_1_1?s=books&ie=UTF8&qid=1405668438&sr=1-1&keywords=calculus+an+intuitive+and+physical+approach

Starts out with a brief history of calculus in chapter 1.

Chapter 2 is derivatives.

Chapter 3 is anti-derivatives

Chapter 4 talks about the geometric importance of the derivative...etc..

Chapter 21 talks about multivariable functions and geometric representation then 22 is over partial differentiation, 23 multiple integrals then an introduction to diff eq.

I don't know if that's what you're looking for.. but its been an excellent read so far, and it tends to be written in layman's terms(great for me) rather than math speak.

I've heard (more than once) that you go to Calculus to finally fail Algebra or Trig. The latter is proving true for me. I've got a copy of this book (my calc class is taught out of Early Transcendentals ... so it pairs up perfectly - check for something similar for your text if you can find it) and it's a good resource for when I can't remember how to do something.

Aside from that, figure out where your weaknesses are. Don't spend valuable time further working on stuff you're already rock solid on - target weak areas and work on them.

Khan Academy, Professor Leonard are both great resources. Wolfram Alpha subscriptions (only a few bucks a month) also has a button that shows step-by-step how they get to the answer, which is frequently useful to me.




Edit: OH! Also, don't go it alone. You've obviously found this place, which is great. Your school probably has a math tutoring center, or math lab, or your professor's office hours, or a TA you can bother or whatever. Those things are all "free" - but in reality they're not. They're included in the price of your tuition. You've already paid for them: USE EM!

Whew, not looking for Stewart or spivak? That's the two ends of the spectrum as far as calculus is concerned.

Maybe check out Morris Kline. Its intuitive and sounds right up your alley (I think)! For vector calc you may need to pick up something more advanced. I hope this helps :)

http://www.amazon.com/gp/aw/review/0486404536/RTE3I14V7OSHN/ref=cm_cr_dp_mb_rvw_1?ie=UTF8&cursor=1

You will likely get a lot out of How to Solve It. It teaches you how to break down problems and how to structure your thinking.

For the math itself, your examples are all covered by pre-algebra and algebra 1 (algebra is a pretty broadly defined area of math, but you want the beginner algebra books that most students would start in 8th-10th grade). I don't have specific book recommendations for that level of math, but taking the Khan Academy classes that the other poster linked would likely be a good start.

Did you know Amazon will donate a portion of every purchase if you shop by going to smile.amazon.com instead? Over $50,000,000 has been raised for charity - all you need to do is change the URL!

Here are your smile-ified links:

https://smile.amazon.com/Math-Book-Pythagoras-Milestones-Mathematics/dp/1402788290/ref=sr_1_2

---

Never forget to smile again | ^^i'm ^^a ^^friendly bot

This is the book I’m using right now in my first proofs class it’s pretty good at explaining the thought processes as well as it can be paired with How to Prove It by Daniel J. Velleman for a more through brake down of them problem types.

You might want to consider reading a book about the history of maths like: https://www.amazon.com/Math-Book-Pythagoras-Milestones-Mathematics/dp/1402788290/ref=sr_1_2?ie=UTF8&qid=1502399281&sr=8-2&keywords=history+of+maths

This will help you understand how math evolved and put the various major math discoveries of the past in context.

Fraleigh is a little bit easier to wrap your head around. Get an old edition (or find it at the library), obviously.

Also, I highly recommend Herstein's Topics in Algebra. Again, try to get it from a university library.

For vector calculus, you might enjoy the less formal British text Div, Grad, Curl, and All That by H. M. Schey; for group theory in brief, consider the free textbook Elements of Abstract and Linear Algebra by Edwin H. Connell.

Alternatives to Schey's book include the much more formal Calculus on Manifolds by Michael Spivak, which does have more exercises than Schey but uses most of them to develop the theory, rather than as the mindless drills that fill an ordinary textbook; Michael E. Corral's free textbook Vector Calculus isn't huge but is written closer to an ordinary textbook.

