= Solution
For a local parametrization $X(u^1,u^2)$, the <first fundamental form> is the Euclidean inner product restricted to each tangent plane:
$$
I=g_{ij}\,du^i\,du^j,\qquad
g_{ij}=X_i\cdot X_j.
$$
Writing $u^1=u,u^2=v$, this is
$$
I=E\,du^2+2F\,du\,dv+G\,dv^2,
\qquad
E=X_u^2,\quad F=X_u\cdot X_v,\quad G=X_v^2.
$$
After choosing an orientation, the <Gauss map> is
$$
N=\frac{X_u\times X_v}{|X_u\times X_v|}.
$$
The <second fundamental form> is $II(v,w)=\langle S(v),w\rangle$, where the shape operator is $S=-dN$. In coordinates,
$$
II=e\,du^2+2f\,du\,dv+g\,dv^2,
\qquad
e=N\cdot X_{uu},\quad
f=N\cdot X_{uv},\quad
g=N\cdot X_{vv}.
$$
The <Gaussian curvature> and <mean curvature> are respectively the determinant and half the trace of $S$:
$$
K=\frac{eg-f^2}{EG-F^2},
\qquad
H=\frac{eG-2fF+gE}{2(EG-F^2)}.
$$
Changing the chosen normal reverses $II$ and $H$ but leaves $K$ unchanged.
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