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Topic: Uniqueness of differential
Replies: 1   Last Post: Feb 27, 2008 1:35 PM

 Rodolfo Medina Posts: 15 Registered: 8/21/06
Re: Uniqueness of differential
Posted: Feb 27, 2008 1:35 PM

rodolfo.medina@gmail.com (Rodolfo Medina) writes:

>>> Can anybody suggest a proper place to ask the following question:
>>> how to demonstrate the uniqueness of the differential of a function?
>>> Given a function f, its differential in P_0 is a linear map L such that
>>>
>>> lim f(P) - f(P_0) - L(P_0 - P)
>>> P->P_0 -------------------------- = 0
>>> |P_0 - P|
>>>
>>>
>>> Thanks for any indication

Christopher Henrich <chenrich@monmouth.com> writes:

>> This has a "homework" look to it, so, here is a leading question.
>>
>> Suppose that, given L as above, there is another linear map L'
>> satisfying a similar equation. Can you say anything about L' - L ?

Rodolfo:

>
> L(P-P_0) - L'(P-P_0)
> lim --------------------- = 0
> P->P_0 |P-P_0|
>
>, but this does not imply L(v) = L'(v) for all vector v.

Christopher Henrich <chenrich@monmouth.com> writes:

> Suppose that for some v L(v) /= L'(v). Consider P on the line through P_0
> that is parallel to the vector v.

This makes me think that P_0 is supposed to be internal to the set - let's call
it X - where f is defined. Isn't it possible to do without this hypothese?

Thanks
Rodolfo