Deduce from that in the definition of directional derivative, , the straight line can be replaced by any curve with initial velocity , that is, such that and

Not quite sure how to proceed. Any help would be appreciated.

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- Feb 9th 2011, 11:59 AMZenniecalculus in euclidean space, working with curves in R3
Deduce from that in the definition of directional derivative, , the straight line can be replaced by any curve with initial velocity , that is, such that and

Not quite sure how to proceed. Any help would be appreciated. - Feb 9th 2011, 03:59 PMTinyboss
Try replacing with and see what you get. In the standard directional derivative definition, you're traveling from p along a straight line in the direction of v, and taking the limit as t goes to zero. They're saying that you could travel t units along the path alpha, and take the same t->0 limit, and get the same answer, provided alpha has the right properties.