1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Guidance to advanced Calculus

  1. Feb 23, 2013 #1
    Hello, to clearly get my point through I think I should give a bit of background info first, or just skip the next paragraph to tl;dr.

    I'm currently in Grade 9 and in a Pre-Calculus 11 course. A few months before the end of Eighth Grade I peered into the back of my school agenda and took a closer look at a physics equation reference sheet which basically triggered me into full math geek. It had basic equations like [itex]F=ma, {F_g}= \frac{G{m_1}{m_2}}{d^2} , a= \frac{{v_f}-{v_i}}{t} [/itex], ect. I had been studying theoretical physics (theories, not math) since age 11, and have always shown prowess in mathematics, especially in middle school. From there on out in a span of about 6 months I learned the rest of the Math 8 curriculum, as well as most of Math 9, Math 10, Math 11, Math 12, and Calculus 12. I had also studied calculus and learned how to do single variable basic/moderate derivatives, integrals (definite/indefinite), and basic separable differential equations. Earlier in this year I managed to show my Science teacher (also to be my math teacher next semester) enough of my math abilities to get him to let me skip Math 9. I scored 100% on the exam, the school then let me do the provincial (haven't gotten the results yet) for Math 10, and a test to determine my class grade. I got 81%, i looked at the test after and recognized my bone-headed mistakes. They swapped my Math 9 for Pre-Calculus 11. Looking at the rest of high school math I want to get back into studying math. the point that im at now is Calculus, specifically multivariate, and differential equations. I also do this to make it easier for me to study physics in my spare time, as i am interested in doing mathematical physics rather then just studying theories. My aspirations for when I grow is going to university for a Ph.D in Theoretical Physics, and perhaps some level of degree in Mathematics. To prepare I want to learn more advanced forms of calculus, for both the sake of my math skills and my physics skills.

    TL;DR I'm in Pre-Calculus 11 and know single variable derivatives, integrals, and seperable differential equations.

    I would very much appreciate if somebody could list everything (important) to learn from basic single variable Calculus to advanced, or at least moderate, multivariate calculus. Please list what topics in calculus I should know to be able to carry out multivariate calculus problems, including any of my previously mentioned capabilities. Sort of like a check list.

    I would very much appreciate this, thank you.
  2. jcsd
  3. Feb 23, 2013 #2
    Go do ALL QUESTIONS in Michael Spivak's "Calculus". Then go do ALL QUESTIONS in "Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach" by John Hubbard and Barbara Burke Hubbard.

    You should be fine and ready to kill Real Analysis. In fact, your level of mathematical maturity could probably take on measure theory as well. You can definitely kill complex analysis if you've gone through both these books.
  4. Feb 23, 2013 #3
    Ok, are those both text books? If so I think ill ask my school's calculus teacher if the school has them. Or is there another way to get those books? Thanks....I, uh.... I'm pretty excited :D
  5. Feb 24, 2013 #4
    Yes, they are textbooks. Spivak is the best textbook on calculus that there is. It has many very challenging exercises. Since you already know the basics of single-variable calculus, then spivak is for you. It basically makes everything in single-variable calculus rigorous. It goes up to complex analysis and proving the fundamental theorem of the algebra.

    Other things you can do is linear algebra. I recommend the excellent "Linear Algebra" by Lang (or the less challenging "Intro to Linear Algebra" by Lang). Linear algebra is very crucial in physics and mathematics. It pops up everywhere. Seeing linear algebra in a rigorous and proofy setting is very much an advantage.
  6. Feb 24, 2013 #5
    Just some advice:

    Please don't say things like this.It shows that you don't really know what theoretical physics is.
    I don't want to call you out. But I can see you writing this on some essays to get admitted to a college. Then things like this might harm you. And it would be very unfortunate.
  7. Feb 24, 2013 #6


