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First, while I was out of town for the holidays, some kind person from TL whom I can't track down by account name (Steve from San Diego) recognized my "6 POOL" license plate in the LAX Terminal 5 parking lot and left a two post-it-note missive saying hello. If you wrote that and happen to read this, hello! I appreciated your note!
Now, on to mathematics.
As anyone who's followed my blog knows, it's been a little over 20 years since I graduated from college with a degree in physics.
While there's a certain conjunction of my college study with what I do today, digital lighting for computer animation, my current field isn't remotely mathematically demanding. Trigonometry, matrix arithmetic, and sometimes seemingly random information from fields like spectroscopy or optics can come in handy. However, all those things come up infrequently.
I don't know why, and I can't honestly say that there's a good reason for it, but I'm finding myself tempted to try to go back and review mathematics to try to get back, at least, to where I was twenty years ago.
This is a bigger task than it might sound. A lot of the mathematics I used routinely as a student is now half a lifetime away. Also, in the course of thinking back over my school experience, I think I missed some very key ideas. For example, a few years ago it occurred to me that I really never learned the technique of partial fraction decomposition of ratios of polynomials. This is a concept that's usually introduced around 10th grade (age 15 for non-US readers) at the latest, but somehow I managed to miss it. In college, it would occasionally come up in physics classes and I'd see how it would be useful but somehow never actually learned to do it except by trial and error.
That realization leaves me wondering how much other material I never quite exactly picked up in second year algebra and pre-calculus. Some of what I wasn't very good at in high school I was forced to master in college, such as trigonometry. Other stuff, like analytic geometry, I tended to look up as needed or just skip over.
Where this leaves me from a practical point of view is that I am not sure how far back to go. I'd love to just review calculus and the more recent stuff, but I feel like I am not on completely solid ground with algebra and that's a bad feeling for someone who actually completed a physics degree at a demanding college. 
Where I hope the community might be able to help out a little bit is in suggesting good, current textbooks for high school algebra, trigonometry, analytic geometry, and pre-calculus. This isn't a simple request though, because a lot of current math writing has been tainted by either teaching for standardized testing (which is of little use to me) or a trend originating in the 1990s toward students trying to synthesize math on their own from first principles. I'm looking for the kind of textbooks that might be used in a college class for math majors, if such students had to take a class on high school material.
So, any recommendations? Also, does anyone have any specific experience learning from the Spivak calculus textbook? Is it worth it? Is anyone able to compare that book with the classic Apostol textbook? (Apostol unfortunately costs 3x as much, but if it's a lot better I'm willing to go there.)
Anyone know where to go for a really comprehensive, mathematically rigorous coverage of trigonometry, analytic geometry, and high school matrix algebra? I'm not put off by rigor but I do value clarity.
I KNOW there must be some math or physical science majors on TL with recent experience looking at some of what's out there in these areas, and I'd love any insights you can share.
  
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In my experience in a maths major you tend to understand things from first principles and there is less of a focus on tools / algorithms like partial fractions (unless its from a computing perspective), its more about gaining mathematical insight i suppose, almost just a more rigorous version of the type of maths you do in a physics course. For example in optics (physics) you might treat Dirac's delta function as an infinitely tall spike with the sifting property but in a mathematics course you might consider it as the limiting case of a a*cos(ax)... ok not a great example.
Are you interested in re-learning the specific toolkit you learned in the past, or in enhancing your understanding? I'd suggest the latter and picking up a real analysis textbook but if you are more interested in algebra, trig, geometry and calculus maybe Khan Academy would be suitable ( https://www.khanacademy.org/ ). It isn't a textbook and I can't vouch for it but have heard great things about the website. Sorry I couldn't suggest a specific book!
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Thanks for the thoughts. When I say I prefer rigor, I am not necessarily pushing for the proof-based approach that would probably be most suitable for a completely rigorous mathematical approach. However, I definitely view the Khan Academy stuff as unacceptably "loose" in terms of its presentation. In any case, I really need my learning materials in written form -- I'm not looking for lecture material, though I might supplement with some of that when I get into certain very problem-solving-oriented areas like ordinary differential equations.
The Dirac delta function is a good example with which to clarify that attitude a bit. In my experience, the Dirac delta function was always defined as the function whose definite integral is 0 for ranges that don't contain the origin and a constant C for ranges that do contain the origin. However, my physics professors always prefaced such discussions with the caveat that its definition in that way is not mathematically rigorous. That's the kind of issue that I don't necessarily feel a need to get into deeply.
