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Calculus advice

  1. Sep 17, 2003 #1

    I want to buy a somewhat advanced calculus textbook to independently study from. The text doesnt need to be too pure and rigorous, but it should be more so than an engineering math text would be because I want to eventually get into theoretical physics, and I dont want to find out that I have learned stuff only half-assed or somewhat incorrectly.
    Do y'all have any recommendations or definite no-no's on what to get?

    I have been looking at Advanced Calculus 5/e by Wilfred Kaplan
    and at Advanced Calculus 3/e by A.E. Taylor and W.R. Mann.

    Does anyone have any experience with either of these? Is one better than the other?
    Kaplans book seems to cover a massive amount so I worried a bit about how detailed it is, but it may be fine.

    If you have info at all I would appreciate it,


  2. jcsd
  3. Sep 17, 2003 #2


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    Perhaps you could tell us what 'advanced calculus' is, since it's not a meaningful term, really. Do you mean multivariable calculus?

    - Warren
  4. Sep 17, 2003 #3
    yeah i kind of agree.
    it seems like the text should address multivariable calculus, vector differential/integral calculus, multiple integrals, sequences and series primarily.
  5. Sep 18, 2003 #4
    one of the texts i used was mathematical analysis by apostle or apostol or something like that. another was mathematical analysis by walter rudin, though i found that rather hard to read at first (and second and third and fourth). it's really skinny but jam packed with info. it goes into mv calc and stokes' theorem, implicit function theorem, and all that jazz.
  6. Sep 19, 2003 #5


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    Sounds like he wants to study the whole broad spectrum of Calculus to me fellas. He did say "independently study from" come on guys help a brother out. I personnally think that there is alot of nice interactive programs out on the web to get you started. eg; visual calculus. I think though you should go re-read algebra and trig before venturing alone into the realm of CALCULUS. These guys are great help for anyone wanting to learn.
    Good Luck!
    Dx :wink:
  7. Sep 21, 2003 #6
    One book I would definitly recommend to ANYONE in the math/science arena is one by Dr. Mary Boas formaly of Depaul Univeristy (not sure if she is still there). It is entitled
    It is the book I used for my Theoretical Physics class. It presuposes only a basic introduction into calculus (integration and Defferentiation), and those are ideas that can easily be learned from resources on the web.

    It has a wide reaching treatment of
    Complex numbers
    vector calculus
    Differential equations
    Partial Differental Equations
    Fourier analysis
    nonelementary functions
    Probability theory
    .. and more I cannot think of right now.

    The book is used to teach the Junior level Theoretical Physics Class at my University, and has a been an excellent resource for my Mechanics and E&M classes as well.

    If you are really looking to get a grounding in the fundamental Mathematics of physics this is the book to purchase

    That's my two cents
  8. Sep 22, 2003 #7
    Yeah, I have looked at Apostol for the calc, and it seems pretty good from what i can tell through the net.
    i think i am going to do linear algebra with anton's elementary linear alg., and then do the calculus with taylor's advanced calculus, and then maybe link the math with a physics geared text like Boas'.
    from what i can tell dx is right; i'd like to get a base in the broad spectrum...not necessarily in just one book.
    when do differential forms and differential geometry come into the picture, and what basics should one have before starting with them?

    thanks for the help.
  9. Sep 23, 2003 #8
    Uggghh...I hate that book. Of course, books work differently with different people but I used Anton's book when I was taking Linear Algebra.

    I won't do you much good in saying that "it sucks." However, I found the book rather ambigious and it constantly refers you to previous examples (and when you are in a hurry, you end up wasting more time trying to go back to previous examples). It was like "in Example 4 of Chapter 3, we did blah blah". When you turn to that example, it goes, "in Example 2 of Chapter 1, we proved blah blah". Then in that example, it was like "in an earlier edition of this book only available in Japan, we showed that..." The layout was decent but the ambigious nature of the worked examples made me to refer to other books when learning Linear Algebra.

    It was really frustrating. Agnew's Linear Algebra is a good book (clear although a bit heavy in the notation) and Lay's Linear Algebra is also a good book for starters (although it refers to the kernel as a null space without mentioning the word "kernel." Even Anton's book (I think) mentioned that they were the same)

    I learned Linear Algebra by going out and purchasing the Schaum's Outlines for the class. It was only about $15.00 so it's not too expensive and I'm strongly convinced that it helped me get an A in the class.
  10. Jul 23, 2004 #9


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    They are expensive now, but the books I recommend for your needs as you described them, are by Richard Courant, Differential and Integral Calculus, volumes 1 and 2.

    They cover one and several variable calculus, with as mucg rigor as anyone needs, but with an honest attention to physical examples as well. They include calculus of variations, and complex variables, and a very clear treatment of infinite series. I still recall clearly after 40 years something like: "In order to show a given series of positive terms converges, the best method is to compare it to series known to converge. This is the so called comparison test. For this purpose the usual example is the geometric series, which gives rise to the ratio test as well as the root test....The other method is to compare with an integral, the so called integral test."

    That is essentially all you need to know about series of constants, and it is not that succinct to my knowledge anywhere else.

    There is not any modern treatment there of linear algebra, but vectors are used in the old fashioned way. Many modern books amount to nothing more than a rewrite of the arguments in Courant.
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