    User Avatar

    Staff: Mentor

    If you have money, you can buy them from amazon.com or wherever. If you have a university nearby, you can probably find them in their library. I would be surprised to find them in a high school library, although I suppose anything is possible nowadays.
  8. Feb 24, 2013 #7
    I'm tired of people saying go learn from Spivak to people who barely started high school... Honestly, it makes NO sense. Are you sure you know everything you need to from calc 1 and 2? Do you know how to expand using taylor series? Do you know your trig subs like the back of your hand? If you were to see an integral, would you know what methods to use as soon as you saw it, or at least where to start? (I.E U-sub, u-sub with a twist, partial fractions, etc). These things are more important than doing "hard" problems from Spivak if you are planning to go into physics (especially for analytic mechanics). They're better math subjects at this point than to learn more about calculus! I'd suggest learning about the methods used in linear algebra.

    But since you asked about calculus, go see if your library has a copy of a book of this book: https://www.amazon.com/Calculus-Analytic-Geometry-6th-Edition/dp/0395869749 and if it does, check it out. Really pound out your integral and derivative rules before moving onto a book like spivak or aristole.

    P.S. In case you're wondering, I've never studied from Spivak, and am currently studying exterior algebra/calculus. So if you do take the spivak path, and find it to be too "hard" and want to do something easier, you'll turn out fine ;D.
    Last edited by a moderator: May 6, 2017
  9. Feb 24, 2013 #8


    User Avatar
    Science Advisor

    I've met enough physics students "studying" differential geometry who have never had a proper course in calculus using apostol or spivak to say this is just not true. Who cares if he just started HS or not? If he has the motivation and the capabilities then there is no reason to tell him otherwise; to the OP if you can handle it then by all means do the book. It is a brilliant book and you will thank yourself for it later. There is no need to worry about the so called integration methods and Taylor expansions as those are unequivocally trivial in comparison to the theory. However it is true that in the future, near or far, you should also look to get into proof - based linear algebra. If it wasn't for micromass convincing me to start studying math properly from rigorous math textbooks, I would be a victim of the poor man's mathematics found in a noticeably large amount of physics texts.
  10. Feb 24, 2013 #9
    However, he isn't interested in theory. You don't need the majority of things you learn in a proof based math class for physics. There hasn't been one time in ANY of the papers ive read, or classes ive taken in physics that they've mentioned an epislon-delta proof. However, when deriving equations in various classes, I've needed needed to use several integration techniques. In several problems ive needed to use perturbation methods. These are things you need in physics.

    Proving that a surface is a minimal surface IFF it can be represented as a translation surface whose generators are isotropic curves (taken from my diff geo notes) has not helped me.

    So to tell a high school student who is interested in physics to go out and learn proof-based calculus, a subject even college students with more mathematical maturity have problems with, is just ridiculous.
  11. Feb 24, 2013 #10


    User Avatar
    Science Advisor

    "the point that im at now is Calculus, specifically multivariate, and differential equations. I also do this to make it easier for me to study physics in my spare time, as i am interested in doing mathematical physics rather then just studying theories. My aspirations for when I grow is going to university for a Ph.D in Theoretical Physics, and perhaps some level of degree in Mathematics. To prepare I want to learn more advanced forms of calculus, for both the sake of my math skills and my physics skills." (words were bolded by me)

    Did you actually read the OP's post or did you just start executing your diatribe impetuously?
  12. Feb 24, 2013 #11
    Proof based calculus is going to help him there how? I'll tell ya my friend, it's not. However, knowing trigs subs, u subs and perpetuation methods will!

    Also, advanced calculus isn't proof based calculus, it's more techniques used in calculus (renyolds transport, special functions, Fourier, etc) lol....

    Also, dunno what diatribe impetuously even means (im guessing it means something around im speaking BS ;D).

    EDIT: I did read his post, if you actually read it you'd see why i keep mention u-subs, trig subs :).
  13. Feb 24, 2013 #12
    This website should be helpful for you, it has everything from calc I, II, and III, to diff eq, some partial differential equations, and linear algebra which will be essential to know.


