A suggestion for the math reader

Discussion in 'Math' started by amilton542, Jun 28, 2013.

  1. amilton542

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  2. studiot

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    Tensor and vector algebra and calculus suffer from the difficulty of chirality.
    So modern physical sciences are tending to move away from them towards differential forms, the exterior calculus and Clifford algebras, which do not suffer from this difficulty.

    Having said that they still have many uses eg the inertia tensor or the stress tensor.

    You should also be versed in the theory of linear algebra to study this stuff and aware of the difference in the mathematical and physics use and definitions of the terms vector and field.

    Having said all that, Borisenko is quite a digestible introduction to the subject, and pretty wide ranging for its size.

    go well
     
    Last edited: Jun 29, 2013
  3. BillO

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    Given the way humans are built, you'd think we'd be okay with that. Instead we seek symmetry. Odd, huh?
     
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  4. amilton542

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    Ha ha ha, I'm impressed Studiot - it is a small book.

    Yes I know, I have to read linear algebra in equal amounts as I do with the aforesaid.

    What books in this subject matter do you consider above an introduction? It all seems to lean towards manifolds (?) and differential geometry at the advanced level?
     
  5. studiot

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    To reply to your question I would need to know your angle.

    That is are you approaching the subject from a physics or pure maths point of view? I realise you have posted in the maths section.

    For an appreciation of the physics side you do not need all the ramifications of linear algebra, but should take away from it

    The notion of a vector and vector space

    The axioms of the vector space

    The idea of (orthogonal) bases

    Incidentally Borisenko (page 109) makes a start on the issue of chirality, from a physics point of view where he introduces pseudotensors and pseudovectors or axial vectors.

    Roughly speaking chirality refers to handedness, parity or both.
    Chemists talk of chiral molecules - left and right handed arrangements of the same atoms.
    Quantum merchanics refers to spin parity in some particles (leptons).

    The additional structure we can apply to some vector spaces, in particular euclidian ones that we use in physics.

    These extra properties include a second operation called multiplication.
    The inner product.
    Linear functionals
    Dual spaces

    The last three above lead on naturally to tensor analysisand analysis by linear forms.

    Note that mathematically tensors are a sub category of vectors (as in the inhabitants of a vector space)

    Whilst physically vectors are a sub category of tensors.
     
    Last edited: Jun 30, 2013
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  6. amilton542

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    Ok I'll hold my hands up, for now I'm approaching these areas from a pure math point of view. The physics side of the coin, I believe, will come later. The truth is, if I pick up an advanced book on rotating electrical machine design, magnetism (that will take it down to the nitty-gritty of quantum mechanics) or a whole book devoted to magnetic hysteresis, I just don't have the math competence in order to interpret the math language.

    You lost me on this idea of chirality. I didn't even know you could apply it to chemistry. But hey, I'm a novice in the making.

    Ok Studiot, thank you for your time.
     
  7. KL7AJ

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    Actually, the chirality of tensors is really handy when working with ionospheric physics...since the Earth's magnetic field gives you a bi-refringent medium. Much easier to keep track of these things with old fashioned tensors.

    I'm not tense about tensors, nor vexed by vectors. :)

    Eric
     
  8. studiot

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    OK then I suggest that you need to keep up a balanced development front of pure maths topics (they are interdependent) and not get too far ahead in any one.

    I suggest Linear algebra and matrix theory by Nering
    or Linear algebra by Hoffman and Kunze

    A first Course in Abstract Algebra by Fraleigh

    An introduction to Algebraic structures by Landin

    Complex Analysis by Stewart and Tall

    Elementary Differential Geometry by O'Neil

    Either Calculus or Mathematical Analysis by Binmore

    Together form a nice rounded set in pure maths.
     
    Last edited: Jul 7, 2013
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  9. amilton542

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    Yes, your mind-map of pure maths virtually coincides with mine - besides the exception of dissimilar authors. But, I am still going to add to the bookshelf.

    I must stress though, I - myself, do get in a lot of trouble associated with the college when my research shoots-off at a tangent. At this precise moment in time, I'm the only student left in the building tying up loose ends?

    Fortunately, it's my three-month summer window. So, with the sun beatin' down on me at 4 a.m in the morning, I can learn new maths 18 hours per day! For 3 months! :)

    Then, when my University entry is confirmed, I go hardcore on electronics and electromagnetism :cool:
     
  10. studiot

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    Binmore offers a particularly clear concise exposition of the transition from 'A' level to university analysis. It doesn't seem much but it represents a whole new way of thinking about pure maths.

    There are bigger heavier and more expensive tomes for all the subjects I mentioned, but I liked to use two books in tandem.
    One a short exposition, picking out the key points (without loss of veracity) that can be used a as road map for heavy study of the second treatise type work.
     
  11. amilton542

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    Hmmm, the University maths on an engineering degree I already know. I've looked up their online modules several times so as to see what I'm in for. I will not be learning anything new in that deparment. However, if I gain access into the University I've considered as my firm choice, then they permit the privelage in choosing two modules on anything you like per semester. So I'll choose maths and physics at degree level. If they expect me to read mechanics at 'A' level and not the Principia level, I'll be p*ssed.
     
  12. BillO

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    +1 for A first Course in Abstract Algebra - Fraleigh

    Other suggestions:

    Linear Algebra - Freidberg, Insel and Spence
    I found this very easy to follow and because of that actually read it cover to cover.

    Calculus Parts 1 and 2 - Kline
    While not an applied course, he uses some applied examples does make a good companion if you are taking physics.

    Advanced Calculus - Olmsted
    A bit more advanced and again with a few applied examples. Another good 'pure math for the physicist' text.

    Ordinary Differential Equations - Platt
    Linear and non-linear. A bit of extra practice. Pure maths, but not too rough going.

    Basic Complex Analysis - Marsden
    As the name suggests, with some physics examples thrown in.
     
  13. BillO

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    Pun intended?
     
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