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Partial fraction decomposition is both boring (there really isn't any interesting idea in it) and pointless - because it is a purely algorithmic things and as such it is something for computers, not people (and any symbolic manipulation package like Mathematica or Maple will do it in a partial fraction of a second). Unfortunately, a lot of mathematics that is being taught in courses for physicists is the same - for some reason, I have spent several years being trained in eficiency of calculations that I really never do by hand now.
Even pure mathematics courses are to some extend plagued by that, but it gets progressively better as you dig deeper into the field. And then that is the place where you reallyy find the interesting concepts - not things that help you trick your way to calculating something quickly on paper, but the real deal. For example, I really liked Lie theory (reprezentations of continuos groups), fucntional analysis (which is essentialy linear algerba on infinite-dimensional spaces, but it's so much deeper) and I always thought I could eventually love agebraic topology if I ever had the time to get started on it (the learning curve is very steep and I never really neede it, so it never happened). But that's all from almost pure curiosity - even though I work in basic elemenatry particle physcics research, I barely ever get to touch anything that is beyond the introductory mathematics for physicists.
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On January 13 2013 22:57 opisska wrote:
Partial fraction decomposition is both boring (there really isn't any interesting idea in it) and pointless - because it is a purely algorithmic things and as such it is something for computers, not people (and any symbolic manipulation package like Mathematica or Maple will do it in a partial fraction of a second).
Thanks for your thoughts, since your experience is very relevant.
I was using partial fractions as an example of a key technique I'd missed along the way early in my math experience, not as an example of something that is typical of the kind of thing I'd like to review. That said, being able to apply that technique quickly without reaching for the computer can be very helpful.
When I was studying college physics the first time, Mathematica was available starting my sophomore year, and I had it and used it extensively, mostly for analyzing large data sets, since its fitting capabilities were far more flexible than other options. However, for routine computation for expressions with manageable numbers of terms, I don't find it very useful. The problem for me as a physics major always was that I'd usually have a preferred way to write a given expression based on what I was doing, and Mathematica almost never would find that for me. It's fantastic when the number of terms in an expression gets out of hand, or when I'm trying to evaluate something that requires a technique (like partial fractions) with which I'm a little weak. It's also great for using complex equations in a numerical context.
I've used recent versions of Mathematica and I do find that recently it's a lot better than it used to be in terms of the form it chooses to present for a result.
The other big problem with using Mathematica in place of doing the work by hand is that there are sometimes useful insights to be had from intermediate results that I don't necessarily see as easily when Mathematica solves an equation for me, even if the relevant intermediate results get printed out. I don't mind using the package to save time if I'm on familiar ground but I believe in doing the work by hand if I'm trying to learn something.
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United States996 Posts
you may already know of https://www.khanacademy.org/ but if not it might be worth a quick check out. they are all videos so it might be a little slow for the pace you are looking for but theres are videos covering all the subjects you listed
similarly, coursera.com has an algebra class starting up in a few days
https://www.coursera.org/course/algebra
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I would recommend that you check out Art of Problem Solving.
http://www.artofproblemsolving.com/Store/index.php
It covers quite a bit of middle school to high school level material with an emphasis on creative problem solving rather than just regurgitating techniques and processes.
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My calculus professor from years ago, whom I have great respect for, swears by Maple. It is math software that typically requires you to pay for a license (cracked copies are available). And this software is sooooo powerful. There are so many different add ons you can use including lessons. It is quite large and cumbersome tho and more oriented toward engineering math than pure math. But its worth looking into if you want something that can do absolutely everything. Its kinda like matlab actually but more user friendly.
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In my opinion, Spivak's calculus is great. (Though it could be argued that its more of an introductory real analysis textbook than a calculus textbook.) If you don't care much for rigor then you might like it less than I did, but it's also full of intuition and the connections between the concepts it introduces. It's very clear though in my opinion.
It depends on what you're looking for. As an example, Spivak proves Intermediate Value Theorem via supremums. On one hand, you can see this as focusing too much on something obvious: if you see it that way, then you probably won't like Spivak very much. In my opinion though, I think even if this seems "obvious", it really elucidates a few things:
1. How the reals are different from the rationals. Intermediate Value Theorem is false if you work with the rationals instead of the reals. But what property of the reals makes it true? The supremum property, and this proof shows you that supremums are in some sense a fundamental difference between the reals and the rationals.
2. How supremums are useful.
3. The continuity/compactness arguments for real intervals in general topology are strongly motivated by this proof.
While you could argue these are all only about rigor, but in my opinion, they're really about how certain concepts work together.
