Solution From the introductions to First and second fundamental forms (https://warwick.ac.uk/fac/sci/maths/people/staff/weiyi_zhang/ma3d92018fall.pdf)
We prefer E,F,and G to the first fundamental form and L,M,N to the second fundamental form, hence;
I=Edu2+2Fdudv+Gdv2, II=Ldu2+2Mdudv+Ndv2. We denote
ϝI=(EFFG),ϝII=(LMMN) From the above concept, we deduced the first, second and third fundamental forms replacing Edu2+2Fdudv+Gdv2 with a11(du1)2+2a12du1du2+a22(du2)2 , II=Ldu2+2Mdudv+Ndv2 with b11(du1)2+2b12du1du2+b22(du2)2 and used dn2=(nu1)2(du1)2+2nu1nu2du1du2+(nu2)2(du2)2 for the third fundamental form
I=dS2=a11(du1)2+2a12du1du2+a22(du2)2aij=(ri,rj);i,j=1,2II=b11(du1)2+2b12du1du2+b22(du2)2bij=(rij,n);i,j=1,2III=dn2=(nu1)2(du1)2+2nu1nu2du1du2+(nu2)2(du2)2((nu1)2nu1nu2nu1nu2(nu2)2)=(b11b12b12b22)⋅(a11a12a12a22)−1⋅(b11b12b12b22)A=(a11a12a12a22)A−1=(Δa22−Δa12−Δa12Δa11)Δ=∣A∣=a11a22−a122
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