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Science and Math For Adults?
Peter Trepan writes "Like most Americans, I made it through high-school and college without a thorough understanding of major scientific and mathematical concepts. I'm trying to remedy this situation both for personal betterment and so I can supplement my *own* kids' education. The problem is, most textbooks are not designed to convey an understanding of the subject, but to squeeze in all the 'facts' required by state law. I'm looking for books that don't just tell me an equation or a concept works, but also explain *why*. Would you please list books that have helped you gain a greater understanding of the basic concepts of algebra, chemistry, calculus, physics, and other core areas of science?" This is similar to an earlier question, but with a broader focus.
Go to the nearest university book store, or even just find the web page for a universities math department and find the text book for the subjects you want and order it online.
I don't think very many text books just give you a equation and say use this. My HS was a poor ass sucky redneck school and didn't do that, we just didn't have much of a variety in subjects. Also I think saying books just do what the states require only applies to states with said systems. Many, maybe most, just say you need to have a class in this that and the other thing.
Also once you get into learning the hows and whys of lots of math you will see why people tend to just want the equation, far less frustrating and confusing for learning. Learning how to do it and then going back for the why is often better for subjects like math. Same for say engineer, it seams a whole lot more fun till your actualy doing it and find out 99% of it sucks big time and is not what you think engineers do.
One book to stay away from if calc. is you game is Thomas Finny, that book sucks beyond belief.
Honestly, you don't have the background to see how deep the Feynman lectures are. His discussion of angular momentum has so much physical insight that you *know* that the writer has a deep understanding of mathematical physics. Never confuse a lack of verbosity or notation for lack of physical insight.
French is one of the poorer intro QM books, IMO (a poor formal treatment of operators and commutators and the like). I perfer Sakurai (although it is hard for the average undergraduate) with Peebles as a supplement for applications (the baby-field theory in the last chapter is brilliant) and Shankar for his treatment of path integrals. Merzbacher and/or Sakurai's second book are good for graduate QM. The standard book for stat mech is Kittel and Kromer. I've taught out of it for over a decade and have never had complaints about the text. The BEST book for intro E/M is Purcell. It is an exceptionally elegant treatment that doesn't shy away from math, but still manages to remain grounded in the physics of the situation. For intermediate E/M, Griffiths is the obvious choice. For advanced E/M, Jackason is the obvious choice. For QFT, Perkins and Schroder is the standard text, although for the very mathematically inclined, Weinberg's series is excellent.
I'm looking for books that don't just tell me an equation or a concept works, but also explain *why*. Would you please list books that have helped you gain a greater understanding of the basic concepts of algebra, chemistry, calculus, physics, and other core areas of science.
This is broad. My own list that you might find useful (or not):
algebra -- a good introduction is Earl Swokowski's "Fundamentals of Algebra and Trigonometry". It's often available in used book stores, campus book sales, etc.. It is a text book, though, and you may or may not enjoy this method of learning. If you want more of an overview of math, take a look at Paulos' "Innumeracy". If you want some lighter reading, try stuff by Martin Gardner.
calculus -- builds upon algebra so you need to know your algebra, especially limits, before you tackle calc. Know the limits well because it will help in many ways. I often refer to Elliot Gootmans' "Calculus" from Barron. For fun, also try "A Tour of the Calculus". Many chapters in "A History of Pi" are interesting (and approachable) also. Stay away from the Dover books until you have a pretty good grasp. They're cheap, but their approach is sometimes a little heavy-handed.
physics -- Feynman's "Six Easy Pieces".
For general reading, also try:
Godel, Escher, Bach (Douglas Hofstadter)
Islands of Truth (??Ivars Peterson??)
BTW, I'm a big proponent of using mathematics software as an addition to traditional study. There are programs such as MuPAD, GnuPLOT, Octave and Maxima that are available for free that can really help in the understanding of concepts. Many people are more visual so a graph is eminently useful.
The math program I was a part of in high school, at Whitney Young Magnet School in Chicago, was called IMP, or Integrated Mathematics program but it could have just as easily stood for Interactive Mathematics Program.
Basically the way it was structured was that instead of the traditional math program where one learns algebra the first year, geometry the second, trig the third and then moves onto precal, we learned a litte bit of each every year.