You should note that I'm very biased; I hated college math until complex analysis (which I thought was kinda cool), liked algebraic topology, and saw algebraic geometry and decided I didn't want to do anything with my life except for algebraic geometry. I'm very much so on the algebraic side of pure mathematics, and I'm very fond of anything that simplifies things conceptually, even if it doesn't help with computations at all. If you're only interested in doing computations, Spivak might no be a good book.
As for software: I'm personally quite fond of SAGE (http://www.sagemath.org/). But SAGE is definitely more on the algebraic side than Mathematica/Maple.
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On January 13 2013 22:47 Lysenko wrote:
Thanks for the thoughts. When I say I prefer rigor, I am not necessarily pushing for the proof-based approach that would probably be most suitable for a completely rigorous mathematical approach. However, I definitely view the Khan Academy stuff as unacceptably "loose" in terms of its presentation. In any case, I really need my learning materials in written form -- I'm not looking for lecture material, though I might supplement with some of that when I get into certain very problem-solving-oriented areas like ordinary differential equations.
The Dirac delta function is a good example with which to clarify that attitude a bit. In my experience, the Dirac delta function was always defined as the function whose definite integral is 0 for ranges that don't contain the origin and a constant C for ranges that do contain the origin. However, my physics professors always prefaced such discussions with the caveat that its definition in that way is not mathematically rigorous. That's the kind of issue that I don't necessarily feel a need to get into deeply.
Well the problem is if someone was mathematically rigorous they would say that the dirac function isn't a function at all - _________ -;
It's a distribution; if you call it a function, bad bad shit happens in the world of real analysis, esp. with regards to the Riemann integral.
Why don't you tell us some subjects you might want to go into. Analysis? Combi/Logic? Control theory? Quantum? There are so many @______@
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I used Spivak in my first year calculus course and very quickly realized that the book has less to do with calculus and more to do with rigorous proving techniques and analysis in general. Seeing as how you're not really into the rigorous proofs and more the computational stuff, Spivak is likely not the best book for you.
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On January 13 2013 22:57 opisska wrote:
Even pure mathematics courses are to some extend plagued by that, but it gets progressively better as you dig deeper into the field. And then that is the place where you reallyy find the interesting concepts - not things that help you trick your way to calculating something quickly on paper, but the real deal. For example, I really liked Lie theory (reprezentations of continuos groups), fucntional analysis (which is essentialy linear algerba on infinite-dimensional spaces, but it's so much deeper) and I always thought I could eventually love agebraic topology if I ever had the time to get started on it (the learning curve is very steep and I never really neede it, so it never happened). But that's all from almost pure curiosity - even though I work in basic elemenatry particle physcics research, I barely ever get to touch anything that is beyond the introductory mathematics for physicists.
D:
Anyone who loves those parts of maths is a freak
(I'm such a freak T_T)
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On January 14 2013 02:57 Nos- wrote:
I used Spivak in my first year calculus course and very quickly realized that the book has less to do with calculus and more to do with rigorous proving techniques and analysis in general. Seeing as how you're not really into the rigorous proofs and more the computational stuff, Spivak is likely not the best book for you.
Hey I'm reading that book right now! All I have to say is, taking specialist math courses is the hardest thing I've ever done, but thankfully I'm muddling through it. Back to inverse functions I go
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On January 14 2013 02:31 Nehsb wrote:
In my opinion, Spivak's calculus is great. (Though it could be argued that its more of an introductory real analysis textbook than a calculus textbook.) If you don't care much for rigor then you might like it less than I did, but it's also full of intuition and the connections between the concepts it introduces. It's very clear though in my opinion.
Your comments actually get me pretty excited about having a look. Thanks so much for posting. I'm not averse to rigor, and in fact I think I'm more inclined to appreciate it than not. I was just trying to communicate that that's not a complete hang-up to me.
Thanks again!
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On January 14 2013 02:43 ymir233 wrote:
Well the problem is if someone was mathematically rigorous they would say that the dirac function isn't a function at all
Yes, that's part of why our physics professors were having that discussion, to let us know to have more care with terminology when speaking with our math professors than they were going to take themselves.
Why don't you tell us some subjects you might want to go into. Analysis? Combi/Logic? Control theory? Quantum? There are so many @______@
Yes to all of the above?
I think once I'm solving undergraduate-level math textbook homework problems like a boss again I'll probably pick an area of physics to review. The three big ones are classical electromagnetism, statistical mechanics/thermodynamics, and quantum mechanics. I suspect getting back up to speed on those three areas will take the rest of my natural life.
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On January 14 2013 02:57 Nos- wrote:
I used Spivak in my first year calculus course and very quickly realized that the book has less to do with calculus and more to do with rigorous proving techniques and analysis in general. Seeing as how you're not really into the rigorous proofs and more the computational stuff, Spivak is likely not the best book for you.