    You don't necessarily need to get some deep proof based book right away. As for actual math books, you might want to take a look at this one which I've had to use for my undergrad physics courses. It sounds like it is at the level you are looking for:

    Last edited by a moderator: May 6, 2017
  14. Feb 24, 2013 #13
    The OP did mention wanting to do mathematical physics, which is why I recommended Spivak. If he does mathematical physics, then he will encounter epsilon-delta proofs.
    If he was just interested in physics, then I wouldn't have recommended Spivak. It's exactly because he's interested in the math behind physics, that I recommended the book.

    Of course, you can argue whether the OP really wants to do mathematical physics. Maybe he doesn't know what it is. But Spivak will show him what math is about and whether he likes it.
  15. Feb 24, 2013 #14


    User Avatar
    Science Advisor

    Take a look at mathematical physics texts;they are extremely proof heavy (e.g. Reed and Simon Functional Analysis). Plus there is nothing wrong in learning proofs; they are much more enlightening than calculations.
    This is assuming too much. This depends totally on what the OP means by advanced calculus. Advanced calculus can just as easily mean calculus on manifolds. There is no uniqueness of the phrase advanced calculus.
    Lol not at all. I'm not claiming you are wrong. I'm saying you are not looking at everything he wants to do and are being impulsive. The OP has knowledge of how much calculus he actually knows and if he has access to the recommended books then he can easily judge for himself if the content is to his liking, interest, and capabilities. Who are we to tell him otherwise?
  16. Feb 24, 2013 #15
    Woudnt working through a

    1) complex variables book
    2) the suggested intro to linear algebra book
    3) A good probability book

    be more useful for an ambitious HS student than working through Spivak/Apostol.

    It seems like it would be better if he learned all the mechanics of math that keeps showing up in physics
    then once he learns this he move to a more advanced book.

    At least he would have a good foundation to build on to take honors/advanced versions of complex/linear alg/analysis classes in college.

    Otherwise you might get student who "self studied" Spivak and decides to skip classes in college prematurely. This could hurt when other students who took a course using Spivak/Apostol/Rudin instead of self studying are competing against him in a course.

    Or get a book on problem solving like
    the art of problem solving.

    To learn useful problem solving skills.

    Basically learn all the supplementary skills that will allow to really succeed in all your math/classes in college rather than just your real analysis class.
  17. Feb 25, 2013 #16
    Everybody here seems to think that Spivak and Apostol are some impossible texts that make people cry. It's true that they are more difficult than the usual calculus textbooks, but the books really are meant for freshmen in college who have some notions of calculus already. I think that fits the OP perfectly. He knows the basics of calculus and he wants to learn more. I know no better book than Spivak for that.

    And the OP is also interested in mathematical physics and pure mathematics. An encounter with Spivak would be ideal because he would already know now whether he would actually like mathematics. If he thinks Spivak is boring and useless, then he knows not to take much math classes in college.

    What Spivak teaches is not easy stuff. Epsilon-delta definitions are very difficult for newbies and it takes much much time to get used to them. It took my a year to really understand them. The earlier you encounter epsilon-delta proofs, the faster you are going to get used to them. If he goes into an actual real analysis class without previous knowledge of epsilon-delta, then things might get very difficult. And this happens all the time.

    Of course it's also important to be comfortable with different kind of substitutions, complex variables, linear algebra, etc. But those are just techniques, there is no conceptual difficulty in it. It would be good if he would work through them now since they pop up everywhere.
    But Spivak has much more concepts which are not easy to grasp. And if he can grasp those things in high school, then he will be much better prepared for math classes.

    And I don't recommend that he skips calculus or analysis classes. That would be a very dumb thing to do.