Furthermore, instead of them just shoving facts down our throat and saying here, memorize these (such as all the proofs from traditional geometry) we were actually guided along in discovering them for ourselves.
Every problem was given to us in word problem format. Each unit, which represented a major concept such as the quadratic equation or some of that other stuff, was presented as one big word problemm and it was broken up into smaller pieces which slowly led up to the solution of the actual problem.
So instead of coming out of it with simply memorizing the quadratic equation, pythagorean theorem, pi, geometric proofs and the like, we were actually able to discover these on our own.
It's just too bad the teachers weren't all that great and the program didn't much fit into the "flash/bang" you need to know this information right now that most high school classes are based around. God forbid students actually understand and can apply the information they are learning.
I also can't seem to recall who published the books we used but I'm sure a bit of googling can solve that.
There are several reasons for wanting to fail students, the most frequently mentioned is that theres "not enough room" in the upper courses. But the real reason is they are simply elitist bastards, they figure, "I had to go through it, you do to." The worst abuse I ever saw was a chemistry course I was in. 250 Students, the teacher spent the entire quarter lecturing about the heart medicine he was working on, and how steel refineries worked (his other interest). No problem -- if the tests are on heart medicines and steel production, but, he gave standardized tests and flunked 90% of the class.
Flunk courses also create some strange strange acedemic relationships. For instance, I was getting 15s and 16s (out of 100) on my physics tests and, with the curve I was getting a nice fat C. The problem with this is two fold ... It sounds great right? get a 15 and get a C? First problem, I'm not getting the education I paid for. Secondly, it encourages cheating because all you have to do is "beat the curve". The thrid and most intriguing problem deserves its own paragraph.
For me to get a C with 15 out of 100 points. That means, about HALF of the students scored worse then me. The students who scored WORSE then me *financed* my C by getting D's and F's. If they weren't the cannon fodder, *I* would have failed the course. Now here's where things get tricky. Sometimes, you are the sacrifical lamb, and sometimes you are the priest. If you are the lamb, you take the course over -- but this time you're the priest because you've taken the course before and it's finally starting to make sense. So the first timers are competing on a curve with people who have taken the course before. This wouldn't be a problem with a normal distribution of scores, but with poor instruction causing scores to center around 15%, that advantadge *REALLY* counts.
So now that I've written a diseratation here, what I really mean is, in your post you assume that mathbooks are even designed to help students, when most of the time, they aren't.
Religion is a gateway psychosis. -- Dave Foley
I highly recommend Cartoon Guide to Statistics and Cartoon Guide to Genetics Despite the titles, they don't sacrifice accuracy for cuteness. If you make it all the way through the Cartoon Guide to Statistics you'll be able to understand common statistical practices like t-tests and confidence intervals, and you'll have a much better chance of recognizing when statistics are being abused.
People go to community college to transfer into a good university and get cheap credits, not get an education.
If they wanted me to focus on an education perhaps they wouldnt make the GPA so damn important.
What is the point of avoiding difficult but important classes simply to preserve your GPA? Are you in school to get an education or to simply achieve some arbitrary GPA? I've been in the position of hiring people for technical positions and I've always been far more impressed by a mediocre GPA in a substantial curriculum then a high GPA in an easy curriculum.
Ok say I do take a few math classes and get a few Cs, well then my GPA goes under 3.0 and I can forget about transfering into a good 4 year university, I can also forget about scholarships and grants which also require a high GPA of above 3.0 or 3.5, I really cannot afford any Cs and I know for a fact that its simply impossible for me to get an A or B in math. I take classes which I know I can/will get an A or B in.
This isnt about the jobs, this is about getting a degree from an elite private university.
I recently returned to school myself, so I do have sympathy with amount of work required to do really well in a course, and I do understand that those planning to continue to a four year school or go on to graduate school need to match minimum requirements, but in my opinion you'll be better served by reducing the number of classes you take in a given term then by trying to ditch the challenging courses.
I never take more than 4 classes per semester, and I never get anything below a B in grades, those are the rules I follow.
Maybe if universities werent so strict and competitive on the GPA issue I could actually focus on learning but right now I have a goal, that goal is to get into Harvard, Tufts, Boston College,Boston University or North Eastern, all which are ELITE private universities which will NOT let you in with a sub 3.0 GPA, you most likely wont get in with a sub 3.5 GPA, so no its not about "learning" right now, its about moving up the ladder, it will be about learning once I get into university, thats when I'll take math clases, get a C or two, and learn something.
