We are asked to prove the following theorem found in page 88 of Differential Geometry: Curves, Surfaces, Manifolds by Wolfgang Kühnel.
Using standard parameters, calculate the Gaussian curvature and the mean curvature of a ruled surface as follows:
$K = -\dfrac {{\lambda}^2} {{{\lambda}^2 +v^2}^2}$
and
$H = -\dfrac {1} {2({\lambda}^2 +v^2)^{3/2}} (Jv^2 + \lambda' v + \lambda (\lambda J + F))$
In standard parameters, a ruled surface is $f(u,v) = c(u) + v X(u)$ and $||X|| = ||X'|| = 1$ and $\langle c', X' \rangle = 0$.
Thus, using standard parameters, a ruled surface is, up to Euclidean motions, uniquely determined by the following quantities:
$F = \langle c', X\rangle$
$\lambda = \langle c' \times X, X' \rangle = \det (c', X, X')$
$J = \langle X'', X \times X' \rangle = \det (X, X', X'')$
Also, the first fundamental form is given as follows:
$I = \begin {pmatrix} \langle c',c' \rangle + v^2 & \langle c', X \rangle \\ \langle c', X \rangle & 1 \end {pmatrix} = \begin {pmatrix} F^2 + {\lambda}^2 + v^2 & F \\ F & 1 \end {pmatrix}$ with $\det (I) = \lambda^2 + v^2$.
So far, I have that
$f_u (u,v) = c' + vX'$
and
$f_v (u,v) = X$
But I don't know how to proceed from there to get the first fundamental form, the normal vector, and the second fundamental form.