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Generalizing l'Hopital's Rule

  1. Sep 2, 2013 #1
    I had a wild thought.

    Out of curiosity, is anyone aware of a kind of generalization for l'Hopital's Rule from analysis for differentiable maps between differentiable manifolds? I'm having trouble formulating if I could do it or not, because (as far as I know), if I have ##f,g:M\to N##, with ##f## and ##g## differentiable and ##M## and ##N## differentiable, ##f(x)/g(x)## is not, in general, defined.

    Again, I don't know if it can be generalized. Ideas are certainly welcome, since I'll probably be stuck thinking about it until I prove something does work or doesn't work.
     
  2. jcsd
  3. Sep 3, 2013 #2

    MathematicalPhysicist

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    Maybe there's something like:

    [itex]\lim_{x\rightarrow 0} \frac{||f(x)||}{||g(x)||} = \frac{||\nabla f(0)||}{||\nabla g(0)||}[/itex]

    But it's just a guess, I didn't confirm this.
    Anyway, you can't define division between two functions unless they are scalars, obviously.

    Edit: Well, you can define some sort of division but it wouldn't be like the case we know from calculus.
     
  4. Sep 3, 2013 #3

    MathematicalPhysicist

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    It seems to me that the next identity should be fulfilled.

    If ##f(x_1,\cdots , x_m)=(f_1,\cdots, f_n)## and ##g(x_1,\cdots,x_m)=(g_1,\cdots,g_n)##, then:

    ##||f||/||g|| = \sqrt{f^2_1+\cdots+f^2_n}/\sqrt{g^2_1+\cdots + g^2_n}##

    Now it seems to me to be plausible that:
    ##\lim_{x\to 0} ||f||/||g|| = \frac{\sqrt{(grad \ f_1)^2+\cdots + (grad \ f_n )^2}}{\sqrt{(grad \ g_1)^2+\cdots + (grad \ g_n )^2}}(at \ x=0)##
     
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