Re: question in spivak's calculus on manifolds
- From: The World Wide Wade <waderameyxiii@xxxxxxxxxxxxxxxxxxxx>
- Date: Sat, 14 Jan 2006 13:48:59 -0800
In article <lZqdnXxXJ8klpFTeRVn-ow@xxxxxxxxxxx>,
"Someonekicked" <someonekicked@xxxxxxxxxxx> wrote:
> 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).
f(a + tx) = f(a) + Df(a)(tx) + E(tx), where E(y)/|y| -> 0 as y ->
0 in R^n. Using the linearity of Df(a), we have [f(a + tx) -
f(a)]/t = Df(a)(x) + E(tx)/t. Now argue that E(tx)/t -> 0 as t ->
0 in R.
.
- References:
- question in spivak's calculus on manifolds
- From: Someonekicked
- question in spivak's calculus on manifolds
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