Problem:
Given a vector $$v_s=4\vec e_1 + \frac \pi4\vec e_2 + \frac \pi4\vec e_3$$ where $\vec e_1, \vec e_2, \vec e_3$ are spherical basis vectors. Convert this vector into terms of cylindrical basis vectors $\vec e^1,\vec e^2, \vec e^3$.
What I've tried:
Looking at my notes for transforming curvilinear systems, the covariant transformation law is $$\vec e^i = \frac {dq_j}{dq^i}*\vec e_j$$ . I cannot see a way to use this transformation law, so I simply converted the spherical coordinates to cylindrical coordinates:
$$v_c = 4*sin \frac {\pi}4 \vec e^1 + \frac{\pi}4\vec e^2 + 4*cos(\frac{\pi}4)\vec e^3$$
This seems incorrect as I am simply converting a coordinate. Would anyone be able to point me in the right direction?
Thanks