    And finally, if the OP was just interested in physics, then I wouldn't say all of those things now. But he's interested in mathematical physics. That means that he will have to do rigorous mathematics once. So I don't see the harm of doing Spivak.
  18. Feb 25, 2013 #17
    No, no, you're right fot calling me out on that. I used 'theoretical physics' in a very misconstrued fasion, I apologize. What I mean was to say that I'd been studying physics theories ( on things like radiation and nuclear decay) that were meant for laymen. Ill try to edit the op appropriatly, thanks
  19. Feb 25, 2013 #18
    Dont think that at all.

    I actually did what you were suggesting as a HS student and worked through Apostol in HS then used Rudin as an undergraduate (I should have added that initially).

    The reason I made those suggestions is not because I think Spivak and Apostol are impossible texts but because in retrospect I would have benefited more as a physics undergraduate if I would of spent that time working through intro level linear algebra,complex variables, and probability books instead of Apostol.

    This would of put the focus on how to be a physics undergraduate instead of focusing on specializing in mathematical physics. You dont get to really specialize until graduate school so its better to prepare for being a general physics student especially if you are talking about HS prep.

    As a physics undergraduate you will work more on problems dealing with linear algebra (Heisenberg formulation of QM); differential equations (Schrodinger formulation of QM); and complex variables (QM) and probability (Stat Mech/QM) than epsilon-delta definitions/real analysis or group theory.

    Prepping for your physics classes seems to make more sense if you want to be a physicist even a theoretical physicist. A lot of theoretical physicist who have done experimental work as undergraduates or master level students say they benefited by not specializing until later.

    That being said I suppose if he is planning to go to math graduate school then I suppose Spivak wont hurt since all my suggestions are for people who are trying to do theoretical physics which typically one does as a Physics PhD.
    Last edited: Feb 25, 2013
  20. Feb 25, 2013 #19
    Well, that's an interesting perspective. Now, I am not saying that doing complex variables, probability or linear algebra are bad things to study. Not at all, they are extremely interesting and useful.

    And again, if the OP is only interested in physics and doesn't want to take math courses, then Spivak is going to be useless. But if the OP is interested in mathematics and is interested in taking math courses in college (or double major), then I think Spivak is a very good choice. And if the OP is thinking about taking pure math courses in college, then doing Spivak now will show him if he likes it. So under these circumstances (which depend on the OP), I think Spivak is a good choice.
  21. Feb 25, 2013 #20
    Don't apologize, lol. I was just giving you some advice.
    Last edited: Feb 25, 2013
  22. Feb 25, 2013 #21
    I would just tell people you are studying "physics" or if you are studying nuclear decay and radiation I would say "basic nuclear physics" because saying you study "physics theories" is not a common usage. In physics people dont usually say they are "studying theory" but the do say they do "theory research" which is mostly working on the extension of current theoretical knowledge or application of theory to make predictions for new experiments.
  23. Apr 14, 2013 #22
    Proofs are nearly essential if you want to fully understand the "how it works" part of math, in my most humble opinion.

    I'm in a similar position, except my interests are mostly in purer math. Thus, I think I can speak with some experience on this. Pre-Calculus geniuses FTW?

    When we get younger maths/physics students here, we should hail them with praise for taking the first step to getting closer to some of the coolest subjects ever, not argue about which textbook is better. He asked for our advice, not our arguments about why everyone else's advice is wrong.

    I shamelessly recommend learning at least some complex analysis. It is possibly the most beautiful form of math out there, and it gives some deep insight into how real analysis works. Linear algebra is also good if you go into theoretical physics, as it will make you a lot more familiar with the concepts of linear operators and matrices, which are widely used in the field.

    As some really personal advice, learn about tensors and try to work with them as much as possible before you have to use them. If you get used to them before you start using them regularly, you will feel a lot better.

    I'd recommend some textbooks as well, but I'm largely self taught. The best advice I can give you is to experiment with stuff on your own and to frequently watch recorded lectures online. YouTube can be your BEST friend when it comes to learning these things.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook