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trj
Posts:
77
From:
Korea
Registered:
2/23/12
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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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