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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



Bacle H
Posts:
283
Registered:
4/8/12


How to Define Derivative of a Vector Field in this Case ( Curve in R^3)?
Posted:
Jan 18, 2013 12:34 AM


Hi, All:
Let C:I>R^3 be a smooth curve, and let Z(s) be a vector field along the curve, parametrized by arclength.
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:
The two vectors Z(s+ds) and Z(s) , are in different tangent spaces 
tangent space at s+ds and s respectively  and , AFAIK, the difference
of vectors in different tangent spaces is not defined, except for cases
where there is a natural isomorphism between the tangent spaces, as in the
case where the tangent spaces are those in R^n itself. Any suggestions,
please?



