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A Frankel: Geometry of Physics

  1. Jun 15, 2017 #1

    JTC

    User Avatar

    Good Day

    Early on, in Frankel's text "The Geometry of Physics" (in the introductory note on differential forms, in fact, on page 3) he writes:

    "We prefer the last expression with the components to the right of the basis vectors."

    Well, I do sort of like this notation and after reading a bit of the text (not easy, struggling, but learning), I am perplexed by one issue: Why?

    Why does he prefer to write the expression for a vector with the components on the right of the basis vectors? It is different from the way others do it. Is there something specific that is gained?

    (And just to stave off any controversy on the text--yes, I know some do not like this text--I am much less interested on whether his preference is bad, and only interested in WHY he prefers it.)
     
    Last edited: Jun 15, 2017
  2. jcsd
  3. Jun 15, 2017 #2

    fresh_42

    Staff: Mentor

    I think it's just a matter of taste. Maybe he likes the way it reads: "In direction v go c steps" rather than "Go c steps - huh? where to? - in direction v - see before". E.g. I knew people who actually liked the Polish notation and find the way all others do it confusing.
     
  4. Jun 15, 2017 #3

    JTC

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    Are you suggesting by this that in any application of this notation (e.g.: 1) rotation matrices of frames or 2) Hamilton's principle for linked systems or 3) something I am not even aware of yet) that there is no difference in whether it is before or after? Are you saying that there is no case where the notation makes things easier to understand or apply?

    I really do like what you wrote, by the way; and it makes sense. And I suppose I can live with it.

    But the flippancy with which Frankel says it, unnerves me just a bit.
     
  5. Jun 15, 2017 #4

    jedishrfu

    Staff: Mentor

    You must be careful in how you read mathematical notation.

    Sometimes its okay to change the order of things and sometimes there's a deeper reason not to.

    It varies with the author and the discipline so you need to always be aware of what is what.
     
  6. Jun 15, 2017 #5

    fresh_42

    Staff: Mentor

    Formally ##M \cdot R## and ##R \cdot M## with a module ##M## and a ring ##R## are two different objects. E.g. if we chose some matrix rings here to be ##M## and ##R## we can really get something different. (In https://www.physicsforums.com/threads/noncommutative-artinian-rings.885151/#post-5572046 is an example.)

    But if we want to write vectors form ##M=V## in coordinates, and have a (commutative) field ##R=F##, we also can swap left and right in all components, so it makes sense to allow ##\alpha \cdot v = v \cdot \alpha\,##. But technically, one has to chose a side.
     
  7. Jun 15, 2017 #6

    JTC

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    I am afraid I do not understand a sufficient amount of algebra to put your response in context. May I implore you to cut to the chase (forgive me for my facetiousness) and explain in a little more common English, what you are saying?

    Are you still suggesting it makes no difference except in so far as we "linguistically" understand: "go in this direction, a certain amount?"
     
  8. Jun 15, 2017 #7

    jedishrfu

    Staff: Mentor

    I looked at Frankels' notation and don't think its a good idea. He even admits that some may confuse it with partial differentiation.

    His reasoning has something to do with getting a 1x1 result and removing the summation symbol kind of like the Einstein summation convention and preserving the notion of v being a column vector.

    https://books.google.com/books?id=gXvHCiUlCgUC&pg=PR31&lpg=PR31&dq=frankel+We+prefer+the+last+expression+with+the+components+to+the+right+of+the+basis+vectors.&source=bl&ots=Km9Scq-Vgl&sig=YxycLRxec5l0tObkl1HJbffnMhY&hl=en&sa=X&ved=0ahUKEwif3eyPssDUAhWL64MKHeOwCnAQ6AEILzAB#v=onepage&q=frankel We prefer the last expression with the components to the right of the basis vectors.&f=false
     
    Last edited: Jun 15, 2017
  9. Jun 15, 2017 #8

    fresh_42

    Staff: Mentor

    If one has less nicely behaving objects than vector spaces and fields, left and right has to be distinguished, because they might have different properties.
    Formally, the need for distinction doesn't go away for vector spaces and fields as scalar domains, and I haven't found (on a very quick search) that the requirement ##\alpha \cdot \vec{v} \stackrel{(*)}{=} \vec{v} \cdot \alpha## for a scalar ##\alpha## had been added to the defining properties of vector spaces. However, they don't produce objects with different properties and can be considered isomorphic (equal). But as long as ##(*)## isn't explicitly required, we have to distinguish the two. On the other hand ##(*)## makes kind of sense, since
    $$
    \alpha \cdot \vec{v} = \alpha \cdot \begin{bmatrix}v_1 \\ v_2\\ \vdots\end{bmatrix} = \begin{bmatrix} \alpha \cdot v_1 \\\alpha \cdot v_2\\ \vdots\end{bmatrix} = \begin{bmatrix}v_1 \cdot \alpha \\ v_2 \cdot \alpha \\ \vdots\end{bmatrix} = \vec{v} \cdot \alpha
    $$
    and thus we may identify left and right vector spaces. We must not if we regard things lilke matrices as operating objects.
     
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