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I confess that I made it through 3 semesters of college calculus and an engineering degree pretty much not understanding the underlying concepts of calculus. It's surprising what you can accomplish by rote. This book was a real forehead-slapper for me, and I can't recommend it highly enough. Many years after graduating, I've finally learned what I should have back then. If it were up to me, this would be the first book anyone learning calculus ever read. I wish Sylvanus Thompson were still alive (I think Calculus Made Easy was published in 1919) so I could give him a big smooch.
I read Isaac Asimov's Realm of Algebra when I was in grade 6, and didn't learn anything beyond it until around grade 10. Actually, I didn't even finish reading Realm of Algebra -- if I did, who knows how many grades worth of math I would have learned in one sitting!
Unfortunately, it is out of print, and has been for some time. I have seen people asking outrageous sums of money for it used, upwards of $300 U.S. This is truly a book that is crying out to be open-sourced/pirated. Maybe someone who owns one would scan it into a tidy little pdf or something. Do the same to Realm of Numbers too.
Mike van Lammeren
It will challenge your head, your brain, and your mind.
In my own experience (from grade school math through grad school math), I have almost always found that the texts aren't terribly helpful until *after* you've learned (at least to some basic level) the mathematics. In one of the posts above, SuperBanana notes this problem, and suggests that you try adult ed courses. I agree that the human interaction with a professor and fellow students can be invaluable. In fact, some of the biggest mathematical ah-ha moments I've had have been when I've been trying to work through an idea with friends. Only then did the stuff in the textbook really make sense.
... and how to find them before the teacher does
Now, that's not to say that there aren't good books out there to help you learn about mathematics. It's just that the ones that are written as textbooks (particularly in the traditional theorem-proof style) don't seem to be written with a learner in mind. By presenting all of the mathematics in a *mathematically* logical progression, many of them end up hiding the kinds of thinking that has to happen in order for someone who doesn't already know the math to learn it. After all, mathematicians don't do their work by smoothly going from stating fixed definitions to giving a theorem with proof- there's a lot of work going on there that we don't see in the formal presentation. I should be careful, though, not to exaggerate. Most textbooks try to give some exposition to help the reader along. However, this usually doesn't do enough to change the fundamental problem of structure that comes with using the mathematically logical sequence to guide the organization of a book intended for learners.
You may find that some of the newer so-called "reform" materials may be closer to what you are looking for. Many of them do make an explicit effort to focus on the ideas and concepts underlying the mathematics (though some complain that they don't focus enough on developing fluency with procedures). The trick with these is that, when used in schools, they generally work best with teachers who themselves have this kind of deep understanding and thus know where the materials are pointing. There has been quite a bit of venom circulating around these newer materials. My suggestion is to try a few different kinds of materials in both the "traditional" and "reform" styles, and see what works for you.
So, here are a few suggestions of books that I found useful in making sense of mathematics, its ways of thinking, and how it can relate to the world. The first several aren't really textbooks, but rather books about mathematics.
Philip J. Davis & Reuben Hersh - The Mathematical Experience
George Polya - How to Solve It
John Allen Paulos - A Mathematician Reads the Newspaper
John H. Conway & Richard K. Guy - The Book of Numbers
Barry Cipra - Misteaks
The Calculus Consortium at Harvard has developed several textbooks, including Functions Modeling Change: A Preparation for Calculus (Eric Connally, Deborah Hughes-Hallett, Andrew Gleason) and Calculus, Single and Multivariable (Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum)
Mr Tompkins in Wonderland and Mr. Tompkins Explores the Atom are both fictional narratives that demonstrate relativity through greatly exaggerated examples-- apparently Mr. George Gamow has written an umber of other physics books as well.
They're fun to read, and definitely helped me in high school AP physics.
I would recommend Mas-Colell, Whinston, and Green's "Microeconomic Theory" and Obstfeld and Rogoff's "Foundations of International Macroeconomics" Both presume only a limited background in mathematics (and economics) and have generous explanations of the mathematical tools being used.
Sig (appended to the end of comments you post, 120 chars)
Danny.