My professor wrote this this book. It is excellent if you already have a memory of PDE's. It is also inexpensive.

Is Herstein out of fashion these days? Excellent book. http://www.amazon.com/Topics-Algebra-2nd-Edition-Herstein/dp/0471010901

Sorry, I went on vacation and totally blanked about posting these for you!

Anyway, here are some books

Linear Algebra Done Right (Undergraduate Texts in Mathematics) https://www.amazon.com/dp/3319110799/ref=cm_sw_r_cp_api_1L8Byb5M5W9D3

This one is actually for analysis but depending on your appetite, it might help greatly with the proof side of your class. You can buy it here: Counterexamples in Analysis (Dover Books on Mathematics) https://www.amazon.com/dp/0486428753/ref=cm_sw_r_cp_api_GS8BybQWYBFXX

But there's also a PDF hosted here: http://www.kryakin.org/am2/_Olmsted.pdf

So I keep pointing students to this one book, and they keep telling me it saved their asses. So I'll point you to it too: Just in Time Algebra and Trig for Early Transcendentals Calculus

It has everything you need, while remaining short and inexpensive. I know of no better preparatory book for calculus, and that includes Stewart, Sullivan, and Blitzer-- all the big names.

The Math Book is full of visuals! Every two pages it has a neat bite-sized idea with a full page picture.

I used this book for my PDE class. It's meant to be supplemented with notes, but it does a decent job explaining material on it's own. That and it's super cheap for a math book.

Note that it has several errors in the answer key.

There are many books that I found helpful in high school for number theory, for example this classic by Niven et al.

http://www.amazon.com/Introduction-Theory-Numbers-Ivan-Niven/dp/0471625469

It's useful for Electromagnetic physics. Surface integrals are used for finding the flux through a Gaussian surfaces so you can use Gauss' Law on non-symmetrical surfaces. Line integrals are used with Ampere's Law to find the magnetic flux. Once you learn the mechanics of working with multivariable calculus, you should read "Div, Grad, Curl and All That"

Where I teach they use Linear Algebra by Lay for the introductory class. I'm not sure what level you need but Linear Algebra Done Right is also commonly recommended; could be more abstract than what you need?

For an introductory text, I recommend Herstein's Topics in Algebra. It slowly walks you through groups, rings, vector spaces, modules, fields, linear transformations and other selected topics.

Has plenty of exercises and doesn't skip over any details.

I recommend http://www.amazon.com/Introduction-Theory-Numbers-Ivan-Niven/dp/0471625469. I'm currently reading through it

Polya tried to answer your question:

How to Solve it

How to Read and Do Proofs by Daniel Solow this book saved my life in Abstract Algebra.

I can't really give a better testimonial other than I read through this book and applied a couple of the concepts and did very well in the course.

One thing to remember, you can always reverse your steps, if you are stuck at some point, then work backwards from the end and you can sometimes meet up to the point you were stuck at.

Also, How to Prove It by Daniel J Velleman is another classic book that can help.

A lot of early math tends to come down to how often you do problems, and computation classes can generally be seen as rote learning. I'd suggest you start doing some proofs, they force you to understand what you are doing, rather then just doing what you've seen. Pick up http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521446635

or, http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&ie=UTF8&qid=1345011596&sr=1-1&keywords=spivaks+calculus

Not knowing random operations as you listed is fine, with time you will get quicker, but don't worry if you need to consider it for a moment.

Analysis: With an Introduction to Proof

The first 8 chapters cover much of the same material as Velleman's book. After that, it's a very reasonable first exposure to real analysis.

If it's a review I suggest a book called Quick Calculus, also look over Paul's math notes trig and algebra review.

Like 50 on amazon but could also try Abebooks and see if there's a cheaper used or international copy.

There is actually a book called An Introduction to Mathematical Reasoning.

Topology by Munkres

Another very accessible book https://www.amazon.com/How-Read-Proofs-Introduction-Mathematical/dp/1118164024