From 'Do carmo Differential Geometry of curves and surfaces'
On page 89, #9.
Show that the parametrized surface S given by
$$ \text{x}(u,v)=(v\cos{u},v\sin{u},au) $$
Compute its normal vector $N(u, v)$ of a tangent plane of $\text{x}$ at $(u,v)$ and show that along the coordinate line $u = u_0$ the tangent plane of x rotates about this line in such a way that the tangent of its angle with the z axis is proportional to the distance from the point $x(u_0, v)$ to the z axis. $$ $$ $$ $$
From my computation, $N(u,v) = \frac{1}{(a^2 + v^2)^\frac{1}{2}}(-a\sin{u},\ a\cos{u},-v)$ Here, I can't understand 'rotate along this line'. I don't think it does rotate along the line, it just rotates along z-axis. And the coordinate line is not a z-axis. What should I do?
Thanks in advance.
