Given $\alpha(s)$ a smooth arc-length parametrized curve, how can you write the equation for the surface $f(s,t) = \alpha(s) + t \alpha'(s)$ component-wise?
That is, I want to write $f = (f_x, f_y, f_z)$.
I can write $\alpha(s) = (x(s), y(s), z(s))$, but that doesn't give much. For instance, I don't think $f_x(s,t) = x(s) + t x'(s)$. Is that right?