I have written over 900 book reviews
I'd add to the math list: 1, 2, 3... Infinity. by George Gamow. Also to the physics list: Einsteins Theory of Relativity by Max Born. A wonderful primer on relativity using nothing more than HS algebra.
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Not everyone by birth is a genius at math, some people must work for YEARS to get the B in math.
"If you can't even get a B in a community college undergraduate math class,"
I'm not a Math person.
"you're not going to make it at Harvard or any truly "ELITE" university, private or not. Sorry."
Thats exactly why I wont major in math or science at Harvard.
"Getting a real education takes work on your part, not simply gaming the system for least effort per credit or slapping the right label on a bogus degree. It's not something other people do to you, it's something you do for yourself."
I am working, but I also know the system is not a very fair system, and the system does not reward hard work, it rewards those who "game" the system. So yeah I could learn math, get a C in math, have a bad GPA and never get into an elite private university, or I can get a good GPA, find some way into an elite university, and then take the math classes when I'm there.
I see no reason why I should take them now and get bad grades now when my grades actually matter when I can get bad grades later. And what you said doesnt make any sense, you act like a person must get a B in every single class they ever took in college, we all know that this is very unlikely as most people are humans who have strengths and weaknesses. I might get a C in Algebra and Calculus, but I'll never have to take those two classes again once I actually go ahead and do it, so for you to tell me that because I cant get a B in calculus that I'll never be able to handle university is pretty ignorant, I mean sure if I were majoring in math and science you'd be right, but I suppose you didnt do a good job looking at the list of possible majors which do not require you take tons of math classes.
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http://www.math.com/
http://homeschooling.about.com/cs/math/index.htm?t erms=math
http://homeschooling.about.com/cs/science/
http://physics.about.com/
What is Science?
Even on the off chance that the About network doesn't have all the information you need, they have a large number of links to sites with relevant information across the Web, so there's a very good chance that you will be able to use them to find what you are looking for.
Also...although these are not strictly an answer to your question, I would still heartily encourage you to follow the links to these (listed in a suggested order of reading...my probably misguided opinion only) text files, web pages, and books, as I think they could be of enormous benefit to both your children and yourself...indeed, anyone who wishes to read them. Although I understand that several of these could possibly only be understood at tertiary level, they also as far as I know are not normally included in *general* curriculums, and IMHO they should be.
It used to be in the past that the education systems of most nations didn't want us to know the why (philosophy, religion, history, political theory) of life, but were content enough to let us know the how. (Science without analysis, numeracy and literacy skills, etc) Now however we are seeing that primarily in America, but also in other places, government education departments no longer even want to allow people to know the how.
Mathematics is part of the how - a means to an end, a way of solving problems - but it is not a destination in itself. The material I've given you links to in my second section is concerned with finding out *why* - "Why am I here? Who am I? How do I know what reality is? What do I want to do with my life? What moral values do I believe in?"
The answers to these questions are far more important than becoming merely literate or mathematically capable for their own sake. Figure out what your purpose is first, and the rest, although still requiring work, will be relatively easy. That is what the links in the second list will help you do, and it's not something you'll be taught to do in any contemporary public school, either...Governments consider people with purpose to be highly dangerous.
Barbara Lee Bleau Ph.D. are excellent books. I was in a similar situation in that I decided to go back to college at age 32. Being that I was educated in Louisiana (worst in the nation) I never was properly taught many math principles. I was very fortunate when friend pointed me to these books. Both book start under the assumption that your math understanding is at an elementary level (basic addition, subtraction, multiplication, and division.) It is a truly great teaching guide and workbook which was so successful for me that I passed the math placement test at The Univ. of North Texas and will be taking Pre-Calc this semester. As for physics, I have seen several great books recommended so far. I'm reading Dr. Hawking's book right now.
There is nothing inherently safe about liberty. That's why so many people died protecting it.
E=mc^2 by David Bodanis is an easy to read book and explains that equation we all "know" so that even cameron diaz can understand it. www.davidbodanis.com has got lots of physics related stuff for the interested too.
