= Solution
For the <ruled surface>, $r_{vv}=0$, so $N=0$. Also
$$
M=\frac{(b\times a)\cdot a'}{|b\times a|}=\frac{\gamma'\cdot(a\times a')}{|b\times a|},\qquad EG-F^2=|b\times a|^2.
$$
Thus
$$
\boxed{K=-\frac{[\gamma'\cdot(a\times a')]^2}{|b\times a|^4}.}
$$
The denominator is positive at every regular point, proving that zero <Gaussian curvature> is equivalent to the vanishing of the stated scalar triple product. In particular the <Gaussian curvature> of a regular <ruled surface> is always nonpositive.
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