If you've read my subsequent comments you'll probably get the sense that I'm ambivalent on this. I would LOVE to improve the rigor of my understanding of that material. I just don't want to miss out on refreshing my computational skills at the same time.
The comment one of the earlier posters made about Spivak being more of a real analysis book actually makes it more appealing to me. Later in my college career I took a junior level real analysis course and it was a lot of fun at the time, though much of it has since escaped me.
Thanks so much for the insightful thoughts! I appreciate it all.
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I love Spivak, but it's not what you want if reviewing calculus is your goal. If understanding calculus is the only goal, sure, knock yourself out, but any of the standard (Stewart, Briggs, etc) undergrad calc textbooks will do a better job giving you a nice overview of how calc works in practice. Later on if you're interested in the guts of how and why this all makes sense, check out something like Spivak or Rudin.
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Might be a bit beyond what you had in mind, and they are a bit pricey, but i think the Princeton series in Analysis, by Stein and Shakarchi is a really good set of "self-study" books. They cover a lot of ground, and are relatively verbose about explaining the how and why about the methods - at least as math textbooks go. They also have a pretty good number of applications for the concepts throughout, which may or may not be what you're looking for.
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about to head into calculus, I haven't got a clue what partial fraction decomposition is D:
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On January 14 2013 14:07 UniversalSnip wrote:
about to head into calculus, I haven't got a clue what partial fraction decomposition is D:
i've taken 3 semesters and it never came up from what i remember. so i think you're good.
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Is partial fraction decomposition that technique where they split an integral of a fraction with some function as the denominator into two or more fractions with easier functions as the denominators?
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Just about any widely used calculus/algebra*/geometry books for high school and freshmen college students are enough for you to understand the concept. Whether or not you can master or at least be competent in the subject depends on whether or not you're willing to solve problems and exercises on your own. It's this part that's the most frustrating and time consuming yet equally rewarding as well.
Too many students simply read the concept portion and examples given, thinking so long as they understand the concept, then they've mastered the concept, skipping the problems at the end of the chapter. Absolutely stupid.
Once you've done some problems on your own and have a good grasp of the concept you've learned, you're then better off using computational software like Mathematica, Mathlab, and MAPLE to solve such problems in the future.
As for looking into references with more rigor, specifically analysis as a more rigorous form of calculus, I would caution you that while you'll be more challenged and have more solid foundation, such could be excessive if your goal is to simply learn and be proficient enough to apply calculus/geometry/algebra to other things that don't require such rigor and critical analysis. It's like a C++ programmer trying to understand the electrical circuits of the CPU in the hopes that he or she will program better in C++.
*Be extremely careful of selecting an algebra book meant for college. The two standard algebra subjects (linear and abstract) in college is much different from the standard algebra in high school
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Hello sir! I have just finished my undergraduate degree in Mathematics and like you I know that I am missing out on alot of usefull things I could have learnt previously. The very first time I encountered logic , sets and proofs was in 2nd year of my university degree (in high school we did not cover sets or proofs at all!) and to further add to the injury they just told us:
"Right, here is a proof, this is how its done make sure you learn it because it will be examinable" and that is it, no how it was created nor why  . So I have decide to familiarise myself with logic, set theory and proofs using this book http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521675995 . I highly recommend this piece of knowledge giving contraption, it actually explains why, how and when to use particular proof techniques and starts off with introducing basic logic and set theory.
I am not sure if this particular topic is of interest to you (perhaps you already know how to proof stuff very well) but if this is something of interest for you then I highly recommend buying it (note: for Australian people, buy it online if you can since this book costs upwards of $200+ at Dymmocks/Kinokunya!!! blasphemy!)
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On January 14 2013 16:12 targ wrote:
Is partial fraction decomposition that technique where they split an integral of a fraction with some function as the denominator into two or more fractions with easier functions as the denominators?
Partial fraction decomposition is taking the ratio of two polynomials and factoring either the numerator or denominator to write the expression with lower-order polynomials. It's a basic algebraic technique, though it can be useful as a step in integration.
Regarding the comment about Spivak that it's not that great for learning computation, I suspect my memory of the techniques will come back pretty fast, and a more rigorous approach might be just what I need to push myself beyond where I was before in understanding.
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On January 14 2013 15:54 Carbonyl wrote:Show nested quote +On January 14 2013 14:07 UniversalSnip wrote:
about to head into calculus, I haven't got a clue what partial fraction decomposition is D: i've taken 3 semesters and it never came up from what i remember. so i think you're good.
It comes in handy for some integrations and laplace transforms mostly, iirc.