If I read the grandparent correctly, he wasn't saying that problems were dumb. He was saying that understanding the idea behind the problems before attempting them is a better way of learning for him. When I was back in high school, Calculus came pretty easy for me, but I remember some of the kids having problems doing the examples. The problem was that after they got the hang of one type of problem, they would get the next type of problem (say, differentiation), and not understand what to do. Since my teacher focused on rote techniques to teach things, when we moved from simple problems like differentiating x^2 to relatively nasty ones like x^3*sin(x)/(2*x). Since they didn't really understand *what* they were doing with the simple examples, they got way over their heads when they hit the uglier stuff. The best thing for me was having a friend that explained what I was *actually* doing when I did the techniques, so that when I got to the nastier stuff, I still understood what was going on.
I'm a lawyer, but not yours. I wouldn't represent someone who thinks taking legal advice from Slashdot is a good idea.
Any of his non-fiction books, and there's a ton. All subjects, from algebra to the brain to chemistry. (He even wrote about the Bible...)
As an avid Asimov fan (fiction and non-fiction) I concur - his science books are fascinating.
They would make great ebooks - especially since most are collections of short essays. I suggested that to one ebook vendor of his SF stories, and they said they'd look into it. Never saw them offer them, however. Guess I'll have to dig up my old paperbacks hen i get home.
That's one problem with libraries - you read a lot of great books, and when you can finally afford to buy some of them, they're out of print.
I'm a consultant - I convert gibberish into cash-flow.
Guidelines:
1. If you really want to understand mathematics, stay away from suggestions made my engineers; in particular, eschew books that dumb down mathematical theory in favor of the 'this is how you compute the solution' approach. Silvanus Thompson I find to be especially egrigious in this regard (those who try to learn calculus from Thompson will never understand the rigorous notion of a Limit, which is hardly pedantic since the derivative is itself a limit and the Riemann integral is the limit of a Riemann sum).
2. Be patiant with yourself. Geometry, Analysis (which includes what is called calculus) and Algebra have required centuries of constant effort to develop. If you go for the 'fast and cheap' approach to learning it, you will aquire nothing more than skills, when what you really want is knowledge.
Books:
Preliminary topics: Before you can think, you must memorize certain things and learn other things by rote. This will be hard and painful, but these fundamental topics are to mathematics as the alphabet and grammar is to Shakespeare, Milton, and Joyce. They are: the notion of a function, the laws of exponents, elementary trigonometry (sine, cosine, tangent, and their inverses), the binomial theorem, the definition of a polynomial, factoring polynomials, setting up applied problems in algebra, linear equations and their graphs, simple nonlinear equations and their graphs, slope and area, the Pythagorean Theorem. Most of these basic noitions are covered in Forgotten Algebra (which is published by Barrons for people just like us, and College Algebra, by Michael Sullivan.
Fundamental Notions:
By fundamental notions I mean ideas that form the basis for other ideas. Mathematics is all about definitions, and definitions are all about ideas; you cannot learn complicated ideas without understanding basic ideas (if you don't believe me, try explaining why every vector space has a basis to someone who doesn't understand what linear independence is). Unlike preliminary topics, fundamental notions are actually fun to learn--you get to think instead of just memorize and drill! I know of one wonderful book for this sort of thing, for someone in your position:
1. A Tour of the Calculus, by David Berlinski. This will make you think about what 'continuity' is. Good preparation for calculus, which is all about continuous functions, and good because it presents mathematics as a branch of philosophy (which it is).
Single Variable Calculus
Single variable calculus is where you will find most of the major concepts in the subject; the next time you will think this much is in linear algebra, when you study why the derivative for a n-dimensional vector space is actually representable in terms of matrix multiplication (the derivative is a linear map.) Here are some good books on calculus:
1. Calculus, Thomas and Finney. This text features a superb fusion of theory and application. The exercises are challenging, but doable for an independent student, and solution guides are available (these are indispensable as you search, at 2AM, for the mistake in your integration by partial fractions problem that required nine pages and is off by a constant).
2. Calculus, by Michael Spivak. My favorite calculus book. A brilliant synthesis of upper division real analysis and run-of-the-mill calculus. Reading it is like feeling awestruck by the beauty of someone you have known for years and years. This also has a solution manual (which you will need, because here there are proofs).
