= Solution
Differentiating the arc-length identity gives $f'f''+h'h''=0$. Multiply by $f'$ and use $(f')^2+(h')^2=1$ to obtain
$$
(f'h''-f''h')h'=f'h'h''-f''(h')^2=-f''\bigl((f')^2+(h')^2\bigr)=-f''.
$$
This derivation does not divide by $h'$, so it remains valid where $h'=0$. Inserting the <first fundamental form> and <second fundamental form> computed above gives the <Gaussian curvature of a surface of revolution>:
$$
\boxed{K=\frac{(f'h''-f''h')fh'}{f^2}=-\frac{f''}{f}.}
$$
Back to article page