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On January 14 2013 16:16 BirdKiller wrote:
Just about any widely used calculus/algebra*/geometry books for high school and freshmen college students are enough for you to understand the concept. Whether or not you can master or at least be competent in the subject depends on whether or not you're willing to solve problems and exercises on your own. It's this part that's the most frustrating and time consuming yet equally rewarding as well.
Excellent point, and yes, I'm mainly looking to have problems to solve.
Too many students simply read the concept portion and examples given, thinking so long as they understand the concept, then they've mastered the concept, skipping the problems at the end of the chapter. Absolutely stupid.
I don't have a lot of sympathy for their point of view, myself.
As for looking into references with more rigor, specifically analysis as a more rigorous form of calculus, I would caution you that while you'll be more challenged and have more solid foundation, such could be excessive if your goal is to simply learn and be proficient enough to apply calculus/geometry/algebra to other things that don't require such rigor and critical analysis. It's like a C++ programmer trying to understand the electrical circuits of the CPU in the hopes that he or she will program better in C++.
I think it's safe to say that my goal is simply to go back through the material that I learned in college and strive for a better understanding than I had the first time.
As I was coming up on graduation, I achieved about a 60th percentile on the physics GRE. This tells me that I was doing a little better than average for physics majors serious about graduate study, but it also highlights that I had weak areas and I'm interested in trying to get back into those in more depth.
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On January 14 2013 14:07 UniversalSnip wrote:
about to head into calculus, I haven't got a clue what partial fraction decomposition is D:
was the same, no worries.
I actually first learned about partial fraction decomp in calc ii, so far it's not a big deal for undergrad math requirements for engineering majors :/
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I can't say I have a serious answer but here's an idea I've been experimenting with:
When I see something that I don't understand, but feel like I should, I review that exact technique. Then I try to think if there are some obvious related ideas I might want to go over. After I'm finished I make a mental note to go over these ideas again in a few days (spaced repetition). Review should mean writing out the material without using a reference.
For example I might see something like sin(a/2) = +-sqrt((1-cosa)/2)
I realize that I didn't remember this, so I do a google search for trig identities, and find that it comes from the identity
cos2x = cos^2x-sin^2x = 1 - 2sin^2x. Substituting a/2 = x gives the first formula.
This is nice, but I'm sure I'll forget it in a week, if not faster. So the plan is to learn the half-angle formulas, addition and difference formulas for sine and cosine (I actually still remember these so no extra work), double angle formulas for cosine (know this too) and the fact that they can be rewritten using the pythagorean theorem.
So I have 4 formulas to learn (sin(a/2), cos(a/2), cos2x with RHS written only in terms of cos^2x or only in terms of sin^2x) and a few pieces of connections.
A review should look something like writing:
sin(a/2)=
cos(a/2)=
cos2x= (something with cos^2x)
cos2x= (something with sin^2x)
and finally the derivation of the first two lines from the last 2. This should take less than 3 minutes if you actually know what you're doing. Note that I'm writing the formula before the derivation because I want to remember it without having to derive it every time.
BTW, if there's any misunderstanding, like where does the +- come from or what happens if the value under the square root is negative (it never is, do you see why?), those can be included in the review cycle too.
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On January 14 2013 03:02 BrTarolg wrote:Show nested quote +On January 13 2013 22:57 opisska wrote:
Even pure mathematics courses are to some extend plagued by that, but it gets progressively better as you dig deeper into the field. And then that is the place where you reallyy find the interesting concepts - not things that help you trick your way to calculating something quickly on paper, but the real deal. For example, I really liked Lie theory (reprezentations of continuos groups), fucntional analysis (which is essentialy linear algerba on infinite-dimensional spaces, but it's so much deeper) and I always thought I could eventually love agebraic topology if I ever had the time to get started on it (the learning curve is very steep and I never really neede it, so it never happened). But that's all from almost pure curiosity - even though I work in basic elemenatry particle physcics research, I barely ever get to touch anything that is beyond the introductory mathematics for physicists. D: Anyone who loves those parts of maths is a freak (I'm such a freak T_T)
No you're not. Perfectly normal to love what is beautiful to you eyes. First quoted post here summarized what is imo the most beautiful and interesting maths.
Edit : OP you know that MIT has a moutain of free videos of lecture from 1st year to 3rd year in maths ? You should give it a shot. For litterature sadly the books we use in France are often unknown outside, as much as I don't know who the hell is Spivak nor what you do in calc i, calc ii, etc. A book of Rudin 'principle of mathematic analysis" is quite famous on basic real anlysis now that I think of it. I don't know what you're searching for in Algebra, but if you're giving me more informations I can search in my bibliography.
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EPT