Advanced Mathematics
Don't stop learning math just because you
"Oh, the tragedy of math gone wrong. I can't even talk about it." -Wil Wheaton http://www.wilwheaton.net
These basic proof techniques are the very basis of mathematics. The reason so many people get through high school with little understanding of math is that they are never forced to do any proofs outside of Geometry.
And the sad fact is that in Texas, that last bastion of logical thought and problem-solving called Geometry is being completely gutted thanks to the new state learning standards called TEKS. Gone are the days of teaching high-school students proofs and reasoning. Instead, geometry is now taught as an extension of algebra, basically "plug and play" with no underlying foundations being taught. The reason? It's very difficult to use standardized testing to test whether a student has mastered the logic of reasoning. The only discipline a high school student in Texas might be introduced to reasoning and proofs is calculus, but this is not a required course, and very few students see the need to take it.
A very sad state of affairs IMO.
Area of a sphere? 4 pi r ^2...no calculus needed ;-)
:-P ::silence::
Of course a (an astute) calculus student would notice that when you derive the volume formula for a sphere (4/3 pi r^3) with respect to the radius you get the area.
My dad is an engineer (I will be too soon...hopefully ) and he has a novel way of find an oddly shaped area.
As long as what you are looking at has a scale of some kind you can actually cut out that area and weigh it on a (sensitive) scale. Then cut out a known square dimension from the same paper. Now you know what that area is relative to a certain weight...well now finding the original area just takes a little knowledge of proportions.
Granted it is not exactly going to score any points in the rigorous category, but it will get the answer with uncanny accuracy, which is the only category engineers have anyway
Yeah I am lucky they don't have -1 geek as a moderation...
--Joey
The best piece of advice I can give anyone trying to learn from a textbook is to tell them to work through the problems. Anyone should be able to pick up many of the textbooks listed below and work though as many of the problems as time allows (limited either by patience or by real life events). Most textbooks provide answers to selected problems, so you can check your progress.
Absolutely, 100%. Nobody is born with the ability to take a triple scalar product or multiply two matrices (both happening in your video card when you're playing Doom!). As a great Calculus teacher once announced to his class through a thick French Canadian accent, "Math is not a spectator sport." (Actually, it came out as "Matt ees not a spectator sport.")
Having said that, Calculus is my favorite kind of math. It's incredibly elegant and probably the most useful advanced math, as it touches everything you do. Consider your car. If you calculate your speed using a watch and the odometer, you have an idea how fast you were going, but your speedometer is actually showing you the value of the derivative at any instantaneous time. Your speedometer shows the rate of change of position (distance travelled) at any instantaneous time. That's calculus.
Don't be afraid. "Calculus" (besides being a formal term for tartar the dentist scrapes off your teeth) means small stones in Latin... small stones as used for counting.
Two *great* books on the subject:
Remember: Do the problems, succeed. Don't do the problems, fail. It's that simple.
Fire and Meat. Yummy.
Math is a very important thing in High School... But not in the way that a lot of you think... There are two uses for math in High School...
1) To teach the concepts of basic math and calculus.
2) *The most important* To exercise the students
brain and to keep them mentally alert.
When a student graduates from school it is a huge shock to them because the world is a lot slower then it is in school (at least it should be if they were working hard). Suddenly you don't have home work every day... You don't have tests every week and there are no such things as exams... Work is very much different. Now some businesses do testing on there employees... but it's not as bad as school...
When you drop math... you drop creativity, the ability to learn other subjects, to stay focused, and most importantly... to stay curious...
--
There I finally was smart enough to save this as plain text lol.
For maths you might want to try the books of Ian Stewart.
The desire to understand the world and the desire to reform it are the two great engines of progress -- Bertrand Russell
Just wonder if there are any good online sites that can help adults who aren't fortunate enough to have the opportunity to properly learn math, science, or whatnots that most think are _basic_skills_ ?
Thank you !
Muchas Gracias, Señor Edward Snowden !
One of the subjects that really put it all together for me was Linear Algebra. It doesn't require calculus so much as a certain mathematical sophistication. The book that made it interesting for me was "linear algebra and its applications" by Gilbert Strang
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I happen to like Stewart's Calculus with Vectors book. Covers from precalc (quick review) all the way through 3-d vector calculus. Lots of problems and decent examples. I used this book as an undergrad to learn calc, but even as a grad student I often find it invaluable as a reference.
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