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Calculus Rigorous calculus textbooks from intro to advanced

  1. Jun 20, 2017 #1
    My ultimate goal is to become a theoretical physicist, a great one at that. I have mastered the prerequisites and I am now looking for rigorous calculus textbooks that make some references to physics, or are more orientated for people who want to become physicist.Thanks for your help.
  2. jcsd
  3. Jun 20, 2017 #2
    Not sure what you mean by rigorous, but for your purpose,I highly recommend Savov.
  4. Jun 21, 2017 #3
    Edwin E Moise: Calculus.

    It is a great balance between applied and pure math. It falls between Stewart and Spivak/Apostol. It is closer to the latter.

    Really wonderful topics that are covered with the insight of a Harvard professor.
  5. Jun 21, 2017 #4
    What other books like Moise's do you know i.e. in between "spivakish" and "stewartish" but still a little bit on the side of Spivak? I tried to glance at the pdf version and it looks a lot better than the standard "thick" books these days. His book is hard to find and old (although it doesn't necessarily mean it's not good).
  6. Jun 21, 2017 #5
    Thanks, but I have heard apostol makes the most references and applications to physics.
  7. Jun 21, 2017 #6
    Simmons (avoid the first edition)
  8. Jun 24, 2017 #7
    If you go on Amazon you can find Moise for $30. This includes book 1 and 2. The cover is green. It is not that expensive compared to todays textbooks.
    Hmm Calculus based books? Only Moise. I found Courant more readable than the other two famous rigorous calculus books, I would not recommend Courant. It is a great book, but maybe too hard.

    For Physics (general intro physics):

    Alonso and Finn: Fundamental University Physics. Really great preparation for upper division physics. If a person finds KK or Purcell to hard. They can first go through Alonso, and get a greater understanding of the above books.

    The problem here is cost. They are not cheap. If you can find all 3 books for less than 150, I would buy it. Although 150 is a lot of money. The amount of knowledge I gained from these 3 books was worth the value. It made me like physics. Try amazon, or even Ebay from time to time. I was gifted the whole set from a professor. You could try getting the indian market ones. But, not sure of print quality.

    I also like Kip: Fundamentals of Electricity and Magnetism. I supplemented this book with Giancoli when I took intro EM.

    For Ordinary Differential Equations:

    I found Simmons book OKish. I much prefer Ross: Differential Equations. Ross can be found for less than 10 dollars shipped. Get an older edition. I have the green colored one.

    There are some more, but for different subjects. Let me know if you need other recommendations.
  9. Jun 24, 2017 #8
    I'm going to answer a little differently in case this would be your first exposure to calculus and since you are self learning. My view is that a lot of rigour is counter productive for a first exposure and it's more important to develop an intuition for what it means and how it might be used. I would recommend a textbook that is heavy on applications, graphs, visuals, intuition and for which solutions are available. These are things that you would normally get from the course instructor as you go but for a self studier they must also be in the textbook.

    I will second Smodak's recommendation of Savov for this purpose. The name of the book really turned me off at first but when I started reading it, I found it to be really good and chock full of the kinds of things a prof or a tutor might tell you ("don't forget that this relates to this other thing" or "people often mistake this for this other thing, here's why they're not the same", etc.).

    The other option I suggest as a more traditional first exposure to Calculus is https://www.amazon.com/Calculus-Early-Transcendentals-James-Stewart/dp/1285741552 This one is used in many universities for their introduction to calculus, and as a result is extremely polished, tries to appeal to a wide audience but is expensive. It has lots of great colourful graphs, visualizations, solution manuals available and provides examples of applications to many different fields. (You can save a lot of money by getting a used, older version. The main difference between them is that the exercises get revised every few years to help profs save time on finding problems whose solutions can't be easily found using google.)

    Once you have the idea of calculus in your head, you may enjoy moving on to something rigorous like Spivak's Calculus text, which is widely praised and loved among people who have already had some basic exposure to Calculus. Going through Spivak would take you more in the direction of how a mathematician views calculus rather than as physicists and engineers tend to see it, which is as a tool.

    The other recommendations for books on differential equations are another good step afterwards. If you've not heard of diff equations before, here's an analogy: in the same way that grade school students first learn how to multiply two expressions, then how to do the inverse, dividing them, and later they learn how to solve equations that incorporate the need to multiply and divide... here it's the same thing. In calculus, you learn how to differentiate expressions and then how to do the inverse, integrate expressions, then in "differential equations" you learn how to solve equations that incorporate the need to differentiate and integrate.
  10. Jun 24, 2017 #9
    it is a waste of time learning from Stewart when there are other great text to learn "applied calculus from."

    Thomas: Calculus with Analytical Geometry 3rd ed comes to mind. Mathwonk strongly recommends this book. There is also Thomas/Finn: Calculus with Analytical Geometry 9th ed. I much prefer the 3rd ed. It is the book I learned Calculus from.

    The ideal study guide for Calculus, If I could go back in time and learn it from the beginning, would be Moise and Thomas Calculus(3rd ed) as a supplement. What ever you do not get after reading Thomas, then look at Paul's Online Calculus Notes.
  11. Jun 24, 2017 #10


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    spivak is for mathematicians. i recomend apostol. you cannot go wrong with that book in my opinion, whether you go to maths or to theoretical physics.
  12. Jun 25, 2017 #11
    Yes! This is a great site and for those who like hard copy, you can even print the content out in textbook form (several giant books!)

    Though I'm puzzled by the strong reaction to the Stewart text. It's almost the defacto standard calculus text for 1st year students and the book sells so well that Stewart became a millionaire many times over.
  13. Jun 25, 2017 #12


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    Remark on Stewart text. It went through several editions, being dumbed down over that process in my opinion as a teacher from it. I rather liked the clarity of the maybe 2nd edition, back when it also was not so expensive. The authors of Edwards and Penney also felt he had essentially cribbed from their successful text as I recall. All these "everyman" calculus texts that get successful are then increased in price and the authors are pressured to increase their audience by reducing the difficulty. If you want Stewart, I would suggest an early edition, maybe 2nd, just as is the case with Thomas. Edwards and Penney is also a nice book in 1st edition especially. Sometime after that they removed some of Keplers laws and made other gradual compromises...
  14. Jun 25, 2017 #13


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  15. Jun 26, 2017 #14
    Thank you, I will be going with apostol though. I want a rigorous book that can develop some mathematical maturity, and make my single and multi variable calculus the best it can be.
  16. Jun 26, 2017 #15
    All the suggestions above are great and while the book I am to recommend you is not for beginners but, I recall Loomis and Sternberg having a great expedition something related to Physics as well with a rigorous application.

    EDIT: It has an entire section on Classical Mechanics done in a highly rigorous fashion. I would recommend it after reading Spivak's Calculus and his Manifolds book.
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