Generalizing l'Hopital's Rule

  • #1
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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.
 

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  • #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.
 
  • #3
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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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