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Topic: How to Define Derivative of a Vector Field in this Case ( Curve in R^3)?
Replies: 3   Last Post: Jan 18, 2013 4:30 AM

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trj

Posts: 89
From: Korea
Registered: 2/23/12
Re: Derivative of a Vector Field
Posted: Jan 18, 2013 4:30 AM
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> On Thu, 17 Jan 2013, cku wrote:
>

> > Let C:I-->R^3 be a smooth curve, and let Z(s) be a
> vector field along
> > the curve, parametrized by arc-length.
> >
> > We define the derivative of a vector field Z along

> the curve to be the quotient:
> >
> > Lim_ds->0 [Z(s+ds)-Z(s)]/ds
> >
> > Now, I don't know how to make sense of the

> difference in the numerator:
>
> Z(s + ds) and Z(s) are two vectors in R^3, as is
> their difference.


That is incorrect. Z(s+ds) is an element of the tangent space at
point C(s+ds), whereas Z(s) is an element of the tangent space at C(s).
They cannot be added or subtracted, since they belong to different
vector spaces.



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