# Thread: calculus in euclidean space, working with curves in R3

1. ## calculus in euclidean space, working with curves in R3

Deduce from $\displaystyle \alpha ' (t) [f] = \frac {d(f(\alpha))}{dt} (t)$ that in the definition of directional derivative, $\displaystyle {\bf v_p}[f] = \frac {d}{dt} (f({\bf p} + t{\bf v}) \mid _{t = 0}$, the straight line $\displaystyle t \to {\bf p} + t{\bf v}$ can be replaced by any curve $\displaystyle \alpha$ with initial velocity $\displaystyle {\bf v_p}$, that is, such that $\displaystyle \alpha (0) = {\bf p}$ and $\displaystyle \alpha ' (0) = {\bf v_p}$

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

2. Try replacing $\displaystyle f(p+tv)$ with $\displaystyle f(\alpha(t))$ 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.