Consider a vector field, tangent to a curve. As a simple example, take the curve
$x(p)=\sin(p)$
$y(p)=\cos(p)$
$z(p)=p$
and its tangent vector field $\mathbf{A}(p)=\left(\begin{array}{c} \cos(p)\\ -\sin(p)\\ 1\\ \end{array}\right)$
Since this vector field lies on a spatial curve, does it have spatial derivatives? What is its curl?
What are the conditions for a vector field such as this one to have spatial derivatives? How does one express the spatial derivatives? Thank you.