This question comes out of the book I have been using to study k-manifolds.
"Consider the helicoid in $R^3$ parametrized as X($u_1,u_2$)=($u_1 $cos($3u_2)$,$u_1 $sin($3u_2)$, $5u_2$)
Where ($u_1,u_2$) $\epsilon$ [$0,5$] x [$0$, $2$$\pi$]
Let S denote the underlying surface of the helicoid and let $\Omega$ be the orientation 2-form defined in terms of X as:
$$\Omega_{X(u_1,u_2)}(\mathbf a,\mathbf b)=\det\begin{bmatrix} -5\sin(3u_2) & a_1 & b_1 \\ 5\cos(3u_2) & a_2 & b_2 \\ -3u_1 & a_3 & b_3 \end{bmatrix},$$
where $\mathbf a=(a_1,a_2,a_3)$ and $\mathbf b=(b_1,b_2,b_3)$ are within the linear span of the tangent vectors to the coordinate curves, T$_{u1}$ and T$_{u2}$"
It was established that the tangent vectors and the parametrization X were incompatible with $\Omega$:
($\Omega$(T$_{u1}$,T$_{u2})$<$0$).
The exercise asks us to:
"Modify X to one having the same underlying surface of S but that is compatible with $\Omega$ "
This is where I am having some difficulties. Anytime I try to modify the parametrization I keep finding that $\Omega$<$0$. Any help here would be appreciated.