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Calculus Hubbard and Hubbard for multivariate calculus

  1. May 3, 2016 #1
    I've taken the single and multivariate calculus classes at my school, (college class offered in my high school for accelerated students). I'm currently a junior, but in the summer of my senior year, I plan to read a book on proofs, brush up on math glossed over by the american education system (combinatorics and geometry), and do spivak's calculus book and hubbards vector calc book simultaneously.

    Will spivak's book and hubbard's book cover all that would be covered in a standard college calculus track (this is rather vague, I'm not sure how to express this in words. What i am trying to ask is whether or not i could potentially replace an entire multivariate calculus course in college with hubbard's book or if his book is good, but lacking in content in some areas)

    The book is arrive some time in the next two weeks, and I'll compare the table of contents to a college syllabus when i get time. for now i rely on all you helpful peeps in physics forums.

    any comparisons with spivak's calculus on manifolds, and trombas vector calculus book would be helpful too. I am planning on majoring in ee and at least minoring in physics, but I do not mind focusing on the pure math side of calculus, as i find math equally as interesting.
  2. jcsd
  3. May 4, 2016 #2
    I can't help you in reference to the Hubbard book, but I picked up Calculus With Analytic Geometry by George Simmons and Intro to Modern Linear Algebra by Poole, and both give out what students typically study in freshman/prep years.

    Simmons goes over which topics form the core of the material and gives out a few options and personal reservations as to which materials he would focus on. Poole gives out a simple "plug and play" chapters and sections that would typically fit in 1 semester - and which material would constitute a more in-depth 2 semester course.

    Calculus courses can and do vary from school to school, but the rules are all the same..

    Essentially, so far I have found that I just like going through the book in its entirety and focus on understanding the concepts and being able to apply them to the problem sets properly. Of course I encounter many challenges and use Khan Academy and Youtube to help me through some examples - as sometimes the way ideas are written down in a book don't always make immediate sense to me.

    I see more value in really learning the material for what it is than really setting a bar in terms of what College/University covers. With the subjects you mentioned, the more math you do the better. Any more mathematic education is only another tool in the box.

    I would really recommend understanding Single Variable Calc properly before diving into Multivariable. High school Calc might not be quite the same as college course.
  4. May 4, 2016 #3
    As long as you pay special attention to translating 3d and 2d vector calculus problems back and forth between the language of differential forms and the language of vectors, div, grad, and curl, then the answer is yes. In fact, you'll learn a lot more from Hubbard & Hubbard than from a typical calc III class, because they cover things like Lebesgue integration, manifolds, and the inverse and implicit function theorems.

    I cannot stress enough, however, that you must practice using the traditional 3d vector calculus methods as well, if you want to understand much of the published material on differential equations, electromagnetism, fluid flow, heat transfer, etc. I recommend getting a cheap copy of any calculus text from the past 50 years or so which covers vector calculus, so you can study the traditional approach and practice translating its problems into the language of differential forms, then translating the answers back into vector notation.

    For example, you can get a used copy of Thomas' Calculus 12th ed. and a used copy of the multivariable solutions manual for that edition for a total of less than $20 after shipping.

    You are on the right path, though. Spivak and Hubbard & Hubbard will give you a much better understanding of "vector calculus," which will serve you very well when solving real, original problems. Also, the methods you learn from them will work in four and more dimensions, and on arbitrary curved surfaces and spaces. Plus, the generalized Stokes theorem replaces all the integral theorems I don't individually remember from Calc III with a few rules that are easy to remember because you use them every time you calculate an integral with differential forms.
  5. May 4, 2016 #4
    Much thanks for both of your lengthy and insightful responses.

    I'll probably check out tromba or some other vector calculus book while I'm reading Hubbard and Hubbard
  6. May 4, 2016 #5
    I read Marsden/Tromba and Hubbard/Hubbard, and I am currently reading Spivak's manifolds book. M/T has a challenging, physics-oriented problems, but the book is very short in proofs as it is primarily designed for applied aspect than theoretical. H/H is fantastic book for a first introduction, and he does cover necessary prerequisites from the analysis; however, you should also pick up Hoffman/Kunze if you intend to learn the linear algebra as well, as H/H is not detailed in linear algebra. I can tell you that you do not need to read Spivak's manifolds if you are going to read H/H. After H/H, read books in differential geometry or other interesting treatments such as Loomis/Sternberg.

    I personally did not read Spivak's Calculus, but I do not like those "transition" books. I think there are better books, such as: Tao's Analysis I-II, Mikula's Foundations of Analysis, Shilov's Elementary R&C Analysis, and Sohrab's Analysis. They all assume no strong background in proof skill, and they are design to teach undergraduate students the analysis (same coverage of topics as Rudin-PMA) and sharpen their proof skills. I also do not think proof books are the best introduction as they usually enforce their proof style, but it is the best to develop your own by trial-and-error.
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