Re: question in spivak's calculus on manifolds
- From: "Someonekicked" <someonekicked@xxxxxxxxxxx>
- Date: Sat, 14 Jan 2006 19:13:37 -0500
thx a lot for the replies
--
Quotes from The Weather Man:
Robert Spritz: Do you know that the harder thing to do, and the right thing
to do, are usually the same thing? "Easy" doesn't enter into grown-up
life... to get anything of value, you have to sacrifice.
"Someonekicked" <someonekicked@xxxxxxxxxxx> wrote in message
news:lZqdnXxXJ8klpFTeRVn-ow@xxxxxxxxxxxxxx
> the notation is cumbersome, thats why if u have the book, maybe better to
> look up the question in the book.
>
> anyway, the question is page 33, 2-29 part c). For some reason, im stuck!
> I cant show why D_x f(a) = Df(a) (x).
>
>
> Let f : R^n -> R
> and let D_x f(a) = limit_(as t -> 0) [ ( f(a + tx) - f(a) ) / t]
> (here x is column vector in R^n).
> the question is to show that if f is differentiable, then D_xf(a) =
> Df(a)(x)
> Df(a) is ,(a linear transformation), the derivative of f at a.
> The associated matrix with Df(a) would be f'(a).
>
>
> By definition, thats how to verify Df(a) (which also previously proven to
> be unique),
>
> lim_(as h -> 0) [ | f(a+h) - f(a) - Df(a)(h) | / |h| ] = 0
>
> Since here the range of f is in R, then norm in the numerator is not
> necessary, so
>
> lim_(as h -> 0) [ ( f(a+h) - f(a) - Df(a)(h) ) / |h| ] = 0
> that is,
> lim_(as h -> 0) [ Df(a)(h) / |h| ] = lim_(as h -> 0) [ ( f(a+h) - f(a) ) /
> |h| ]
>
> so its enough to show that,
> lim_(as h -> 0) [ D_h f(a) / |h| ] = lim_(as h -> 0) [ ( f(a+h) - f(a) ) /
> |h| ] ( I )
>
> now,
> lim_(as h -> 0) [ D_h f(a) / |h| ]
> = lim_(as h -> 0) [ limit_(as t -> 0) [ ( f(a + t h) - f(a) ) / t] / |h| ]
> = lim_(as h -> 0) [ limit_(as t -> 0) [ ( f(a + t h) - f(a) ) / (t
> |h|) ] ]
> now if we let vector K = t h, then we should be able to verify (I) ?
> t |h| is the denominator making it hard for me to do any substitution.
> any suggestions or hints?
>
> in part b), I proved that
> D_(s x) f(a) = s D_x f(a) (here s is constant in R).
> so I can write the expression above as
> lim_(as h -> 0) [ limit_(as t -> 0) [ ( f(a + t h / |h|) - f(a) ) / t ] ]
> but still not sure how to continue.
>
> btw, that question came on my mind, whats the derivative of f: R^n -> R^n
> defined as f(x) = x / |x|
> or more simply, derivative of g: R^n -> R where g(x) = |x|, at some a in
> R^n.
>
> thx in advance
>
>
> --
> Quotes from The Weather Man:
> Robert Spritz: Do you know that the harder thing to do, and the right
> thing
> to do, are usually the same thing? "Easy" doesn't enter into grown-up
> life... to get anything of value, you have to sacrifice.
>
.
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