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In Wikipedia, it is said that[tex]\mathrm dy=\frac{\mathrm

  1. Aug 25, 2011 #1
    In Wikipedia, it is said that
    [tex]\mathrm dy=\frac{\mathrm dy}{\mathrm dx}\mathrm dx.[/tex]
    Can we divide both sides by [itex]\mathrm dx[/itex] and say that the derivative is [itex]\mathrm dy[/itex] divided by [itex]\mathrm dx[/itex]?
     
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  3. Aug 25, 2011 #2

    LCKurtz

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    Re: Differentials

    Yes. That is the usual convention. I think it is less confusing for a calculus student to introduce differentials using the prime notation like this: Given y = f(x) and y' = f'(x), if dx is a small nonzero change in x we define the symbol dy = f'(x) dx. dy can be thought of the change in the y value of the tangent line or the approximate change in y if dx is small. In any case, if you divide both sides by dx you have dy/dx = f'(x).
     
  4. Aug 25, 2011 #3
    Re: Differentials

    It looks like ordinary division of numbers, but [itex]\mathrm dx[/itex] and [itex]\mathrm dy[/itex] are not ordinary numbers. However, we manipulate them symbolically in a way that appears like they are real numbers, for the sake of intuition. But we can do this without loss of precision! A good demonstration of it is using the substitution rule for integrals:

    [tex]\int f'(g(x))g'(x) \mathrm dx[/tex]

    Substituting u = g(x), we get, by the chain rule for derivatives,

    [tex]\int f'(u) \frac{\mathrm du}{\mathrm dx} {\mathrm dx} = f(u) + C = \int f'(u) \mathrm du[/tex]

    The last equality is given by the definition of antiderivative. Though we didn't actually "cancel out" the differentials like fractions, it does turn out that we can safely write [itex]\frac{\mathrm du}{\mathrm dx} {\mathrm dx} = \mathrm du[/itex], by the chain of equalities. Note that the justification for this, however, has nothing to do with fractions.
     
    Last edited: Aug 25, 2011
  5. Aug 25, 2011 #4
    Re: Differentials

    I mean, if we define dy and dx like that, then is it totally correct to say that dy/dx is dy divided by dx?
     
  6. Aug 25, 2011 #5
    Re: Differentials

    Don't you really have to define "divide" (namely, "division") in order to do that? But "divide" typically refers to an operation involving numbers -- and differentials aren't numbers.
     
  7. Aug 25, 2011 #6
    Re: Differentials

    Well, we can define division for differentials, why not?
     
  8. Aug 25, 2011 #7
    Re: Differentials

    Alright, then wouldn't we be interested in defining what a differential is first, so we can define operations on them?

    I think in the elementary, traditional sense, "division" here doesn't really work. I've always thought [itex]\frac{\mathrm dy}{\mathrm dx}[/itex] was cool-looking, but a bit notationally abusive.
     
  9. Aug 26, 2011 #8
    Re: Differentials

    You might do that.. Sometimes there are things that is written in a book, etc. that is really confusing to understand. Maybe a simple explanation might be helpful when you are just starting calculus rather than those with lots of formula. Like those that I red before about the law of derivatives that when i read it. It is so confusing so i just make my own shortcut formula rather do those.
     
  10. Aug 26, 2011 #9

    hunt_mat

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    Re: Differentials

    Hmmm, differentials are really part of of exterior algebra which is a very important tool for differential geometry. So I think you should look into exterior algebra for a firm understanding of a differential.
     
  11. Aug 26, 2011 #10
    Re: Differentials

    Do you have any resources for studying exterior algebra? I'd prefer some online resources since I can't gain access to any bookstores or libraries that have some serious mathematics books (I'm still at junior high school).
     
  12. Aug 26, 2011 #11

    hunt_mat

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    Re: Differentials

    Any set of notes on elementary differential geometry will do but google is your friend...
     
  13. Aug 26, 2011 #12
    Re: Differentials

    Thanks. Just needed to know what to search for (I tried to search for exterior algebra but every result turned out very difficult.
     
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