Solution
= Solution
Differentiate to get
$$
\mathbf r_u=(a\cos v,a\sin v,0),\qquad
\mathbf r_v=(-au\sin v,au\cos v,1).
$$
Their squared lengths are $a^2$ and $a^2u^2+1$, and their <inner product> is zero. Hence the <helicoid> has <first fundamental form>
$$
\boxed{I=a^2du^2+(a^2u^2+1)dv^2.}
$$